--- /srv/rebuilderd/tmp/rebuilderdoQwN9o/inputs/macaulay2-common_1.26.06+ds-5_all.deb +++ /srv/rebuilderd/tmp/rebuilderdoQwN9o/out/macaulay2-common_1.26.06+ds-5_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2026-10-24 01:47:31.000000 debian-binary │ --rw-r--r-- 0 0 0 567932 2026-10-24 01:47:31.000000 control.tar.xz │ --rw-r--r-- 0 0 0 33668792 2026-10-24 01:47:31.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 567888 2026-10-24 01:47:31.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 33670520 2026-10-24 01:47:31.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.26.06+ds-5 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 324507 │ │ │ +Installed-Size: 324509 │ │ │ Depends: fonts-katex (>= 0.16.10+~cs6.1.0), libjs-katex (>= 0.16.10+~cs6.1.0) │ │ │ Section: math │ │ │ Priority: optional │ │ │ Multi-Arch: foreign │ │ │ Homepage: http://macaulay2.com │ │ │ Description: Software system for algebraic geometry research (common files) │ │ │ Macaulay 2 is a software system for algebraic geometry research, written by │ │ ├── ./md5sums │ │ │ ├── ./md5sums │ │ │ │┄ Files differ ├── data.tar.xz │ ├── data.tar │ │ ├── file list │ │ │ @@ -3818,15 +3818,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 3474 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_beilinson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_bgg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2077 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_cohomology__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_direct__Image__Complex_lp__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1294 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_direct__Image__Complex_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_direct__Image__Complex_lp__Module_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2382 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2381 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 456 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_sym__Ext.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_tate__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_universal__Extension.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/___Exterior.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/___Regularity.html │ │ │ @@ -3834,15 +3834,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6717 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_bgg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12329 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_cohomology__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9244 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex_lp__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10236 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13203 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5853 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_projective__Product.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14296 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14295 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7006 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_sym__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7850 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_tate__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_universal__Extension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12502 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9728 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5507 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BIBasis/ │ │ │ @@ -3867,18 +3867,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 77417 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4425 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3108 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2927 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4444 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 289778 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -4615,15 +4615,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__S__Q__Stable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_is__Stable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_koszul__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1957 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 694 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_nonzero__Max.out │ │ │ -rw-r--r-- 0 root (0) root (0) 684 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_prepend__Zero__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 899 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_remove__Zero__Trailing__Terms.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3453 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution_lp__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_scarf__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_substitute_lp__Chain__Complex_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 672 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1333 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_taylor__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1351 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_trivial__Homological__Truncation.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/ │ │ │ @@ -4654,15 +4654,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_is__Stable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4914 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_koszul__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6820 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_nonzero__Max.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6727 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_nonzero__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6108 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_prepend__Zero__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6873 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_remove__Zero__Trailing__Terms.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12471 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7089 2026-10-24 01:47:31.000000 ./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) 9344 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution_lp__Chain__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_scarf__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_substitute_lp__Chain__Complex_cm__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5922 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6958 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6971 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_taylor__Resolution_lp..._cm__Length__Limit_eq_gt..._rp.html │ │ │ @@ -4698,19 +4698,19 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 4376 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1503 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Toric__Variety__Valid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3442 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chern.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2404 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 775 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Toric__Chow__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ --rw-r--r-- 0 root (0) root (0) 341 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2017 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 760 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 666 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 665 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1049 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Multi__Proj__Coord__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Output.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3327 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Segre.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Toric__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_is__Multi__Homogeneous.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_probabilistic_spalgorithm.out │ │ │ @@ -4720,19 +4720,19 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 23815 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/CoincidentRootLoci/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 108157 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CoincidentRootLoci/example-output/___Coincident__Root__Locus_sp_st_sp__Coincident__Root__Locus.out │ │ │ @@ -5089,15 +5089,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 231378 2026-10-24 01:47:31.000000 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│ -rw-r--r-- 0 root (0) root (0) 15235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Complexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Complexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 740220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Complexes/example-output/ │ │ │ @@ -5546,29 +5546,29 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 21402 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16228 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ConformalBlocks/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9600 2026-10-24 01:47:31.000000 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0 root (0) root (0) 13383 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__Jumping__Exponent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12607 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10037 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__Simple__Normal__Crossing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20074 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7138 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/ │ │ │ @@ -7722,15 +7722,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1001 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 732 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 170 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph_lp__G__K__M__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_normal__Toric__Variety_lp__G__K__M__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8062 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8063 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pullback_lp__Equivariant__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pushforward.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_set__Indicator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_trivial__K__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_underlying__Graph_lp__Moment__Graph_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/ │ │ │ @@ -7773,15 +7773,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Flag__Matroid_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8595 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_make__K__Class_lp__G__K__M__Variety_cm__Toric__Divisor_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9369 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_map_lp__G__K__M__Variety_cm__G__K__M__Variety_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8650 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7658 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph_lp__G__K__M__Variety_cm__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph_lp__G__K__M__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7819 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_normal__Toric__Variety_lp__G__K__M__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 20015 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_projective__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7393 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pullback_lp__Equivariant__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7948 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pushforward.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7166 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_set__Indicator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5918 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_trivial__K__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5739 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_underlying__Graph_lp__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/index.html │ │ │ @@ -8841,42 +8841,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 71515 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Graphs/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 57702 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Graphs/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27654 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Graphs/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 56750 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 33163 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 33567 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Constraints.out │ │ │ -rw-r--r-- 0 root (0) root (0) 598 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_find__Weight__Vector.out │ │ │ -rw-r--r-- 0 root (0) root (0) 30368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2709 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_groebner__Stratum.out │ │ │ 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│ │ │ -rw-r--r-- 0 root (0) root (0) 5493 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/___Minimalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9986 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Constraints.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8639 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_find__Weight__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 42308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_groebner__Stratum.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_linear__Part.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23503 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15086 2026-10-24 01:47:31.000000 ./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) 23626 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 24140 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15170 2026-10-24 01:47:31.000000 ./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) 7464 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_smaller__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7835 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_standard__Monomials.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8669 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/_tail__Monomials.html │ │ │ --rw-r--r-- 0 root (0) root (0) 50311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 50715 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11356 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5776 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerStrata/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18071 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 705 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out │ │ │ @@ -8896,28 +8896,28 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 18704 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 123 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Product_lp__List_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1084 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Product_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1159 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_ideal__Of__Projective__Points.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/example-output/_point.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4844 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/___Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/___Point_sp_eq_eq_sp__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5540 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/___Point_sp_st_sp__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4450 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6182 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__Ideal_cm__Z__Z_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6382 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__List_cm__Z__Z_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6416 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__List_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8292 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__Ideal_cm__Ideal_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5953 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7585 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7123 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_ideal__Of__Projective__Points.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5370 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/_point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8581 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Hadamard/html/index.html │ │ │ @@ -9020,15 +9020,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50644 2026-10-24 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4893 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5004 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_diff__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_distraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 768 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_euler__Operators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_gkz.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_is__Torus__Fixed.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4657 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_put__Weyl__Algebra.out │ │ │ -rw-r--r-- 0 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./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Expts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8063 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Expts__Mult.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11336 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12439 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_diff__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_distraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7893 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_euler__Operators.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9171 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_gkz.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_indicial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6606 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_is__Torus__Fixed.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4094 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_nilsson__Support.html │ │ │ @@ -9247,17 +9247,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Frac__P_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 707 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Fractions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1444 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__P__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_idealizer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Keep_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Limit_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 28072 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 28073 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 370 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Variable_eq_gt..._rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2398 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3674 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2595 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ring_cm__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1221 2026-10-24 01:47:31.000000 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│ │ │ -rw-r--r-- 0 root (0) root (0) 12894 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_is__Normal_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9808 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_make__S2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6633 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_make__S2_lp..._cm__Variable_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7958 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/html/_ring__From__Fractions.html │ │ │ @@ -9317,35 +9317,35 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 668 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/InvariantRing/example-output/_finite__Action.out │ │ │ -rw-r--r-- 0 root (0) root (0) 677 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/Jets/html/_jets__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4864 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Max__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7373 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jets__Projection.html │ │ │ @@ -9585,62 +9585,62 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7501 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101880 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1949 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_all__Gradings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2785 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2786 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5835 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_canonical__Homotopies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1036 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2263 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6909 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6911 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_correspondence__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_cox__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9602 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3.html │ │ │ 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root (0) 7292 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution__Twists.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6025 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7697 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_scheme__In__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11028 2026-10-24 01:47:31.000000 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01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__Q__P1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4015 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__R__R.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4048 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Real__X__D.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LLLBases/html/___Threshold.html │ │ │ @@ -10094,35 +10094,35 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 60313 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/LinearTruncations/example-output/_regularity__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_support__Of__Tor.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_comp__Min.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7100 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_diagonal__Multidegrees.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6096 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8581 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10649 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_multigraded__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9034 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_partial__Regularities.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8046 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_regularity__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_support__Of__Tor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25981 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15960 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6334 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/toc.html │ │ │ @@ -10356,15 +10356,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1999 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Expression.out │ │ │ -rw-r--r-- 0 root (0) root (0) 788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Matrix_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1449 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 548 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Flat__Monoid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 340 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_at_at_sp__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 836 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Function_sp_us_sp__Thing.out │ │ │ --rw-r--r-- 0 root (0) root (0) 415 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 416 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 611 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__F.out │ │ │ -rw-r--r-- 0 root (0) root (0) 194 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Rev__Lex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Assign__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 371 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Global__Release__Hook.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6869 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Grassmannian.out │ │ │ @@ -10414,15 +10414,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 248 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_st_st_sp__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 767 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_st_st_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 354 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_us_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 377 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_us_sp__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 484 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_vb_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Matrix_sp_vb_vb_sp__Matrix.out │ │ │ --rw-r--r-- 0 root (0) root (0) 912 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 911 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2509 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 552 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp^_st_st_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 313 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp_pl_pl_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1502 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp_sl_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 299 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Module_sp_st_st_sp__Ring.out │ │ │ @@ -10431,15 +10431,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2458 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 587 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Ideal_sp-_sp__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4407 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Multigraded__Betti__Tally.out │ │ │ -rw-r--r-- 0 root (0) root (0) 218 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-rw-r--r-- 0 root (0) root (0) 205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net_sp_vb_sp__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Net_sp_vb_vb_sp__Net.out │ │ │ -rw-r--r-- 0 root (0) root (0) 339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Numbered__Vertical__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 430 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Option.out │ │ │ -rw-r--r-- 0 root (0) root (0) 571 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Option__Table_sp_gt_gt_sp__Function.out │ │ │ @@ -10471,15 +10471,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring_sp_sl_sp__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1854 2026-10-24 01:47:31.000000 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2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp-_sp__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 157 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_sh_qu_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_st_st_sp__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.1.10.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1721 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.1.8.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.1.9.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.2.13.out │ │ │ @@ -10611,15 +10611,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apply_lp__Z__Z_cm__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1347 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_apropos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ascii.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_asinh.out │ │ │ -rw-r--r-- 0 root (0) root 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773 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_augmented_spassignment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_autoload.out │ │ │ -rw-r--r-- 0 root (0) root (0) 137 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Name.out │ │ │ @@ -10634,25 +10634,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_between.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_binomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_block__Matrix__Form.out │ │ │ -rw-r--r-- 0 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_can__Use__Hilbert__Hint.out │ │ │ --rw-r--r-- 0 root (0) root (0) 588 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 591 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8939 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_capture.out │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Base.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Monomial__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Quotient__Ring_rp.out │ │ │ @@ -10668,15 +10668,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comments.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1539 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1550 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compact__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compare__Exchange.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compose.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compositions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compress.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4287 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ @@ -10764,15 +10764,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1720 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eagon__Northcott.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvalues.out │ │ │ -rw-r--r-- 0 root (0) root (0) 717 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eigenvectors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_eint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 366 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elements.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21213 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 21214 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 782 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3782 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries.out │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_entries_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_equality_spand_spcontainment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 111 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_erfc.out │ │ │ @@ -10801,15 +10801,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out │ │ │ -rw-r--r-- 0 root (0) root (0) 379 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Length.out │ │ │ -rw-r--r-- 0 root (0) root (0) 268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 224 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 82 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Readable.out │ │ │ --rw-r--r-- 0 root (0) root (0) 93 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 92 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 82 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Writable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_fill__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_find__Heft.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_find__Program.out │ │ │ -rw-r--r-- 0 root (0) root (0) 431 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_find__Synonyms_lp__Symbol_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11761 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_fine_spcontrol_spof_spa_sp__Groebner_spbasis_spcomputation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1460 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_finish.out │ │ │ @@ -10914,15 +10914,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1260 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_induced__Map_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1039 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inheritance.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_inputting_spa_spmatrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 592 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_insert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 686 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spassignment_spmethods.out │ │ │ -rw-r--r-- 0 root (0) root (0) 936 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spaugmented_spassignment_spmethods.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integers_spmodulo_spa_spprime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__R__Ri_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ @@ -10957,15 +10957,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Mutable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Open.out │ │ │ -rw-r--r-- 0 root (0) root (0) 290 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Output__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Prime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Primitive.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ready_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Real.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 460 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Skew__Commutative.out │ │ │ @@ -11062,15 +11062,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max.out │ │ │ --rw-r--r-- 0 root (0) root (0) 82 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 83 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out │ │ │ -rw-r--r-- 0 root (0) root (0) 83 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 311 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1637 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1730 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_merge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2673 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 928 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_method__Options_lp__Function_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6785 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out │ │ │ @@ -11078,15 +11078,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 867 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_midpoint.out │ │ │ -rw-r--r-- 0 root (0) root (0) 551 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 310 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 925 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Groebner__Basis_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2181 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 737 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingle.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1805 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1804 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 643 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimize__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 504 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minors_lp__Z__Z_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out │ │ │ @@ -11162,16 +11162,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 681 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pack.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packages.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packing_spmonomials_spfor_spefficiency.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pad.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1059 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pairs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel__Apply.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8673 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8672 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parse.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parsing_spprecedence_cm_spin_spdetail.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partitions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_path.out │ │ │ @@ -11180,15 +11180,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_peek_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permanents.out │ │ │ -rw-r--r-- 0 root (0) root (0) 874 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permutations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffian.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2202 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pi.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pivots_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13709 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13711 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2331 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polarize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polylog.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4055 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polynomial_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 475 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_positions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_powermod.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_precision.out │ │ │ @@ -11249,15 +11249,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_real__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 462 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_realpath.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_recursion__Depth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reduce__Hilbert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reduced__Row__Echelon__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regex__Quote.out │ │ │ --rw-r--r-- 0 root (0) root (0) 712 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 659 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regularity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 238 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regularized__Beta.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regularized__Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_relativize__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 631 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 713 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remainder_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove.out │ │ │ @@ -11320,15 +11320,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_shuffle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sign.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2452 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_simple_sp__Groebner_spbasis_spcomputations_spover_spvarious_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_singular__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 294 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_size2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3173 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_smith__Normal__Form_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4508 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 891 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_some__Terms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort__Columns.out │ │ │ -rw-r--r-- 0 root (0) root (0) 495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_span.out │ │ │ @@ -11396,15 +11396,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 687 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_terms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 730 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_tests.out │ │ │ -rw-r--r-- 0 root (0) root (0) 214 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_tex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 151 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_tex__Math.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3455 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_the_spdebugger.out │ │ │ -rw-r--r-- 0 root (0) root (0) 235 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_thread__Local.out │ │ │ -rw-r--r-- 0 root (0) root (0) 134 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_throw.out │ │ │ --rw-r--r-- 0 root (0) root (0) 143 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 142 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 186 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_timing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 141 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__Absolute__Path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 296 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__C__C.out │ │ │ -rw-r--r-- 0 root (0) root (0) 428 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__External__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 593 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__Field_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 94 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_to__Lower.out │ │ │ @@ -11431,15 +11431,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_unique.out │ │ │ -rw-r--r-- 0 root (0) root (0) 268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_unique__Permutations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 122 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_unlock.out │ │ │ -rw-r--r-- 0 root (0) root (0) 196 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_unsequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 168 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_url__Encode.out │ │ │ -rw-r--r-- 0 root (0) root (0) 413 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_use_lp__Package_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 519 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_use_lp__Ring_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1173 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1174 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 586 2026-10-24 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2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___G__G.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5148 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp^_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_bs_bs_sp__Multiprojective__Variety.out │ │ │ @@ -14756,42 +14756,42 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179 2026-10-24 01:47:31.000000 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│ │ -rw-r--r-- 0 root (0) root (0) 809 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_entries_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 239 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_euler_lp__Multiprojective__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_factor_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1459 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_fiber__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 673 2026-10-24 01:47:31.000000 ./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) 1856 2026-10-24 01:47:31.000000 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root (0) 934 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 803 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1407 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 386 2026-10-24 01:47:31.000000 ./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) 754 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_tangent__Cone_lp__Embedded__Projective__Variety_cm__Embedded__Projective__Variety_rp.out │ │ │ @@ -14828,15 +14828,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 631 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_⋂.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_⋃.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 641 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 14292 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/Nauty/html/_generate__Graphs.html │ │ │ @@ -15376,48 +15376,48 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8306 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Nauty/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 131317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.out │ │ │ -rw-r--r-- 0 root (0) root (0) 493 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1405 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1406 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 826 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_add__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_are__Isomorphic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_build__Graph__Filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 124 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ --rw-r--r-- 0 root (0) root (0) 499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 323 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__To__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 217 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_is__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 849 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_neighborhood__Complements.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_new__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 247 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_only__Planar.out │ │ │ -rw-r--r-- 0 root (0) root (0) 322 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Bipartite.out │ │ │ -rw-r--r-- 0 root (0) root (0) 381 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_relabel__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Isomorphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 203 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6397 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5908 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8749 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8750 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7835 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13355 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8002 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8282 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9334 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6672 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph6__To__Sparse6.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7935 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7934 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__To__String.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6131 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_is__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_neighborhood__Complements.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6592 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_new__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6503 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_only__Planar.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7157 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Bipartite.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9231 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_relabel__Graph.html │ │ │ @@ -15495,15 +15495,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179808 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 431 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Dependent__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op.out │ │ │ -rw-r--r-- 0 root (0) root (0) 553 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2349 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2350 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_amult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 931 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_coordinate__Change__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 354 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_differential__Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 349 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_eliminating__Dual.out │ │ │ -rw-r--r-- 0 root (0) root (0) 270 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_evaluate_lp__Diff__Op_cm__Abstract__Point_rp.out │ │ │ @@ -15536,15 +15536,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 75 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Dependent__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9234 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op_sp__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5207 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Sampler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7481 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8027 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8017 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Tolerance_sp_lp__Noetherian__Operators_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6940 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_amult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5516 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_colon.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8646 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_coordinate__Change__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4322 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6720 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op_lp__Matrix_rp.html │ │ │ @@ -15616,15 +15616,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 20057 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NonPrincipalTestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16625 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NonPrincipalTestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6404 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NonPrincipalTestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 609279 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 3513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3515 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3270 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1905 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 908 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 840 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_st_st_sp__Normal__Toric__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1922 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1977 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___O__O_sp__Toric__Divisor.out │ │ │ @@ -15688,17 +15688,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1271 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_lattice__Points_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Simplicial_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1568 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Smooth_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1011 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_map_lp__Normal__Toric__Variety_cm__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 912 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_matrix_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_max_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1557 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1560 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2228 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_nef__Generators_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1001 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3171 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__List_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1599 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Normal__Toric__Variety_rp.out │ │ │ @@ -15726,15 +15726,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vector_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vertices_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1650 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weighted__Projective__Space_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 452 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 985 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Toric__Map_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 68 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 13066 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13068 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___H__H^__Z__Z_lp__Normal__Toric__Variety_cm__Coherent__Sheaf_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27816 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety.html │ │ │ 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./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Simplicial_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13861 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Smooth_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9792 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_making_spnormal_sptoric_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11657 2026-10-24 01:47:31.000000 ./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) 12713 2026-10-24 01:47:31.000000 ./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) 11308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_matrix_lp__Toric__Map_rp.html │ │ │ 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(0) root (0) 19370 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10007 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12299 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10951 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Normal__Toric__Variety_rp.html │ │ │ @@ -16123,40 +16123,40 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 40622 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24531 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6381 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 146200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 1058 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1060 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Numerical__Interpolation__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Pseudo__Witness__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 443 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_is__On__Image.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1224 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1222 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ -rw-r--r-- 0 root (0) root (0) 240 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Degree.out │ │ │ --rw-r--r-- 0 root (0) root (0) 573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 576 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 712 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9533 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10027 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Sample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Nullity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Source__Sample.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18968 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_pseudo__Witness__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11048 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_pseudo__Witness__Set_lp..._cm__Software_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8997 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34829 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalImplicitization/html/index.html │ │ │ @@ -16377,18 +16377,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 665 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_lead__Term_lp__Vector__In__Width_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 750 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_make__Free__O__I__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 445 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_make__Polynomial__O__I__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2211 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_minimize__O__I__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Free__O__I__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 312 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Free__O__I__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 462 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__Module__In__Width_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 397 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 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2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out │ │ │ -rw-r--r-- 0 root (0) root (0) 509 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_restricted__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 655 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_terms_lp__Vector__In__Width_rp.out │ │ │ @@ -16447,18 +16447,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_lead__Term_lp__Vector__In__Width_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8446 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_make__Free__O__I__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6668 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_make__Polynomial__O__I__Algebra.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9163 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6178 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Free__O__I__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5593 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Free__O__I__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5829 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Module__In__Width_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 5989 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5988 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5534 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Polynomial__O__I__Algebra_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5969 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__Vector__In__Width_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8512 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6547 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Orbit.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8699 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10079 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Syz.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_restricted__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6530 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_terms_lp__Vector__In__Width_rp.html │ │ │ @@ -16490,21 +16490,21 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 567 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Graded__Module_sp_st_st_sp__Graded__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 554 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Graded__Module_sp_st_st_sp__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 971 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___H__H^__Z__Z_sp__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___H__H_sp__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 537 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___H__H_us__Z__Z_sp__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1686 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Hom_lp__Module_cm__Chain__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 703 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Ring_sp^_sp__Betti__Tally.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1814 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp__Graded__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_chain__Complex_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_complete_lp__Chain__Complex_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_cone_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dd.out │ │ │ -rw-r--r-- 0 root (0) root (0) 536 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_dual_lp__Chain__Complex__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1335 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extend_lp__Chain__Complex_cm__Chain__Complex_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1130 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_extracting_spinformation_spfrom_spchain_spcomplexes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_free_spresolutions_spof_spmodules.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_graded__Module__Map.out │ │ │ @@ -16569,25 +16569,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5804 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___H__H_sp__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4946 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___H__H_sp__Chain__Complex__Map.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/___H__H_us__Z__Z_sp__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5159 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1080 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.1_co_spunique_spgraph_spon_sp8_spvertices_spwith_spexotic_spsolutions_spand_spno_spinduced_spcycle_spof_splength_spat_spleast_sp5.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2543 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2081 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.3_co_spexamples_spof_spgluing_sptwo_spcycles_spalong_span_spedge.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.4_co_sp__The_spsquare_spwithin_spa_spsquare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Generation_spof_spall_sp__S__C__T_sp_lpsimple_cm_spconnected_cm_sp2-connected_rp_spgraphs_spon_spsmall_spnumbers_spof_spvertices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 16187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___Oscillators.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3814 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1682 2026-10-24 01:47:31.000000 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01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/html/_osc__Jacobian.html │ │ │ @@ -16989,15 +16989,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12822 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Parsing/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9410 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Parsing/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Parsing/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 135847 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2841 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2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_histogram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1300 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_maximal__Entry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1502 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_normalize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5291 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Simplicial__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1143 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1163 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5439 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Discrete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4846 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___With__L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Zero__Mean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_disturb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7432 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_histogram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_maximal__Entry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7072 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14710 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Simplicial__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8245 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13046 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4959 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52982 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/ │ │ │ @@ -19045,25 +19045,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5165 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 86199 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Alexander__Probability.out │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Finding_sp__Extreme_sp__Examples.out │ │ │ --rw-r--r-- 0 root (0) root (0) 447 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 445 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__Chain__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 906 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 479 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Elements__From__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 287 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 384 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 359 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Pure__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal__Chain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Sparse__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out │ │ │ @@ -19080,28 +19080,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5977 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_is__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7962 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Addition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7486 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7888 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6753 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7981 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Elements__From__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7846 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Ideal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6462 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6456 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7888 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Pure__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7555 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7614 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shellable__Ideal__Chain.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9884 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Sparse__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7646 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18182 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Toric__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5925 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_reg__Seq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6209 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5930 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_square__Free_lp__Z__Z_cm__Ring_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 25798 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 25796 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17675 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 151393 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomMonomialIdeals/example-output/___C__M__Stats.out │ │ │ @@ -19213,15 +19213,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 96387 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 356 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Max__Coordinates__To__Replace.out │ │ │ -rw-r--r-- 0 root (0) root (0) 268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Point__Check__Attempts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2427 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/___Replacement.out │ │ │ -rw-r--r-- 0 root (0) root (0) 292 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1455 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1454 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 379 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_find__A__Non__Zero__Minor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1379 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_generic__Projection.out │ │ │ -rw-r--r-- 0 root (0) root (0) 841 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_get__Random__Linear__Forms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_projection__To__Hypersurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 805 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Coordinate__Change.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/ │ │ │ @@ -19231,15 +19231,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6079 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Dimension__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6221 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Extend__Field.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Max__Coordinates__To__Replace.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5123 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Num__Threads__To__Use.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6241 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Point__Check__Attempts.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/___Replacement.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9043 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html │ │ │ --rw-r--r-- 0 root (0) root (0) 13620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13619 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11137 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomPoints/html/_find__A__Non__Zero__Minor.html │ │ │ -rw-r--r-- 0 root 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│ │ │ -rw-r--r-- 0 root (0) root (0) 7649 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2051 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_generic__Polynomials.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5201 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3139 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6760 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1902 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/Resultants/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20915 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7538 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Resultants/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37681 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 415 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ --rw-r--r-- 0 root (0) root (0) 5992 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/RunExternalM2/html/_is__External__M2__Parent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7798 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ --rw-r--r-- 0 root (0) root (0) 23723 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5849 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2__Return__Answer.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6510 2026-10-24 01:47:31.000000 ./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) 7746 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RunExternalM2/html/_suggestions_spfor_spusing_sp__Run__External__M2.html │ │ │ 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./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12984 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15064 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_are__Pseudo__Inverses.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_check__S__V__D__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10109 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14657 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14656 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14339 2026-10-24 01:47:31.000000 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root (0) root (0) 4223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimpleDoc/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialComplexes/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialComplexes/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 389652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialComplexes/dump/rawdocumentation.dump │ │ │ @@ -21065,15 +21065,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10013 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_coimage_lp__Simplicial__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 27943 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_cokernel_lp__Simplicial__Module__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 28194 2026-10-24 01:47:31.000000 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60726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_induced__Map_lp__Simplicial__Module_cm__Simplicial__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 12255 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_is__Commutative_lp__Simplicial__Module__Map_rp.out │ │ │ @@ -21136,15 +21136,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 36316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_components_lp__Simplicial__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_components_lp__Simplicial__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11527 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_degree_lp__Simplicial__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35440 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/SimplicialModules/html/_face_sldegeneracy_spmaps_spof_spa_spsimplicial_spmodule.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5827 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Complex_lp..._cm__Remember__Summands_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16703 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Complex_lp__Simplicial__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8334 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Degeneracy_lp__Simplicial__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7153 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_id_sp_us_sp__Simplicial__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 45212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SimplicialModules/html/_image_lp__Simplicial__Module__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6926 2026-10-24 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./usr/share/doc/Macaulay2/SparseResultants/example-output/___Sparse__Resultant_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Discriminant_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 385 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_char_lp__Sparse__Resultant_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1023 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 930 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1364 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 814 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 931 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1362 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 973 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dim_lp__Multidimensional__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 494 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8386 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize_lp__Hodge__Special__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6716 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold_lp__String_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7886 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9392 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5826 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surfaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_swap.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6169 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__External__String_lp__Hodge__Special__Fourfold_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10948 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5579 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_trisecant__Flop.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7138 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 40566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35967 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15426 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpectralSequences/ │ │ │ @@ -22252,15 +22252,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_no__Packed__All__Subs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_no__Packed__Sub.out │ │ │ -rw-r--r-- 0 root (0) root (0) 174 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_squarefree__Gens.out │ │ │ -rw-r--r-- 0 root (0) root (0) 169 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_squarefree__In__Codim.out │ │ │ -rw-r--r-- 0 root (0) root (0) 651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symb__Power__Prime__Pos__Char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Defect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 193 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Polyhedron.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1211 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 180 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power__Join.out │ │ │ -rw-r--r-- 0 root (0) root (0) 320 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_waldschmidt.out │ │ │ -rw-r--r-- 0 root (0) root (0) 200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_waldschmidt_lp..._cm__Sample__Size_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 593 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 15 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4808 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/___A_spquick_spintroduction_spto_spthis_sppackage.html │ │ │ @@ -22293,15 +22293,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6158 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_no__Packed__All__Subs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5903 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_no__Packed__Sub.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5895 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_squarefree__Gens.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6043 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_squarefree__In__Codim.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6682 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symb__Power__Prime__Pos__Char.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6725 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Defect.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Polyhedron.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9864 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9863 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power__Join.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8118 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_waldschmidt.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7210 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/_waldschmidt_lp..._cm__Sample__Size_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22559 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20056 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11262 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymbolicPowers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SymmetricPolynomials/ │ │ │ @@ -22558,15 +22558,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 166859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 7476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_action__On__Direct__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1183 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 926 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Bundle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 632 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Contraction.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1692 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2581 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_bgg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1129 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_coarse__Multigraded__Regularity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2746 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_cohomology__Hash__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 837 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_cohomology__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 46631 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_composed__Functions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8706 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_corner__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_direct__Image__Complex.out │ │ │ @@ -22600,15 +22600,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/___Rings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5425 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/___Sub__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3987 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/___Tate__Data.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_action__On__Direct__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10364 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9213 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Contraction.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8993 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8994 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11253 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_bgg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_coarse__Multigraded__Regularity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12770 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_cohomology__Hash__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9999 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_cohomology__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 69237 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_composed__Functions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5768 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_contraction__Data.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23344 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_corner__Complex.html │ │ │ @@ -22749,25 +22749,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 806 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_compatible__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_decompose__Fraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_descend__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Preimage.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1117 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Trace__On__Canonical__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 488 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1817 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1823 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Pure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1492 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1514 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2756 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Ascent__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4556 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__C__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__Domain.html │ │ │ @@ -22800,25 +22800,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9322 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_compatible__Ideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_decompose__Fraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_descend__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10782 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12579 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5408 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16914 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16916 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8725 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/TestIdeals/html/_multiplicative__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8061 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_parameter__Test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Element.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15165 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37686 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34923 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12593 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Text/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Text/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 137618 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -23020,30 +23020,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 27964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11775 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24955 2026-10-24 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./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1034 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1155 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8702 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Lineage__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6899 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6818 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7172 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13549 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21706 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6872 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4771 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Topcom/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Topcom/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 67031 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump │ │ │ @@ -23382,15 +23382,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7386 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 166062 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Chirotope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 815 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulation.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2275 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2276 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 18048 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_all__Triangulations_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_bistellar__Flip.out │ │ │ -rw-r--r-- 0 root (0) root (0) 440 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_chirotope.out │ │ │ -rw-r--r-- 0 root (0) root (0) 388 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_degree__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_delaunay__Subdivision.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_delaunay__Weights.out │ │ │ -rw-r--r-- 0 root (0) root (0) 383 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_fine__Star__Triangulation.out │ │ │ @@ -23450,15 +23450,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11148 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Triangulation__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_secondary__Cone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_some__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6793 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_vectors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_volume__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_wall__Circuits.html │ │ │ --rw-r--r-- 0 root (0) root (0) 39758 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 39759 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49559 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/___Betti1_lp__Triplet_rp.out │ │ │ @@ -24412,15 +24412,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_flattening__Relations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Flat__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 110 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__From__Disk.out │ │ │ -rw-r--r-- 0 root (0) root (0) 990 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Range__Of__One__Parameter__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1616 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Smoothing__Family__With__Versal__Deformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 842 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_give1683__Format.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_hilbert__Burch__Matrices.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1293 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Range.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1291 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Range.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3279 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Unfolding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1617 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_prune__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 15838 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_solving__Flattening__Relations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12188 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_test__Bound.html │ │ │ @@ -24513,15 +24513,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_map__To__Projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_non__Cartier__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 780 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_positive__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2026-10-24 01:47:31.000000 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01:47:31.000000 ./usr/share/info/SymmetricPolynomials.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 14494 2026-10-24 01:47:31.000000 ./usr/share/info/TSpreadIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 30234 2026-10-24 01:47:31.000000 ./usr/share/info/Tableaux.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 1559 2026-10-24 01:47:31.000000 ./usr/share/info/TangentCone.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 47061 2026-10-24 01:47:31.000000 ./usr/share/info/TateOnProducts.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 47062 2026-10-24 01:47:31.000000 ./usr/share/info/TateOnProducts.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 22580 2026-10-24 01:47:31.000000 ./usr/share/info/TensorComplexes.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2548 2026-10-24 01:47:31.000000 ./usr/share/info/TerraciniLoci.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 2115 2026-10-24 01:47:31.000000 ./usr/share/info/TestAudit.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 31168 2026-10-24 01:47:31.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 31171 2026-10-24 01:47:31.000000 ./usr/share/info/TestIdeals.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 15163 2026-10-24 01:47:31.000000 ./usr/share/info/Text.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 23201 2026-10-24 01:47:31.000000 ./usr/share/info/ThinSincereQuivers.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8350 2026-10-24 01:47:31.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8563 2026-10-24 01:47:31.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13976 2026-10-24 01:47:31.000000 ./usr/share/info/Topcom.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7724 2026-10-24 01:47:31.000000 ./usr/share/info/TorAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6967 2026-10-24 01:47:31.000000 ./usr/share/info/ToricHigherDirectImages.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4013 2026-10-24 01:47:31.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4015 2026-10-24 01:47:31.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9922 2026-10-24 01:47:31.000000 ./usr/share/info/ToricTopology.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 31419 2026-10-24 01:47:31.000000 ./usr/share/info/ToricVectorBundles.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2026-10-24 01:47:31.000000 ./usr/share/info/TriangularSets.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33244 2026-10-24 01:47:31.000000 ./usr/share/info/Triangulations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33247 2026-10-24 01:47:31.000000 ./usr/share/info/Triangulations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7304 2026-10-24 01:47:31.000000 ./usr/share/info/Triplets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 11010 2026-10-24 01:47:31.000000 ./usr/share/info/Tropical.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10699 2026-10-24 01:47:31.000000 ./usr/share/info/TropicalToric.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5763 2026-10-24 01:47:31.000000 ./usr/share/info/Truncations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8693 2026-10-24 01:47:31.000000 ./usr/share/info/Units.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4688 2026-10-24 01:47:31.000000 ./usr/share/info/VNumber.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8937 2026-10-24 01:47:31.000000 ./usr/share/info/Valuations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 44396 2026-10-24 01:47:31.000000 ./usr/share/info/Varieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19350 2026-10-24 01:47:31.000000 ./usr/share/info/VectorFields.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 51845 2026-10-24 01:47:31.000000 ./usr/share/info/VectorGraphics.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 41360 2026-10-24 01:47:31.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 41359 2026-10-24 01:47:31.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13242 2026-10-24 01:47:31.000000 ./usr/share/info/VirtualResolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10436 2026-10-24 01:47:31.000000 ./usr/share/info/Visualize.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33371 2026-10-24 01:47:31.000000 ./usr/share/info/WeierstrassSemigroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 38095 2026-10-24 01:47:31.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33369 2026-10-24 01:47:31.000000 ./usr/share/info/WeierstrassSemigroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 38097 2026-10-24 01:47:31.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10618 2026-10-24 01:47:31.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33219 2026-10-24 01:47:31.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14718 2026-10-24 01:47:31.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33224 2026-10-24 01:47:31.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14719 2026-10-24 01:47:31.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19544 2026-10-24 01:47:31.000000 ./usr/share/info/WittVectors.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8865 2026-10-24 01:47:31.000000 ./usr/share/info/XML.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 49515 2026-10-24 01:47:31.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 10748 2026-10-23 21:15:50.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ ├── ./usr/share/doc/Macaulay2/A1BrouwerDegrees/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ Z2V0VHJhY2U= │ │ │ #:len=2312 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIHRyYWNlIG92ZXIg │ │ │ JGskIGZvciBhbiBlbGVtZW50IGluIGEgZmluaXRlIGRpbWVuc2lvbmFsICRrJCAtYWxnZWJyYSIs │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ aGFzTWluaW1hbE11bHQoUmluZyxJbmZpbml0ZU51bWJlcik= │ │ │ #:len=285 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU1OCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaGFzTWluaW1hbE11bHQsUmluZyxJbmZpbml0ZU51 │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ @@ -10,25 +10,25 @@ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ - -- 2.44098s elapsed │ │ │ + -- 1.33458s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ - -- 2.39145s elapsed │ │ │ + -- 1.66109s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ @@ -95,28 +95,28 @@ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i3 : elapsedTime burkeResolution(M, 7, Check => false)
│ │ │ - -- 2.44098s elapsed
│ │ │ + -- 1.33458s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o3 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o3 : Complex
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i4 : elapsedTime burkeResolution(M, 7, Check => true)
│ │ │ - -- 2.39145s elapsed
│ │ │ + -- 1.66109s elapsed
│ │ │  
│ │ │        1      3      9      27      81      243      729      2187
│ │ │  o4 = R  <-- R  <-- R  <-- R   <-- R   <-- R    <-- R    <-- R
│ │ │                                                               
│ │ │       0      1      2      3       4       5        6        7
│ │ │  
│ │ │  o4 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -23,24 +23,24 @@ │ │ │ │ i2 : M = coker vars R │ │ │ │ │ │ │ │ o2 = cokernel | a b c | │ │ │ │ │ │ │ │ 1 │ │ │ │ o2 : R-module, quotient of R │ │ │ │ i3 : elapsedTime burkeResolution(M, 7, Check => false) │ │ │ │ - -- 2.44098s elapsed │ │ │ │ + -- 1.33458s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o3 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o3 : Complex │ │ │ │ i4 : elapsedTime burkeResolution(M, 7, Check => true) │ │ │ │ - -- 2.39145s elapsed │ │ │ │ + -- 1.66109s elapsed │ │ │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ ZmFjZXRzKEFic3RyYWN0U2ltcGxpY2lhbENvbXBsZXgp │ │ │ #:len=368 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmYWNldHMsQWJzdHJhY3RTaW1wbGljaWFsQ29tcGxl │ │ ├── ./usr/share/doc/Macaulay2/AbstractToricVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ QWJzdHJhY3RUb3JpY1ZhcmlldGllcw== │ │ │ #:len=379 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibGlua3MgYWJzdHJhY3Qgc2ltcGxpY2lh │ │ │ bCAobm9ybWFsKSB0b3JpYyB2YXJpZXRpZXMgdG8gU2NodWJlcnQyIiwgRGVzY3JpcHRpb24gPT4g │ │ ├── ./usr/share/doc/Macaulay2/AdjointIdeal/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=25 │ │ │ dHJhY2VNYXRyaXgoSWRlYWwsTWF0cml4KQ== │ │ │ #:len=268 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE3MCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodHJhY2VNYXRyaXgsSWRlYWwsTWF0cml4KSwidHJh │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ YWRqdW5jdGlvblByb2Nlc3MoSWRlYWwsWlop │ │ │ #:len=310 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjc4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGp1bmN0aW9uUHJvY2VzcyxJZGVhbCxaWiksImFk │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ o8 : BettiTally │ │ │ │ │ │ i9 : c=codim I │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ i10 : elapsedTime fI=res I │ │ │ - -- .0257838s elapsed │ │ │ + -- .0460455s elapsed │ │ │ │ │ │ 1 14 33 28 8 │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o10 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ @@ -87,30 +87,30 @@ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : phi=map(P2,Pn,H); │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ i15 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .609185s elapsed │ │ │ + -- .589313s elapsed │ │ │ │ │ │ 0 1 │ │ │ o15 = total: 1 11 │ │ │ 0: 1 . │ │ │ 1: . 3 │ │ │ 2: . 8 │ │ │ │ │ │ o15 : BettiTally │ │ │ │ │ │ i16 : I'== I │ │ │ │ │ │ o16 = true │ │ │ │ │ │ i17 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 6.98144s elapsed │ │ │ + -- 5.98644s elapsed │ │ │ │ │ │ i18 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o18 = Tally{(1, 1, total: 1 2) => 5} │ │ │ 0: 1 2 │ │ │ 0 1 │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ @@ -79,40 +79,40 @@ │ │ │ 1: . . │ │ │ 2: . . │ │ │ 3: . 8 │ │ │ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : elapsedTime sub(I,H) │ │ │ - -- .0200526s elapsed │ │ │ + -- .0125973s elapsed │ │ │ │ │ │ o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) │ │ │ │ │ │ o14 : Ideal of P2 │ │ │ │ │ │ i15 : phi=map(P2,Pn,H); │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ i16 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .0565566s elapsed │ │ │ + -- .0612869s elapsed │ │ │ │ │ │ 0 1 │ │ │ o16 = total: 1 12 │ │ │ 0: 1 . │ │ │ 1: . 12 │ │ │ │ │ │ o16 : BettiTally │ │ │ │ │ │ i17 : I'== I │ │ │ │ │ │ o17 = true │ │ │ │ │ │ i18 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 1.96041s elapsed │ │ │ + -- 1.73307s elapsed │ │ │ │ │ │ i19 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o19 = Tally{(0, 34, total: 1 15) => 1} │ │ │ 0: 1 . │ │ │ 1: . . │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ @@ -154,15 +154,15 @@ │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i10 : elapsedTime fI=res I
│ │ │ - -- .0257838s elapsed
│ │ │ + -- .0460455s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn
│ │ │                                            
│ │ │        0       1        2        3        4
│ │ │  
│ │ │  o10 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2: . 12 │ │ │ │ │ │ │ │ o8 : BettiTally │ │ │ │ i9 : c=codim I │ │ │ │ │ │ │ │ o9 = 4 │ │ │ │ i10 : elapsedTime fI=res I │ │ │ │ - -- .0257838s elapsed │ │ │ │ + -- .0460455s elapsed │ │ │ │ │ │ │ │ 1 14 33 28 8 │ │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o10 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ @@ -222,15 +222,15 @@ │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .609185s elapsed
│ │ │ + -- .589313s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │ @@ -243,15 +243,15 @@
│ │ │  
│ │ │  o16 = true
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 6.98144s elapsed
│ │ │ + -- 5.98644s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                            0 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -110,28 +110,28 @@
│ │ │ │            6: . 7
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o14 : RingMap P2 <-- Pn
│ │ │ │  i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .609185s elapsed
│ │ │ │ + -- .589313s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o15 = total: 1 11
│ │ │ │            0: 1  .
│ │ │ │            1: .  3
│ │ │ │            2: .  8
│ │ │ │  
│ │ │ │  o15 : BettiTally
│ │ │ │  i16 : I'== I
│ │ │ │  
│ │ │ │  o16 = true
│ │ │ │  i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 6.98144s elapsed
│ │ │ │ + -- 5.98644s elapsed
│ │ │ │  i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                            0 1
│ │ │ │  o18 = Tally{(1, 1, total: 1 2) => 5}
│ │ │ │                         0: 1 2
│ │ │ │                            0 1
│ │ │ │              (1, 3, total: 1 3) => 8
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o13 : BettiTally
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : elapsedTime sub(I,H)
│ │ │ - -- .0200526s elapsed
│ │ │ + -- .0125973s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2
│ │ │ │ │ │ │ │ │ │ │ │ @@ -215,15 +215,15 @@ │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .0565566s elapsed
│ │ │ + -- .0612869s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally
│ │ │ @@ -235,15 +235,15 @@ │ │ │ │ │ │ o17 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 1.96041s elapsed
│ │ │ + -- 1.73307s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │  
│ │ │                             0  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,36 +82,36 @@
│ │ │ │            0: 1 .
│ │ │ │            1: . .
│ │ │ │            2: . .
│ │ │ │            3: . 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : elapsedTime sub(I,H)
│ │ │ │ - -- .0200526s elapsed
│ │ │ │ + -- .0125973s elapsed
│ │ │ │  
│ │ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │ │  
│ │ │ │  o14 : Ideal of P2
│ │ │ │  i15 : phi=map(P2,Pn,H);
│ │ │ │  
│ │ │ │  o15 : RingMap P2 <-- Pn
│ │ │ │  i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ │ - -- .0565566s elapsed
│ │ │ │ + -- .0612869s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 1.96041s elapsed
│ │ │ │ + -- 1.73307s elapsed
│ │ │ │  i19 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                             0  1
│ │ │ │  o19 = Tally{(0, 34, total: 1 15) => 1}
│ │ │ │                          0: 1  .
│ │ │ │                          1: .  .
│ │ │ │                          2: .  .
│ │ ├── ./usr/share/doc/Macaulay2/AlgebraicSplines/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c3RhbmxleVJlaXNuZXIoTGlzdCxMaXN0KQ==
│ │ │  #:len=286
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjI0NCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3RhbmxleVJlaXNuZXIsTGlzdCxMaXN0KSwic3Rh
│ │ ├── ./usr/share/doc/Macaulay2/AllMarkovBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  cmFuZG9tTWFya292KE1hdHJpeCk=
│ │ │  #:len=265
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyYW5kb21NYXJrb3YsTWF0cml4KSwicmFuZG9tTWFy
│ │ ├── ./usr/share/doc/Macaulay2/AnalyzeSheafOnP1/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  YW5hbHl6ZShNb2R1bGUp
│ │ │  #:len=251
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbmFseXplLE1vZHVsZSksImFuYWx5emUoTW9kdWxl
│ │ ├── ./usr/share/doc/Macaulay2/AssociativeAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c2VxdWVuY2VUb1ZhcmlhYmxlU3ltYm9scw==
│ │ │  #:len=228
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgsICJ1bmRvY3VtZW50ZWQiID0+IHRy
│ │ │  dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7InNlcXVlbmNl
│ │ ├── ./usr/share/doc/Macaulay2/BGG/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  dGF0ZVJlc29sdXRpb24=
│ │ │  #:len=2005
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluaXRlIHBpZWNlIG9mIHRoZSBUYXRl
│ │ │  IHJlc29sdXRpb24iLCAibGluZW51bSIgPT4gNzUzLCBJbnB1dHMgPT4ge1NQQU57VFR7Im0ifSwi
│ │ ├── ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out
│ │ │ @@ -114,26 +114,26 @@
│ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │  
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │  
│ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.450017s (cpu); 0.376863s (thread); 0s (gc)
│ │ │ + -- used 0.592468s (cpu); 0.41681s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │  o14 : BettiTally
│ │ │  
│ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.468351s (cpu); 0.403732s (thread); 0s (gc)
│ │ │ + -- used 0.487778s (cpu); 0.398918s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html
│ │ │ @@ -258,15 +258,15 @@
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.450017s (cpu); 0.376863s (thread); 0s (gc)
│ │ │ + -- used 0.592468s (cpu); 0.41681s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │ @@ -277,15 +277,15 @@
│ │ │          

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

│ │ │          

│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,27 +192,27 @@
│ │ │ │  o18 : ActionOnComplex
│ │ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │ │  
│ │ │ │  o19 = Complex with 6 actors
│ │ │ │  
│ │ │ │  o19 : ActionOnComplex
│ │ │ │  i20 : elapsedTime a1 = character A1
│ │ │ │ - -- 1.06712s elapsed
│ │ │ │ + -- .573334s elapsed
│ │ │ │  
│ │ │ │  o20 = Character over kk
│ │ │ │  
│ │ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │ │                   |                4    2         4    2
│ │ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │ │        (2, {13})  |  1   1  1  1            1                  1
│ │ │ │  
│ │ │ │  o20 : Character
│ │ │ │  i21 : elapsedTime a2 = character A2
│ │ │ │ - -- 36.3539s elapsed
│ │ │ │ + -- 22.2743s elapsed
│ │ │ │  
│ │ │ │  o21 = Character over kk
│ │ │ │  
│ │ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │ │                   |                4    2         4    2
│ │ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ │ @@ -319,15 +319,15 @@
│ │ │ │  i30 : M = Is2 / I2;
│ │ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │ │  
│ │ │ │  o31 = Module with 6 actors
│ │ │ │  
│ │ │ │  o31 : ActionOnGradedModule
│ │ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ │ - -- 16.4277s elapsed
│ │ │ │ + -- 9.52784s elapsed
│ │ │ │  
│ │ │ │  o32 = Character over kk
│ │ │ │  
│ │ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │ │  
│ │ │ │  o32 : Character
│ │ │ │  i33 : b/T
│ │ ├── ./usr/share/doc/Macaulay2/BinomialEdgeIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
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│ │ │  #:format=standard
│ │ │  # End of header
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│ │ │  YmVp
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│ │ ├── ./usr/share/doc/Macaulay2/Binomials/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
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│ │ │  #:format=standard
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│ │ ├── ./usr/share/doc/Macaulay2/BoijSoederberg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
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│ │ ├── ./usr/share/doc/Macaulay2/Book3264Examples/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
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│ │ ├── ./usr/share/doc/Macaulay2/BooleanGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  Z2JCb29sZWFuKElkZWFsKQ==
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│ │ ├── ./usr/share/doc/Macaulay2/Brackets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  R0NBbGdlYnJh
│ │ │  #:len=1521
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiAoRElWe1BBUkF7VEVYeyJBbiBvYmpl
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│ │ ├── ./usr/share/doc/Macaulay2/Browse/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  QnJvd3Nl
│ │ │  #:len=397
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBtZXRob2QgZm9yIGJyb3dzaW5nIGFu
│ │ │  ZCBleGFtaW5pbmcgTWFjYXVsYXkyIGRhdGEgc3RydWN0dXJlcyIsIERlc2NyaXB0aW9uID0+ICgi
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  aXNTeXp5Z3koTW9kdWxlLFpaKQ==
│ │ │  #:len=228
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc1N5enlneSxNb2R1bGUsWlopLCJpc1N5enlneShN
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/example-output/_bruns.out
│ │ │ @@ -230,15 +230,15 @@
│ │ │            0: 1 . . . .
│ │ │            1: . 4 2 . .
│ │ │            2: . 1 6 5 1
│ │ │  
│ │ │  o22 : BettiTally
│ │ │  
│ │ │  i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.402268s (cpu); 0.246249s (thread); 0s (gc)
│ │ │ + -- used 0.364543s (cpu); 0.233221s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
│ │ │  
│ │ │  i24 : betti res j
│ │ │  
│ │ │               0 1 2 3 4
│ │ │  o24 = total: 1 3 6 5 1
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html
│ │ │ @@ -385,15 +385,15 @@
│ │ │  
│ │ │  o22 : BettiTally
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          

│ │ │                
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.468351s (cpu); 0.403732s (thread); 0s (gc)
│ │ │ + -- used 0.487778s (cpu); 0.398918s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │ ├── html2text {}
│ │ │ │ @@ -161,30 +161,30 @@
│ │ │ │        | -30a-29b -29a-24b -47a-39b 38a+2b   |
│ │ │ │        | 19a+19b  -38a-16b -18a-13b 16a+22b  |
│ │ │ │        | -10a-29b 39a+21b  -43a-15b 45a-34b  |
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o13 : Matrix A  <-- A
│ │ │ │  i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ │ - -- used 0.450017s (cpu); 0.376863s (thread); 0s (gc)
│ │ │ │ + -- used 0.592468s (cpu); 0.41681s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o14 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ │ │  
│ │ │ │  o14 : BettiTally
│ │ │ │  With the form pureResolution(p,q,D) we can directly create the situation of
│ │ │ │  pureResolution(M,D) where M is generic product(m_i+1) x #D-1+sum(m_i) matrix of
│ │ │ │  linear forms defined over a ring with product(m_i+1) * #D-1+sum(m_i) variables
│ │ │ │  of characteristic p, created by the script. For a given number of variables in
│ │ │ │  A this runs much faster than taking a random matrix M.
│ │ │ │  i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ │ - -- used 0.468351s (cpu); 0.403732s (thread); 0s (gc)
│ │ │ │ + -- used 0.487778s (cpu); 0.398918s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o15 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BIBasis/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  YmlCYXNpcyguLi4sdG9Hcm9lYm5lcj0+Li4uKQ==
│ │ │  #:len=240
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODMsIHN5bWJvbCBEb2N1bWVudFRhZyA9
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│ │ ├── ./usr/share/doc/Macaulay2/BeginningMacaulay2/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  QmVnaW5uaW5nTWFjYXVsYXky
│ │ │  #:len=29304
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│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  QmVuY2htYXJr
│ │ │  #:len=347
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│ │ │  bWFya3MiLCBEZXNjcmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoaXMgcGFja2FnZSBwcm92
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
│ │ │ @@ -1,10 +1,10 @@
│ │ │  -- -*- M2-comint -*- hash: 1330545576567
│ │ │  
│ │ │  i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sat Oct 24 03:20:06 UTC 2026
│ │ │ --- Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Thu Oct 29 12:11:12 UTC 2026
│ │ │ +-- Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2100.028  
│ │ │  -- Macaulay2 1.26.06, compiled with gcc 16.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .30315 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .147426 seconds
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
│ │ │ @@ -80,19 +80,19 @@
│ │ │          

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

│ │ │          

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

│ │ │                
i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sat Oct 24 03:20:06 UTC 2026
│ │ │ --- Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Thu Oct 29 12:11:12 UTC 2026
│ │ │ +-- Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2100.028  
│ │ │  -- Macaulay2 1.26.06, compiled with gcc 16.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .30315 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .147426 seconds
│ │ │              

│ │ │        
│ │ │        

│ │ │          

│ │ │            
For the programmer

│ │ │ ├── html2text {}
│ │ │ │ @@ -23,18 +23,18 @@
│ │ │ │  "gb3445" -- gb of an ideal with elements of degree 3,4,4,5 in 8 variables
│ │ │ │  "gbB148" -- gb of Bayesian graph ideal #148
│ │ │ │  "res39" -- res of a generic 3 by 9 matrix over ZZ/101
│ │ │ │  "resG25" -- res of the coordinate ring of Grassmannian(2,5)
│ │ │ │  "yang-gb1" -- an example of Yang-Hui He arising in string theory
│ │ │ │  "yang-subring" -- an example of Yang-Hui He
│ │ │ │  i1 : runBenchmarks "res39"
│ │ │ │ --- beginning computation Sat Oct 24 03:20:06 UTC 2026
│ │ │ │ --- Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1
│ │ │ │ -(2026-10-15) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250
│ │ │ │ +-- beginning computation Thu Oct 29 12:11:12 UTC 2026
│ │ │ │ +-- Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2100.028
│ │ │ │  -- Macaulay2 1.26.06, compiled with gcc 16.2.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .30315 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .147426 seconds
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_u_n_B_e_n_c_h_m_a_r_k_s is a _c_o_m_m_a_n_d.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Benchmark.m2:319:0.
│ │ ├── ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:file=rawdocumentation-dcba-8.db
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│ │ │  #:format=standard
│ │ │  # End of header
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│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  bHVlICIsIi0xIn0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
│ │ │  YmVydGluaVplcm9EaW1Tb2x2ZSxWZXJib3NlXSwiYmVydGluaVplcm9EaW1Tb2x2ZSguLi4sVmVy
│ │ │  Ym9zZT0+Li4uKSIsIkJlcnRpbmkifSwiVmVyYm9zZSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │              
  • HomVariableGroup => ..., default value {}, an option to group variables and use multihomogeneous homotopies
    • 
│ │ │              
  • M2Precision (missing documentation)
│ │ │   => ..., default value 53, 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23669-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-30102-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list whose entries are lists of solutions for each target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-23669-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-30102-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list whose entries are lists of solutions for each
│ │ │ │              target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │   => ..., default value 53, 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │              
  • RandomReal (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23669-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-30102-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S0, a list, a list of solutions to the target system
      • 
│ │ │            
    
│ │ │          
    • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │              value {},
│ │ │ │            o HomVariableGroup (missing documentation) => ..., default value {},
│ │ │ │            o M2Precision (missing documentation) => ..., default value 53,
│ │ │ │            o OutputStyle (missing documentation) => ..., default value
│ │ │ │              "OutPoints",
│ │ │ │            o RandomComplex (missing documentation) => ..., default value {},
│ │ │ │            o RandomReal (missing documentation) => ..., default value {},
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-23669-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-30102-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S0, a _l_i_s_t, a list of solutions to the target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This method calls Bertini to track a user-defined homotopy. The user needs to
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │   => ..., default value "main_data", 
│ │ │              
  • NameSolutionsFile (missing documentation)
│ │ │   => ..., default value "raw_solutions", 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23669-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-30102-0/0", Option to change directory for file storage.
    • 
│ │ │              
  • UseRegeneration (missing documentation)
│ │ │   => ..., default value -1, 
    • 
│ │ │              
  • Verbose => ..., default value false, Option to silence additional output
    • 
│ │ │            
│ │ │          
│ │ │          
  • Outputs:          
      
│ │ │              
    • S, a list, a list of points that are contained in the variety of F
      • 
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │              "OutPoints",
│ │ │ │            o _R_a_n_d_o_m_C_o_m_p_l_e_x => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │            o _R_a_n_d_o_m_R_e_a_l => ..., default value {}, an option which designates
│ │ │ │              symbols/strings/variables that will be set to be a random real
│ │ │ │              number or random complex number
│ │ │ │ -          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-23669-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-30102-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o UseRegeneration (missing documentation) => ..., default value -1,
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list of points that are contained in the variety of F
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  YWN0aW9uKE1vZHVsZSxMaXN0KQ==
│ │ │  #:len=290
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzgzNywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYWN0aW9uLE1vZHVsZSxMaXN0KSwiYWN0aW9uKE1v
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out
│ │ │ @@ -76,15 +76,15 @@
│ │ │  i8 : A = action(RI,S7)
│ │ │  
│ │ │  o8 = Complex with 15 actors
│ │ │  
│ │ │  o8 : ActionOnComplex
│ │ │  
│ │ │  i9 : elapsedTime c = character A
│ │ │ - -- .449182s elapsed
│ │ │ + -- .304193s elapsed
│ │ │  
│ │ │  o9 = Character over QQ
│ │ │        
│ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out
│ │ │ @@ -100,15 +100,15 @@
│ │ │  i6 : A=action(RI,S6)
│ │ │  
│ │ │  o6 = Complex with 11 actors
│ │ │  
│ │ │  o6 : ActionOnComplex
│ │ │  
│ │ │  i7 : elapsedTime c=character A
│ │ │ - -- 1.06265s elapsed
│ │ │ + -- .430787s elapsed
│ │ │  
│ │ │  o7 = Character over QQ
│ │ │        
│ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out
│ │ │ @@ -187,28 +187,28 @@
│ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │  
│ │ │  o19 = Complex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
│ │ │  
│ │ │  i20 : elapsedTime a1 = character A1
│ │ │ - -- 1.06712s elapsed
│ │ │ + -- .573334s elapsed
│ │ │  
│ │ │  o20 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │                   |                4    2         4    2
│ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │        (2, {13})  |  1   1  1  1            1                  1
│ │ │  
│ │ │  o20 : Character
│ │ │  
│ │ │  i21 : elapsedTime a2 = character A2
│ │ │ - -- 36.3539s elapsed
│ │ │ + -- 22.2743s elapsed
│ │ │  
│ │ │  o21 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │                   |                4    2         4    2
│ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ @@ -308,15 +308,15 @@
│ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │  
│ │ │  o31 = Module with 6 actors
│ │ │  
│ │ │  o31 : ActionOnGradedModule
│ │ │  
│ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ - -- 16.4277s elapsed
│ │ │ + -- 9.52784s elapsed
│ │ │  
│ │ │  o32 = Character over kk
│ │ │         
│ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │  
│ │ │  o32 : Character
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html
│ │ │ @@ -167,15 +167,15 @@
│ │ │  
│ │ │  o8 : ActionOnComplex
    • 
│ │ │              
    
│ │ │                
    i9 : elapsedTime c = character A
│ │ │ - -- .449182s elapsed
│ │ │ + -- .304193s elapsed
│ │ │  
│ │ │  o9 = Character over QQ
│ │ │        
│ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ │ ├── html2text {}
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │  o7 : List
│ │ │ │  i8 : A = action(RI,S7)
│ │ │ │  
│ │ │ │  o8 = Complex with 15 actors
│ │ │ │  
│ │ │ │  o8 : ActionOnComplex
│ │ │ │  i9 : elapsedTime c = character A
│ │ │ │ - -- .449182s elapsed
│ │ │ │ + -- .304193s elapsed
│ │ │ │  
│ │ │ │  o9 = Character over QQ
│ │ │ │  
│ │ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o6 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime c=character A
│ │ │ - -- 1.06265s elapsed
│ │ │ + -- .430787s elapsed
│ │ │  
│ │ │  o7 = Character over QQ
│ │ │        
│ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ │ ├── html2text {}
│ │ │ │ @@ -113,15 +113,15 @@
│ │ │ │  o5 : List
│ │ │ │  i6 : A=action(RI,S6)
│ │ │ │  
│ │ │ │  o6 = Complex with 11 actors
│ │ │ │  
│ │ │ │  o6 : ActionOnComplex
│ │ │ │  i7 : elapsedTime c=character A
│ │ │ │ - -- 1.06265s elapsed
│ │ │ │ + -- .430787s elapsed
│ │ │ │  
│ │ │ │  o7 = Character over QQ
│ │ │ │  
│ │ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o19 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime a1 = character A1
│ │ │ - -- 1.06712s elapsed
│ │ │ + -- .573334s elapsed
│ │ │  
│ │ │  o20 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │                   |                4    2         4    2
│ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │ @@ -331,15 +331,15 @@
│ │ │  
│ │ │  o20 : Character
    
│ │ │              
    
│ │ │                
    i21 : elapsedTime a2 = character A2
│ │ │ - -- 36.3539s elapsed
│ │ │ + -- 22.2743s elapsed
│ │ │  
│ │ │  o21 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │                   |                4    2         4    2
│ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ @@ -483,15 +483,15 @@
│ │ │  
│ │ │  o31 : ActionOnGradedModule
    
│ │ │              
    
│ │ │                
    i32 : elapsedTime b = character(B,21)
│ │ │ - -- 16.4277s elapsed
│ │ │ + -- 9.52784s elapsed
│ │ │  
│ │ │  o32 = Character over kk
│ │ │         
│ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │  
│ │ │  o32 : Character
    
│ │ │              
    
│ │ │                
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.402268s (cpu); 0.246249s (thread); 0s (gc)
│ │ │ + -- used 0.364543s (cpu); 0.233221s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
    
│ │ │              
    
│ │ │                
    i24 : betti res j
│ │ │ ├── html2text {}
│ │ │ │ @@ -230,15 +230,15 @@
│ │ │ │  o22 = total: 1 5 8 5 1
│ │ │ │            0: 1 . . . .
│ │ │ │            1: . 4 2 . .
│ │ │ │            2: . 1 6 5 1
│ │ │ │  
│ │ │ │  o22 : BettiTally
│ │ │ │  i23 : time j=bruns F.dd_3;
│ │ │ │ - -- used 0.402268s (cpu); 0.246249s (thread); 0s (gc)
│ │ │ │ + -- used 0.364543s (cpu); 0.233221s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of S
│ │ │ │  i24 : betti res j
│ │ │ │  
│ │ │ │               0 1 2 3 4
│ │ │ │  o24 = total: 1 3 6 5 1
│ │ │ │            0: 1 . . . .
│ │ ├── ./usr/share/doc/Macaulay2/CellularResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  c3ViY29tcGxleChDZWxsQ29tcGxleCxMaXN0KQ==
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU2OCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ViY29tcGxleCxDZWxsQ29tcGxleCxMaXN0KSwi
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  RUtSZXNvbHV0aW9uKE1vbm9taWFsSWRlYWwp
│ │ │  #:len=292
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA3MSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoRUtSZXNvbHV0aW9uLE1vbm9taWFsSWRlYWwpLCJF
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize_lp__Chain__Complex_rp.out
│ │ │ @@ -63,15 +63,15 @@
│ │ │  o11 : ChainComplex
│ │ │  
│ │ │  i12 : isMinimalChainComplex E
│ │ │  
│ │ │  o12 = false
│ │ │  
│ │ │  i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.296573s (cpu); 0.217064s (thread); 0s (gc)
│ │ │ + -- used 0.303686s (cpu); 0.224105s (thread); 0s (gc)
│ │ │  
│ │ │  i14 : isQuasiIsomorphism m
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : E[1] == source m
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out
│ │ │ @@ -27,18 +27,18 @@
│ │ │  i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5);
│ │ │  
│ │ │  i6 : mods = for i from 0 to max C list pushForward(f, C_i);
│ │ │  
│ │ │  i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S));
│ │ │  
│ │ │  i8 : time m = resolutionOfChainComplex C;
│ │ │ - -- used 0.0995083s (cpu); 0.0987657s (thread); 0s (gc)
│ │ │ + -- used 0.0918956s (cpu); 0.0933198s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.113261s (cpu); 0.1149s (thread); 0s (gc)
│ │ │ + -- used 0.112808s (cpu); 0.115634s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │            0: 1  3  3   1   .   .   .   .
│ │ │            1: .  .  1   3   3   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html
│ │ │ @@ -186,15 +186,15 @@
│ │ │          
    
│ │ │            
    Now we minimize the result. The free summand we added to the end maps to zero, and thus is part of the minimization.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -306,15 +306,15 @@
│ │ │  
│ │ │  o21 : A
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -160,15 +160,15 @@
│ │ │ │  
│ │ │ │                2              2
│ │ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │ │                0 3    1 2 4   2 5
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time csmK=CSM(A,K)
│ │ │ │ - -- used 0.399939s (cpu); 0.310473s (thread); 0s (gc)
│ │ │ │ + -- used 1.05252s (cpu); 0.349864s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2     2         2    2            2
│ │ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │ │  
│ │ │ │  o15 : A
│ │ │ │  i16 : csmKHash= CSM(A,K,Output=>HashForm)
│ │ │ │ @@ -199,15 +199,15 @@
│ │ │ │  
│ │ │ │          2 2     2         2     2             2
│ │ │ │  o21 = 9h h  + 9h h  + 9h h  + 3h  + 7h h  + 3h  + 3h  + 2h
│ │ │ │          1 2     1 2     1 2     1     1 2     2     1     2
│ │ │ │  
│ │ │ │  o21 : A
│ │ │ │  i22 : time CSM(A,K,m)
│ │ │ │ - -- used 0.182686s (cpu); 0.107681s (thread); 0s (gc)
│ │ │ │ + -- used 0.194621s (cpu); 0.0881138s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2     2         2    2            2
│ │ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │ │  
│ │ │ │  o22 : A
│ │ │ │  In the case where the ambient space is a toric variety which is not a product
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.296573s (cpu); 0.217064s (thread); 0s (gc)
    
│ │ │ + -- used 0.303686s (cpu); 0.224105s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i14 : isQuasiIsomorphism m
│ │ │  
│ │ │  o14 = true
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,15 +81,15 @@
│ │ │ │  o11 : ChainComplex
│ │ │ │  i12 : isMinimalChainComplex E
│ │ │ │  
│ │ │ │  o12 = false
│ │ │ │  Now we minimize the result. The free summand we added to the end maps to zero,
│ │ │ │  and thus is part of the minimization.
│ │ │ │  i13 : time m = minimize (E[1]);
│ │ │ │ - -- used 0.296573s (cpu); 0.217064s (thread); 0s (gc)
│ │ │ │ + -- used 0.303686s (cpu); 0.224105s (thread); 0s (gc)
│ │ │ │  i14 : isQuasiIsomorphism m
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : E[1] == source m
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  i16 : E' = target m
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html
│ │ │ @@ -134,21 +134,21 @@
│ │ │                  		
│ │ │                
    i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S));
    
│ │ │              
    
│ │ │                
    i8 : time m = resolutionOfChainComplex C;
│ │ │ - -- used 0.0995083s (cpu); 0.0987657s (thread); 0s (gc)
    
│ │ │ + -- used 0.0918956s (cpu); 0.0933198s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.113261s (cpu); 0.1149s (thread); 0s (gc)
    
│ │ │ + -- used 0.112808s (cpu); 0.115634s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,17 +49,17 @@
│ │ │ │  
│ │ │ │  o4 : RingMap R <-- S
│ │ │ │  i5 : C = res(R^1/(ideal vars R))**(R^1/(ideal vars R)^5);
│ │ │ │  i6 : mods = for i from 0 to max C list pushForward(f, C_i);
│ │ │ │  i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-
│ │ │ │  1),mods_i,substitute(matrix C.dd_i,S));
│ │ │ │  i8 : time m = resolutionOfChainComplex C;
│ │ │ │ - -- used 0.0995083s (cpu); 0.0987657s (thread); 0s (gc)
│ │ │ │ + -- used 0.0918956s (cpu); 0.0933198s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.113261s (cpu); 0.1149s (thread); 0s (gc)
│ │ │ │ + -- used 0.112808s (cpu); 0.115634s (thread); 0s (gc)
│ │ │ │  i10 : betti source m
│ │ │ │  
│ │ │ │               0  1  2   3   4   5   6   7
│ │ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │ │            0: 1  3  3   1   .   .   .   .
│ │ │ │            1: .  .  1   3   3   .   .   .
│ │ │ │            2: .  1  3   3   2   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexOperations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  c3ltMihDb21wbGV4KQ==
│ │ │  #:len=259
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW0yLENvbXBsZXgpLCJzeW0yKENvbXBsZXgpIiwi
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  TXVsdGlQcm9qQ29vcmRSaW5n
│ │ │  #:len=2206
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQSBxdWljayB3YXkgdG8gYnVpbGQgdGhl
│ │ │  IGNvb3JkaW5hdGUgcmluZyBvZiBhIHByb2R1Y3Qgb2YgcHJvamVjdGl2ZSBzcGFjZXMiLCAibGlu
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out
│ │ │ @@ -83,15 +83,15 @@
│ │ │                2              2
│ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │                0 3    1 2 4   2 5
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.399939s (cpu); 0.310473s (thread); 0s (gc)
│ │ │ + -- used 1.05252s (cpu); 0.349864s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
│ │ │  
│ │ │ @@ -124,15 +124,15 @@
│ │ │          2 2     2         2     2             2
│ │ │  o21 = 9h h  + 9h h  + 9h h  + 3h  + 7h h  + 3h  + 3h  + 2h
│ │ │          1 2     1 2     1 2     1     1 2     2     1     2
│ │ │  
│ │ │  o21 : A
│ │ │  
│ │ │  i22 : time CSM(A,K,m)
│ │ │ - -- used 0.182686s (cpu); 0.107681s (thread); 0s (gc)
│ │ │ + -- used 0.194621s (cpu); 0.0881138s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out
│ │ │ @@ -9,28 +9,28 @@
│ │ │  i2 : U = toricProjectiveSpace 7
│ │ │  
│ │ │  o2 = U
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │  
│ │ │  i3 : time CSM U
│ │ │ - -- used 0.272841s (cpu); 0.176546s (thread); 0s (gc)
│ │ │ + -- used 0.257079s (cpu); 0.173932s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │  o3 : -----------------------------------------------------------------------------------------------
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
│ │ │  
│ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.414662s (cpu); 0.322186s (thread); 0s (gc)
│ │ │ + -- used 0.371629s (cpu); 0.288751s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out
│ │ │ @@ -18,29 +18,29 @@
│ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │  
│ │ │  i4 : I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.553623s (cpu); 0.383064s (thread); 0s (gc)
│ │ │ + -- used 0.860943s (cpu); 0.358135s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o5 : ------
│ │ │          6
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.31313s (cpu); 1.85451s (thread); 0s (gc)
│ │ │ + -- used 1.95658s (cpu); 1.72724s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -53,29 +53,29 @@
│ │ │  i8 : S=QQ[s_0..s_3];
│ │ │  
│ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │  
│ │ │  o9 : Ideal of S
│ │ │  
│ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.302112s (cpu); 0.205175s (thread); 0s (gc)
│ │ │ + -- used 0.259878s (cpu); 0.181526s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0879468s (cpu); 0.087955s (thread); 0s (gc)
│ │ │ + -- used 0.0860362s (cpu); 0.0860412s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out
│ │ │ @@ -21,20 +21,20 @@
│ │ │               2                                                        2
│ │ │       - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x )
│ │ │               3         0 4        1 4         2 4         3 4         4
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0376966s (cpu); 0.0369951s (thread); 0s (gc)
│ │ │ + -- used 0.0856686s (cpu); 0.05264s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.257354s (cpu); 0.164445s (thread); 0s (gc)
│ │ │ + -- used 0.255586s (cpu); 0.155415s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
│ │ │  
│ │ │  i6 : EulerIHash=Euler(I,Output=>HashForm);
│ │ │  
│ │ │  i7 : A=ring EulerIHash#"CSM"
│ │ │  
│ │ │ @@ -62,20 +62,20 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x )
│ │ │          0 3
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.0792323s (cpu); 0.0754648s (thread); 0s (gc)
│ │ │ + -- used 0.146322s (cpu); 0.0793595s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.195562s (cpu); 0.103705s (thread); 0s (gc)
│ │ │ + -- used 0.201422s (cpu); 0.0849565s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │  
│ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out
│ │ │ @@ -13,12 +13,12 @@
│ │ │              2    2    2
│ │ │  o3 = ideal(x  + x  + x  - 1)
│ │ │              1    2    3
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time EulerAffine I
│ │ │ - -- used 0.047108s (cpu); 0.0467349s (thread); 0s (gc)
│ │ │ + -- used 0.0696694s (cpu); 0.0541544s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 1.72196s (cpu); 1.16792s (thread); 0s (gc)
│ │ │ + -- used 3.89676s (cpu); 1.14809s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │  o3 : ----------
│ │ │          3   3
│ │ │        (h , h )
│ │ │          1   2
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 1.40622s (cpu); 1.05667s (thread); 0s (gc)
│ │ │ + -- used 3.83569s (cpu); 1.09282s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out
│ │ │ @@ -3,43 +3,43 @@
│ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 0.618754s (cpu); 0.466376s (thread); 0s (gc)
│ │ │ + -- used 0.828408s (cpu); 0.42103s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.033987s (cpu); 0.0334213s (thread); 0s (gc)
│ │ │ + -- used 0.062296s (cpu); 0.0336924s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.0322268s (cpu); 0.0317709s (thread); 0s (gc)
│ │ │ + -- used 0.0587522s (cpu); 0.0339297s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM I
│ │ │ - -- used 1.19911s (cpu); 0.877209s (thread); 0s (gc)
│ │ │ + -- used 2.36867s (cpu); 0.932237s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          7
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.310562s (cpu); 0.233341s (thread); 0s (gc)
│ │ │ + -- used 0.551817s (cpu); 0.218256s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html
│ │ │ @@ -239,15 +239,15 @@
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.399939s (cpu); 0.310473s (thread); 0s (gc)
│ │ │ + -- used 1.05252s (cpu); 0.349864s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o15 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o15 : A
    
│ │ │              
    
│ │ │                
    i22 : time CSM(A,K,m)
│ │ │ - -- used 0.182686s (cpu); 0.107681s (thread); 0s (gc)
│ │ │ + -- used 0.194621s (cpu); 0.0881138s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2    2            2
│ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │  
│ │ │  o22 : A
    
│ │ │              
    
│ │ │                
    i3 : time CSM U
│ │ │ - -- used 0.272841s (cpu); 0.176546s (thread); 0s (gc)
│ │ │ + -- used 0.257079s (cpu); 0.173932s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ @@ -93,15 +93,15 @@
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
    
│ │ │              
    
│ │ │                
    i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.414662s (cpu); 0.322186s (thread); 0s (gc)
│ │ │ + -- used 0.371629s (cpu); 0.288751s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,30 +16,30 @@
│ │ │ │  o1 : Package
│ │ │ │  i2 : U = toricProjectiveSpace 7
│ │ │ │  
│ │ │ │  o2 = U
│ │ │ │  
│ │ │ │  o2 : NormalToricVariety
│ │ │ │  i3 : time CSM U
│ │ │ │ - -- used 0.272841s (cpu); 0.176546s (thread); 0s (gc)
│ │ │ │ + -- used 0.257079s (cpu); 0.173932s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ │ │  o3 : --------------------------------------------------------------------------
│ │ │ │  ---------------------
│ │ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x ,
│ │ │ │  - x  + x , - x  + x )
│ │ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5
│ │ │ │  0    6     0    7
│ │ │ │  i4 : time CSM(U,CheckSmooth=>false)
│ │ │ │ - -- used 0.414662s (cpu); 0.322186s (thread); 0s (gc)
│ │ │ │ + -- used 0.371629s (cpu); 0.288751s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.553623s (cpu); 0.383064s (thread); 0s (gc)
│ │ │ + -- used 0.860943s (cpu); 0.358135s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -114,15 +114,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.31313s (cpu); 1.85451s (thread); 0s (gc)
│ │ │ + -- used 1.95658s (cpu); 1.72724s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o9 : Ideal of S
    
│ │ │              
    
│ │ │                
    i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.302112s (cpu); 0.205175s (thread); 0s (gc)
│ │ │ + -- used 0.259878s (cpu); 0.181526s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │ @@ -164,15 +164,15 @@
│ │ │          h
│ │ │           1
    
│ │ │              
    
│ │ │                
    i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0879468s (cpu); 0.087955s (thread); 0s (gc)
│ │ │ + -- used 0.0860362s (cpu); 0.0860412s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o11 = 3H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │           4
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,28 +32,28 @@
│ │ │ │  using the regenerative cascade implemented in Bertini. This is done by choosing
│ │ │ │  the option bertini, provided Bertini is _i_n_s_t_a_l_l_e_d_ _a_n_d_ _c_o_n_f_i_g_u_r_e_d.
│ │ │ │  i3 : R=ZZ/32749[v_0..v_5];
│ │ │ │  i4 : I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 0.553623s (cpu); 0.383064s (thread); 0s (gc)
│ │ │ │ + -- used 0.860943s (cpu); 0.358135s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │ │         1      1      1      1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o5 : ------
│ │ │ │          6
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ │ - -- used 2.31313s (cpu); 1.85451s (thread); 0s (gc)
│ │ │ │ + -- used 1.95658s (cpu); 1.72724s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         5      4      3      2
│ │ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          6
│ │ │ │ @@ -62,28 +62,28 @@
│ │ │ │  
│ │ │ │  o7 = 2
│ │ │ │  i8 : S=QQ[s_0..s_3];
│ │ │ │  i9 : K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
│ │ │ │  
│ │ │ │  o9 : Ideal of S
│ │ │ │  i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ │ - -- used 0.302112s (cpu); 0.205175s (thread); 0s (gc)
│ │ │ │ + -- used 0.259878s (cpu); 0.181526s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.0879468s (cpu); 0.087955s (thread); 0s (gc)
│ │ │ │ + -- used 0.0860362s (cpu); 0.0860412s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o11 = 3H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html
│ │ │ @@ -130,23 +130,23 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0376966s (cpu); 0.0369951s (thread); 0s (gc)
│ │ │ + -- used 0.0856686s (cpu); 0.05264s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : time Euler I
│ │ │ - -- used 0.257354s (cpu); 0.164445s (thread); 0s (gc)
│ │ │ + -- used 0.255586s (cpu); 0.155415s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │                
    i6 : EulerIHash=Euler(I,Output=>HashForm);
    
│ │ │ @@ -194,23 +194,23 @@
│ │ │          
    
│ │ │            
    Note that the ideal J above is a complete intersection, thus we may change the method option which may speed computation in some cases. We may also note that the ideal generated by the first 2 generators of I defines a smooth scheme and input this information into the method. This may also improve computation speed.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.0792323s (cpu); 0.0754648s (thread); 0s (gc)
│ │ │ + -- used 0.146322s (cpu); 0.0793595s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    
│ │ │              
    
│ │ │                
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.195562s (cpu); 0.103705s (thread); 0s (gc)
│ │ │ + -- used 0.201422s (cpu); 0.0849565s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Now consider an example in \PP^2 \times \PP^2.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -74,19 +74,19 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │               2                                                        2
│ │ │ │       - 14254x  - 11226x x  + 2653x x  + 12365x x  - 10226x x  - 12696x )
│ │ │ │               3         0 4        1 4         2 4         3 4         4
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.0376966s (cpu); 0.0369951s (thread); 0s (gc)
│ │ │ │ + -- used 0.0856686s (cpu); 0.05264s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : time Euler I
│ │ │ │ - -- used 0.257354s (cpu); 0.164445s (thread); 0s (gc)
│ │ │ │ + -- used 0.255586s (cpu); 0.155415s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 4
│ │ │ │  i6 : EulerIHash=Euler(I,Output=>HashForm);
│ │ │ │  i7 : A=ring EulerIHash#"CSM"
│ │ │ │  
│ │ │ │  o7 = A
│ │ │ │  
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  o9 : Ideal of R
│ │ │ │  Note that the ideal J above is a complete intersection, thus we may change the
│ │ │ │  method option which may speed computation in some cases. We may also note that
│ │ │ │  the ideal generated by the first 2 generators of I defines a smooth scheme and
│ │ │ │  input this information into the method. This may also improve computation
│ │ │ │  speed.
│ │ │ │  i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ │ - -- used 0.0792323s (cpu); 0.0754648s (thread); 0s (gc)
│ │ │ │ + -- used 0.146322s (cpu); 0.0793595s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 2
│ │ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ │ - -- used 0.195562s (cpu); 0.103705s (thread); 0s (gc)
│ │ │ │ + -- used 0.201422s (cpu); 0.0849565s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = 2
│ │ │ │  Now consider an example in \PP^2 \times \PP^2.
│ │ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │ │  
│ │ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │              
    
│ │ │                
    i4 : time EulerAffine I
│ │ │ - -- used 0.047108s (cpu); 0.0467349s (thread); 0s (gc)
│ │ │ + -- used 0.0696694s (cpu); 0.0541544s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Observe that the algorithm is a probabilistic algorithm and may give a wrong answer with a small but nonzero probability. Read more under probabilistic algorithm.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  
│ │ │ │              2    2    2
│ │ │ │  o3 = ideal(x  + x  + x  - 1)
│ │ │ │              1    2    3
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time EulerAffine I
│ │ │ │ - -- used 0.047108s (cpu); 0.0467349s (thread); 0s (gc)
│ │ │ │ + -- used 0.0696694s (cpu); 0.0541544s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  Observe that the algorithm is a probabilistic algorithm and may give a wrong
│ │ │ │  answer with a small but nonzero probability. Read more under _p_r_o_b_a_b_i_l_i_s_t_i_c
│ │ │ │  _a_l_g_o_r_i_t_h_m.
│ │ │ │  ********** WWaayyss ttoo uussee EEuulleerrAAffffiinnee:: **********
│ │ │ │      * EulerAffine(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 1.72196s (cpu); 1.16792s (thread); 0s (gc)
│ │ │ + -- used 3.89676s (cpu); 1.14809s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ @@ -92,15 +92,15 @@
│ │ │        (h , h )
│ │ │          1   2
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 1.40622s (cpu); 1.05667s (thread); 0s (gc)
│ │ │ + -- used 3.83569s (cpu); 1.09282s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,28 +16,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ │ - -- used 1.72196s (cpu); 1.16792s (thread); 0s (gc)
│ │ │ │ + -- used 3.89676s (cpu); 1.14809s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ │ │  o3 : ----------
│ │ │ │          3   3
│ │ │ │        (h , h )
│ │ │ │          1   2
│ │ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ │ - -- used 1.40622s (cpu); 1.05667s (thread); 0s (gc)
│ │ │ │ + -- used 3.83569s (cpu); 1.09282s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 0.618754s (cpu); 0.466376s (thread); 0s (gc)
│ │ │ + -- used 0.828408s (cpu); 0.42103s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -88,15 +88,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.033987s (cpu); 0.0334213s (thread); 0s (gc)
│ │ │ + -- used 0.062296s (cpu); 0.0336924s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -110,15 +110,15 @@
│ │ │          
    
│ │ │            
    Note that one could, equivalently, use the command Chern instead in this case.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -438,26 +438,26 @@
│ │ │  
│ │ │  o26 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -182,15 +182,15 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, -1, -1}}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ │ - -- 3.09141s elapsed
│ │ │ │ + -- 2.79897s elapsed
│ │ │ │  
│ │ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -262,42 +262,42 @@
│ │ │ │                         {2, 2, 3, 1, -4, -6} => {{0, 1, 0, 0, 0}, {{1, 1x1*x2}}}
│ │ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │ │  
│ │ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │ │  
│ │ │ │  o22 : List
│ │ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ │ - -- .3809s elapsed
│ │ │ │ + -- .43603s elapsed
│ │ │ │  
│ │ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o23 : List
│ │ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X
│ │ │ │  (0,0,1,2,0,-1))
│ │ │ │ - -- 11.5175s elapsed
│ │ │ │ + -- 8.12029s elapsed
│ │ │ │  
│ │ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o24 : List
│ │ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │ │  
│ │ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ │ - -- .397928s elapsed
│ │ │ │ + -- .428707s elapsed
│ │ │ │  
│ │ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j
│ │ │ │  (X, OO_X(0,0,1,2,-2,-1))
│ │ │ │ - -- .553978s elapsed
│ │ │ │ - -- .554009s elapsed
│ │ │ │ + -- .380703s elapsed
│ │ │ │ + -- .380729s elapsed
│ │ │ │  
│ │ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │ │  
│ │ │ │  o28 : List
│ │ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ │ │  _c_o_h_o_m_C_a_l_g computes cohomology vectors by calling CohomCalg. It also stashes
│ │ │ │  it's results in the toric variety's cache table, so computations need not be
│ │ ├── ./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  Y29tcGxleHJhbmsoUmluZ0VsZW1lbnQp
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjb21wbGV4cmFuayxSaW5nRWxlbWVudCksImNvbXBs
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  RWlzZW5idWRTaGFtYXNoVG90YWwoTW9kdWxlKQ==
│ │ │  #:len=349
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTE2OSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoRWlzZW5idWRTaGFtYXNoVG90YWwsTW9kdWxlKSwi
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : len = 10
│ │ │  
│ │ │  o6 = 10
│ │ │  
│ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.68392s (cpu); 5.01853s (thread); 0s (gc)
│ │ │ + -- used 6.45886s (cpu); 5.02278s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -140,37 +140,37 @@
│ │ │  i19 : R1 = R/ideal ff
│ │ │  
│ │ │  o19 = R1
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.232858s (cpu); 0.156146s (thread); 0s (gc)
│ │ │ + -- used 0.173287s (cpu); 0.106511s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
│ │ │  
│ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 0.980622s (cpu); 0.754248s (thread); 0s (gc)
│ │ │ + -- used 0.939527s (cpu); 0.74235s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │  
│ │ │  o21 : Complex
│ │ │  
│ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.994666s (cpu); 0.734783s (thread); 0s (gc)
│ │ │ + -- used 0.954247s (cpu); 0.731707s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out
│ │ │ @@ -2,21 +2,21 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.547511s (cpu); 0.41022s (thread); 0s (gc)
│ │ │ + -- used 0.647333s (cpu); 0.403323s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.106844s (cpu); 0.10641s (thread); 0s (gc)
│ │ │ + -- used 0.134381s (cpu); 0.111476s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 4.058e-06s (cpu); 3.486e-06s (thread); 0s (gc)
│ │ │ + -- used 3.552e-06s (cpu); 2.425e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out
│ │ │ @@ -2,23 +2,23 @@
│ │ │  
│ │ │  i1 : setRandomSeed 0
│ │ │   -- setting random seed to 0
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : twoMonomials(2,3)
│ │ │ - -- used 1.04089s (cpu); 0.839487s (thread); 0s (gc)
│ │ │ + -- used 1.01943s (cpu); 0.628818s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.552679s (cpu); 0.395293s (thread); 0s (gc)
│ │ │ + -- used 0.561911s (cpu); 0.374722s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.10407s (cpu); 0.103794s (thread); 0s (gc)
│ │ │ + -- used 0.202248s (cpu); 0.145293s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o6 = 10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time Chern I
│ │ │ - -- used 0.0322268s (cpu); 0.0317709s (thread); 0s (gc)
│ │ │ + -- used 0.0587522s (cpu); 0.0339297s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o5 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,42 +9,42 @@
│ │ │ │  input ideal is known to define a smooth subscheme setting this option to true
│ │ │ │  will speed up computations (it is set to false by default).
│ │ │ │  i1 : R = ZZ/32749[x_0..x_4];
│ │ │ │  i2 : I=ideal(random(2,R),random(2,R),random(1,R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 0.618754s (cpu); 0.466376s (thread); 0s (gc)
│ │ │ │ + -- used 0.828408s (cpu); 0.42103s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.033987s (cpu); 0.0334213s (thread); 0s (gc)
│ │ │ │ + -- used 0.062296s (cpu); 0.0336924s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o4 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o4 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  Note that one could, equivalently, use the command _C_h_e_r_n instead in this case.
│ │ │ │  i5 : time Chern I
│ │ │ │ - -- used 0.0322268s (cpu); 0.0317709s (thread); 0s (gc)
│ │ │ │ + -- used 0.0587522s (cpu); 0.0339297s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o5 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 1.19911s (cpu); 0.877209s (thread); 0s (gc)
│ │ │ + -- used 2.36867s (cpu); 0.932237s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -92,15 +92,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.310562s (cpu); 0.233341s (thread); 0s (gc)
│ │ │ + -- used 0.551817s (cpu); 0.218256s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,28 +18,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM I
│ │ │ │ - -- used 1.19911s (cpu); 0.877209s (thread); 0s (gc)
│ │ │ │ + -- used 2.36867s (cpu); 0.932237s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o3 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          7
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ │ - -- used 0.310562s (cpu); 0.233341s (thread); 0s (gc)
│ │ │ │ + -- used 0.551817s (cpu); 0.218256s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
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│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  UmluZ01hcCBDaG9yZGFsTmV0
│ │ │  #:len=1424
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYXBwbHkgcmluZyBtYXAgdG8gYSBjaG9y
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│ │ ├── ./usr/share/doc/Macaulay2/Classic/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bW9ub21pYWxJZGVhbChTdHJpbmcp
│ │ │  #:len=1235
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│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  VmFuaXNoaW5nSWRlYWw=
│ │ │  #:len=1237
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmFuaXNoaW5nIGlkZWFsIG9mIGFuIGV2
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│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  Y29ob21DYWxnKE5vcm1hbFRvcmljVmFyaWV0eSk=
│ │ │  #:len=2221
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibG9jYWxseSBzdGFzaGVkIGNvaG9tb2xv
│ │ │  Z3kgdmVjdG9ycyBmcm9tIENvaG9tQ2FsZyIsICJsaW5lbnVtIiA9PiAyODIsIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out
│ │ │ @@ -184,15 +184,15 @@
│ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, -1, -1}}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.09141s elapsed
│ │ │ + -- 2.79897s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -265,45 +265,45 @@
│ │ │  i22 : degree(X_3 + X_7 + X_8)
│ │ │  
│ │ │  o22 = {0, 0, 1, 2, 0, -1}
│ │ │  
│ │ │  o22 : List
│ │ │  
│ │ │  i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .3809s elapsed
│ │ │ + -- .43603s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
│ │ │  
│ │ │  i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 11.5175s elapsed
│ │ │ + -- 8.12029s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
│ │ │  
│ │ │  i25 : assert(cohomvec1 == cohomvec2)
│ │ │  
│ │ │  i26 : degree(X_3 + X_7 - X_8)
│ │ │  
│ │ │  o26 = {0, 0, 1, 2, -2, -1}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .397928s elapsed
│ │ │ + -- .428707s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .553978s elapsed
│ │ │ - -- .554009s elapsed
│ │ │ + -- .380703s elapsed
│ │ │ + -- .380729s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
│ │ │  
│ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/html/index.html
│ │ │ @@ -314,15 +314,15 @@
│ │ │  
│ │ │  o19 : List
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.09141s elapsed
│ │ │ + -- 2.79897s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -404,25 +404,25 @@
│ │ │  
│ │ │  o22 : List
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .3809s elapsed
│ │ │ + -- .43603s elapsed
│ │ │  
│ │ │  o23 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o23 : List
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
│ │ │ - -- 11.5175s elapsed
│ │ │ + -- 8.12029s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .397928s elapsed
│ │ │ + -- .428707s elapsed
│ │ │  
│ │ │  o27 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
│ │ │ - -- .553978s elapsed
│ │ │ - -- .554009s elapsed
│ │ │ + -- .380703s elapsed
│ │ │ + -- .380729s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
    
│ │ │              
    
│ │ │                
    i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.68392s (cpu); 5.01853s (thread); 0s (gc)
│ │ │ + -- used 6.45886s (cpu); 5.02278s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -300,28 +300,28 @@
│ │ │  
│ │ │  o19 : QuotientRing
    
│ │ │              
    
│ │ │                
    i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.232858s (cpu); 0.156146s (thread); 0s (gc)
│ │ │ + -- used 0.173287s (cpu); 0.106511s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        0       1       2        3        4
│ │ │  
│ │ │  o20 : Complex
    
│ │ │              
    
│ │ │                
    i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ - -- used 0.980622s (cpu); 0.754248s (thread); 0s (gc)
│ │ │ + -- used 0.939527s (cpu); 0.74235s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │ @@ -333,15 +333,15 @@
│ │ │          
    
│ │ │            
    The function also deals correctly with complexes F where min F is not 0:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.994666s (cpu); 0.734783s (thread); 0s (gc)
│ │ │ + -- used 0.954247s (cpu); 0.731707s (thread); 0s (gc)
│ │ │  
│ │ │          1       6       18       38       66
│ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │                                           
│ │ │        -2      -1      0        1        2
│ │ │  
│ │ │  o22 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │  o5 = R
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : len = 10
│ │ │ │  
│ │ │ │  o6 = 10
│ │ │ │  i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ │ - -- used 6.68392s (cpu); 5.01853s (thread); 0s (gc)
│ │ │ │ + -- used 6.45886s (cpu); 5.02278s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S
│ │ │ │  \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /
│ │ │ │  S   \68     /    S   \76
│ │ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |-------
│ │ │ │  -|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |-
│ │ │ │  -------|   <-- |--------|
│ │ │ │ @@ -165,36 +165,36 @@
│ │ │ │  o18 : Matrix R  <-- R
│ │ │ │  i19 : R1 = R/ideal ff
│ │ │ │  
│ │ │ │  o19 = R1
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : FF = time Shamash(R1,F,4)
│ │ │ │ - -- used 0.232858s (cpu); 0.156146s (thread); 0s (gc)
│ │ │ │ + -- used 0.173287s (cpu); 0.106511s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o20 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        0       1       2        3        4
│ │ │ │  
│ │ │ │  o20 : Complex
│ │ │ │  i21 : GG = time EisenbudShamash(ff,F,4)
│ │ │ │ - -- used 0.980622s (cpu); 0.754248s (thread); 0s (gc)
│ │ │ │ + -- used 0.939527s (cpu); 0.74235s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │ │  
│ │ │ │        0         1         2          3          4
│ │ │ │  
│ │ │ │  o21 : Complex
│ │ │ │  The function also deals correctly with complexes F where min F is not 0:
│ │ │ │  i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ │ - -- used 0.994666s (cpu); 0.734783s (thread); 0s (gc)
│ │ │ │ + -- used 0.954247s (cpu); 0.731707s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        -2      -1      0        1        2
│ │ │ │  
│ │ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html
│ │ │ @@ -84,23 +84,23 @@
│ │ │  
│ │ │  o1 = 0
│ │ │              
    
│ │ │                
    i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.547511s (cpu); 0.41022s (thread); 0s (gc)
│ │ │ + -- used 0.647333s (cpu); 0.403323s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.106844s (cpu); 0.10641s (thread); 0s (gc)
│ │ │ + -- used 0.134381s (cpu); 0.111476s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 4.058e-06s (cpu); 3.486e-06s (thread); 0s (gc)
│ │ │ + -- used 3.552e-06s (cpu); 2.425e-06s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{}
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,23 +18,23 @@
│ │ │ │  appropriate syzygy M of M0 = R/(m1+m2) where m1 and m2 are monomials of the
│ │ │ │  same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : sumTwoMonomials(2,3)
│ │ │ │ - -- used 0.547511s (cpu); 0.41022s (thread); 0s (gc)
│ │ │ │ + -- used 0.647333s (cpu); 0.403323s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │ │  
│ │ │ │ - -- used 0.106844s (cpu); 0.10641s (thread); 0s (gc)
│ │ │ │ + -- used 0.134381s (cpu); 0.111476s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  
│ │ │ │ - -- used 4.058e-06s (cpu); 3.486e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.552e-06s (cpu); 2.425e-06s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ssuummTTwwooMMoonnoommiiaallss:: **********
│ │ │ │      * sumTwoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html
│ │ │ @@ -88,25 +88,25 @@
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │                
    i2 : twoMonomials(2,3)
│ │ │ - -- used 1.04089s (cpu); 0.839487s (thread); 0s (gc)
│ │ │ + -- used 1.01943s (cpu); 0.628818s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.552679s (cpu); 0.395293s (thread); 0s (gc)
│ │ │ + -- used 0.561911s (cpu); 0.374722s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.10407s (cpu); 0.103794s (thread); 0s (gc)
│ │ │ + -- used 0.202248s (cpu); 0.145293s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,25 +20,25 @@
│ │ │ │  that is, for an appropriate syzygy M of M0 = R/(m1, m2) where m1 and m2 are
│ │ │ │  monomials of the same degree.
│ │ │ │  i1 : setRandomSeed 0
│ │ │ │   -- setting random seed to 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : twoMonomials(2,3)
│ │ │ │ - -- used 1.04089s (cpu); 0.839487s (thread); 0s (gc)
│ │ │ │ + -- used 1.01943s (cpu); 0.628818s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{1, 1}} => 2        }
│ │ │ │        {{2, 2}, {1, 2}} => 4
│ │ │ │  
│ │ │ │ - -- used 0.552679s (cpu); 0.395293s (thread); 0s (gc)
│ │ │ │ + -- used 0.561911s (cpu); 0.374722s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │ │        {{3, 3}, {2, 3}} => 1
│ │ │ │  
│ │ │ │ - -- used 0.10407s (cpu); 0.103794s (thread); 0s (gc)
│ │ │ │ + -- used 0.202248s (cpu); 0.145293s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ttwwooMMoonnoommiiaallss:: **********
│ │ │ │      * twoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  cHJ1bmVDb21wbGV4KC4uLixQcnVuaW5nTWFwPT4uLi4p
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzQ4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1twcnVuZUNvbXBsZXgsUHJ1bmluZ01hcF0sInBydW5l
│ │ ├── ./usr/share/doc/Macaulay2/ConformalBlocks/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  Y2Fub25pY2FsRGl2aXNvck0wbmJhcg==
│ │ │  #:len=1172
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY2xhc3Mgb2YgdGhl
│ │ │  IGNhbm9uaWNhbCBkaXZpc29yIG9uIHRoZSBtb2R1bGkgc3BhY2Ugb2Ygc3RhYmxlIG4tcG9pbnRl
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=39
│ │ │  aXNFcHNpbG9uRmFjdG9yaXplZChNYXRyaXgsUmluZ0VsZW1lbnQp
│ │ │  #:len=334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0Vwc2lsb25GYWN0b3JpemVkLE1hdHJpeCxSaW5n
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.out
│ │ │ @@ -27,18 +27,18 @@
│ │ │       - ϵ*z*dy + 2ϵ  - ϵ, x*dx + y*dy + z*dz - 2ϵ)
│ │ │  
│ │ │  o7 : Ideal of D
│ │ │  
│ │ │  i8 : assert(holonomicRank I == 4)
│ │ │  
│ │ │  i9 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- 4.39641s elapsed
│ │ │ + -- 2.53916s elapsed
│ │ │  
│ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.46882s elapsed
│ │ │ + -- 3.54734s elapsed
│ │ │  
│ │ │  i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o11 = || 2ϵ/x                                                                                                                                                                                                                                                                                                                                                                                                                                    -y/x                                                                                                                                                                                                                                                                                                                                                                                                                                                                   -z/x                                                                                                                                                                                                                                                                                                                                                                                                                                                  0                                                                                                                             |                                                                                                                                                                                                                                                                                                                                     |
│ │ │        || (4x2y2ϵ^2+4xy2zϵ^2-2x2z2ϵ^2-2y2z2ϵ^2-4xz3ϵ^2+x3zϵ-3xy2zϵ+2xz3ϵ)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                               (2x3y2ϵ-2x2y3ϵ+2x3yzϵ-2xy3zϵ-x3z2ϵ+x2yz2ϵ-xy2z2ϵ+y3z2ϵ-2x3yz+2xy3z)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                          (-2x2y2zϵ-x3z2ϵ-3xy2z2ϵ+x2z3ϵ+y2z3ϵ+4xz4ϵ+2xy2z2-2xz4)/(2x4y2+2x3y3+x4yz+2x3y2z+x2y3z-x4z2-x3yz2-x2y2z2-xy3z2-x3z3-2x2yz3-xy2z3)                                                                                                                                                                                                                                                                                                                      (-xyz+xz2+yz2-z3)/(2x2y+2xy2-x2z-2xyz-y2z)                                                                                    |                                                                                                                                                                                                                                                                                                                                     |
│ │ │        || (-2xyz2ϵ^2-2y2z2ϵ^2-4yz3ϵ^2+2x2y2ϵ+x2yzϵ+xy2zϵ+2y2z2ϵ+2yz3ϵ)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                 (x2yz2ϵ+2xy2z2ϵ+y3z2ϵ+2xyz3ϵ+2y2z3ϵ-2x2y3-x2y2z-xy3z-x2yz2-y3z2-xyz3-y2z3)/(2x3y2z+x3yz2+x2y2z2-x3z3-xy2z3-x2z4-xyz4)                                                                                                                                                                                                                                                                                                                                                  (2x2y2ϵ+x2yzϵ+xy2zϵ-2x2z2ϵ+xyz2ϵ+y2z2ϵ-2xz3ϵ+2yz3ϵ-2x2y2-x2yz-xy2z+x2z2-y2z2+xz3-yz3)/(2x3y2+x3yz+x2y2z-x3z2-xy2z2-x2z3-xyz3)                                                                                                                                                                                                                                                                                                                         (-yz+z2)/(2xy-xz-yz)                                                                                                          |                                                                                                                                                                                                                                                                                                                                     |
│ │ │ @@ -56,24 +56,24 @@
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  
│ │ │  i12 : F = baseFractionField D;
│ │ │  
│ │ │  i13 : B = {1_D,dx,dy,dx*dy};
│ │ │  
│ │ │  i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .606974s elapsed
│ │ │ + -- .47013s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
│ │ │  
│ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.30033s elapsed
│ │ │ + -- .992488s elapsed
│ │ │  
│ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .857943s elapsed
│ │ │ + -- .741178s elapsed
│ │ │  
│ │ │  i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o17 = || 0                            1                      0                                         0                                                      |                                                                         |
│ │ │        || (-2ϵ^2+ϵ)/(x2-z2)            (3xϵ+zϵ-2x)/(x2-z2)    (yϵ+zϵ)/(x2-z2)                           (-y-z)/(x-z)                                           |                                                                         |
│ │ │        || 0                            0                      0                                         1                                                      |                                                                         |
│ │ │ @@ -96,18 +96,18 @@
│ │ │                {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
│ │ │                {0, 0, 0, -(x+y)*(x+z)*(y+z)}});
│ │ │  
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
│ │ │  
│ │ │  i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .393587s elapsed
│ │ │ + -- .293345s elapsed
│ │ │  
│ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- 1.06124s elapsed
│ │ │ + -- .591629s elapsed
│ │ │  
│ │ │  i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |                                     |
│ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |                                     |
│ │ │        || 0       0           ϵ/(x+z) (-yϵ+zϵ)/(x2+xy+xz+yz) |                                     |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/example-output/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.out
│ │ │ @@ -16,18 +16,18 @@
│ │ │  
│ │ │                     2
│ │ │  o6 = {1, dx, dy, dy }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- .32932s elapsed
│ │ │ + -- .19475s elapsed
│ │ │  
│ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .168768s elapsed
│ │ │ + -- .146281s elapsed
│ │ │  
│ │ │  i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o9 = || 0                                                       1                                               0                                                                     0                                                                      ||
│ │ │       || 0                                                       -1/x                                            1/x                                                                   y/x                                                                    ||
│ │ │       || -1/2xy                                                  -1/y                                            (-x-3y+1)/2xy                                                         (-x-y+1)/2x                                                            ||
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html
│ │ │ @@ -123,21 +123,21 @@
│ │ │          
    
│ │ │            
    Then, we compute the system in connection form and verify that it meets the integrability conditions.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,15 +82,15 @@
│ │ │ │                2                 2
│ │ │ │       - a*c + e         - b*c + f
│ │ │ │       ----------*v, x + ----------*v)
│ │ │ │        d*e - a*f         d*e - a*f
│ │ │ │  
│ │ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i6 : time phi^** q
│ │ │ │ - -- used 0.368428s (cpu); 0.207459s (thread); 0s (gc)
│ │ │ │ + -- used 0.325119s (cpu); 0.188176s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                  e        d        c        b        a
│ │ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │ │                  f        f        f        f        f
│ │ │ │  
│ │ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i7 : oo == p
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html
│ │ │ @@ -139,59 +139,59 @@
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
    
│ │ │                
    i9 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- 4.39641s elapsed
    
│ │ │ + -- 2.53916s elapsed
│ │ │              
    
│ │ │                
    i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.46882s elapsed
    
│ │ │ + -- 3.54734s elapsed
│ │ │              
    
│ │ │                
    i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -172,30 +172,30 @@
│ │ │              
    		
│ │ │                
    i13 : B = {1_D,dx,dy,dx*dy};
    
│ │ │              
    
│ │ │                
    i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .606974s elapsed
│ │ │ + -- .47013s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.30033s elapsed
    
│ │ │ + -- .992488s elapsed
│ │ │              
    
│ │ │                
    i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .857943s elapsed
    
│ │ │ + -- .741178s elapsed
│ │ │              
    
│ │ │                
    i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -232,21 +232,21 @@
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .393587s elapsed
    
│ │ │ + -- .293345s elapsed
│ │ │              
    
│ │ │                
    i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- 1.06124s elapsed
    
│ │ │ + -- .591629s elapsed
│ │ │              
    
│ │ │                
    i21 : netList A2
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,17 +41,17 @@
│ │ │ │  
│ │ │ │  o7 : Ideal of D
│ │ │ │  First, we check that the system has finite holonomic rank using _h_o_l_o_n_o_m_i_c_R_a_n_k.
│ │ │ │  i8 : assert(holonomicRank I == 4)
│ │ │ │  Then, we compute the system in connection form and verify that it meets the
│ │ │ │  integrability conditions.
│ │ │ │  i9 : elapsedTime A = pfaffianSystem I;
│ │ │ │ - -- 4.39641s elapsed
│ │ │ │ + -- 2.53916s elapsed
│ │ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ │ - -- 5.46882s elapsed
│ │ │ │ + -- 3.54734s elapsed
│ │ │ │  i11 : netList A
│ │ │ │  
│ │ │ │        +----------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │  -----------------------------------------------------------------------------------------------------------------
│ │ │ │ @@ -227,22 +227,22 @@
│ │ │ │  -----------------------------------------------------------------------------------+
│ │ │ │  Next, we use _g_a_u_g_e_ _m_a_t_r_i_x for changing base to a base given by suitable set of
│ │ │ │  standard monomials, and compute the _g_a_u_g_e_ _t_r_a_n_s_f_o_r_m with respect to this gauge
│ │ │ │  matrix.
│ │ │ │  i12 : F = baseFractionField D;
│ │ │ │  i13 : B = {1_D,dx,dy,dx*dy};
│ │ │ │  i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ │ - -- .606974s elapsed
│ │ │ │ + -- .47013s elapsed
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o14 : Matrix F  <-- F
│ │ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ │ - -- 1.30033s elapsed
│ │ │ │ + -- .992488s elapsed
│ │ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ │ - -- .857943s elapsed
│ │ │ │ + -- .741178s elapsed
│ │ │ │  i17 : netList A1
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  --------------------------------------------------------------------------+
│ │ │ │  o17 = || 0                            1                      0
│ │ │ │  0                                                      |
│ │ │ │ @@ -300,17 +300,17 @@
│ │ │ │                {0, ϵ*(x^2-z^2), 0, ϵ*(x+y)*(x+z)},
│ │ │ │                {0, 0, ϵ*(y^2-z^2), ϵ*(x+y)*(y+z)},
│ │ │ │                {0, 0, 0, -(x+y)*(x+z)*(y+z)}});
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o18 : Matrix F  <-- F
│ │ │ │  i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ │ - -- .393587s elapsed
│ │ │ │ + -- .293345s elapsed
│ │ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ │ - -- 1.06124s elapsed
│ │ │ │ + -- .591629s elapsed
│ │ │ │  i21 : netList A2
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------+
│ │ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |
│ │ │ │  |
│ │ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html
│ │ │ @@ -105,21 +105,21 @@
│ │ │          
    
│ │ │            
    Finally, we can compute the connection matrices.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- .32932s elapsed
    
│ │ │ + -- .19475s elapsed
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .168768s elapsed
    
│ │ │ + -- .146281s elapsed
│ │ │              
    
│ │ │                
    i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,17 +20,17 @@
│ │ │ │  
│ │ │ │                     2
│ │ │ │  o6 = {1, dx, dy, dy }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  Finally, we can compute the connection matrices.
│ │ │ │  i7 : elapsedTime A = pfaffianSystem I;
│ │ │ │ - -- .32932s elapsed
│ │ │ │ + -- .19475s elapsed
│ │ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ │ - -- .168768s elapsed
│ │ │ │ + -- .146281s elapsed
│ │ │ │  i9 : netList A
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------+
│ │ │ │  o9 = || 0                                                       1
│ │ ├── ./usr/share/doc/Macaulay2/ConvexInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGVzY3JpYmUgYSByYXRpb25hbCBtYXAi
│ │ │  LCAibGluZW51bSIgPT4gOTgzLCBJbnB1dHMgPT4ge1NQQU57VFR7InBoaSJ9LCIsICIsU1BBTnsi
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
│ │ │ @@ -13,27 +13,27 @@
│ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
│ │ │  
│ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.26956s (cpu); 0.894031s (thread); 0s (gc)
│ │ │ + -- used 1.12596s (cpu); 0.854789s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
│ │ │  
│ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.46902s (cpu); 1.00309s (thread); 0s (gc)
│ │ │ + -- used 1.16897s (cpu); 0.819846s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -62,27 +62,27 @@
│ │ │          0,2 1,3    0,1 2,3
│ │ │  
│ │ │                  ZZ
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i9 : time ChernClass G
│ │ │ - -- used 0.110298s (cpu); 0.110305s (thread); 0s (gc)
│ │ │ + -- used 0.119362s (cpu); 0.119364s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
│ │ │  
│ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0113359s (cpu); 0.0107235s (thread); 0s (gc)
│ │ │ + -- used 0.0233192s (cpu); 0.0117052s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out
│ │ │ @@ -1,56 +1,56 @@
│ │ │  -- -*- M2-comint -*- hash: 10433409267944421825
│ │ │  
│ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │  
│ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00558593s (cpu); 0.00558112s (thread); 0s (gc)
│ │ │ + -- used 0.00490964s (cpu); 0.00490733s (thread); 0s (gc)
│ │ │  
│ │ │              ZZ              ZZ                3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o2 = map (------[t ..t ], ------[x ..x ], {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   300007  0   9       2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
│ │ │  
│ │ │  i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.0591643s (cpu); 0.0591742s (thread); 0s (gc)
│ │ │ + -- used 0.0460566s (cpu); 0.0460589s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │                  ZZ
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
│ │ │  
│ │ │  i4 : time degreeMap phi
│ │ │ - -- used 0.0354136s (cpu); 0.0354203s (thread); 0s (gc)
│ │ │ + -- used 0.0284222s (cpu); 0.0284253s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.683938s (cpu); 0.537126s (thread); 0s (gc)
│ │ │ + -- used 0.46581s (cpu); 0.397063s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.223023s (cpu); 0.137433s (thread); 0s (gc)
│ │ │ + -- used 0.0592977s (cpu); 0.0593036s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00354109s (cpu); 0.00354235s (thread); 0s (gc)
│ │ │ + -- used 0.00223669s (cpu); 0.0022397s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -59,15 +59,15 @@
│ │ │                                                                             ------[x ..x ]
│ │ │                 ZZ                                                          300007  0   9
│ │ │  o7 : RingMap ------[t ..t ] <-- ----------------------------------------------------------------------------------------------------
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i8 : time psi = inverseMap phi
│ │ │ - -- used 0.743183s (cpu); 0.662105s (thread); 0s (gc)
│ │ │ + -- used 0.505302s (cpu); 0.406599s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -76,32 +76,32 @@
│ │ │                                                          ------[x ..x ]
│ │ │                                                          300007  0   9                                                   ZZ
│ │ │  o8 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[t ..t ]
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.016456s (cpu); 0.0164579s (thread); 0s (gc)
│ │ │ + -- used 0.0101755s (cpu); 0.0101884s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.376507s (cpu); 0.274317s (thread); 0s (gc)
│ │ │ + -- used 0.153949s (cpu); 0.153956s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 5.65221s (cpu); 4.79159s (thread); 0s (gc)
│ │ │ + -- used 4.79568s (cpu); 4.30587s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00224049s (cpu); 0.00224156s (thread); 0s (gc)
│ │ │ + -- used 0.00239542s (cpu); 0.00239948s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -147,15 +147,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │  
│ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.0523877s (cpu); 0.0523973s (thread); 0s (gc)
│ │ │ + -- used 0.0513685s (cpu); 0.0513715s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -217,15 +217,15 @@
│ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │                        }
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i14 : time phi^(-1)
│ │ │ - -- used 0.586241s (cpu); 0.446401s (thread); 0s (gc)
│ │ │ + -- used 0.511826s (cpu); 0.418254s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -275,71 +275,71 @@
│ │ │                         x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x
│ │ │                          6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │                        }
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
│ │ │  
│ │ │  i15 : time degrees phi^(-1)
│ │ │ - -- used 0.624809s (cpu); 0.460082s (thread); 0s (gc)
│ │ │ + -- used 0.279261s (cpu); 0.239719s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.0257756s (cpu); 0.0252268s (thread); 0s (gc)
│ │ │ + -- used 0.0281718s (cpu); 0.0173267s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00459023s (cpu); 0.00459072s (thread); 0s (gc)
│ │ │ + -- used 0.0038676s (cpu); 0.00387328s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i18 : time describe phi^(-1)
│ │ │ - -- used 0.0138045s (cpu); 0.0138079s (thread); 0s (gc)
│ │ │ + -- used 0.010034s (cpu); 0.0100393s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │        number of minimal representatives: 1
│ │ │        dimension base locus: 4
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
│ │ │  
│ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.0132166s (cpu); 0.0131906s (thread); 0s (gc)
│ │ │ + -- used 0.0101159s (cpu); 0.0101206s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i21 : time degrees f
│ │ │ - -- used 1.60372s (cpu); 1.18114s (thread); 0s (gc)
│ │ │ + -- used 1.12259s (cpu); 0.837973s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 2.0168e-05s (cpu); 1.9606e-05s (thread); 0s (gc)
│ │ │ + -- used 1.5474e-05s (cpu); 1.4716e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.00171686s (cpu); 0.00171795s (thread); 0s (gc)
│ │ │ + -- used 0.00132095s (cpu); 0.00132446s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out
│ │ │ @@ -3,18 +3,18 @@
│ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │  
│ │ │                  ZZ
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │  
│ │ │  i2 : time EulerCharacteristic I
│ │ │ - -- used 0.590242s (cpu); 0.241339s (thread); 0s (gc)
│ │ │ + -- used 0.473545s (cpu); 0.199432s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0127473s (cpu); 0.0122967s (thread); 0s (gc)
│ │ │ + -- used 0.0700004s (cpu); 0.0180693s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 10
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │  
│ │ │  o3 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i4 : time phi! ;
│ │ │ - -- used 0.0572395s (cpu); 0.0568639s (thread); 0s (gc)
│ │ │ + -- used 0.114779s (cpu); 0.063545s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │  
│ │ │  i5 : describe phi
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 2
│ │ │       source variety: PP^5
│ │ │ @@ -37,15 +37,15 @@
│ │ │  
│ │ │  o8 = rational map defined by forms of degree 2
│ │ │       source variety: PP^4
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
│ │ │  
│ │ │  i9 : time phi! ;
│ │ │ - -- used 0.0401527s (cpu); 0.03977s (thread); 0s (gc)
│ │ │ + -- used 0.0954699s (cpu); 0.0414719s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i10 : describe phi
│ │ │  
│ │ │  o10 = rational map defined by forms of degree 2
│ │ │        source variety: PP^4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp^_st_st_sp__Ideal.out
│ │ │ @@ -67,15 +67,15 @@
│ │ │       - a*c + e         - b*c + f
│ │ │       ----------*v, x + ----------*v)
│ │ │        d*e - a*f         d*e - a*f
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │  
│ │ │  i6 : time phi^** q
│ │ │ - -- used 0.368428s (cpu); 0.207459s (thread); 0s (gc)
│ │ │ + -- used 0.325119s (cpu); 0.188176s (thread); 0s (gc)
│ │ │  
│ │ │                  e        d        c        b        a
│ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │                  f        f        f        f        f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out
│ │ │ @@ -47,50 +47,50 @@
│ │ │                                                                            P7
│ │ │  o3 : Ideal of -------------------------------------------------------------------------------------------------------------------------
│ │ │                 2 2                2 2                                        2 2                                                    2 2
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │  
│ │ │  i4 : time SegreClass X
│ │ │ - -- used 0.885027s (cpu); 0.51192s (thread); 0s (gc)
│ │ │ + -- used 0.874277s (cpu); 0.513743s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.568535s (cpu); 0.343331s (thread); 0s (gc)
│ │ │ + -- used 0.415801s (cpu); 0.286306s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0216529s (cpu); 0.0213527s (thread); 0s (gc)
│ │ │ + -- used 0.0499798s (cpu); 0.0222984s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
│ │ │  
│ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0973043s (cpu); 0.0970462s (thread); 0s (gc)
│ │ │ + -- used 0.280271s (cpu); 0.151321s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -98,22 +98,22 @@
│ │ │  
│ │ │  i8 : o4 == o6 and o5 == o7
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ -= ------[x ..x ]
│ │ │ -  100003  0   6
│ │ │ +o9 =   ZZ
│ │ │ + ------[x ..x ]
│ │ │ + 100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
│ │ │  
│ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.237104s (cpu); 0.125625s (thread); 0s (gc)
│ │ │ + -- used 0.0661419s (cpu); 0.0661437s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -122,15 +122,15 @@
│ │ │                                                           ------[y ..y ]
│ │ │                                                           100003  0   9                                                   ZZ
│ │ │  o10 : RingMap ---------------------------------------------------------------------------------------------------- <-- ------[x ..x ]
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │  
│ │ │  i11 : time SegreClass phi
│ │ │ - -- used 0.385145s (cpu); 0.250766s (thread); 0s (gc)
│ │ │ + -- used 0.374685s (cpu); 0.2463s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -150,27 +150,27 @@
│ │ │                                                            100003  0   9
│ │ │  o12 : Ideal of ----------------------------------------------------------------------------------------------------
│ │ │                 (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │  
│ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.405977s (cpu); 0.267353s (thread); 0s (gc)
│ │ │ + -- used 0.391422s (cpu); 0.267298s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
│ │ │  
│ │ │  i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.63119s (cpu); 0.951264s (thread); 0s (gc)
│ │ │ + -- used 1.40422s (cpu); 0.83095s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out
│ │ │ @@ -17,32 +17,32 @@
│ │ │  
│ │ │  o3 = QQ[u ..u ]
│ │ │           0   5
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000476825s (cpu); 0.000472506s (thread); 0s (gc)
│ │ │ + -- used 0.000395736s (cpu); 0.000391753s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │  
│ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.1194s (cpu); 0.119423s (thread); 0s (gc)
│ │ │ + -- used 0.119041s (cpu); 0.119046s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.532769s (cpu); 0.401923s (thread); 0s (gc)
│ │ │ + -- used 0.424402s (cpu); 0.338794s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -113,48 +113,48 @@
│ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │                   ZZ
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
│ │ │  
│ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.0383176s (cpu); 0.0383202s (thread); 0s (gc)
│ │ │ + -- used 0.040632s (cpu); 0.0406162s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 2.18663s (cpu); 1.5847s (thread); 0s (gc)
│ │ │ + -- used 2.2766s (cpu); 1.56941s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 3.0007e-05s (cpu); 2.9705e-05s (thread); 0s (gc)
│ │ │ + -- used 2.6532e-05s (cpu); 2.4366e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 3.67928s (cpu); 2.6019s (thread); 0s (gc)
│ │ │ + -- used 3.51385s (cpu); 2.49866s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
│ │ │  
│ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │  
│ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │ @@ -169,15 +169,15 @@
│ │ │  i20 : T q
│ │ │  
│ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │  o20 : List
│ │ │  
│ │ │  i21 : time f = rationalMap T
│ │ │ - -- used 3.12656s (cpu); 2.217s (thread); 0s (gc)
│ │ │ + -- used 2.98878s (cpu); 2.11239s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out
│ │ │ @@ -54,15 +54,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.559174s (cpu); 0.374887s (thread); 0s (gc)
│ │ │ + -- used 0.444139s (cpu); 0.300791s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │  
│ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.367939s (cpu); 0.280942s (thread); 0s (gc)
│ │ │ + -- used 0.249231s (cpu); 0.164507s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i6 : assert(psi == psi')
│ │ │  
│ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-3*x_6*x_7+2*x_0*x_8+11*x_1*x_8-37*x_3*x_8-23*x_4*x_8-33*x_6*x_8+8*x_0*x_9+10*x_1*x_9-25*x_2*x_9-9*x_3*x_9+3*x_4*x_9+24*x_5*x_9-27*x_6*x_9-5*x_0*x_10+28*x_1*x_10+37*x_2*x_10+9*x_4*x_10+27*x_6*x_10-25*x_0*x_11+9*x_2*x_11+27*x_4*x_11-27*x_5*x_11,x_2^2+17*x_2*x_3-14*x_3^2-13*x_2*x_4+34*x_3*x_4+44*x_0*x_5-30*x_2*x_5+27*x_3*x_5+31*x_2*x_6-36*x_3*x_6-x_0*x_7+13*x_1*x_7+8*x_3*x_7+9*x_5*x_7+46*x_6*x_7+41*x_0*x_8-7*x_1*x_8-34*x_3*x_8-9*x_4*x_8-46*x_6*x_8-17*x_0*x_9+32*x_1*x_9-8*x_2*x_9-35*x_3*x_9-46*x_4*x_9+26*x_5*x_9+17*x_6*x_9+15*x_0*x_10+35*x_1*x_10+34*x_2*x_10+20*x_4*x_10+14*x_0*x_11+36*x_1*x_11+35*x_2*x_11-17*x_4*x_11,x_1*x_2-40*x_2*x_3+28*x_3^2-x_0*x_4+5*x_2*x_4-16*x_3*x_4+5*x_0*x_5-36*x_2*x_5+37*x_3*x_5+48*x_2*x_6-5*x_1*x_7-5*x_3*x_7+x_5*x_7+20*x_6*x_7+10*x_0*x_8+34*x_1*x_8+41*x_3*x_8-x_4*x_8+x_6*x_8+40*x_0*x_9-32*x_1*x_9+5*x_2*x_9-11*x_3*x_9-20*x_4*x_9+45*x_5*x_9-14*x_6*x_9-25*x_0*x_10+45*x_1*x_10-41*x_2*x_10-46*x_4*x_10+8*x_6*x_10-28*x_0*x_11+11*x_2*x_11+14*x_4*x_11-8*x_5*x_11),{t_4^2+t_0*t_5+t_1*t_5+35*t_2*t_5+10*t_3*t_5+25*t_4*t_5-5*t_5^2-14*t_0*t_6-14*t_1*t_6-5*t_2*t_6-13*t_4*t_6+37*t_5*t_6+22*t_6^2-31*t_3*t_7+26*t_4*t_7+12*t_5*t_7-45*t_6*t_7-46*t_3*t_8+37*t_4*t_8+28*t_5*t_8+33*t_6*t_8,t_3*t_4+4*t_0*t_5+39*t_1*t_5-40*t_2*t_5+40*t_3*t_5+26*t_4*t_5-20*t_5^2+41*t_0*t_6+36*t_1*t_6-22*t_2*t_6+36*t_4*t_6-30*t_5*t_6-13*t_6^2-25*t_3*t_7+5*t_4*t_7-35*t_5*t_7+10*t_6*t_7+11*t_3*t_8+46*t_4*t_8+29*t_5*t_8+28*t_6*t_8,t_2*t_4-5*t_0*t_5-40*t_1*t_5+12*t_2*t_5+47*t_3*t_5+37*t_4*t_5+25*t_5^2-27*t_0*t_6-22*t_1*t_6+27*t_2*t_6-23*t_4*t_6+5*t_5*t_6-13*t_6^2-39*t_3*t_7-29*t_4*t_7+9*t_5*t_7+39*t_6*t_7+36*t_3*t_8+13*t_4*t_8+26*t_5*t_8+37*t_6*t_8,t_0*t_4-t_0*t_5-8*t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+15*t_0*t_6+15*t_1*t_6+5*t_2*t_6+15*t_4*t_6-38*t_5*t_6-22*t_6^2+31*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_2*t_3-t_0*t_5-t_1*t_5-35*t_2*t_5-10*t_3*t_5-33*t_4*t_5+5*t_5^2+14*t_0*t_6+14*t_1*t_6+5*t_2*t_6+14*t_4*t_6-31*t_5*t_6-24*t_6^2+32*t_3*t_7-25*t_4*t_7-19*t_5*t_7+47*t_6*t_7+46*t_3*t_8-36*t_4*t_8-35*t_5*t_8-31*t_6*t_8,t_1*t_3-7*t_1*t_5+t_1*t_6+t_4*t_6-7*t_5*t_6+2*t_6^2-t_3*t_7,t_0*t_3-46*t_0*t_5-39*t_1*t_5-43*t_2*t_5-41*t_3*t_5-26*t_4*t_5-28*t_5^2-35*t_0*t_6-36*t_1*t_6+20*t_2*t_6-36*t_4*t_6+9*t_5*t_6+15*t_6^2+26*t_3*t_7-5*t_4*t_7+35*t_5*t_7-10*t_6*t_7-10*t_3*t_8-46*t_4*t_8+47*t_5*t_8-25*t_6*t_8,t_2^2-46*t_1*t_4-33*t_0*t_5-45*t_1*t_5-39*t_2*t_5-39*t_3*t_5-46*t_4*t_5-29*t_5^2-48*t_0*t_6-38*t_1*t_6-30*t_2*t_6+19*t_4*t_6-44*t_5*t_6-47*t_6^2-36*t_0*t_7-46*t_1*t_7+t_2*t_7-44*t_3*t_7+48*t_4*t_7-14*t_5*t_7+4*t_6*t_7-36*t_0*t_8-46*t_1*t_8+47*t_2*t_8-34*t_3*t_8-24*t_4*t_8-12*t_5*t_8-47*t_6*t_8+47*t_7*t_8,t_1*t_2+6*t_1*t_5+5*t_0*t_6-2*t_1*t_6-t_4*t_6-t_5*t_6+5*t_0*t_7+t_1*t_7-2*t_2*t_7-7*t_5*t_7+2*t_6*t_7-2*t_1*t_8+3*t_7*t_8,t_0*t_2+t_1*t_4+5*t_0*t_5+32*t_1*t_5-20*t_2*t_5-47*t_3*t_5-37*t_4*t_5-25*t_5^2+19*t_0*t_6+22*t_1*t_6-25*t_2*t_6+25*t_4*t_6-5*t_5*t_6+13*t_6^2+5*t_0*t_7+t_1*t_7+39*t_3*t_7+28*t_4*t_7-9*t_5*t_7-39*t_6*t_7+4*t_0*t_8+t_1*t_8-36*t_3*t_8-14*t_4*t_8-26*t_5*t_8-37*t_6*t_8,t_0*t_1-39*t_1*t_4+40*t_1*t_5-37*t_0*t_6-39*t_1*t_6+19*t_4*t_6-39*t_5*t_6-38*t_0*t_7+39*t_1*t_7+19*t_2*t_7+18*t_5*t_7-19*t_6*t_7+19*t_1*t_8+20*t_7*t_8,t_0^2+12*t_1*t_4+20*t_0*t_5+27*t_1*t_5-8*t_2*t_5+37*t_3*t_5+28*t_4*t_5+30*t_5^2-46*t_0*t_6+24*t_1*t_6-40*t_2*t_6+25*t_4*t_6+16*t_5*t_6-35*t_6^2+29*t_0*t_7+12*t_1*t_7-35*t_2*t_7-8*t_3*t_7-18*t_4*t_7+42*t_5*t_7-12*t_6*t_7-6*t_0*t_8+12*t_1*t_8-15*t_3*t_8+9*t_4*t_8+20*t_5*t_8-30*t_6*t_8+4*t_7*t_8})
│ │ │  
│ │ │ @@ -192,15 +192,15 @@
│ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │  
│ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.7613s (cpu); 1.89303s (thread); 0s (gc)
│ │ │ + -- used 1.99287s (cpu); 1.62692s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -258,15 +258,15 @@
│ │ │  
│ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.89555s (cpu); 2.92238s (thread); 0s (gc)
│ │ │ + -- used 2.8432s (cpu); 2.39173s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out
│ │ │ @@ -9,27 +9,27 @@
│ │ │                                   2                  2                             2                                       2                                                2                                                           2                                                                       2                                                                              2                                                                                            2         2                 2                             2                                       2                                              2                                                           2                                                                   2                                                                               2                                                                                          2        2                   2                            2                                      2                                                  2                                                          2                                                                      2                                                                               2                                                                                            2        2                   2                             2                                      2                                                 2                                                          2                                                                    2                                                                               2                                                                                          2         2                2                          2                                      2                                                 2                                                           2                                                                       2                                                                            2                                                                                          2        2                  2                         2                                       2                                                2                                                           2                                                                      2                                                                               2                                                                                        2       2                  2                             2                                    2                                                2                                                          2                                                                    2                                                                               2                                                                                       2       2                 2                           2                                      2                                                 2                                                            2                                                                       2                                                                                2                                                                                           2        2                   2                             2                                       2                                                  2                                                            2                                                                   2                                                                            2                                                                                          2      2                 2                            2                                       2                                                2                                                         2                                                                        2                                                                               2                                                                                          2     2                   2                             2                                      2                                                   2                                                          2                                                                     2                                                                               2                                                                                          2         2                2                            2                                       2                                                 2                                                           2                                                                      2                                                                                  2                                                                                              2      2                  2                            2                                    2                                                2                                                            2                                                                    2                                                                                2                                                                                          2       2                  2                            2                                    2                                                   2                                                        2                                                                         2                                                                               2                                                                                           2       2                  2                             2                                       2                                                 2                                                          2                                                                       2                                                                               2                                                                                       2
│ │ │  o4 = map (ringP8, ringP14, {- 95x  + 181x x  + 1028x  - 1384x x  - 1455x x  + 559x  - 502x x  + 1264x x  - 162x x  + 1209x  - 180x x  - 504x x  - 1168x x  - 676x x  + 501x  + 73x x  + 1263x x  + 1035x x  + 844x x  + 1593x x  + 785x  + 982x x  - 412x x  + 1335x x  + 1136x x  + 826x x  + 1078x x  + 1158x  + 335x x  - 982x x  - 1479x x  - 15x x  + 1363x x  + 1397x x  - 575x x  - 71x  + 1255x x  - 1138x x  - 1590x x  + 604x x  + 1182x x  - 63x x  - 1382x x  - 1255x x  - 613x , - 1444x  + 575x x  + 767x  - 1495x x  + 1631x x  - 217x  - 294x x  - 1511x x  - 504x x  - 1284x  - 1459x x  + 152x x  + 141x x  - 10x x  - 95x  + 1056x x  + 654x x  + 1397x x  - 930x x  + 578x x  - 696x  + 759x x  + 733x x  + 505x x  - 609x x  + 526x x  - 659x x  + 846x  + 1253x x  - 1519x x  + 635x x  + 576x x  + 54x x  - 1261x x  - 822x x  - 257x  - 986x x  + 356x x  - 1488x x  - 1561x x  - 850x x  - 85x x  - 1350x x  - 783x x  - 1335x , - 871x  + 1006x x  - 1399x  - 1636x x  - 699x x  - 769x  - 307x x  - 1645x x  - 502x x  - 719x  + 1405x x  + 870x x  - 1133x x  + 425x x  - 1203x  - 1601x x  + 117x x  - 382x x  + 318x x  - 117x x  - 560x  + 1135x x  + 1468x x  + 869x x  - 943x x  - 335x x  - 1218x x  + 201x  - 11x x  + 540x x  - 710x x  - 489x x  + 1605x x  + 1663x x  - 423x x  + 1246x  + 97x x  - 644x x  + 1655x x  + 1219x x  + 1476x x  + 1355x x  + 1594x x  + 893x x  + 1150x , - 143x  + 1240x x  - 1042x  + 1649x x  + 1024x x  + 794x  + 1442x x  - 1263x x  + 537x x  - 82x  - 734x x  - 1569x x  - 798x x  - 366x x  + 1289x  - 569x x  - 254x x  + 237x x  - 1234x x  - 807x x  + 264x  - 202x x  - 616x x  + 44x x  + 1465x x  + 685x x  + 1630x x  - 406x  - 123x x  - 4x x  + 1583x x  + 1235x x  + 162x x  + 1034x x  - 1035x x  + 737x  + 660x x  + 1459x x  - 359x x  - 1291x x  + 1638x x  - 325x x  - 631x x  + 73x x  - 1471x , - 1340x  + 31x x  - 994x  - 880x x  - 89x x  + 574x  + 760x x  - 1054x x  + 772x x  - 239x  - 443x x  + 1240x x  + 637x x  - 1423x x  + 320x  - 1363x x  - 1139x x  - 158x x  - 325x x  - 1578x x  + 32x  + 695x x  + 305x x  + 1012x x  + 1492x x  + 1290x x  + 1579x x  - 342x  - 83x x  - 104x x  + 998x x  - 92x x  + 1554x x  + 201x x  - 237x x  + 160x  - 228x x  - 543x x  - 1147x x  - 376x x  + 1313x x  + 603x x  + 106x x  - 1361x x  + 699x , - 228x  - 1510x x  + 277x  - 4x x  - 22x x  - 1526x  + 234x x  + 969x x  + 1253x x  - 1426x  - 1474x x  + 947x x  + 194x x  - 316x x  - 988x  - 1211x x  + 1087x x  + 536x x  - 491x x  + 870x x  - 659x  + 1490x x  - 469x x  + 1190x x  + 807x x  + 650x x  + 448x x  - 1353x  - 218x x  + 759x x  - 253x x  + 830x x  - 1080x x  - 143x x  - 1313x x  - 374x  - 180x x  + 741x x  + 742x x  - 1254x x  + 458x x  - 345x x  + 597x x  + 1567x x  - 31x , 1120x  + 709x x  - 1538x  - 1048x x  - 162x x  - 1518x  - 73x x  + 380x x  + 533x x  - 286x  + 1374x x  - 74x x  - 22x x  + 1535x x  - 1071x  - 839x x  - 560x x  + 928x x  + 335x x  - 1008x x  + 810x  - 448x x  - 357x x  - 107x x  + 40x x  + 784x x  - 1423x x  + 1276x  + 147x x  + 443x x  - 598x x  - 1077x x  - 1214x x  + 322x x  - 1408x x  + 72x  - 63x x  - 1513x x  - 791x x  + 11x x  + 77x x  + 836x x  - 1100x x  + 1637x x  - 788x , 1331x  + 318x x  - 704x  + 51x x  + 275x x  + 1149x  + 1526x x  + 768x x  + 414x x  - 782x  - 262x x  + 686x x  - 380x x  + 1377x x  + 1077x  + 1650x x  - 1129x x  - 508x x  + 846x x  + 1513x x  + 460x  - 1626x x  - 1024x x  + 862x x  + 1352x x  - 188x x  - 1382x x  - 650x  + 55x x  - 326x x  + 1037x x  + 705x x  - 667x x  + 1483x x  + 1661x x  - 1652x  - 1052x x  - 692x x  - 542x x  + 162x x  + 582x x  - 1369x x  + 934x x  + 1392x x  + 1227x , - 346x  + 1408x x  - 1225x  - 1536x x  - 1028x x  - 985x  - 210x x  - 1312x x  + 915x x  + 1633x  - 202x x  - 1636x x  - 1653x x  - 480x x  - 1260x  - 813x x  - 1623x x  - 1429x x  + 1094x x  - 747x x  + 955x  + 898x x  - 795x x  - 35x x  - 566x x  + 1631x x  - 324x x  + 926x  - 132x x  - 9x x  - 1290x x  - 543x x  + 902x x  + 735x x  - 342x x  - 400x  + 900x x  - 463x x  + 694x x  - 1262x x  - 1449x x  - 448x x  - 1402x x  - 731x x  - 996x , 301x  + 166x x  - 955x  - 739x x  - 1199x x  - 319x  + 1047x x  - 532x x  + 902x x  + 1195x  - 663x x  + 1215x x  - 534x x  - 332x x  - 973x  + 772x x  - 308x x  + 315x x  - 454x x  - 483x x  - 239x  - 1313x x  - 419x x  - 1340x x  - 1388x x  - 1340x x  - 1665x x  - 333x  - 465x x  - 1084x x  + 676x x  - 1612x x  - 288x x  + 11x x  - 1170x x  - 189x  + 498x x  - 889x x  + 693x x  + 1460x x  - 473x x  - 414x x  - 122x x  - 1659x x  - 1421x , 14x  - 1049x x  + 1506x  + 1235x x  + 642x x  - 1034x  + 460x x  + 150x x  + 760x x  - 1246x  - 1407x x  + 1570x x  + 1403x x  - 1610x x  - 431x  + 574x x  + 893x x  - 657x x  + 417x x  + 1362x x  + 224x  + 268x x  + 1097x x  + 1132x x  + 148x x  + 1331x x  - 77x x  - 756x  + 228x x  + 136x x  - 1484x x  - 1478x x  - 13x x  + 1620x x  - 701x x  - 769x  - 760x x  - 492x x  - 1077x x  - 1249x x  - 834x x  - 395x x  - 1358x x  - 988x x  + 113x , - 1634x  - 13x x  + 805x  - 21x x  - 1655x x  + 1479x  - 1510x x  - 646x x  + 225x x  - 1411x  + 1227x x  - 1108x x  + 1291x x  - 59x x  - 142x  + 586x x  - 676x x  + 655x x  - 1476x x  + 453x x  - 1076x  - 1152x x  + 1373x x  - 1191x x  - 416x x  + 699x x  + 317x x  + 825x  - 1560x x  - 488x x  - 1035x x  - 1561x x  - 644x x  - 1178x x  - 1320x x  + 158x  + 889x x  + 1444x x  - 1486x x  - 1211x x  + 1269x x  - 1228x x  + 568x x  + 1591x x  + 1207x , 105x  - 538x x  - 1222x  - 277x x  + 716x x  - 1067x  - 428x x  + 154x x  - 469x x  + 77x  + 538x x  - 179x x  + 921x x  - 223x x  + 1093x  - 262x x  + 1299x x  + 631x x  + 1486x x  - 1280x x  - 121x  - 50x x  - 978x x  - 694x x  - 531x x  + 505x x  + 1412x x  - 1061x  + 1202x x  + 448x x  - 187x x  + 1276x x  - 121x x  + 1361x x  + 697x x  + 682x  + 1592x x  + 705x x  - 227x x  - 7x x  - 1423x x  - 1446x x  - 1578x x  + 1511x x  + 917x , 1270x  - 391x x  - 1116x  - 287x x  + 653x x  + 1643x  + 1623x x  + 514x x  - 14x x  - 90x  + 1232x x  - 1434x x  + 1296x x  + 1522x x  + 136x  - 623x x  - 607x x  + 18x x  + 896x x  - 29x x  + 1059x  - 1053x x  + 1643x x  + 1652x x  - 1190x x  - 1073x x  + 1470x x  - 944x  - 93x x  - 187x x  - 994x x  - 1415x x  - 229x x  - 796x x  + 1642x x  + 1600x  - 344x x  + 905x x  + 1032x x  - 538x x  - 891x x  + 1243x x  + 1290x x  + 490x x  - 1148x , 1613x  + 175x x  - 1346x  - 1000x x  - 1217x x  - 729x  - 1296x x  + 1456x x  + 745x x  + 539x  + 525x x  - 811x x  + 753x x  + 1362x x  + 1629x  - 840x x  + 513x x  + 429x x  + 842x x  + 1414x x  - 308x  + 1415x x  - 1461x x  - 1135x x  + 701x x  + 766x x  + 785x x  + 1503x  + 147x x  + 929x x  - 1220x x  - 853x x  + 493x x  + 226x x  + 1416x x  + 280x  - 7x x  + 1632x x  + 520x x  + 1259x x  + 157x x  + 1596x x  + 655x x  - 42x x  - 586x })
│ │ │                                   0       0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4       1 4        2 4       3 4       4      0 5        1 5        2 5       3 5        4 5       5       0 6       1 6        2 6        3 6       4 6        5 6        6       0 7       1 7        2 7      3 7        4 7        5 7       6 7      7        0 8        1 8        2 8       3 8        4 8      5 8        6 8        7 8       8         0       0 1       1        0 2        1 2       2       0 3        1 3       2 3        3        0 4       1 4       2 4      3 4      4        0 5       1 5        2 5       3 5       4 5       5       0 6       1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8      5 8        6 8       7 8        8        0        0 1        1        0 2       1 2       2       0 3        1 3       2 3       3        0 4       1 4        2 4       3 4        4        0 5       1 5       2 5       3 5       4 5       5        0 6        1 6       2 6       3 6       4 6        5 6       6      0 7       1 7       2 7       3 7        4 7        5 7       6 7        7      0 8       1 8        2 8        3 8        4 8        5 8        6 8       7 8        8        0        0 1        1        0 2        1 2       2        0 3        1 3       2 3      3       0 4        1 4       2 4       3 4        4       0 5       1 5       2 5        3 5       4 5       5       0 6       1 6      2 6        3 6       4 6        5 6       6       0 7     1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8       2 8        3 8        4 8       5 8       6 8      7 8        8         0      0 1       1       0 2      1 2       2       0 3        1 3       2 3       3       0 4        1 4       2 4        3 4       4        0 5        1 5       2 5       3 5        4 5      5       0 6       1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7      3 7        4 7       5 7       6 7       7       0 8       1 8        2 8       3 8        4 8       5 8       6 8        7 8       8        0        0 1       1     0 2      1 2        2       0 3       1 3        2 3        3        0 4       1 4       2 4       3 4       4        0 5        1 5       2 5       3 5       4 5       5        0 6       1 6        2 6       3 6       4 6       5 6        6       0 7       1 7       2 7       3 7        4 7       5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8      8       0       0 1        1        0 2       1 2        2      0 3       1 3       2 3       3        0 4      1 4      2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5       0 6       1 6       2 6      3 6       4 6        5 6        6       0 7       1 7       2 7        3 7        4 7       5 7        6 7      7      0 8        1 8       2 8      3 8      4 8       5 8        6 8        7 8       8       0       0 1       1      0 2       1 2        2        0 3       1 3       2 3       3       0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5        4 5       5        0 6        1 6       2 6        3 6       4 6        5 6       6      0 7       1 7        2 7       3 7       4 7        5 7        6 7        7        0 8       1 8       2 8       3 8       4 8        5 8       6 8        7 8        8        0        0 1        1        0 2        1 2       2       0 3        1 3       2 3        3       0 4        1 4        2 4       3 4        4       0 5        1 5        2 5        3 5       4 5       5       0 6       1 6      2 6       3 6        4 6       5 6       6       0 7     1 7        2 7       3 7       4 7       5 7       6 7       7       0 8       1 8       2 8        3 8        4 8       5 8        6 8       7 8       8      0       0 1       1       0 2        1 2       2        0 3       1 3       2 3        3       0 4        1 4       2 4       3 4       4       0 5       1 5       2 5       3 5       4 5       5        0 6       1 6        2 6        3 6        4 6        5 6       6       0 7        1 7       2 7        3 7       4 7      5 7        6 7       7       0 8       1 8       2 8        3 8       4 8       5 8       6 8        7 8        8     0        0 1        1        0 2       1 2        2       0 3       1 3       2 3        3        0 4        1 4        2 4        3 4       4       0 5       1 5       2 5       3 5        4 5       5       0 6        1 6        2 6       3 6        4 6      5 6       6       0 7       1 7        2 7        3 7      4 7        5 7       6 7       7       0 8       1 8        2 8        3 8       4 8       5 8        6 8       7 8       8         0      0 1       1      0 2        1 2        2        0 3       1 3       2 3        3        0 4        1 4        2 4      3 4       4       0 5       1 5       2 5        3 5       4 5        5        0 6        1 6        2 6       3 6       4 6       5 6       6        0 7       1 7        2 7        3 7       4 7        5 7        6 7       7       0 8        1 8        2 8        3 8        4 8        5 8       6 8        7 8        8      0       0 1        1       0 2       1 2        2       0 3       1 3       2 3      3       0 4       1 4       2 4       3 4        4       0 5        1 5       2 5        3 5        4 5       5      0 6       1 6       2 6       3 6       4 6        5 6        6        0 7       1 7       2 7        3 7       4 7        5 7       6 7       7        0 8       1 8       2 8     3 8        4 8        5 8        6 8        7 8       8       0       0 1        1       0 2       1 2        2        0 3       1 3      2 3      3        0 4        1 4        2 4        3 4       4       0 5       1 5      2 5       3 5      4 5        5        0 6        1 6        2 6        3 6        4 6        5 6       6      0 7       1 7       2 7        3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8       4 8        5 8        6 8       7 8        8       0       0 1        1        0 2        1 2       2        0 3        1 3       2 3       3       0 4       1 4       2 4        3 4        4       0 5       1 5       2 5       3 5        4 5       5        0 6        1 6        2 6       3 6       4 6       5 6        6       0 7       1 7        2 7       3 7       4 7       5 7        6 7       7     0 8        1 8       2 8        3 8       4 8        5 8       6 8      7 8       8
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
│ │ │  
│ │ │  i5 : time degreeMap phi
│ │ │ - -- used 0.0497248s (cpu); 0.0497258s (thread); 0s (gc)
│ │ │ + -- used 0.0479509s (cpu); 0.0479511s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have degree equal to deg(G(1,5))=14)
│ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │  
│ │ │                                   2                  2                           2                                      2                                                 2                                                           2                                                                   2                                                                              2                                                                                          2        2                  2                              2                                       2                                                2                                                             2                                                                  2                                                                              2                                                                                            2        2                  2                             2                                       2                                                2                                                           2                                                                      2                                                                              2                                                                                         2         2                 2                            2                                       2                                                  2                                                             2                                                                    2                                                                                2                                                                                             2       2                   2                            2                                     2                                                2                                                          2                                                                  2                                                                                   2                                                                                            2        2                2                           2                                      2                                                  2                                                            2                                                                      2                                                                                 2                                                                                          2   2                   2                           2                                     2                                                  2                                                           2                                                                    2                                                                              2                                                                                         2      2                  2                           2                                      2                                                  2                                                             2                                                                       2                                                                              2                                                                                          2         2                  2                            2                                     2                                                 2                                                              2                                                                    2                                                                               2                                                                                        2
│ │ │  o6 = map (ringP8, ringP8, {- 780x  - 506x x  + 1537x  - 132x x  - 928x x  + 386x  - 102x x  + 422x x  + 725x x  - 1073x  - 905x x  - 830x x  + 1500x x  + 276x x  + 1533x  - 653x x  + 1558x x  + 939x x  - 1432x x  + 462x x  - 329x  - 92x x  + 661x x  - 1298x x  - 684x x  + 70x x  - 715x x  + 1093x  + 581x x  + 329x x  + 454x x  - 911x x  - 84x x  - 1452x x  - 809x x  + 1202x  + 1353x x  + 1503x x  + 482x x  + 893x x  - 643x x  + 598x x  + 110x x  + 1064x x  - 472x , - 522x  - 583x x  + 1339x  + 1535x x  - 1317x x  + 1113x  - 169x x  + 1440x x  - 1657x x  + 721x  + 40x x  - 1576x x  - 367x x  + 257x x  - 1454x  + 1612x x  + 1529x x  - 1068x x  + 560x x  - 1441x x  + 608x  - 92x x  - 1006x x  + 285x x  + 102x x  - 397x x  + 66x x  - 643x  - 38x x  + 1380x x  + 1069x x  - 426x x  + 1147x x  + 982x x  + 10x x  - 662x  + 16x x  + 1561x x  + 1597x x  + 512x x  + 1288x x  - 1253x x  + 1317x x  + 1481x x  - 354x , - 640x  - 1551x x  + 469x  + 1482x x  - 1593x x  - 986x  + 471x x  + 612x x  + 1228x x  + 1156x  - 731x x  + 1503x x  - 628x x  + 674x x  - 799x  + 1137x x  + 844x x  + 589x x  - 666x x  + 829x x  - 1024x  - 170x x  + 450x x  + 1497x x  + 1204x x  - 907x x  + 1621x x  - 417x  + 1297x x  + 1444x x  + 4x x  + 398x x  + 996x x  - 1031x x  + 239x x  + 303x  + 1215x x  - 83x x  + 1571x x  - 1543x x  - 925x x  - 694x x  + 151x x  - 520x x  + 880x , - 1210x  - 222x x  + 185x  + 245x x  + 1059x x  - 322x  + 238x x  + 962x x  + 1260x x  - 1581x  + 50x x  + 1352x x  - 1465x x  + 1555x x  + 1333x  + 1362x x  + 1365x x  + 1168x x  - 1401x x  + 149x x  - 652x  + 1378x x  - 557x x  - 112x x  + 26x x  + 315x x  + 111x x  + 1592x  - 283x x  - 1454x x  + 907x x  + 212x x  + 400x x  + 1049x x  - 882x x  - 1429x  - 183x x  + 1571x x  - 1286x x  - 1179x x  + 1319x x  + 240x x  - 1100x x  + 1500x x  - 348x , 1051x  - 1325x x  + 1354x  - 346x x  - 1532x x  - 466x  + 163x x  - 659x x  - 291x x  + 966x  + 789x x  + 393x x  + 403x x  - 1199x x  - 570x  - 93x x  - 492x x  - 418x x  + 713x x  - 1323x x  - 1384x  - 830x x  - 54x x  - 306x x  + 709x x  + 421x x  - 954x x  - 299x  + 1053x x  - 1080x x  + 686x x  + 170x x  - 1272x x  - 1661x x  + 1235x x  + 1553x  - 1454x x  - 1411x x  - 1195x x  - 962x x  + 737x x  - 390x x  + 957x x  + 1538x x  + 1234x , - 509x  + 9x x  - 1563x  - 710x x  - 642x x  + 541x  + 220x x  - 1214x x  - 16x x  + 1008x  - 1088x x  + 755x x  - 886x x  - 1433x x  + 1154x  + 1627x x  - 1547x x  - 951x x  + 866x x  + 163x x  - 1142x  - 668x x  + 1361x x  + 1324x x  - 490x x  + 282x x  - 1133x x  - 612x  + 805x x  - 126x x  + 1296x x  - 973x x  + 1271x x  - 1646x x  + 844x x  + 1073x  - 1452x x  - 1112x x  - 141x x  + 176x x  - 1579x x  - 78x x  + 848x x  - 1365x x  + 711x , x  + 1543x x  - 1076x  + 493x x  - 526x x  + 868x  - 582x x  - 996x x  + 206x x  - 419x  + 1258x x  - 391x x  + 1002x x  - 1539x x  + 931x  - 1504x x  + 810x x  + 324x x  + 1356x x  + 313x x  + 772x  + 299x x  + 1186x x  + 718x x  + 407x x  - 64x x  - 828x x  - 1393x  + 94x x  - 290x x  - 766x x  + 950x x  - 640x x  + 265x x  - 1640x x  - 1403x  - 126x x  + 891x x  - 1519x x  - 927x x  - 1335x x  - 1448x x  - x x  - 1103x x  - 1152x , 821x  + 558x x  - 1174x  - 168x x  + 986x x  + 790x  + 549x x  + 817x x  + 1396x x  + 695x  + 1211x x  + 878x x  - 1061x x  - 1244x x  - 880x  + 1409x x  - 567x x  + 1240x x  + 1126x x  - 1262x x  + 490x  + 1553x x  + 1276x x  + 805x x  + 576x x  - 1076x x  + 1617x x  - 495x  - 750x x  - 277x x  + 544x x  + 1479x x  - 784x x  - 64x x  - 1203x x  + 405x  + 1013x x  + 604x x  + 1301x x  + 1003x x  + 235x x  + 696x x  + 939x x  - 714x x  - 879x , - 1452x  + 727x x  - 1159x  + 449x x  - 1169x x  + 732x  + 575x x  - 600x x  + 924x x  - 837x  + 1298x x  - 860x x  + 1010x x  + 774x x  + 319x  + 1087x x  - 1120x x  + 1439x x  + 1175x x  - 1648x x  + 985x  - 1317x x  - 878x x  + 399x x  - 1339x x  + 70x x  - 463x x  + 470x  - 628x x  - 907x x  + 748x x  + 98x x  + 1150x x  + 1140x x  + 1308x x  + 621x  + 369x x  - 991x x  - 1186x x  + 61x x  - 907x x  - 681x x  - 1528x x  + 717x x  + 854x })
│ │ │                                   0       0 1        1       0 2       1 2       2       0 3       1 3       2 3        3       0 4       1 4        2 4       3 4        4       0 5        1 5       2 5        3 5       4 5       5      0 6       1 6        2 6       3 6      4 6       5 6        6       0 7       1 7       2 7       3 7      4 7        5 7       6 7        7        0 8        1 8       2 8       3 8       4 8       5 8       6 8        7 8       8        0       0 1        1        0 2        1 2        2       0 3        1 3        2 3       3      0 4        1 4       2 4       3 4        4        0 5        1 5        2 5       3 5        4 5       5      0 6        1 6       2 6       3 6       4 6      5 6       6      0 7        1 7        2 7       3 7        4 7       5 7      6 7       7      0 8        1 8        2 8       3 8        4 8        5 8        6 8        7 8       8        0        0 1       1        0 2        1 2       2       0 3       1 3        2 3        3       0 4        1 4       2 4       3 4       4        0 5       1 5       2 5       3 5       4 5        5       0 6       1 6        2 6        3 6       4 6        5 6       6        0 7        1 7     2 7       3 7       4 7        5 7       6 7       7        0 8      1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1       1       0 2        1 2       2       0 3       1 3        2 3        3      0 4        1 4        2 4        3 4        4        0 5        1 5        2 5        3 5       4 5       5        0 6       1 6       2 6      3 6       4 6       5 6        6       0 7        1 7       2 7       3 7       4 7        5 7       6 7        7       0 8        1 8        2 8        3 8        4 8       5 8        6 8        7 8       8       0        0 1        1       0 2        1 2       2       0 3       1 3       2 3       3       0 4       1 4       2 4        3 4       4      0 5       1 5       2 5       3 5        4 5        5       0 6      1 6       2 6       3 6       4 6       5 6       6        0 7        1 7       2 7       3 7        4 7        5 7        6 7        7        0 8        1 8        2 8       3 8       4 8       5 8       6 8        7 8        8        0     0 1        1       0 2       1 2       2       0 3        1 3      2 3        3        0 4       1 4       2 4        3 4        4        0 5        1 5       2 5       3 5       4 5        5       0 6        1 6        2 6       3 6       4 6        5 6       6       0 7       1 7        2 7       3 7        4 7        5 7       6 7        7        0 8        1 8       2 8       3 8        4 8      5 8       6 8        7 8       8   0        0 1        1       0 2       1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5       2 5        3 5       4 5       5       0 6        1 6       2 6       3 6      4 6       5 6        6      0 7       1 7       2 7       3 7       4 7       5 7        6 7        7       0 8       1 8        2 8       3 8        4 8        5 8    6 8        7 8        8      0       0 1        1       0 2       1 2       2       0 3       1 3        2 3       3        0 4       1 4        2 4        3 4       4        0 5       1 5        2 5        3 5        4 5       5        0 6        1 6       2 6       3 6        4 6        5 6       6       0 7       1 7       2 7        3 7       4 7      5 7        6 7       7        0 8       1 8        2 8        3 8       4 8       5 8       6 8       7 8       8         0       0 1        1       0 2        1 2       2       0 3       1 3       2 3       3        0 4       1 4        2 4       3 4       4        0 5        1 5        2 5        3 5        4 5       5        0 6       1 6       2 6        3 6      4 6       5 6       6       0 7       1 7       2 7      3 7        4 7        5 7        6 7       7       0 8       1 8        2 8      3 8       4 8       5 8        6 8       7 8       8
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
│ │ │  
│ │ │  i7 : time degreeMap phi'
│ │ │ - -- used 1.46278s (cpu); 0.843714s (thread); 0s (gc)
│ │ │ + -- used 1.2641s (cpu); 0.680409s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_force__Image.out
│ │ │ @@ -5,14 +5,14 @@
│ │ │  o2 : Ideal of P6
│ │ │  
│ │ │  i3 : Phi = rationalMap(X,Dominant=>2);
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000667762s (cpu); 0.00065607s (thread); 0s (gc)
│ │ │ + -- used 0.000627751s (cpu); 0.000625436s (thread); 0s (gc)
│ │ │  
│ │ │  i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │                        - x  + x x
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0166208s (cpu); 0.0163389s (thread); 0s (gc)
│ │ │ + -- used 0.0809438s (cpu); 0.0234439s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │                                    ZZ                                 ZZ
│ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
│ │ │                                  190181  0   1   2   3   4          190181  0   1   2   3   4   5
│ │ │ @@ -173,15 +173,15 @@
│ │ │  i8 : projectiveDegrees p2
│ │ │  
│ │ │  o8 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : time g = graph p2;
│ │ │ - -- used 0.0319997s (cpu); 0.0316715s (thread); 0s (gc)
│ │ │ + -- used 0.097417s (cpu); 0.0423217s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │  
│ │ │  i11 : g_1;
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_ideal_lp__Rational__Map_rp.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │                        x  - x x
│ │ │                         1    0 3
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │  
│ │ │  i3 : time ideal phi
│ │ │ - -- used 0.00352782s (cpu); 0.00352276s (thread); 0s (gc)
│ │ │ + -- used 0.00376922s (cpu); 0.00376915s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -108,15 +108,15 @@
│ │ │                        y
│ │ │                         4
│ │ │                       }
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
│ │ │  
│ │ │  i6 : time ideal phi'
│ │ │ - -- used 0.281847s (cpu); 0.150706s (thread); 0s (gc)
│ │ │ + -- used 0.253056s (cpu); 0.143947s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │                        w w  - w w  + w w
│ │ │                         2 4    1 5    0 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │  
│ │ │  i2 : time psi = inverseMap phi
│ │ │ - -- used 0.0833126s (cpu); 0.0833315s (thread); 0s (gc)
│ │ │ + -- used 0.0820946s (cpu); 0.0820953s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -158,15 +158,15 @@
│ │ │  o4 = map (QQ[w ..w  ], QQ[w ..w  ], {w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   - w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   + w  w   - w w  , w w   - w w   + w w   + w w   + w w  , w  w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   + w  w   - w w  , w w   - w w   + w w   + w w   + w w  , w  w   - w  w   - w  w   + w  w   - w w  , w w   - w w   - w w   + w w   + w w  , w  w   - w  w   - w  w   + w  w   - w w  , w  w   - w  w   - w  w   + w  w   - w w  , w w   - w w   - w w   + w w   + w w  , w w   - w w   - w w   + w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   + w w   - w w  , w w   - w w   + w w   + w w   - w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   - w w   + w w  , w w   - w w   - w w   + w w   - w w  , w w   - w w   - w w   + w w   - w w  , w w  - w w  - w w  + w w  - w w })
│ │ │                0   26       0   26     21 22    20 23    15 24    10 25    0 26   19 22    18 23    16 24    11 25    1 26   19 20    18 21    17 24    12 25    2 26   15 19    16 21    17 23    13 25    3 26   10 19    11 21    12 23    13 24    4 26   0 19    1 21    2 23    3 24    4 25   15 18    16 20    17 22    14 25    5 26   10 18    11 20    12 22    14 24    6 26   0 18    1 20    2 22    5 24    6 25   12 16    11 17    13 18    14 19    7 26   2 16    1 17    3 18    5 19    7 25   12 15    10 17    13 20    14 21    8 26   11 15    10 16    13 22    14 23    9 26   2 15    0 17    3 20    5 21    8 25   1 15    0 16    3 22    5 23    9 25   5 13    3 14    7 15    8 16    9 17   5 12    2 14    6 17    8 18    7 20   3 12    2 13    4 17    8 19    7 21   5 11    1 14    6 16    9 18    7 22   3 11    1 13    4 16    9 19    7 23   2 11    1 12    4 18    6 19    7 24   7 10    8 11    9 12    6 13    4 14   5 10    0 14    6 15    9 20    8 22   3 10    0 13    4 15    9 21    8 23   2 10    0 12    4 20    6 21    8 24   1 10    0 11    4 22    6 23    9 24   4 5    3 6    0 7    1 8    2 9
│ │ │  
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ │  
│ │ │  i5 : time psi = inverseMap phi
│ │ │ - -- used 0.300391s (cpu); 0.208058s (thread); 0s (gc)
│ │ │ + -- used 0.280211s (cpu); 0.1851s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   + w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w  , w w   - w w   + w w   + w  w   - w  w  , - w w   + w w   + w  w   + w w   - w w  , - w w   + w w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , w  w   - w  w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w  - w w  - w w   + w w   - w w  , - w w  + w w  + w w   - w w   + w w  , w w  - w w  - w w  + w w   - w w  })
│ │ │                0   26       0   26       5 22    8 23    14 24    13 25    0 26     5 18    8 19    14 20    10 25    1 26     5 16    8 17    13 20    10 24    2 26     5 15    14 17    13 19    10 23    3 26     5 21    20 23    19 24    17 25    4 26     8 15    14 16    13 18    10 22    6 26     8 21    20 22    18 24    16 25    7 26   17 18    16 19    15 20    10 21    9 26     13 21    17 22    16 23    15 24    11 26     14 21    19 22    18 23    15 25    12 26   0 21    4 22    7 23    12 24    11 25     4 18    7 19    12 20    1 21    9 25     4 16    7 17    11 20    2 21    9 24     4 15    12 17    11 19    3 21    9 23     7 15    12 16    11 18    6 21    9 22   12 13    11 14    0 15    3 22    6 23   10 12    9 14    1 15    3 18    6 19   10 11    9 13    2 15    3 16    6 17   8 9    7 10    1 16    2 18    6 20   5 9    4 10    1 17    2 19    3 20   8 11    7 13    0 16    2 22    6 24   5 11    4 13    0 17    2 23    3 24   8 12    7 14    0 18    1 22    6 25   5 12    4 14    0 19    1 23    3 25   5 7    4 8    0 20    1 24    2 25     5 6    3 8    0 10    1 13    2 14   4 6    3 7    0 9    1 11    2 12
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out
│ │ │ @@ -28,15 +28,15 @@
│ │ │                        - -------x  + ---------x x  + ------------x x  - ----------x x  - -----x  - -----------x x  + -------------x x x  + -------------x x x  - --------x x  - ----------x x  + -------------x x x  - ----------x x  - -----------x x  + ----------x x  + ------x  + -----------x x  + ----------x x x  - -----------x x x  - -------x x  + -------------x x x  + ------------x x x x  - -----------x x x  + -----------x x x  - ------------x x x  + ----------x x  - -----------x x  - ------------x x x  - ---------x x  - ------------x x x  - -----------x x x  + -----------x x  - ----------x x  + -------x x  + --------x x  + ------x  + ---------x x  - ------------x x x  - -------------x x x  - ----------x x  + --------------x x x  + -------------x x x x  - ------------x x x  + -------------x x x  + ------------x x x  + ----------x x  + -----------x x x  - -------------x x x x  - ----------x x x  + --------------x x x x  - -------------x x x x  + -------------x x x  - ------------x x x  + ---------x x x  - ------------x x x  + ---------x x  - ---------x x  - -----------x x x  - ----------x x  + -----------x x x  + -----------x x x  + ----------x x  - -----------x x x  - -----------x x x  - ------------x x x  - ----------x x  + ---------x x  - ------x x  - --------x x  - ----------x x  - -----x
│ │ │                           290304 0    3888000  0 1    2939328000  0 1    163296000 0 1   20250 1    228614400  0 2    41150592000  0 1 2    41150592000  0 1 2    3888000 1 2     3572100  0 2    10287648000  0 1 2    342921600 1 2    114307200  0 2    63504000  1 2    25200 2     76204800  0 3    42336000  0 1 3    428652000  0 1 3    212625 1 3     5334336000  0 2 3    9601804800  0 1 2 3    489888000  1 2 3    222264000  0 2 3    12002256000 1 2 3    66679200  2 3    666792000  0 3     666792000  0 1 3    47628000 1 3    1333584000  0 2 3    444528000  1 2 3    777924000  2 3    55566000  0 3    105840 1 3    3472875 2 3    11025 3    4665600  0 4    2939328000  0 1 4     4898880000  0 1 4    29160000  1 4     41150592000  0 2 4    20575296000  0 1 2 4    4898880000  1 2 4    20575296000  0 2 4    1371686400  1 2 4    95256000  2 4     40824000  0 3 4     8573040000  0 1 3 4    11664000  1 3 4     24004512000  0 2 3 4    34292160000  1 2 3 4    12002256000  2 3 4     333396000  0 3 4    5292000  1 3 4    1333584000  2 3 4    3969000  3 4    6804000  0 4    272160000  0 1 4    58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4    476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4    30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │  
│ │ │  i3 : time inverse phi
│ │ │ - -- used 0.0578597s (cpu); 0.0578645s (thread); 0s (gc)
│ │ │ + -- used 0.0563582s (cpu); 0.05636s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out
│ │ │ @@ -40,18 +40,18 @@
│ │ │                        - t  + t t
│ │ │                           3    2 4
│ │ │                       }
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │  
│ │ │  i3 : time isBirational phi
│ │ │ - -- used 0.0202901s (cpu); 0.0202894s (thread); 0s (gc)
│ │ │ + -- used 0.0191121s (cpu); 0.0191143s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0145752s (cpu); 0.014156s (thread); 0s (gc)
│ │ │ + -- used 0.0635394s (cpu); 0.0197596s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
│ │ │  
│ │ │  i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.33576s (cpu); 2.59272s (thread); 0s (gc)
│ │ │ + -- used 2.20394s (cpu); 1.90339s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : P7 = ZZ/101[x_0..x_7];
│ │ │  
│ │ │  i5 : -- hyperelliptic curve of genus 3
│ │ │       C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7-23*x_1*x_7+8*x_2*x_7-22*x_3*x_7+20*x_4*x_7-15*x_5*x_7,x_2*x_5+47*x_5^2-40*x_0*x_6+37*x_1*x_6-25*x_2*x_6-22*x_3*x_6-8*x_4*x_6+27*x_5*x_6+15*x_6^2-23*x_0*x_7-42*x_1*x_7+27*x_2*x_7+35*x_3*x_7+39*x_4*x_7+24*x_5*x_7,x_1*x_5+15*x_5^2+49*x_0*x_6+8*x_1*x_6-31*x_2*x_6+9*x_3*x_6+38*x_4*x_6-36*x_5*x_6-30*x_6^2-33*x_0*x_7+26*x_1*x_7+32*x_2*x_7+27*x_3*x_7+6*x_4*x_7+36*x_5*x_7,x_0*x_5+30*x_5^2-11*x_0*x_6-38*x_1*x_6+13*x_2*x_6-32*x_3*x_6-30*x_4*x_6+4*x_5*x_6-28*x_6^2-30*x_0*x_7-6*x_1*x_7-45*x_2*x_7+34*x_3*x_7+20*x_4*x_7+48*x_5*x_7,x_3*x_4+46*x_5^2-37*x_0*x_6+27*x_1*x_6+33*x_2*x_6+8*x_3*x_6-32*x_4*x_6+42*x_5*x_6-34*x_6^2-37*x_0*x_7-28*x_1*x_7+10*x_2*x_7-27*x_3*x_7-42*x_4*x_7-8*x_5*x_7,x_2*x_4-25*x_5^2-4*x_0*x_6+2*x_1*x_6-31*x_2*x_6-5*x_3*x_6+16*x_4*x_6-24*x_5*x_6+31*x_6^2-30*x_0*x_7+32*x_1*x_7+12*x_2*x_7-40*x_3*x_7+3*x_4*x_7-28*x_5*x_7,x_0*x_4+15*x_5^2+48*x_0*x_6-50*x_1*x_6+46*x_2*x_6-48*x_3*x_6-23*x_4*x_6-28*x_5*x_6+39*x_6^2+38*x_1*x_7-5*x_3*x_7+5*x_4*x_7-34*x_5*x_7,x_3^2-31*x_5^2+41*x_0*x_6-30*x_1*x_6-4*x_2*x_6+43*x_3*x_6+23*x_4*x_6+7*x_5*x_6+31*x_6^2-19*x_0*x_7+25*x_1*x_7-49*x_2*x_7-16*x_3*x_7-45*x_4*x_7+25*x_5*x_7,x_2*x_3+13*x_5^2-45*x_0*x_6-22*x_1*x_6+33*x_2*x_6-26*x_3*x_6-21*x_4*x_6+34*x_5*x_6-21*x_6^2-47*x_0*x_7-10*x_1*x_7+29*x_2*x_7-46*x_3*x_7-x_4*x_7+20*x_5*x_7,x_1*x_3+22*x_5^2+4*x_0*x_6+3*x_1*x_6+45*x_2*x_6+37*x_3*x_6+17*x_4*x_6+36*x_5*x_6-2*x_6^2-31*x_0*x_7+3*x_1*x_7-12*x_2*x_7+19*x_3*x_7+28*x_4*x_7+30*x_5*x_7,x_0*x_3-47*x_5^2-43*x_0*x_6+6*x_1*x_6-40*x_2*x_6+21*x_3*x_6+26*x_4*x_6-5*x_5*x_6-5*x_6^2+4*x_0*x_7-15*x_1*x_7+18*x_2*x_7-31*x_3*x_7+50*x_4*x_7-46*x_5*x_7,x_2^2+4*x_5^2+31*x_0*x_6+41*x_1*x_6+31*x_2*x_6+28*x_3*x_6+42*x_4*x_6-28*x_5*x_6-4*x_6^2-7*x_0*x_7+15*x_1*x_7-9*x_2*x_7+31*x_3*x_7+3*x_4*x_7+7*x_5*x_7,x_1*x_2-46*x_5^2-6*x_0*x_6-50*x_1*x_6+32*x_2*x_6-10*x_3*x_6+42*x_4*x_6+33*x_5*x_6+18*x_6^2-9*x_0*x_7-20*x_1*x_7+45*x_2*x_7-9*x_3*x_7+10*x_4*x_7-8*x_5*x_7,x_0*x_2-9*x_5^2+34*x_0*x_6-45*x_1*x_6+19*x_2*x_6+24*x_3*x_6+23*x_4*x_6-37*x_5*x_6-44*x_6^2+24*x_0*x_7-33*x_2*x_7+41*x_3*x_7-40*x_4*x_7+4*x_5*x_7,x_1^2+x_1*x_4+x_4^2-28*x_5^2-33*x_0*x_6-17*x_1*x_6+11*x_3*x_6+20*x_4*x_6+25*x_5*x_6-21*x_6^2-22*x_0*x_7+24*x_1*x_7-14*x_2*x_7+5*x_3*x_7-39*x_4*x_7-18*x_5*x_7,x_0*x_1-47*x_5^2-5*x_0*x_6-9*x_1*x_6-45*x_2*x_6+48*x_3*x_6+45*x_4*x_6-29*x_5*x_6+3*x_6^2+29*x_0*x_7+40*x_1*x_7+46*x_2*x_7+27*x_3*x_7-36*x_4*x_7-39*x_5*x_7,x_0^2-31*x_5^2+36*x_0*x_6-30*x_1*x_6-10*x_2*x_6+42*x_3*x_6+9*x_4*x_6+34*x_5*x_6-6*x_6^2+48*x_0*x_7-47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7);
│ │ │ @@ -21,12 +21,12 @@
│ │ │  
│ │ │  i6 : phi = rationalMap(C,3,2);
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from PP^7 to PP^7)
│ │ │  
│ │ │  i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 4.05735s (cpu); 2.55688s (thread); 0s (gc)
│ │ │ + -- used 3.54851s (cpu); 2.46268s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out
│ │ │ @@ -6,23 +6,23 @@
│ │ │  o1 = map (QQ[x ..x ], QQ[y ..y  ], {- 5x x  + x x  + x x  + 35x x  - 7x x  + x x  - x x  - 49x  - 5x x  + 2x x  - x x  + 27x x  - 4x  + x x  - 7x x  + 2x x  - 2x x  + 14x x  - 4x x , - x x  - 6x x  - 5x x  + 2x x  + x x  + x x  - 5x x  - x x  + 2x x  + 7x x  - 2x x  + 2x x  - 3x x , - 25x  + 9x x  + 10x x  - 2x x  - x  + 29x x  - x x  - 7x x  - 13x x  + 3x x  + x x  - x x  + 2x x  - x x  + 7x x  - 2x x  - 8x x  + 2x x  - 3x x , x x  + x x  + x  + 7x x  - 9x x  + 12x x  - 4x  + 2x x  + 2x x  - 14x x  + 4x x  + x x  - x x  - 14x x  + x x , - 5x x  + x x  - 7x x  + 8x x  - 5x x  + 2x x  - x x  + x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , x x  + x  - 7x x  - 8x x  + x x  + x x  + 2x x  - x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  + x  - 8x x  + x x  + 6x x  - 2x  + x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  - 7x x  + x x  + x x  - 7x x  + 2x  - x x , - 4x x  + x x  - x  - 7x x  + 8x x  + x x  - x x  - 6x x  + 2x  - x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , - 5x x  + 2x  + x x  - x  - x x  + 8x x  - 10x x  + 2x x  + 2x x  - 2x x  + 14x x  - 4x x  + 5x x  - 3x x  - 2x x  + 7x x  - 2x x  - 3x x , - 5x x  + x x  + x x  - 4x x  - x x  + x x  + x x , x x  - x x  + 5x x  + x x  - 14x x  - x x  - 8x x  - 8x x  + 2x x  + 4x x  + 2x x  + 4x x  + 3x x  - 7x x  + 2x x  - 3x x })
│ │ │                0   8       0   11        0 3    2 4    3 4      0 5     2 5    3 5    4 5      5     0 6     2 6    4 6      5 6     6    4 7     5 7     6 7     4 8      5 8     6 8     1 2     1 5     0 6     1 6    4 6    5 6     0 7    1 7     2 7     5 7     6 7     1 8     7 8       0     0 2      0 4     2 4    4      0 5    2 5     4 5      0 6     4 6    5 6    0 7     2 7    4 7     5 7     6 7     0 8     4 8     7 8   2 4    3 4    4     2 5     4 5      5 6     6     3 7     4 7      5 7     6 7    3 8    4 8      5 8    6 8      0 4    2 4     2 5     4 5     0 6     2 6    4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   0 4    4     1 5     4 5    0 6    1 6     4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   2 3    4     4 5    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8   1 3     1 5    1 6    4 6     5 6     6    3 7      0 3    3 4    4     0 5     4 5    0 6    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8      0 2     2    2 4    4    2 5     4 5      0 6     5 6     2 7     4 7      5 7     6 7     0 8     2 8     4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7   0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2 7     0 8     1 8     5 8     6 8     7 8
│ │ │  
│ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │                   0   8          0   11
│ │ │  
│ │ │  i2 : time kernel(phi,1)
│ │ │ - -- used 0.0180149s (cpu); 0.0180127s (thread); 0s (gc)
│ │ │ + -- used 0.0193662s (cpu); 0.0193675s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 1.06451s (cpu); 0.555512s (thread); 0s (gc)
│ │ │ + -- used 0.926832s (cpu); 0.437835s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_parametrize_lp__Ideal_rp.out
│ │ │ @@ -26,15 +26,15 @@
│ │ │                8           9
│ │ │  
│ │ │                   ZZ
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i3 : time parametrize L
│ │ │ - -- used 0.00443973s (cpu); 0.00443525s (thread); 0s (gc)
│ │ │ + -- used 0.00501935s (cpu); 0.00501783s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -116,15 +116,15 @@
│ │ │               5 9           6 9           7 9           8 9           9
│ │ │  
│ │ │                   ZZ
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
│ │ │  
│ │ │  i5 : time parametrize Q
│ │ │ - -- used 0.76754s (cpu); 0.460282s (thread); 0s (gc)
│ │ │ + -- used 0.548412s (cpu); 0.385579s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 3560583829489988690
│ │ │  
│ │ │  i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331);
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
│ │ │  
│ │ │  i2 : time p = point source f
│ │ │ - -- used 0.298623s (cpu); 0.198159s (thread); 0s (gc)
│ │ │ + -- used 0.260566s (cpu); 0.154493s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -20,12 +20,12 @@
│ │ │                                                             -----[y ..y  ]
│ │ │                                                             33331  0   11
│ │ │  o2 : Ideal of -------------------------------------------------------------------------------------------------------
│ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │  i3 : time p == f^* f p
│ │ │ - -- used 0.176944s (cpu); 0.176946s (thread); 0s (gc)
│ │ │ + -- used 0.0955232s (cpu); 0.0955261s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out
│ │ │ @@ -8,15 +8,15 @@
│ │ │                       0   4              0   5       1    0 2     1 2    0 3     2    1 3     1 3    0 4     2 3    1 4     3    2 4
│ │ │  
│ │ │  o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ]
│ │ │                          0   4                 0   5
│ │ │  
│ │ │  i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0278401s (cpu); 0.0272526s (thread); 0s (gc)
│ │ │ + -- used 0.0265372s (cpu); 0.0152986s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : psi=inverseMap(toMap(phi,Dominant=>infinity))
│ │ │  
│ │ │ @@ -30,15 +30,15 @@
│ │ │                           0   5
│ │ │  o4 : RingMap ------------------ <-- GF 109561[t ..t ]
│ │ │               x x  - x x  + x x                 0   4
│ │ │                2 3    1 4    0 5
│ │ │  
│ │ │  i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.019353s (cpu); 0.0187436s (thread); 0s (gc)
│ │ │ + -- used 0.0718317s (cpu); 0.0182071s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : -- Cremona transformation of P^6 defined by the quadrics through a rational octic surface
│ │ │       phi = map specialCremonaTransformation(7,ZZ/300007)
│ │ │ @@ -48,21 +48,21 @@
│ │ │            300007  0   6   300007  0   6     2 4    1 5          0 4          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6   2 3    0 5          1 3          1 4          4         0 5          1 5         2 5          4 5         5          3 6         4 6         5 6        0 3         1 4         3 4         4          0 5         1 5         2 5          3 5          4 5         5         3 6          4 6         5 6          0 1          1         0 2          1 2         2          1 4          1 5         2 5          0 6         1 6         2 6         0          1         0 2         1 2         2         1 4          4         0 5         1 5          2 5          4 5         5         0 6         1 6          2 6          3 6         4 6         5 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o6 : RingMap ------[x ..x ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   6
│ │ │  
│ │ │  i7 : time projectiveDegrees phi
│ │ │ - -- used 5.2489e-05s (cpu); 4.7148e-05s (thread); 0s (gc)
│ │ │ + -- used 5.6289e-05s (cpu); 5.0428e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.626e-05s (cpu); 2.6219e-05s (thread); 0s (gc)
│ │ │ + -- used 2.2005e-05s (cpu); 2.1699e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
│ │ │  
│ │ │                  ZZ
│ │ │  o2 : Ideal of -----[x ..x ]
│ │ │                33331  0   6
│ │ │  
│ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.117734s (cpu); 0.117709s (thread); 0s (gc)
│ │ │ + -- used 0.101826s (cpu); 0.101827s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ring_cm__Tally_rp.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │                     0         1         2         3        4         5
│ │ │  
│ │ │  o4 : Ideal of X
│ │ │  
│ │ │  i5 : D = new Tally from {H => 2,C => 1};
│ │ │  
│ │ │  i6 : time phi = rationalMap D
│ │ │ - -- used 0.029123s (cpu); 0.029123s (thread); 0s (gc)
│ │ │ + -- used 0.0317304s (cpu); 0.0317266s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -123,13 +123,13 @@
│ │ │                        x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
│ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │                       }
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
│ │ │  
│ │ │  i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.23167s (cpu); 0.77726s (thread); 0s (gc)
│ │ │ + -- used 1.46964s (cpu); 0.596782s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cremona__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1330846641081
│ │ │  
│ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.3736s (cpu); 1.13714s (thread); 0s (gc)
│ │ │ + -- used 1.00193s (cpu); 0.911121s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cubic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1730018912715498288
│ │ │  
│ │ │  i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.102301s (cpu); 0.102302s (thread); 0s (gc)
│ │ │ + -- used 0.081544s (cpu); 0.0815448s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -62,15 +62,15 @@
│ │ │                        8x x  - 12x x  + 24x  - 11x x  + 17x x x  - 24x x  - 10x x  + 11x x  - 3x  - 6x x  + 28x x x  - 70x x  - 21x x x  + 47x x x  - 13x x  - 14x x  + 66x x  - 22x x  - 20x  + 2x x  - 2x x x  - 10x x  - 11x x x  + 8x x x  - 5x x  + 3x x x  + 23x x x  - 11x x x  - 12x x  + 3x x  - 3x x  - 2x x  + 3x x  + x  - 11x x  + 14x x x  + 34x x  - 6x x x  - 16x x x  + 3x x  - 15x x x  - 66x x x  + 12x x x  + 30x x  - 19x x x  + 2x x x  - 5x x x  - 2x x x  - 7x x  + 6x x  + 21x x  - 3x x  - 21x x  + x x  + 5x  - 8x x  + 7x x x  - 32x x  - 13x x x  + 28x x x  - 9x x  + 70x x x  - 27x x x  - 36x x  + x x x  + 4x x x  - 7x x x  - 2x x x  + 3x x  - 25x x x  - 23x x x  + 4x x x  + 27x x x  - 14x x x  - 9x x  - 2x x  + 10x x  - 6x x  - 10x x  + 3x x  - 2x x
│ │ │                          0 1      0 1      1      0 2      0 1 2      1 2      0 2      1 2     2     0 3      0 1 3      1 3      0 2 3      1 2 3      2 3      0 3      1 3      2 3      3     0 4     0 1 4      1 4      0 2 4     1 2 4     2 4     0 3 4      1 3 4      2 3 4      3 4     0 4     1 4     2 4     3 4    4      0 5      0 1 5      1 5     0 2 5      1 2 5     2 5      0 3 5      1 3 5      2 3 5      3 5      0 4 5     1 4 5     2 4 5     3 4 5     4 5     0 5      1 5     2 5      3 5    4 5     5     0 6     0 1 6      1 6      0 2 6      1 2 6     2 6      1 3 6      2 3 6      3 6    0 4 6     1 4 6     2 4 6     3 4 6     4 6      0 5 6      1 5 6     2 5 6      3 5 6      4 5 6     5 6     0 6      1 6     2 6      3 6     4 6     5 6
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.0245428s (cpu); 0.0245442s (thread); 0s (gc)
│ │ │ + -- used 0.0173718s (cpu); 0.017373s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_special__Quadratic__Transformation.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1729200582376678705
│ │ │  
│ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0879444s (cpu); 0.0879457s (thread); 0s (gc)
│ │ │ + -- used 0.0735507s (cpu); 0.0735503s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -50,15 +50,15 @@
│ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │                       }
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │  
│ │ │  i2 : time describe oo
│ │ │ - -- used 0.00847425s (cpu); 0.00847698s (thread); 0s (gc)
│ │ │ + -- used 0.00699743s (cpu); 0.00699696s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/_to__External__String_lp__Rational__Map_rp.out
│ │ │ @@ -7,34 +7,34 @@
│ │ │  i2 : str = toExternalString phi;
│ │ │  
│ │ │  i3 : #str
│ │ │  
│ │ │  o3 = 6927
│ │ │  
│ │ │  i4 : time phi' = value str;
│ │ │ - -- used 0.0227997s (cpu); 0.022798s (thread); 0s (gc)
│ │ │ + -- used 0.0238765s (cpu); 0.0238763s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.00554153s (cpu); 0.00554197s (thread); 0s (gc)
│ │ │ + -- used 0.00607877s (cpu); 0.0060834s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │       number of minimal representatives: 1
│ │ │       dimension base locus: 1
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
│ │ │  
│ │ │  i6 : time describe inverse phi'
│ │ │ - -- used 0.00455456s (cpu); 0.00455575s (thread); 0s (gc)
│ │ │ + -- used 0.00512393s (cpu); 0.0051286s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html
│ │ │ @@ -102,30 +102,30 @@
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │              
    
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.26956s (cpu); 0.894031s (thread); 0s (gc)
│ │ │ + -- used 1.12596s (cpu); 0.854789s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o3 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o3 : -----
│ │ │          5
│ │ │         H
    
│ │ │              
    
│ │ │                
    i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 1.46902s (cpu); 1.00309s (thread); 0s (gc)
│ │ │ + -- used 1.16897s (cpu); 0.819846s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -172,30 +172,30 @@
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i9 : time ChernClass G
│ │ │ - -- used 0.110298s (cpu); 0.110305s (thread); 0s (gc)
│ │ │ + -- used 0.119362s (cpu); 0.119364s (thread); 0s (gc)
│ │ │  
│ │ │          9      8      7      6      5      4     3
│ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o9 : -----
│ │ │         10
│ │ │        H
    
│ │ │              
    
│ │ │                
    i10 : time ChernClass(G,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0113359s (cpu); 0.0107235s (thread); 0s (gc)
│ │ │ + -- used 0.0233192s (cpu); 0.0117052s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,26 +39,26 @@
│ │ │ │                 2                           2
│ │ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │ │                          0   4
│ │ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ │ - -- used 1.26956s (cpu); 0.894031s (thread); 0s (gc)
│ │ │ │ + -- used 1.12596s (cpu); 0.854789s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o3 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o3 : -----
│ │ │ │          5
│ │ │ │         H
│ │ │ │  i4 : time ChernSchwartzMacPherson(C,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 1.46902s (cpu); 1.00309s (thread); 0s (gc)
│ │ │ │ + -- used 1.16897s (cpu); 0.819846s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         4     3     2
│ │ │ │  o4 = 3H  + 5H  + 3H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          5
│ │ │ │ @@ -88,26 +88,26 @@
│ │ │ │          0,2 1,3    0,1 2,3
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i9 : time ChernClass G
│ │ │ │ - -- used 0.110298s (cpu); 0.110305s (thread); 0s (gc)
│ │ │ │ + -- used 0.119362s (cpu); 0.119364s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          9      8      7      6      5      4     3
│ │ │ │  o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o9 : -----
│ │ │ │         10
│ │ │ │        H
│ │ │ │  i10 : time ChernClass(G,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0113359s (cpu); 0.0107235s (thread); 0s (gc)
│ │ │ │ + -- used 0.0233192s (cpu); 0.0117052s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6      5      4     3
│ │ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o10 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html
│ │ │ @@ -90,24 +90,24 @@
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i2 : time EulerCharacteristic I
│ │ │ - -- used 0.590242s (cpu); 0.241339s (thread); 0s (gc)
│ │ │ + -- used 0.473545s (cpu); 0.199432s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
    
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0127473s (cpu); 0.0122967s (thread); 0s (gc)
│ │ │ + -- used 0.0700004s (cpu); 0.0180693s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 10
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -31,20 +31,20 @@
│ │ │ │  i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p
│ │ │ │  ]
│ │ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
│ │ │ │  i2 : time EulerCharacteristic I
│ │ │ │ - -- used 0.590242s (cpu); 0.241339s (thread); 0s (gc)
│ │ │ │ + -- used 0.473545s (cpu); 0.199432s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0127473s (cpu); 0.0122967s (thread); 0s (gc)
│ │ │ │ + -- used 0.0700004s (cpu); 0.0180693s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = 10
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  No test is made to see if the projective variety is smooth.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_u_l_e_r_(_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- topological Euler characteristic of a
│ │ │ │        (smooth) projective variety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i4 : time phi! ;
│ │ │ - -- used 0.0572395s (cpu); 0.0568639s (thread); 0s (gc)
│ │ │ + -- used 0.114779s (cpu); 0.063545s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
    
│ │ │              
    
│ │ │                
    i5 : describe phi
│ │ │ @@ -132,15 +132,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i9 : time phi! ;
│ │ │ - -- used 0.0401527s (cpu); 0.03977s (thread); 0s (gc)
│ │ │ + -- used 0.0954699s (cpu); 0.0414719s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
    
│ │ │              
    
│ │ │                
    i10 : describe phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i3 : describe phi
│ │ │ │  
│ │ │ │  o3 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i4 : time phi! ;
│ │ │ │ - -- used 0.0572395s (cpu); 0.0568639s (thread); 0s (gc)
│ │ │ │ + -- used 0.114779s (cpu); 0.063545s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
│ │ │ │  i5 : describe phi
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^5
│ │ │ │       target variety: PP^5
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  i8 : describe phi
│ │ │ │  
│ │ │ │  o8 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^5
│ │ │ │       coefficient ring: QQ
│ │ │ │  i9 : time phi! ;
│ │ │ │ - -- used 0.0401527s (cpu); 0.03977s (thread); 0s (gc)
│ │ │ │ + -- used 0.0954699s (cpu); 0.0414719s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │ │  i10 : describe phi
│ │ │ │  
│ │ │ │  o10 = rational map defined by forms of degree 2
│ │ │ │        source variety: PP^4
│ │ │ │        target variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_st_st_sp__Ideal.html
│ │ │ @@ -158,15 +158,15 @@
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i6 : time phi^** q
│ │ │ - -- used 0.368428s (cpu); 0.207459s (thread); 0s (gc)
│ │ │ + -- used 0.325119s (cpu); 0.188176s (thread); 0s (gc)
│ │ │  
│ │ │                  e        d        c        b        a
│ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │                  f        f        f        f        f
│ │ │  
│ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i4 : time SegreClass X
│ │ │ - -- used 0.885027s (cpu); 0.51192s (thread); 0s (gc)
│ │ │ + -- used 0.874277s (cpu); 0.513743s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i5 : time SegreClass lift(X,P7)
│ │ │ - -- used 0.568535s (cpu); 0.343331s (thread); 0s (gc)
│ │ │ + -- used 0.415801s (cpu); 0.286306s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o5 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i6 : time SegreClass(X,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0216529s (cpu); 0.0213527s (thread); 0s (gc)
│ │ │ + -- used 0.0499798s (cpu); 0.0222984s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5       4      3
│ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          8
│ │ │         H
    
│ │ │              
    
│ │ │                
    i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0973043s (cpu); 0.0970462s (thread); 0s (gc)
│ │ │ + -- used 0.280271s (cpu); 0.151321s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -208,25 +208,25 @@
│ │ │          
    
│ │ │          
    The method also accepts as input a ring map phi representing a rational map $\Phi:X\dashrightarrow Y$ between projective varieties. In this case, the method returns the push-forward to the Chow ring of the ambient projective space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it basically computes SegreClass ideal matrix phi. In the next example, we compute the Segre class of the base locus of a birational map $\mathbb{G}(1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ -= ------[x ..x ]
│ │ │ -  100003  0   6
│ │ │ +o9 =   ZZ
│ │ │ + ------[x ..x ]
│ │ │ + 100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.237104s (cpu); 0.125625s (thread); 0s (gc)
│ │ │ + -- used 0.0661419s (cpu); 0.0661437s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -238,15 +238,15 @@
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i11 : time SegreClass phi
│ │ │ - -- used 0.385145s (cpu); 0.250766s (thread); 0s (gc)
│ │ │ + -- used 0.374685s (cpu); 0.2463s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -272,30 +272,30 @@
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.405977s (cpu); 0.267353s (thread); 0s (gc)
│ │ │ + -- used 0.391422s (cpu); 0.267298s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o13 : -----
│ │ │          10
│ │ │         H
    
│ │ │              
    
│ │ │                
    i14 : -- Segre class of B in P^9
│ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ - -- used 1.63119s (cpu); 0.951264s (thread); 0s (gc)
│ │ │ + -- used 1.40422s (cpu); 0.83095s (thread); 0s (gc)
│ │ │  
│ │ │             9       8       7      6     5
│ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o14 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,47 +81,47 @@
│ │ │ │                 2 2                2 2                                        2
│ │ │ │  2                                                    2 2
│ │ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x
│ │ │ │  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1
│ │ │ │  6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │ │  i4 : time SegreClass X
│ │ │ │ - -- used 0.885027s (cpu); 0.51192s (thread); 0s (gc)
│ │ │ │ + -- used 0.874277s (cpu); 0.513743s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o4 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o4 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i5 : time SegreClass lift(X,P7)
│ │ │ │ - -- used 0.568535s (cpu); 0.343331s (thread); 0s (gc)
│ │ │ │ + -- used 0.415801s (cpu); 0.286306s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o5 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o5 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i6 : time SegreClass(X,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0216529s (cpu); 0.0213527s (thread); 0s (gc)
│ │ │ │ + -- used 0.0499798s (cpu); 0.0222984s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5       4      3
│ │ │ │  o6 = 3240H  - 1188H  + 396H  - 114H  + 24H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o6 : -----
│ │ │ │          8
│ │ │ │         H
│ │ │ │  i7 : time SegreClass(lift(X,P7),Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0973043s (cpu); 0.0970462s (thread); 0s (gc)
│ │ │ │ + -- used 0.280271s (cpu); 0.151321s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o7 : -----
│ │ │ │          8
│ │ │ │ @@ -134,22 +134,22 @@
│ │ │ │  method returns the push-forward to the Chow ring of the ambient projective
│ │ │ │  space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it
│ │ │ │  basically computes SegreClass ideal matrix phi. In the next example, we compute
│ │ │ │  the Segre class of the base locus of a birational map $\mathbb{G}
│ │ │ │  (1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
│ │ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │ │  
│ │ │ │ -o9     ZZ
│ │ │ │ -= ------[x ..x ]
│ │ │ │ -  100003  0   6
│ │ │ │ +o9 =   ZZ
│ │ │ │ + ------[x ..x ]
│ │ │ │ + 100003  0   6
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},
│ │ │ │  {x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ │ - -- used 0.237104s (cpu); 0.125625s (thread); 0s (gc)
│ │ │ │ + -- used 0.0661419s (cpu); 0.0661437s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                          ZZ
│ │ │ │                                                        ------[y ..y ]
│ │ │ │                                                        100003  0   9
│ │ │ │  ZZ              2                              2
│ │ │ │  o10 = map (--------------------------------------------------------------------
│ │ │ │  --------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │  o10 : RingMap -----------------------------------------------------------------
│ │ │ │  ----------------------------------- <-- ------[x ..x ]
│ │ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y
│ │ │ │  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6
│ │ │ │  1 7    0 8   2 3    1 4    0 5
│ │ │ │  i11 : time SegreClass phi
│ │ │ │ - -- used 0.385145s (cpu); 0.250766s (thread); 0s (gc)
│ │ │ │ + -- used 0.374685s (cpu); 0.2463s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │          10
│ │ │ │ @@ -198,26 +198,26 @@
│ │ │ │  ------------------------------------
│ │ │ │                 (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y
│ │ │ │  - y y  + y y , y y  - y y  + y y )
│ │ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2
│ │ │ │  6    1 7    0 8   2 3    1 4    0 5
│ │ │ │  i13 : -- Segre class of B in G(1,4)
│ │ │ │        time SegreClass B
│ │ │ │ - -- used 0.405977s (cpu); 0.267353s (thread); 0s (gc)
│ │ │ │ + -- used 0.391422s (cpu); 0.267298s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6     5
│ │ │ │  o13 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o13 : -----
│ │ │ │          10
│ │ │ │         H
│ │ │ │  i14 : -- Segre class of B in P^9
│ │ │ │        time SegreClass lift(B,ambient ring B)
│ │ │ │ - -- used 1.63119s (cpu); 0.951264s (thread); 0s (gc)
│ │ │ │ + -- used 1.40422s (cpu); 0.83095s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             9       8       7      6     5
│ │ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o14 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000476825s (cpu); 0.000472506s (thread); 0s (gc)
│ │ │ + -- used 0.000395736s (cpu); 0.000391753s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │ @@ -124,23 +124,23 @@
│ │ │            
    
│ │ │          
    Now we compute first the degree of the forms defining the abstract map psi and then the corresponding concrete rational map.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.1194s (cpu); 0.119423s (thread); 0s (gc)
│ │ │ + -- used 0.119041s (cpu); 0.119046s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
    
│ │ │                
    i6 : time rationalMap psi
│ │ │ - -- used 0.532769s (cpu); 0.401923s (thread); 0s (gc)
│ │ │ + -- used 0.424402s (cpu); 0.338794s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -238,15 +238,15 @@
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
    
│ │ │              
    
│ │ │                
    i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.0383176s (cpu); 0.0383202s (thread); 0s (gc)
│ │ │ + -- used 0.040632s (cpu); 0.0406162s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -258,26 +258,26 @@
│ │ │            
    
│ │ │          
    The degree of the forms defining the abstract map T can be obtained by the following command:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 2.18663s (cpu); 1.5847s (thread); 0s (gc)
│ │ │ + -- used 2.2766s (cpu); 1.56941s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself is defined by linear forms:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i16 : time T2 = T * T
│ │ │ - -- used 3.0007e-05s (cpu); 2.9705e-05s (thread); 0s (gc)
│ │ │ + -- used 2.6532e-05s (cpu); 2.4366e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -286,15 +286,15 @@
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
    
│ │ │              
    
│ │ │                
    i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 3.67928s (cpu); 2.6019s (thread); 0s (gc)
│ │ │ + -- used 3.51385s (cpu); 2.49866s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself leaves a random point fixed:
    
│ │ │          
│ │ │ @@ -327,15 +327,15 @@
│ │ │            
│ │ │          
    
│ │ │          
    We now compute the concrete rational map corresponding to T:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i21 : time f = rationalMap T
│ │ │ - -- used 3.12656s (cpu); 2.217s (thread); 0s (gc)
│ │ │ + -- used 2.98878s (cpu); 2.11239s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,32 +35,32 @@
│ │ │ │  i3 : P5 := QQ[u_0..u_5]
│ │ │ │  
│ │ │ │  o3 = QQ[u ..u ]
│ │ │ │           0   5
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ │ - -- used 0.000476825s (cpu); 0.000472506s (thread); 0s (gc)
│ │ │ │ + -- used 0.000395736s (cpu); 0.000391753s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: given by a function
│ │ │ │  
│ │ │ │  o4 : AbstractRationalMap (rational map from PP^4 to PP^5)
│ │ │ │  Now we compute first the degree of the forms defining the abstract map psi and
│ │ │ │  then the corresponding concrete rational map.
│ │ │ │  i5 : time projectiveDegrees(psi,3)
│ │ │ │ - -- used 0.1194s (cpu); 0.119423s (thread); 0s (gc)
│ │ │ │ + -- used 0.119041s (cpu); 0.119046s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.532769s (cpu); 0.401923s (thread); 0s (gc)
│ │ │ │ + -- used 0.424402s (cpu); 0.338794s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = -- rational map --
│ │ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │ │                        0   1   2   3   4   5
│ │ │ │       defining forms: {
│ │ │ │ @@ -139,48 +139,48 @@
│ │ │ │  o13 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                  1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o13 : Ideal of -----[x ..x ]
│ │ │ │                 65521  0   3
│ │ │ │  i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ │ - -- used 0.0383176s (cpu); 0.0383202s (thread); 0s (gc)
│ │ │ │ + -- used 0.040632s (cpu); 0.0406162s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  The degree of the forms defining the abstract map T can be obtained by the
│ │ │ │  following command:
│ │ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ │ - -- used 2.18663s (cpu); 1.5847s (thread); 0s (gc)
│ │ │ │ + -- used 2.2766s (cpu); 1.56941s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3
│ │ │ │  We verify that the composition of T with itself is defined by linear forms:
│ │ │ │  i16 : time T2 = T * T
│ │ │ │ - -- used 3.0007e-05s (cpu); 2.9705e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.6532e-05s (cpu); 2.4366e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ │ - -- used 3.67928s (cpu); 2.6019s (thread); 0s (gc)
│ │ │ │ + -- used 3.51385s (cpu); 2.49866s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 1
│ │ │ │  We verify that the composition of T with itself leaves a random point fixed:
│ │ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │ │  
│ │ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │ @@ -193,15 +193,15 @@
│ │ │ │  i20 : T q
│ │ │ │  
│ │ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │  o20 : List
│ │ │ │  We now compute the concrete rational map corresponding to T:
│ │ │ │  i21 : time f = rationalMap T
│ │ │ │ - -- used 3.12656s (cpu); 2.217s (thread); 0s (gc)
│ │ │ │ + -- used 2.98878s (cpu); 2.11239s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.559174s (cpu); 0.374887s (thread); 0s (gc)
│ │ │ + -- used 0.444139s (cpu); 0.300791s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -205,15 +205,15 @@
│ │ │            
    
│ │ │                
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.367939s (cpu); 0.280942s (thread); 0s (gc)
│ │ │ + -- used 0.249231s (cpu); 0.164507s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i6 : assert(psi == psi')
    
│ │ │ @@ -300,15 +300,15 @@
│ │ │            
    
│ │ │                
    i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.7613s (cpu); 1.89303s (thread); 0s (gc)
│ │ │ + -- used 1.99287s (cpu); 1.62692s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -372,15 +372,15 @@
│ │ │              
    		
│ │ │                
    i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.89555s (cpu); 2.92238s (thread); 0s (gc)
│ │ │ + -- used 2.8432s (cpu); 2.39173s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = -- rational map --
│ │ │                                   ZZ
│ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │                {
│ │ │                                   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  -- approximateInverseMap: step 4 of 10
│ │ │ │  -- approximateInverseMap: step 5 of 10
│ │ │ │  -- approximateInverseMap: step 6 of 10
│ │ │ │  -- approximateInverseMap: step 7 of 10
│ │ │ │  -- approximateInverseMap: step 8 of 10
│ │ │ │  -- approximateInverseMap: step 9 of 10
│ │ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ │ - -- used 0.559174s (cpu); 0.374887s (thread); 0s (gc)
│ │ │ │ + -- used 0.444139s (cpu); 0.300791s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                    ZZ
│ │ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │ │                                  ZZ
│ │ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -252,15 +252,15 @@
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i4 : assert(phi * psi == 1 and psi * phi == 1)
│ │ │ │  i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 0.367939s (cpu); 0.280942s (thread); 0s (gc)
│ │ │ │ + -- used 0.249231s (cpu); 0.164507s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i6 : assert(psi == psi')
│ │ │ │  A more complicated example is the following (here _i_n_v_e_r_s_e_M_a_p takes a lot of
│ │ │ │  time!).
│ │ │ │  i7 : phi = rationalMap map(P8,ZZ/97[x_0..x_11]/ideal(x_1*x_3-8*x_2*x_3+25*x_3^2-25*x_2*x_4-
│ │ │ │  22*x_3*x_4+x_0*x_5+13*x_2*x_5+41*x_3*x_5-x_0*x_6+12*x_2*x_6+25*x_1*x_7+25*x_3*x_7+23*x_5*x_7-
│ │ │ │ @@ -418,15 +418,15 @@
│ │ │ │  
│ │ │ │  o7 : RationalMap (quadratic rational map from PP^8 to 8-dimensional subvariety of PP^11)
│ │ │ │  i8 : -- without the option 'CodimBsInv=>4', it takes about triple time
│ │ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │ - -- used 2.7613s (cpu); 1.89303s (thread); 0s (gc)
│ │ │ │ + -- used 1.99287s (cpu); 1.62692s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = -- rational map --
│ │ │ │                                  ZZ
│ │ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │               {
│ │ │ │                                  2
│ │ │ │ @@ -526,15 +526,15 @@
│ │ │ │  o9 = false
│ │ │ │  i10 : -- in this case we can remedy enabling the option Certify
│ │ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │ │  -- approximateInverseMap: step 1 of 3
│ │ │ │  -- approximateInverseMap: step 2 of 3
│ │ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 3.89555s (cpu); 2.92238s (thread); 0s (gc)
│ │ │ │ + -- used 2.8432s (cpu); 2.39173s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = -- rational map --
│ │ │ │                                   ZZ
│ │ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │                {
│ │ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    
│ │ │              
    
│ │ │                
    i5 : time degreeMap phi
│ │ │ - -- used 0.0497248s (cpu); 0.0497258s (thread); 0s (gc)
│ │ │ + -- used 0.0479509s (cpu); 0.0479511s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    
│ │ │              
    
│ │ │                
    i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
    
│ │ │              
    
│ │ │                
    i7 : time degreeMap phi'
│ │ │ - -- used 1.46278s (cpu); 0.843714s (thread); 0s (gc)
│ │ │ + -- used 1.2641s (cpu); 0.680409s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 14
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -266,15 +266,15 @@
│ │ │ │  4       0 5       1 5       2 5       3 5        4 5       5        0 6
│ │ │ │  1 6        2 6       3 6       4 6       5 6        6       0 7       1 7
│ │ │ │  2 7       3 7       4 7       5 7        6 7       7     0 8        1 8       2
│ │ │ │  8        3 8       4 8        5 8       6 8      7 8       8
│ │ │ │  
│ │ │ │  o4 : RingMap ringP8 <-- ringP14
│ │ │ │  i5 : time degreeMap phi
│ │ │ │ - -- used 0.0497248s (cpu); 0.0497258s (thread); 0s (gc)
│ │ │ │ + -- used 0.0479509s (cpu); 0.0479511s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 1
│ │ │ │  i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a
│ │ │ │  general subspace of P^14
│ │ │ │       -- of dimension 5 (so that the composition phi':P^8--->P^8 must have
│ │ │ │  degree equal to deg(G(1,5))=14)
│ │ │ │       phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14))
│ │ │ │ @@ -418,15 +418,15 @@
│ │ │ │  0 5        1 5        2 5        3 5        4 5       5        0 6       1 6
│ │ │ │  2 6        3 6      4 6       5 6       6       0 7       1 7       2 7      3
│ │ │ │  7        4 7        5 7        6 7       7       0 8       1 8        2 8
│ │ │ │  3 8       4 8       5 8        6 8       7 8       8
│ │ │ │  
│ │ │ │  o6 : RingMap ringP8 <-- ringP8
│ │ │ │  i7 : time degreeMap phi'
│ │ │ │ - -- used 1.46278s (cpu); 0.843714s (thread); 0s (gc)
│ │ │ │ + -- used 1.2641s (cpu); 0.680409s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = 14
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map
│ │ │ │      * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s -- projective degrees of a rational map between
│ │ │ │        projective varieties
│ │ │ │  ********** WWaayyss ttoo uussee ddeeggrreeeeMMaapp:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_force__Image.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000667762s (cpu); 0.00065607s (thread); 0s (gc)
    
│ │ │ + -- used 0.000627751s (cpu); 0.000625436s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i5 : Phi;
│ │ │  
│ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  
│ │ │ │  o2 : Ideal of P6
│ │ │ │  i3 : Phi = rationalMap(X,Dominant=>2);
│ │ │ │  
│ │ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ │ - -- used 0.000667762s (cpu); 0.00065607s (thread); 0s (gc)
│ │ │ │ + -- used 0.000627751s (cpu); 0.000625436s (thread); 0s (gc)
│ │ │ │  i5 : Phi;
│ │ │ │  
│ │ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional
│ │ │ │  subvariety of PP^9)
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  If the declaration is false, nonsensical answers may result.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_graph.html
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0166208s (cpu); 0.0163389s (thread); 0s (gc)
    
│ │ │ + -- used 0.0809438s (cpu); 0.0234439s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │ @@ -277,15 +277,15 @@
│ │ │            
    
│ │ │          
    When the source of the rational map is a multi-projective variety, the method returns all the projections.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │  
│ │ │ │                        w w  - w w  + w w
│ │ │ │                         2 4    1 5    0 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │ │  i2 : time psi = inverseMap phi
│ │ │ │ - -- used 0.0833126s (cpu); 0.0833315s (thread); 0s (gc)
│ │ │ │ + -- used 0.0820946s (cpu); 0.0820953s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = -- rational map --
│ │ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w
│ │ │ │  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13
│ │ │ │  14   15   16   17   18   19   20
│ │ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w
│ │ │ │ @@ -216,15 +216,15 @@
│ │ │ │  15    9 20    8 22   3 10    0 13    4 15    9 21    8 23   2 10    0 12    4
│ │ │ │  20    6 21    8 24   1 10    0 11    4 22    6 23    9 24   4 5    3 6    0 7
│ │ │ │  1 8    2 9
│ │ │ │  
│ │ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │ │                   0   26          0   26
│ │ │ │  i5 : time psi = inverseMap phi
│ │ │ │ - -- used 0.300391s (cpu); 0.208058s (thread); 0s (gc)
│ │ │ │ + -- used 0.280211s (cpu); 0.1851s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  ,
│ │ │ │  - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   -
│ │ │ │  w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   -
│ │ │ │  w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   +
│ │ │ │  w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   +
│ │ │ │  w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse_lp__Rational__Map_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time g = graph p2;
│ │ │ - -- used 0.0319997s (cpu); 0.0316715s (thread); 0s (gc)
    
│ │ │ + -- used 0.097417s (cpu); 0.0423217s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i10 : g_0;
│ │ │  
│ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │                        - x  + x x
│ │ │ │                           3    2 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in
│ │ │ │  PP^5)
│ │ │ │  i3 : time (p1,p2) = graph phi;
│ │ │ │ - -- used 0.0166208s (cpu); 0.0163389s (thread); 0s (gc)
│ │ │ │ + -- used 0.0809438s (cpu); 0.0234439s (thread); 0s (gc)
│ │ │ │  i4 : p1
│ │ │ │  
│ │ │ │  o4 = -- rational map --
│ │ │ │                                    ZZ                                 ZZ
│ │ │ │       source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y
│ │ │ │  , y , y , y , y ]) defined by
│ │ │ │                                  190181  0   1   2   3   4          190181  0
│ │ │ │ @@ -192,15 +192,15 @@
│ │ │ │  
│ │ │ │  o8 = {51, 28, 14, 6, 2}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  When the source of the rational map is a multi-projective variety, the method
│ │ │ │  returns all the projections.
│ │ │ │  i9 : time g = graph p2;
│ │ │ │ - -- used 0.0319997s (cpu); 0.0316715s (thread); 0s (gc)
│ │ │ │ + -- used 0.097417s (cpu); 0.0423217s (thread); 0s (gc)
│ │ │ │  i10 : g_0;
│ │ │ │  
│ │ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ │ │  of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │ │  i11 : g_1;
│ │ │ │  
│ │ │ │  o11 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_ideal_lp__Rational__Map_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │              
    
│ │ │                
    i3 : time ideal phi
│ │ │ - -- used 0.00352782s (cpu); 0.00352276s (thread); 0s (gc)
│ │ │ + -- used 0.00376922s (cpu); 0.00376915s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -200,15 +200,15 @@
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
    
│ │ │              
    
│ │ │                
    i6 : time ideal phi'
│ │ │ - -- used 0.281847s (cpu); 0.150706s (thread); 0s (gc)
│ │ │ + -- used 0.253056s (cpu); 0.143947s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal 1
│ │ │  
│ │ │                                                                                                              QQ[x ..x , y ..y ]
│ │ │                                                                                                                  0   5   0   4
│ │ │  o6 : Ideal of --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
│ │ │                                                                                                                                                                                                       2
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │                         2
│ │ │ │                        x  - x x
│ │ │ │                         1    0 3
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i3 : time ideal phi
│ │ │ │ - -- used 0.00352782s (cpu); 0.00352276s (thread); 0s (gc)
│ │ │ │ + -- used 0.00376922s (cpu); 0.00376915s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2                                     2
│ │ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2
│ │ │ │       x x , x  - x x )
│ │ │ │ @@ -121,15 +121,15 @@
│ │ │ │                        y
│ │ │ │                         4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of
│ │ │ │  PP^5 x PP^4 to PP^4)
│ │ │ │  i6 : time ideal phi'
│ │ │ │ - -- used 0.281847s (cpu); 0.150706s (thread); 0s (gc)
│ │ │ │ + -- used 0.253056s (cpu); 0.143947s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal 1
│ │ │ │  
│ │ │ │  
│ │ │ │  QQ[x ..x , y ..y ]
│ │ │ │  
│ │ │ │  0   5   0   4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
│ │ │ @@ -157,15 +157,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
    
│ │ │              
    
│ │ │                
    i2 : time psi = inverseMap phi
│ │ │ - -- used 0.0833126s (cpu); 0.0833315s (thread); 0s (gc)
│ │ │ + -- used 0.0820946s (cpu); 0.0820953s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = -- rational map --
│ │ │       source: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       target: Proj(QQ[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  , w  ])
│ │ │                        0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
│ │ │       defining forms: {
│ │ │ @@ -255,15 +255,15 @@
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i5 : time psi = inverseMap phi
│ │ │ - -- used 0.300391s (cpu); 0.208058s (thread); 0s (gc)
│ │ │ + -- used 0.280211s (cpu); 0.1851s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   + w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w  , w w   - w w   + w w   + w  w   - w  w  , - w w   + w w   + w  w   + w w   - w w  , - w w   + w w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , - w w   - w  w   + w  w   + w w   - w w  , w  w   - w  w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w  w   - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w  - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w   - w w   + w w   - w w   + w w  , w w  - w w  - w w   + w w   - w w  , - w w  + w w  + w w   - w w   + w w  , w w  - w w  - w w  + w w   - w w  })
│ │ │                0   26       0   26       5 22    8 23    14 24    13 25    0 26     5 18    8 19    14 20    10 25    1 26     5 16    8 17    13 20    10 24    2 26     5 15    14 17    13 19    10 23    3 26     5 21    20 23    19 24    17 25    4 26     8 15    14 16    13 18    10 22    6 26     8 21    20 22    18 24    16 25    7 26   17 18    16 19    15 20    10 21    9 26     13 21    17 22    16 23    15 24    11 26     14 21    19 22    18 23    15 25    12 26   0 21    4 22    7 23    12 24    11 25     4 18    7 19    12 20    1 21    9 25     4 16    7 17    11 20    2 21    9 24     4 15    12 17    11 19    3 21    9 23     7 15    12 16    11 18    6 21    9 22   12 13    11 14    0 15    3 22    6 23   10 12    9 14    1 15    3 18    6 19   10 11    9 13    2 15    3 16    6 17   8 9    7 10    1 16    2 18    6 20   5 9    4 10    1 17    2 19    3 20   8 11    7 13    0 16    2 22    6 24   5 11    4 13    0 17    2 23    3 24   8 12    7 14    0 18    1 22    6 25   5 12    4 14    0 19    1 23    3 25   5 7    4 8    0 20    1 24    2 25     5 6    3 8    0 10    1 13    2 14   4 6    3 7    0 9    1 11    2 12
│ │ │  
│ │ │  o5 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i3 : time inverse phi
│ │ │ - -- used 0.0578597s (cpu); 0.0578645s (thread); 0s (gc)
│ │ │ + -- used 0.0563582s (cpu); 0.05636s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │                        0   1   2   3   4
│ │ │       defining forms: {
│ │ │ ├── html2text {}
│ │ │ │ @@ -290,15 +290,15 @@
│ │ │ │  58320000  1 4    190512000  0 2 4    4898880000 1 2 4    190512000 2 4
│ │ │ │  476280000  0 3 4    204120000  1 3 4    2857680000  2 3 4    23814000  3 4
│ │ │ │  30618000 0 4    46656 1 4   12757500 2 4    51030000  3 4   30375 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │ │  i3 : time inverse phi
│ │ │ │ - -- used 0.0578597s (cpu); 0.0578645s (thread); 0s (gc)
│ │ │ │ + -- used 0.0563582s (cpu); 0.05636s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Birational.html
│ │ │ @@ -128,24 +128,24 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i3 : time isBirational phi
│ │ │ - -- used 0.0202901s (cpu); 0.0202894s (thread); 0s (gc)
│ │ │ + -- used 0.0191121s (cpu); 0.0191143s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │                
    i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0145752s (cpu); 0.014156s (thread); 0s (gc)
│ │ │ + -- used 0.0635394s (cpu); 0.0197596s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,20 +58,20 @@
│ │ │ │                        - t  + t t
│ │ │ │                           3    2 4
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in
│ │ │ │  PP^5)
│ │ │ │  i3 : time isBirational phi
│ │ │ │ - -- used 0.0202901s (cpu); 0.0202894s (thread); 0s (gc)
│ │ │ │ + -- used 0.0191121s (cpu); 0.0191143s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0145752s (cpu); 0.014156s (thread); 0s (gc)
│ │ │ │ + -- used 0.0635394s (cpu); 0.0197596s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_D_o_m_i_n_a_n_t -- whether a rational map is dominant
│ │ │ │  ********** WWaayyss ttoo uussee iissBBiirraattiioonnaall:: **********
│ │ │ │      * isBirational(RationalMap)
│ │ │ │      * isBirational(RingMap)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.33576s (cpu); 2.59272s (thread); 0s (gc)
│ │ │ + -- used 2.20394s (cpu); 1.90339s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : P7 = ZZ/101[x_0..x_7];
    │ │ │ @@ -120,15 +120,15 @@ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 4.05735s (cpu); 2.55688s (thread); 0s (gc)
│ │ │ + -- used 3.54851s (cpu); 2.46268s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ i1 : P8 = ZZ/101[x_0..x_8]; │ │ │ │ i2 : phi = rationalMap ideal jacobian ideal det matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7}, │ │ │ │ {x_4..x_8}}; │ │ │ │ │ │ │ │ o2 : RationalMap (rational map from PP^8 to PP^8) │ │ │ │ i3 : time isDominant(phi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 3.33576s (cpu); 2.59272s (thread); 0s (gc) │ │ │ │ + -- used 2.20394s (cpu); 1.90339s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ i4 : P7 = ZZ/101[x_0..x_7]; │ │ │ │ i5 : -- hyperelliptic curve of genus 3 │ │ │ │ C = ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6- │ │ │ │ 26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5- │ │ │ │ 24*x_5^2+21*x_0*x_6+x_1*x_6+46*x_3*x_6+27*x_4*x_6+5*x_5*x_6+35*x_6^2+20*x_0*x_7- │ │ │ │ @@ -65,15 +65,15 @@ │ │ │ │ │ │ │ │ o5 : Ideal of P7 │ │ │ │ i6 : phi = rationalMap(C,3,2); │ │ │ │ │ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ i7 : time isDominant(phi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 4.05735s (cpu); 2.55688s (thread); 0s (gc) │ │ │ │ + -- used 3.54851s (cpu); 2.46268s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = false │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_B_i_r_a_t_i_o_n_a_l -- whether a rational map is birational │ │ │ │ ********** WWaayyss ttoo uussee iissDDoommiinnaanntt:: ********** │ │ │ │ * isDominant(RationalMap) │ │ │ │ * isDominant(RingMap) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_kernel_lp__Ring__Map_cm__Z__Z_rp.html │ │ │ @@ -93,26 +93,26 @@ │ │ │ o1 : RingMap QQ[x ..x ] <-- QQ[y ..y ] │ │ │ 0 8 0 11 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.0180149s (cpu); 0.0180127s (thread); 0s (gc)
│ │ │ + -- used 0.0193662s (cpu); 0.0193675s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time kernel(phi,2)
│ │ │ - -- used 1.06451s (cpu); 0.555512s (thread); 0s (gc)
│ │ │ + -- used 0.926832s (cpu); 0.437835s (thread); 0s (gc)
│ │ │  
│ │ │                             2                                                
│ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                             
│ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │  4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7
│ │ │ │  0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2
│ │ │ │  7     0 8     1 8     5 8     6 8     7 8
│ │ │ │  
│ │ │ │  o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
│ │ │ │                   0   8          0   11
│ │ │ │  i2 : time kernel(phi,1)
│ │ │ │ - -- used 0.0180149s (cpu); 0.0180127s (thread); 0s (gc)
│ │ │ │ + -- used 0.0193662s (cpu); 0.0193675s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 1.06451s (cpu); 0.555512s (thread); 0s (gc)
│ │ │ │ + -- used 0.926832s (cpu); 0.437835s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                             2
│ │ │ │  o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
│ │ │ │               2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │  
│ │ │ │       4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_parametrize_lp__Ideal_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time parametrize L
│ │ │ - -- used 0.00443973s (cpu); 0.00443525s (thread); 0s (gc)
│ │ │ + -- used 0.00501935s (cpu); 0.00501783s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -205,15 +205,15 @@
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time parametrize Q
│ │ │ - -- used 0.76754s (cpu); 0.460282s (thread); 0s (gc)
│ │ │ + -- used 0.548412s (cpu); 0.385579s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = -- rational map --
│ │ │                       ZZ
│ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │                    10000019  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │       - 849671x  + 3034137x )
│ │ │ │                8           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o2 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i3 : time parametrize L
│ │ │ │ - -- used 0.00443973s (cpu); 0.00443525s (thread); 0s (gc)
│ │ │ │ + -- used 0.00501935s (cpu); 0.00501783s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │       1211601x x  - 2168594x x  - 1801762x x  + 3022242x x  + 3618789x )
│ │ │ │               5 9           6 9           7 9           8 9           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o4 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i5 : time parametrize Q
│ │ │ │ - -- used 0.76754s (cpu); 0.460282s (thread); 0s (gc)
│ │ │ │ + -- used 0.548412s (cpu); 0.385579s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │       source: Proj(--------[t , t , t , t , t , t , t ])
│ │ │ │                    10000019  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │       target: Proj(--------[x , x , x , x , x , x , x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_point_lp__Quotient__Ring_rp.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time p = point source f
│ │ │ - -- used 0.298623s (cpu); 0.198159s (thread); 0s (gc)
│ │ │ + -- used 0.260566s (cpu); 0.154493s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
│ │ │               10        11   9         11   8        11   7        11   6  
│ │ │       ------------------------------------------------------------------------
│ │ │       13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
│ │ │             11   5        11   4        11   3         11   2        11   1  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -104,15 +104,15 @@
│ │ │                (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
│ │ │                  6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time p == f^* f p
│ │ │ - -- used 0.176944s (cpu); 0.176946s (thread); 0s (gc)
│ │ │ + -- used 0.0955232s (cpu); 0.0955261s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ _p_o_i_n_t_(_M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_). │ │ │ │ Below we verify the birationality of a rational map. │ │ │ │ i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331); │ │ │ │ │ │ │ │ o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to │ │ │ │ PP^8) │ │ │ │ i2 : time p = point source f │ │ │ │ - -- used 0.298623s (cpu); 0.198159s (thread); 0s (gc) │ │ │ │ + -- used 0.260566s (cpu); 0.154493s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = ideal (y - 9235y , y + 11075y , y - 5847y , y + 7396y , y + │ │ │ │ 10 11 9 11 8 11 7 11 6 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 13530y , y + 4359y , y - 2924y , y + 13040y , y + 6904y , y - │ │ │ │ 11 5 11 4 11 3 11 2 11 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -40,15 +40,15 @@ │ │ │ │ o2 : Ideal of ----------------------------------------------------------------- │ │ │ │ -------------------------------------- │ │ │ │ (y y - y y + y y , y y - y y + y y , y y - y y + y y , y │ │ │ │ y - y y + y y , y y - y y + y y ) │ │ │ │ 6 7 5 8 4 11 3 7 2 8 1 11 3 5 2 6 0 11 │ │ │ │ 3 4 1 6 0 8 2 4 1 5 0 7 │ │ │ │ i3 : time p == f^* f p │ │ │ │ - -- used 0.176944s (cpu); 0.176946s (thread); 0s (gc) │ │ │ │ + -- used 0.0955232s (cpu); 0.0955261s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t -- pick a random K rational point on the scheme X │ │ │ │ defined by I │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * point(PolynomialRing) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ 0 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0278401s (cpu); 0.0272526s (thread); 0s (gc)
│ │ │ + -- used 0.0265372s (cpu); 0.0152986s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -122,15 +122,15 @@ │ │ │ 2 3 1 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.019353s (cpu); 0.0187436s (thread); 0s (gc)
│ │ │ + -- used 0.0718317s (cpu); 0.0182071s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -148,25 +148,25 @@ │ │ │ o6 : RingMap ------[x ..x ] <-- ------[x ..x ] │ │ │ 300007 0 6 300007 0 6 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time projectiveDegrees phi
│ │ │ - -- used 5.2489e-05s (cpu); 4.7148e-05s (thread); 0s (gc)
│ │ │ + -- used 5.6289e-05s (cpu); 5.0428e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.626e-05s (cpu); 2.6219e-05s (thread); 0s (gc)
│ │ │ + -- used 2.2005e-05s (cpu); 2.1699e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = {4, 1}
│ │ │  
│ │ │  o8 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -53,15 +53,15 @@ │ │ │ │ 0 4 0 5 1 0 2 1 2 0 3 │ │ │ │ 2 1 3 1 3 0 4 2 3 1 4 3 2 4 │ │ │ │ │ │ │ │ o2 : RingMap GF 109561[t ..t ] <-- GF 109561[x ..x ] │ │ │ │ 0 4 0 5 │ │ │ │ i3 : time projectiveDegrees(phi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 0.0278401s (cpu); 0.0272526s (thread); 0s (gc) │ │ │ │ + -- used 0.0265372s (cpu); 0.0152986s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = {1, 2, 4, 4, 2} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : psi=inverseMap(toMap(phi,Dominant=>infinity)) │ │ │ │ │ │ │ │ GF 109561[x ..x ] │ │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ GF 109561[x ..x ] │ │ │ │ 0 5 │ │ │ │ o4 : RingMap ------------------ <-- GF 109561[t ..t ] │ │ │ │ x x - x x + x x 0 4 │ │ │ │ 2 3 1 4 0 5 │ │ │ │ i5 : time projectiveDegrees(psi,Certify=>true) │ │ │ │ Certify: output certified! │ │ │ │ - -- used 0.019353s (cpu); 0.0187436s (thread); 0s (gc) │ │ │ │ + -- used 0.0718317s (cpu); 0.0182071s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = {2, 4, 4, 2, 1} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : -- Cremona transformation of P^6 defined by the quadrics through a │ │ │ │ rational octic surface │ │ │ │ phi = map specialCremonaTransformation(7,ZZ/300007) │ │ │ │ @@ -119,21 +119,21 @@ │ │ │ │ 4 5 5 0 6 1 6 2 6 3 6 4 6 │ │ │ │ 5 6 │ │ │ │ │ │ │ │ ZZ ZZ │ │ │ │ o6 : RingMap ------[x ..x ] <-- ------[x ..x ] │ │ │ │ 300007 0 6 300007 0 6 │ │ │ │ i7 : time projectiveDegrees phi │ │ │ │ - -- used 5.2489e-05s (cpu); 4.7148e-05s (thread); 0s (gc) │ │ │ │ + -- used 5.6289e-05s (cpu); 5.0428e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {1, 2, 4, 8, 8, 4, 1} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : time projectiveDegrees(phi,NumDegrees=>1) │ │ │ │ - -- used 2.626e-05s (cpu); 2.6219e-05s (thread); 0s (gc) │ │ │ │ + -- used 2.2005e-05s (cpu); 2.1699e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = {4, 1} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ Another way to use this method is by passing an integer i as second argument. │ │ │ │ However, this is equivalent to first projectiveDegrees(phi,NumDegrees=>i) and │ │ │ │ generally it is not faster. │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.html │ │ │ @@ -93,15 +93,15 @@ │ │ │ o2 : Ideal of -----[x ..x ] │ │ │ 33331 0 6 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.117734s (cpu); 0.117709s (thread); 0s (gc)
│ │ │ + -- used 0.101826s (cpu); 0.101827s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                      ZZ
│ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │                    33331  0   1   2   3   4   5   6
│ │ │                      ZZ
│ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  , y  ])
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-
│ │ │ │  x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o2 : Ideal of -----[x ..x ]
│ │ │ │                33331  0   6
│ │ │ │  i3 : time phi = rationalMap(V,3,2)
│ │ │ │ - -- used 0.117734s (cpu); 0.117709s (thread); 0s (gc)
│ │ │ │ + -- used 0.101826s (cpu); 0.101827s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                      ZZ
│ │ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │ │                    33331  0   1   2   3   4   5   6
│ │ │ │                      ZZ
│ │ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  ,
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ring_cm__Tally_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │              
│ │ │                
    i5 : D = new Tally from {H => 2,C => 1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time phi = rationalMap D
│ │ │ - -- used 0.029123s (cpu); 0.029123s (thread); 0s (gc)
│ │ │ + -- used 0.0317304s (cpu); 0.0317266s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -224,15 +224,15 @@
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.23167s (cpu); 0.77726s (thread); 0s (gc)
│ │ │ + -- used 1.46964s (cpu); 0.596782s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │       hypersurfaces of degree 2
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    See also the package WeilDivisors, which provides general tools for working with divisors.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  
│ │ │ │  o4 = ideal(- 32646x  - 28377x  + 26433x  - 29566x  + 3783x  + 26696x )
│ │ │ │                     0         1         2         3        4         5
│ │ │ │  
│ │ │ │  o4 : Ideal of X
│ │ │ │  i5 : D = new Tally from {H => 2,C => 1};
│ │ │ │  i6 : time phi = rationalMap D
│ │ │ │ - -- used 0.029123s (cpu); 0.029123s (thread); 0s (gc)
│ │ │ │ + -- used 0.0317304s (cpu); 0.0317266s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = -- rational map --
│ │ │ │                                    ZZ
│ │ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │ │                                  65521  0   1   2   3   4   5
│ │ │ │               {
│ │ │ │                   2                  2
│ │ │ │ @@ -169,15 +169,15 @@
│ │ │ │                                                    2                         2
│ │ │ │                        x x x  + x x x  + x x x  + x x  + x x x  - 2x x x  + x x
│ │ │ │                         0 1 5    0 2 5    1 2 5    2 5    1 4 5     2 4 5    4 5
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
│ │ │ │  i7 : time ? image(phi,"F4")
│ │ │ │ - -- used 1.23167s (cpu); 0.77726s (thread); 0s (gc)
│ │ │ │ + -- used 1.46964s (cpu); 0.596782s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │ │       hypersurfaces of degree 2
│ │ │ │  See also the package _W_e_i_l_D_i_v_i_s_o_r_s, which provides general tools for working
│ │ │ │  with divisors.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_t_i_o_n_a_l_M_a_p -- makes a rational map
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.3736s (cpu); 1.13714s (thread); 0s (gc)
│ │ │ + -- used 1.00193s (cpu); 0.911121s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │        source variety: PP^3                      
│ │ │        target variety: PP^3                      
│ │ │        dominance: true                           
│ │ │        birationality: true                       
│ │ │        projective degrees: {1, 3, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │              K, according to the classification given in Table 1 of _S_p_e_c_i_a_l
│ │ │ │              _c_u_b_i_c_ _C_r_e_m_o_n_a_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _P_6_ _a_n_d_ _P_7.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  A Cremona transformation is said to be special if the base locus scheme is
│ │ │ │  smooth and irreducible. To ensure this condition, the field K must be large
│ │ │ │  enough but no check is made.
│ │ │ │  i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ │ - -- used 1.3736s (cpu); 1.13714s (thread); 0s (gc)
│ │ │ │ + -- used 1.00193s (cpu); 0.911121s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │ │        source variety: PP^3
│ │ │ │        target variety: PP^3
│ │ │ │        dominance: true
│ │ │ │        birationality: true
│ │ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cubic__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.102301s (cpu); 0.102302s (thread); 0s (gc)
│ │ │ + -- used 0.081544s (cpu); 0.0815448s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.0245428s (cpu); 0.0245442s (thread); 0s (gc)
│ │ │ + -- used 0.0173718s (cpu); 0.017373s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 3
│ │ │       source variety: PP^6
│ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special cubic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 2 of _S_p_e_c_i_a_l_ _c_u_b_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s_ _o_f_ _p_r_o_j_e_c_t_i_v_e
│ │ │ │              _s_p_a_c_e_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialCubicTransformation 9
│ │ │ │ - -- used 0.102301s (cpu); 0.102302s (thread); 0s (gc)
│ │ │ │ + -- used 0.081544s (cpu); 0.0815448s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6
│ │ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -323,15 +323,15 @@
│ │ │ │  6     4 6      0 5 6      1 5 6     2 5 6      3 5 6      4 5 6     5 6     0 6
│ │ │ │  1 6     2 6      3 6     4 6     5 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.0245428s (cpu); 0.0245442s (thread); 0s (gc)
│ │ │ │ + -- used 0.0173718s (cpu); 0.017373s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^6
│ │ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Quadratic__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0879444s (cpu); 0.0879457s (thread); 0s (gc)
│ │ │ + -- used 0.0735507s (cpu); 0.0735503s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.00847425s (cpu); 0.00847698s (thread); 0s (gc)
│ │ │ + -- used 0.00699743s (cpu); 0.00699696s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = rational map defined by forms of degree 2
│ │ │       source variety: PP^8
│ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │       dominance: true
│ │ │       birationality: true
│ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │            o a _r_a_t_i_o_n_a_l_ _m_a_p, an example of special quadratic birational
│ │ │ │              transformation over K, according to the classification given in
│ │ │ │              Table 1 of _E_x_a_m_p_l_e_s_ _o_f_ _s_p_e_c_i_a_l_ _q_u_a_d_r_a_t_i_c_ _b_i_r_a_t_i_o_n_a_l_ _t_r_a_n_s_f_o_r_m_a_t_i_o_n_s
│ │ │ │              _i_n_t_o_ _c_o_m_p_l_e_t_e_ _i_n_t_e_r_s_e_c_t_i_o_n_s_ _o_f_ _q_u_a_d_r_i_c_s.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The field K is required to be large enough.
│ │ │ │  i1 : time specialQuadraticTransformation 4
│ │ │ │ - -- used 0.0879444s (cpu); 0.0879457s (thread); 0s (gc)
│ │ │ │ + -- used 0.0735507s (cpu); 0.0735503s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4   5   6   7   8
│ │ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ])
│ │ │ │  defined by
│ │ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │                                                     2
│ │ │ │                        x x  - x x  + x x  - x x  - x  - x x
│ │ │ │                         0 1    0 4    3 6    4 6    6    5 7
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
│ │ │ │  i2 : time describe oo
│ │ │ │ - -- used 0.00847425s (cpu); 0.00847698s (thread); 0s (gc)
│ │ │ │ + -- used 0.00699743s (cpu); 0.00699696s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^8
│ │ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html
│ │ │ @@ -93,23 +93,23 @@
│ │ │  
│ │ │  o3 = 6927
    
│ │ │              
    
│ │ │                
    i4 : time phi' = value str;
│ │ │ - -- used 0.0227997s (cpu); 0.022798s (thread); 0s (gc)
│ │ │ + -- used 0.0238765s (cpu); 0.0238763s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │              
    
│ │ │                
    i5 : time describe phi'
│ │ │ - -- used 0.00554153s (cpu); 0.00554197s (thread); 0s (gc)
│ │ │ + -- used 0.00607877s (cpu); 0.0060834s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ @@ -118,15 +118,15 @@
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
    
│ │ │              
    
│ │ │                
    i6 : time describe inverse phi'
│ │ │ - -- used 0.00455456s (cpu); 0.00455575s (thread); 0s (gc)
│ │ │ + -- used 0.00512393s (cpu); 0.0051286s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,32 +19,32 @@
│ │ │ │  
│ │ │ │  o1 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i2 : str = toExternalString phi;
│ │ │ │  i3 : #str
│ │ │ │  
│ │ │ │  o3 = 6927
│ │ │ │  i4 : time phi' = value str;
│ │ │ │ - -- used 0.0227997s (cpu); 0.022798s (thread); 0s (gc)
│ │ │ │ + -- used 0.0238765s (cpu); 0.0238763s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.00554153s (cpu); 0.00554197s (thread); 0s (gc)
│ │ │ │ + -- used 0.00607877s (cpu); 0.0060834s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^3
│ │ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ │       number of minimal representatives: 1
│ │ │ │       dimension base locus: 1
│ │ │ │       degree base locus: 5
│ │ │ │       coefficient ring: ZZ/33331
│ │ │ │  i6 : time describe inverse phi'
│ │ │ │ - -- used 0.00455456s (cpu); 0.00455575s (thread); 0s (gc)
│ │ │ │ + -- used 0.00512393s (cpu); 0.0051286s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = rational map defined by forms of degree 2
│ │ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │ │       target variety: PP^3
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/index.html
│ │ │ @@ -63,29 +63,29 @@
│ │ │              
    		
│ │ │                
    i1 : ZZ/300007[t_0..t_6];
    
│ │ │              
    
│ │ │                
    i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00558593s (cpu); 0.00558112s (thread); 0s (gc)
│ │ │ + -- used 0.00490964s (cpu); 0.00490733s (thread); 0s (gc)
│ │ │  
│ │ │              ZZ              ZZ                3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o2 = map (------[t ..t ], ------[x ..x ], {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   300007  0   9       2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │  
│ │ │                 ZZ                 ZZ
│ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │               300007  0   6      300007  0   9
    
│ │ │              
    
│ │ │                
    i3 : time J = kernel(phi,2)
│ │ │ - -- used 0.0591643s (cpu); 0.0591742s (thread); 0s (gc)
│ │ │ + -- used 0.0460566s (cpu); 0.0460589s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │ @@ -93,43 +93,43 @@
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
    
│ │ │              
    
│ │ │                
    i4 : time degreeMap phi
│ │ │ - -- used 0.0354136s (cpu); 0.0354203s (thread); 0s (gc)
│ │ │ + -- used 0.0284222s (cpu); 0.0284253s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.683938s (cpu); 0.537126s (thread); 0s (gc)
│ │ │ + -- used 0.46581s (cpu); 0.397063s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.223023s (cpu); 0.137433s (thread); 0s (gc)
│ │ │ + -- used 0.0592977s (cpu); 0.0593036s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00354109s (cpu); 0.00354235s (thread); 0s (gc)
│ │ │ + -- used 0.00223669s (cpu); 0.0022397s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -141,15 +141,15 @@
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i8 : time psi = inverseMap phi
│ │ │ - -- used 0.743183s (cpu); 0.662105s (thread); 0s (gc)
│ │ │ + -- used 0.505302s (cpu); 0.406599s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -161,44 +161,44 @@
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.016456s (cpu); 0.0164579s (thread); 0s (gc)
│ │ │ + -- used 0.0101755s (cpu); 0.0101884s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time degreeMap psi
│ │ │ - -- used 0.376507s (cpu); 0.274317s (thread); 0s (gc)
│ │ │ + -- used 0.153949s (cpu); 0.153956s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │              
    
│ │ │                
    i11 : time projectiveDegrees psi
│ │ │ - -- used 5.65221s (cpu); 4.79159s (thread); 0s (gc)
│ │ │ + -- used 4.79568s (cpu); 4.30587s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │          
    We repeat the example using the type RationalMap and using deterministic methods.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │        2
│ │ │ │  o6 = y T
│ │ │ │          1,2
│ │ │ │  
│ │ │ │  o6 : R[T   ..T   ]
│ │ │ │          1,1   1,3
│ │ │ │  i7 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.0161623s (cpu); 0.0151328s
│ │ │ │ +Finding easy relations           :  -- used 0.0346552s (cpu); 0.0165132s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = H
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html
│ │ │ @@ -134,15 +134,15 @@
│ │ │  
│ │ │  o5 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00224049s (cpu); 0.00224156s (thread); 0s (gc)
│ │ │ + -- used 0.00239542s (cpu); 0.00239948s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -247,15 +247,15 @@
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
    
│ │ │              
    
│ │ │                
    i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.0523877s (cpu); 0.0523973s (thread); 0s (gc)
│ │ │ + -- used 0.0513685s (cpu); 0.0513715s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -320,15 +320,15 @@
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i14 : time phi^(-1)
│ │ │ - -- used 0.586241s (cpu); 0.446401s (thread); 0s (gc)
│ │ │ + -- used 0.511826s (cpu); 0.418254s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -381,49 +381,49 @@
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
    
│ │ │              
    
│ │ │                
    i15 : time degrees phi^(-1)
│ │ │ - -- used 0.624809s (cpu); 0.460082s (thread); 0s (gc)
│ │ │ + -- used 0.279261s (cpu); 0.239719s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │                
    i16 : time degrees phi
│ │ │ - -- used 0.0257756s (cpu); 0.0252268s (thread); 0s (gc)
│ │ │ + -- used 0.0281718s (cpu); 0.0173267s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : time describe phi
│ │ │ - -- used 0.00459023s (cpu); 0.00459072s (thread); 0s (gc)
│ │ │ + -- used 0.0038676s (cpu); 0.00387328s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = rational map defined by forms of degree 3
│ │ │        source variety: PP^6
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i18 : time describe phi^(-1)
│ │ │ - -- used 0.0138045s (cpu); 0.0138079s (thread); 0s (gc)
│ │ │ + -- used 0.010034s (cpu); 0.0100393s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ @@ -432,41 +432,41 @@
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.0132166s (cpu); 0.0131906s (thread); 0s (gc)
│ │ │ + -- used 0.0101159s (cpu); 0.0101206s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i21 : time degrees f
│ │ │ - -- used 1.60372s (cpu); 1.18114s (thread); 0s (gc)
│ │ │ + -- used 1.12259s (cpu); 0.837973s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │              
    
│ │ │                
    i22 : time degree f
│ │ │ - -- used 2.0168e-05s (cpu); 1.9606e-05s (thread); 0s (gc)
│ │ │ + -- used 1.5474e-05s (cpu); 1.4716e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │              
    
│ │ │                
    i23 : time describe f
│ │ │ - -- used 0.00171686s (cpu); 0.00171795s (thread); 0s (gc)
│ │ │ + -- used 0.00132095s (cpu); 0.00132446s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
│ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        dominance: true
│ │ │        birationality: true
│ │ │        projective degrees: {904, 508, 268, 130, 56, 20, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  map) from a list of $m+1$ homogeneous elements of the same degree in $K
│ │ │ │  [x_0,...,x_n]/I$.
│ │ │ │  Below is an example using the methods provided by this package, dealing with a
│ │ │ │  birational transformation $\Phi:\mathbb{P}^6 \dashrightarrow \mathbb{G}
│ │ │ │  (2,4)\subset\mathbb{P}^9$ of bidegree $(3,3)$.
│ │ │ │  i1 : ZZ/300007[t_0..t_6];
│ │ │ │  i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.00558593s (cpu); 0.00558112s (thread); 0s (gc)
│ │ │ │ + -- used 0.00490964s (cpu); 0.00490733s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              ZZ              ZZ                3                2    2
│ │ │ │  2        2                      2                  2    2                 2
│ │ │ │  3                2    2                2                                 2
│ │ │ │  2    2                                  2        2                      2
│ │ │ │  2                        2                         2    2                 2
│ │ │ │  3                2    2
│ │ │ │ @@ -52,43 +52,43 @@
│ │ │ │  0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4
│ │ │ │  3 4 5    2 5    3 6    2 4 6
│ │ │ │  
│ │ │ │                 ZZ                 ZZ
│ │ │ │  o2 : RingMap ------[t ..t ] <-- ------[x ..x ]
│ │ │ │               300007  0   6      300007  0   9
│ │ │ │  i3 : time J = kernel(phi,2)
│ │ │ │ - -- used 0.0591643s (cpu); 0.0591742s (thread); 0s (gc)
│ │ │ │ + -- used 0.0460566s (cpu); 0.0460589s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x
│ │ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o3 : Ideal of ------[x ..x ]
│ │ │ │                300007  0   9
│ │ │ │  i4 : time degreeMap phi
│ │ │ │ - -- used 0.0354136s (cpu); 0.0354203s (thread); 0s (gc)
│ │ │ │ + -- used 0.0284222s (cpu); 0.0284253s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.683938s (cpu); 0.537126s (thread); 0s (gc)
│ │ │ │ + -- used 0.46581s (cpu); 0.397063s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.223023s (cpu); 0.137433s (thread); 0s (gc)
│ │ │ │ + -- used 0.0592977s (cpu); 0.0593036s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.00354109s (cpu); 0.00354235s (thread); 0s (gc)
│ │ │ │ + -- used 0.00223669s (cpu); 0.0022397s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                                         ZZ
│ │ │ │                                                                       ------[x
│ │ │ │  ..x ]
│ │ │ │              ZZ                                                       300007  0
│ │ │ │  9                                                  3                2    2
│ │ │ │  2        2                      2                  2    2                 2
│ │ │ │ @@ -123,15 +123,15 @@
│ │ │ │  o7 : RingMap ------[t ..t ] <-- -----------------------------------------------
│ │ │ │  -----------------------------------------------------
│ │ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  -
│ │ │ │  x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5
│ │ │ │  2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ │  i8 : time psi = inverseMap phi
│ │ │ │ - -- used 0.743183s (cpu); 0.662105s (thread); 0s (gc)
│ │ │ │ + -- used 0.505302s (cpu); 0.406599s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                         ZZ
│ │ │ │                                                       ------[x ..x ]
│ │ │ │                                                       300007  0   9
│ │ │ │  ZZ              3                2               2    2
│ │ │ │  2                          2     2        2                               2
│ │ │ │  2               2             2                       3
│ │ │ │ @@ -164,31 +164,31 @@
│ │ │ │  o8 : RingMap ------------------------------------------------------------------
│ │ │ │  ---------------------------------- <-- ------[t ..t ]
│ │ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x
│ │ │ │  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │ │  1 6    0 8   2 4    1 5    0 7
│ │ │ │  i9 : time isInverseMap(phi,psi)
│ │ │ │ - -- used 0.016456s (cpu); 0.0164579s (thread); 0s (gc)
│ │ │ │ + -- used 0.0101755s (cpu); 0.0101884s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.376507s (cpu); 0.274317s (thread); 0s (gc)
│ │ │ │ + -- used 0.153949s (cpu); 0.153956s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 5.65221s (cpu); 4.79159s (thread); 0s (gc)
│ │ │ │ + -- used 4.79568s (cpu); 4.30587s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  We repeat the example using the type _R_a_t_i_o_n_a_l_M_a_p and using deterministic
│ │ │ │  methods.
│ │ │ │  i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ │ - -- used 0.00224049s (cpu); 0.00224156s (thread); 0s (gc)
│ │ │ │ + -- used 0.00239542s (cpu); 0.00239948s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o12 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                       ZZ
│ │ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ │ @@ -233,15 +233,15 @@
│ │ │ │                            3                2    2
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
│ │ │ │  i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ │ - -- used 0.0523877s (cpu); 0.0523973s (thread); 0s (gc)
│ │ │ │ + -- used 0.0513685s (cpu); 0.0513715s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │ │                     300007  0   1   2   3   4   5   6
│ │ │ │                                     ZZ
│ │ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │ @@ -304,15 +304,15 @@
│ │ │ │                         - t  + 2t t t  - t t  - t t  + t t t
│ │ │ │                            4     3 4 5    2 5    3 6    2 4 6
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of
│ │ │ │  PP^9)
│ │ │ │  i14 : time phi^(-1)
│ │ │ │ - -- used 0.586241s (cpu); 0.446401s (thread); 0s (gc)
│ │ │ │ + -- used 0.511826s (cpu); 0.418254s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                                     ZZ
│ │ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x
│ │ │ │  ]) defined by
│ │ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │ │                {
│ │ │ │ @@ -373,67 +373,67 @@
│ │ │ │                          6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6
│ │ │ │  9     4 6 9    2 7 9    4 7 9     0 9
│ │ │ │                        }
│ │ │ │  
│ │ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9
│ │ │ │  to PP^6)
│ │ │ │  i15 : time degrees phi^(-1)
│ │ │ │ - -- used 0.624809s (cpu); 0.460082s (thread); 0s (gc)
│ │ │ │ + -- used 0.279261s (cpu); 0.239719s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.0257756s (cpu); 0.0252268s (thread); 0s (gc)
│ │ │ │ + -- used 0.0281718s (cpu); 0.0173267s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00459023s (cpu); 0.00459072s (thread); 0s (gc)
│ │ │ │ + -- used 0.0038676s (cpu); 0.00387328s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = rational map defined by forms of degree 3
│ │ │ │        source variety: PP^6
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {1, 3, 9, 17, 21, 15, 5}
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i18 : time describe phi^(-1)
│ │ │ │ - -- used 0.0138045s (cpu); 0.0138079s (thread); 0s (gc)
│ │ │ │ + -- used 0.010034s (cpu); 0.0100393s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 = rational map defined by forms of degree 3
│ │ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │        target variety: PP^6
│ │ │ │        dominance: true
│ │ │ │        birationality: true (the inverse map is already calculated)
│ │ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ │        number of minimal representatives: 1
│ │ │ │        dimension base locus: 4
│ │ │ │        degree base locus: 24
│ │ │ │        coefficient ring: ZZ/300007
│ │ │ │  i19 : time (f,g) = graph phi^-1; f;
│ │ │ │ - -- used 0.0132166s (cpu); 0.0131906s (thread); 0s (gc)
│ │ │ │ + -- used 0.0101159s (cpu); 0.0101206s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety
│ │ │ │  of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
│ │ │ │  i21 : time degrees f
│ │ │ │ - -- used 1.60372s (cpu); 1.18114s (thread); 0s (gc)
│ │ │ │ + -- used 1.12259s (cpu); 0.837973s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 2.0168e-05s (cpu); 1.9606e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.5474e-05s (cpu); 1.4716e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.00171686s (cpu); 0.00171795s (thread); 0s (gc)
│ │ │ │ + -- used 0.00132095s (cpu); 0.00132446s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = rational map defined by multiforms of degree {1, 0}
│ │ │ │        source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20
│ │ │ │  hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1,
│ │ │ │  1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2,
│ │ │ │  0},{2, 0})
│ │ │ │        target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5
│ │ ├── ./usr/share/doc/Macaulay2/Cyclotomic/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  Q3ljbG90b21pYw==
│ │ │  #:len=478
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY3ljbG90b21pYyBmaWVsZHMiLCBEZXNj
│ │ │  cmlwdGlvbiA9PiAoRU17IkN5Y2xvdG9taWMifSwiIGlzIGEgcGFja2FnZSBmb3IgY3ljbG90b21p
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  c291cmNlKERHQWxnZWJyYU1hcCk=
│ │ │  #:len=1484
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIHNvdXJjZSBvZiBhIERHIGFsZ2Vi
│ │ │  cmEgbWFwIiwgImxpbmVudW0iID0+IDkyNTUsIElucHV0cyA9PiB7U1BBTntUVHsicGhpIn0sIiwg
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out
│ │ │ @@ -155,15 +155,15 @@
│ │ │                                    2       2       2         2 2       2 2        2 2        2 2       2 2            2 2           2 2            2           2           2
│ │ │         Differential => {a, b, c, a T   , b T   , c T   , a*b c T   , b c T   , -a b T   , -a c T   , b c T   T   , -a c T   T   , b c T   T   , -a T   T   , c T   T   , b T   T   }
│ │ │                                      1,1     1,2     1,3         1,1       2,1        2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3 2,4     1,2 2,4
│ │ │  
│ │ │  o16 : DGAlgebra
│ │ │  
│ │ │  i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0164909s (cpu); 0.0155111s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0239318s (cpu); 0.0141738s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │        Underlying algebra => R[S   ..S   ]
│ │ │                                 1,1   1,4
│ │ │        Differential => {a, b, c, d}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0198089s (cpu); 0.0185677s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.203315s (cpu); 0.042911s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i6 : describe HB
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │                                      2
│ │ │        Differential => {a, b, c, d, a S   }
│ │ │                                        1,1
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │  
│ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.019996s (cpu); 0.0184306s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0308054s (cpu); 0.0192565s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │        Underlying algebra => R[T   ..T   ]
│ │ │                                 1,1   1,3
│ │ │        Differential => {a, b, c}
│ │ │  
│ │ │  o2 : DGAlgebra
│ │ │  
│ │ │  i3 : HA = HH A
│ │ │ -Finding easy relations           :  -- used 0.0541819s (cpu); 0.0514376s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.351496s (cpu); 0.0657411s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = HA
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out
│ │ │ @@ -28,15 +28,15 @@
│ │ │  
│ │ │  o4 = map (R[Y   ..Y   ], R[T   ..T   ], {Y   , Y   , a, b, c})
│ │ │               1,1   1,3      1,1   1,2     1,2   1,3
│ │ │  
│ │ │  o4 : DGAlgebraMap
│ │ │  
│ │ │  i5 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0165828s (cpu); 0.015481s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0370027s (cpu); 0.0166315s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │        Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
│ │ │                                     |     0    |  |     0    |
│ │ │                                     |     0    |  |     0    |
│ │ │  
│ │ │  o4 : DGModule
│ │ │  
│ │ │  i5 : HM = homology M
│ │ │ -Finding easy relations           :  -- used 0.14103s (cpu); 0.0653344s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.235005s (cpu); 0.0527513s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ │  
│ │ │                                                  4
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module__Map.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │        Natural => {0, 0} | 1 0 0 |
│ │ │                   {2, 2} | 0 1 0 |
│ │ │                   {2, 2} | 0 0 1 |
│ │ │  
│ │ │  o4 : DGModuleMap
│ │ │  
│ │ │  i5 : h = homology idM
│ │ │ -Finding easy relations           :  -- used 0.172027s (cpu); 0.0676646s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.304952s (cpu); 0.0618236s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │       {3, 4} | 0 1 0 0 |
│ │ │       {3, 4} | 0 0 1 0 |
│ │ │       {3, 4} | 0 0 0 1 |
│ │ │  
│ │ │  o5 : Matrix
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │                                1                                                             {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |     3
│ │ │                                                                                      
│ │ │                                                                                     2
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.0242787s (cpu); 0.0231763s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.174174s (cpu); 0.0344958s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i8 : ideal HKR
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │  i9 : R' = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-c^2*d^2}
│ │ │  
│ │ │  o9 = R'
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.691391s (cpu); 0.592007s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.673352s (cpu); 0.659403s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_cycles.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra A
│ │ │ -Finding easy relations           :  -- used 0.0210175s (cpu); 0.0187759s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.146629s (cpu); 0.0265698s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Deg__N__Module.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │        Underlying algebra => R[T   ..T   ]
│ │ │                                 1,1   1,3
│ │ │        Differential => {x, y, z}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HA = HH KR;
│ │ │ -Finding easy relations           :  -- used 0.0155194s (cpu); 0.01441s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.179068s (cpu); 0.0408131s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : H0 = zerothHomology KR
│ │ │  
│ │ │  o6 = H0
│ │ │  
│ │ │  o6 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0223745s (cpu); 0.0212429s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0303154s (cpu); 0.0194258s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}
│ │ │  
│ │ │ @@ -46,15 +46,15 @@
│ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0919266s (cpu); 0.08901s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0975287s (cpu); 0.0872175s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
│ │ │  
│ │ │  i9 : numgens HA
│ │ │  
│ │ │ @@ -122,15 +122,15 @@
│ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o15 = {1, 7, 7, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0612159s (cpu); 0.0591286s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.06157s (cpu); 0.0512631s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : R = ZZ/101[a,b,c,d]
│ │ │  
│ │ │ @@ -159,14 +159,14 @@
│ │ │         Underlying algebra => S[T   ..T   ]
│ │ │                                  1,1   1,4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
│ │ │  
│ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.020179s (cpu); 0.0185169s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.191469s (cpu); 0.0421566s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out
│ │ │ @@ -43,15 +43,15 @@
│ │ │  o6 = y T
│ │ │          1,2
│ │ │  
│ │ │  o6 : R[T   ..T   ]
│ │ │          1,1   1,3
│ │ │  
│ │ │  i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0161623s (cpu); 0.0151328s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0346552s (cpu); 0.0165132s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │  
│ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │                                    
│ │ │       0      1      2      3      4
│ │ │  
│ │ │  o5 : Complex
│ │ │  
│ │ │  i6 : HKR = HH(KR)
│ │ │ - -- used 0.114372s (cpu); 0.108291s (thread); 0s (gc)
│ │ │ + -- used 0.122198s (cpu); 0.101508s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
│ │ │  
│ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product.out
│ │ │ @@ -68,15 +68,15 @@
│ │ │                 2
│ │ │  o9 = (true, x y T   T   T    - x x y T   T   T   )
│ │ │               2 2 1,1 1,2 1,3    1 2 2 1,2 1,3 1,4
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.596064s (cpu); 0.508805s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.887842s (cpu); 0.609206s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T   T   T   T
│ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │  
│ │ │  o10 : R[T   ..T   ]
│ │ │           1,1   1,5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -27,15 +27,15 @@
│ │ │                                 1,1   1,4
│ │ │        Differential => {t , t , t , t }
│ │ │                          1   2   3   4
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.174343s (cpu); 0.170772s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.183311s (cpu); 0.172616s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_tor__Algebra_lp__Ring_cm__Ring_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ -Finding easy relations           :  -- used 0.532566s (cpu); 0.419747s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.644486s (cpu); 0.539534s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html
│ │ │ @@ -294,15 +294,15 @@
│ │ │          
    
│ │ │            
    One can also obtain the map on homology induced by a DGAlgebra map.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0164909s (cpu); 0.0155111s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0239318s (cpu); 0.0141738s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -218,15 +218,15 @@
│ │ │ │                                      1,1     1,2     1,3         1,1       2,1
│ │ │ │  2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3
│ │ │ │  2,4     1,2 2,4
│ │ │ │  
│ │ │ │  o16 : DGAlgebra
│ │ │ │  One can also obtain the map on homology induced by a DGAlgebra map.
│ │ │ │  i17 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0164909s (cpu); 0.0155111s
│ │ │ │ +Finding easy relations           :  -- used 0.0239318s (cpu); 0.0141738s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                            ZZ
│ │ │ │                           ---[a..c]
│ │ │ │              ZZ           101
│ │ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │             101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html
│ │ │ @@ -118,15 +118,15 @@
│ │ │          
    
│ │ │            
    One can compute the homology algebra of a DGAlgebra using the homology (or HH) command.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -179,15 +179,15 @@
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │                                 1,1   1,4
│ │ │ │        Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  One can compute the homology algebra of a DGAlgebra using the homology (or HH)
│ │ │ │  command.
│ │ │ │  i5 : HB = HH B
│ │ │ │ -Finding easy relations           :  -- used 0.0198089s (cpu); 0.0185677s
│ │ │ │ +Finding easy relations           :  -- used 0.203315s (cpu); 0.042911s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │ @@ -87,15 +87,15 @@
│ │ │ │                                 1,1   1,4   2,1
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a S   }
│ │ │ │                                        1,1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ -Finding easy relations           :  -- used 0.019996s (cpu); 0.0184306s
│ │ │ │ +Finding easy relations           :  -- used 0.0308054s (cpu); 0.0192565s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │         ZZ
│ │ │ │  o10 = ---[X ..X ]
│ │ │ │        101  1   3
│ │ │ │  
│ │ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o2 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o2 = {Ring => R                          }
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,3
│ │ │ │        Differential => {a, b, c}
│ │ │ │  
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : HA = HH A
│ │ │ │ -Finding easy relations           :  -- used 0.0541819s (cpu); 0.0514376s
│ │ │ │ +Finding easy relations           :  -- used 0.351496s (cpu); 0.0657411s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = HA
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html
│ │ │ @@ -120,15 +120,15 @@
│ │ │  
│ │ │  o4 : DGAlgebraMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -255,15 +255,15 @@
│ │ │  
│ │ │  o15 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0198089s (cpu); 0.0185677s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.203315s (cpu); 0.042911s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.019996s (cpu); 0.0184306s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0308054s (cpu); 0.0192565s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i3 : HA = HH A
│ │ │ -Finding easy relations           :  -- used 0.0541819s (cpu); 0.0514376s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.351496s (cpu); 0.0657411s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = HA
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
    
│ │ │                
    i5 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0165828s (cpu); 0.015481s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0370027s (cpu); 0.0166315s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  i4 : g = dgAlgebraMap(K1, K2, matrix{{Y_(1,2), Y_(1,3)}})
│ │ │ │  
│ │ │ │  o4 = map (R[Y   ..Y   ], R[T   ..T   ], {Y   , Y   , a, b, c})
│ │ │ │               1,1   1,3      1,1   1,2     1,2   1,3
│ │ │ │  
│ │ │ │  o4 : DGAlgebraMap
│ │ │ │  i5 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0165828s (cpu); 0.015481s
│ │ │ │ +Finding easy relations           :  -- used 0.0370027s (cpu); 0.0166315s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o4 : DGModule
    
│ │ │              
    
│ │ │                
    i5 : HM = homology M
│ │ │ -Finding easy relations           :  -- used 0.14103s (cpu); 0.0653344s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.235005s (cpu); 0.0527513s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ │  
│ │ │                                                  4
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,15 +57,15 @@
│ │ │ │        Generator degrees => {{0, 0}, {2, 2}, {2, 2}}
│ │ │ │        Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
│ │ │ │                                     |     0    |  |     0    |
│ │ │ │                                     |     0    |  |     0    |
│ │ │ │  
│ │ │ │  o4 : DGModule
│ │ │ │  i5 : HM = homology M
│ │ │ │ -Finding easy relations           :  -- used 0.14103s (cpu); 0.0653344s
│ │ │ │ +Finding easy relations           :  -- used 0.235005s (cpu); 0.0527513s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module__Map.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o4 : DGModuleMap
    
│ │ │              
    
│ │ │                
    i5 : h = homology idM
│ │ │ -Finding easy relations           :  -- used 0.172027s (cpu); 0.0676646s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.304952s (cpu); 0.0618236s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │       {3, 4} | 0 1 0 0 |
│ │ │       {3, 4} | 0 0 1 0 |
│ │ │       {3, 4} | 0 0 0 1 |
│ │ │  
│ │ │  o5 : Matrix
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,15 +59,15 @@
│ │ │ │                      1,1   1,2
│ │ │ │        Natural => {0, 0} | 1 0 0 |
│ │ │ │                   {2, 2} | 0 1 0 |
│ │ │ │                   {2, 2} | 0 0 1 |
│ │ │ │  
│ │ │ │  o4 : DGModuleMap
│ │ │ │  i5 : h = homology idM
│ │ │ │ -Finding easy relations           :  -- used 0.172027s (cpu); 0.0676646s
│ │ │ │ +Finding easy relations           :  -- used 0.304952s (cpu); 0.0618236s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │ │       {3, 4} | 0 1 0 0 |
│ │ │ │       {3, 4} | 0 0 1 0 |
│ │ │ │       {3, 4} | 0 0 0 1 |
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -143,15 +143,15 @@
│ │ │          
    
│ │ │            
    Since the Koszul complex is a DG algebra, its homology is itself an algebra.  One can obtain this algebra using the command homology, homologyAlgebra, or HH (all commands work).  This algebra structure can detect whether or not the ring is a complete intersection or Gorenstein.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │  
│ │ │ │  o6 : Complex
│ │ │ │  Since the Koszul complex is a DG algebra, its homology is itself an algebra.
│ │ │ │  One can obtain this algebra using the command homology, homologyAlgebra, or HH
│ │ │ │  (all commands work). This algebra structure can detect whether or not the ring
│ │ │ │  is a complete intersection or Gorenstein.
│ │ │ │  i7 : HKR = HH KR
│ │ │ │ -Finding easy relations           :  -- used 0.0242787s (cpu); 0.0231763s
│ │ │ │ +Finding easy relations           :  -- used 0.174174s (cpu); 0.0344958s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = HKR
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i8 : ideal HKR
│ │ │ │  
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  i9 : R' = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-
│ │ │ │  c^2*d^2}
│ │ │ │  
│ │ │ │  o9 = R'
│ │ │ │  
│ │ │ │  o9 : QuotientRing
│ │ │ │  i10 : HKR' = HH koszulComplexDGA R'
│ │ │ │ -Finding easy relations           :  -- used 0.691391s (cpu); 0.592007s
│ │ │ │ +Finding easy relations           :  -- used 0.673352s (cpu); 0.659403s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = HKR'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_cycles.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra A
│ │ │ │ -Finding easy relations           :  -- used 0.0210175s (cpu); 0.0187759s
│ │ │ │ +Finding easy relations           :  -- used 0.146629s (cpu); 0.0265698s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Deg__N__Module.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.0242787s (cpu); 0.0231763s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.174174s (cpu); 0.0344958s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.691391s (cpu); 0.592007s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.673352s (cpu); 0.659403s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra A
│ │ │ -Finding easy relations           :  -- used 0.0210175s (cpu); 0.0187759s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.146629s (cpu); 0.0265698s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i5 : HA = HH KR;
│ │ │ -Finding easy relations           :  -- used 0.0155194s (cpu); 0.01441s (thread); 0s (gc)
    
│ │ │ +Finding easy relations           :  -- used 0.179068s (cpu); 0.0408131s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : H0 = zerothHomology KR
│ │ │  
│ │ │  o6 = H0
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o4 = {Ring => R                          }
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,3
│ │ │ │        Differential => {x, y, z}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : HA = HH KR;
│ │ │ │ -Finding easy relations           :  -- used 0.0155194s (cpu); 0.01441s
│ │ │ │ +Finding easy relations           :  -- used 0.179068s (cpu); 0.0408131s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  i6 : H0 = zerothHomology KR
│ │ │ │  
│ │ │ │  o6 = H0
│ │ │ │  
│ │ │ │  o6 : QuotientRing
│ │ │ │  i7 : M1 = getDegNModule(1, H0, HA)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0223745s (cpu); 0.0212429s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0303154s (cpu); 0.0194258s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o7 : List
│ │ │              
    
│ │ │                
    i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0919266s (cpu); 0.08901s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0975287s (cpu); 0.0872175s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │              
    
│ │ │                
    i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0612159s (cpu); 0.0591286s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.06157s (cpu); 0.0512631s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
    
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │              
    
│ │ │                
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.020179s (cpu); 0.0185169s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.191469s (cpu); 0.0421566s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0223745s (cpu); 0.0212429s
│ │ │ │ +Finding easy relations           :  -- used 0.0303154s (cpu); 0.0194258s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  Note that HA is a graded commutative polynomial ring (i.e. an exterior algebra)
│ │ │ │  since R is a complete intersection.
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  o6 : DGAlgebra
│ │ │ │  i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o7 = {1, 5, 10, 10, 4}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  i8 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0919266s (cpu); 0.08901s
│ │ │ │ +Finding easy relations           :  -- used 0.0975287s (cpu); 0.0872175s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │ @@ -130,15 +130,15 @@
│ │ │ │  o14 : DGAlgebra
│ │ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o15 = {1, 7, 7, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0612159s (cpu); 0.0591286s
│ │ │ │ +Finding easy relations           :  -- used 0.06157s (cpu); 0.0512631s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = HA
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  One can check that HA has Poincare duality since R is Gorenstein.
│ │ │ │  If your DGAlgebra has generators in even degrees, then one must specify the
│ │ │ │ @@ -166,15 +166,15 @@
│ │ │ │  o20 = {Ring => S                          }
│ │ │ │         Underlying algebra => S[T   ..T   ]
│ │ │ │                                  1,1   1,4
│ │ │ │         Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o20 : DGAlgebra
│ │ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ │ -Finding easy relations           :  -- used 0.020179s (cpu); 0.0185169s
│ │ │ │ +Finding easy relations           :  -- used 0.191469s (cpu); 0.0421566s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = HB
│ │ │ │  
│ │ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  ********** WWaayyss ttoo uussee hhoommoollooggyyAAllggeebbrraa:: **********
│ │ │ │      * homologyAlgebra(DGAlgebra)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html
│ │ │ @@ -140,15 +140,15 @@
│ │ │  o6 : R[T   ..T   ]
│ │ │          1,1   1,3
    
│ │ │              
    
│ │ │                
    i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0161623s (cpu); 0.0151328s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0346552s (cpu); 0.0165132s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.114372s (cpu); 0.108291s (thread); 0s (gc)
│ │ │ + -- used 0.122198s (cpu); 0.101508s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │        1      4      6      4      1
│ │ │ │  o5 = R  <-- R  <-- R  <-- R  <-- R
│ │ │ │  
│ │ │ │       0      1      2      3      4
│ │ │ │  
│ │ │ │  o5 : Complex
│ │ │ │  i6 : HKR = HH(KR)
│ │ │ │ - -- used 0.114372s (cpu); 0.108291s (thread); 0s (gc)
│ │ │ │ + -- used 0.122198s (cpu); 0.101508s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o6 = HKR
│ │ │ │  
│ │ │ │  o6 : QuotientRing
│ │ │ │  The following is the graded canonical module of R:
│ │ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product.html
│ │ │ @@ -197,15 +197,15 @@
│ │ │          
    
│ │ │            
    Given cycles z1,z2,z3 such that z1*z2 and z2*z3 are boundaries, the Massey triple product of the homology classes represented by z1,z2 and z3 is the homology class of lift12*z3 + z1*lift23.  To see this, we compute and check:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,4
│ │ │ │        Differential => {t , t , t , t }
│ │ │ │                          1   2   3   4
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.174343s (cpu); 0.170772s
│ │ │ │ +Finding easy relations           :  -- used 0.183311s (cpu); 0.172616s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = H
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_tor__Algebra_lp__Ring_cm__Ring_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  i2 : M = coker matrix {{a^3*b^3*c^3*d^3}};
│ │ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ │ -Finding easy relations           :  -- used 0.532566s (cpu); 0.419747s
│ │ │ │ +Finding easy relations           :  -- used 0.644486s (cpu); 0.539534s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DecomposableSparseSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  c29sdmVEZWNvbXBvc2FibGVTeXN0ZW0oTGlzdCk=
│ │ │  #:len=338
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzb2x2ZURlY29tcG9zYWJsZVN5c3RlbSxMaXN0KSwi
│ │ ├── ./usr/share/doc/Macaulay2/Depth/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  U2VlZA==
│ │ │  #:len=219
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTZWVkIiwiU2VlZCIsIkRlcHRoIn0sIFByaW1hcnlU
│ │ ├── ./usr/share/doc/Macaulay2/DeterminantalRepresentations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  Z2VuZXJhbGl6ZWRNaXhlZERpc2NyaW1pbmFudChMaXN0KQ==
│ │ │  #:len=367
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODYwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnZW5lcmFsaXplZE1peGVkRGlzY3JpbWluYW50LExp
│ │ ├── ./usr/share/doc/Macaulay2/DiffAlg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  bGluZWFyQ29tYg==
│ │ │  #:len=2390
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2VuZXJpYyBsaW5lYXIgY29tYmluYXRp
│ │ │  b24gb2YgZWxlbWVudHMiLCAibGluZW51bSIgPT4gODYzLCBJbnB1dHMgPT4ge1NQQU57VFR7Ikwi
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  dGFsbHlTdW1tYW5kcw==
│ │ │  #:len=2086
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGFsbHkgYSBsaXN0IG9mIG1vZHVsZXMg
│ │ │  dXAgdG8gaXNvbW9ycGhpc20iLCAibGluZW51bSIgPT4gNjgwLCBJbnB1dHMgPT4ge1NQQU57VFR7
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/example-output/___Direct__Summands.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i3 : F = frobeniusPushforward(1, OO_X);
│ │ │  
│ │ │  i4 : elapsedTime tallySummands summands F
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │ - -- 3.35396s elapsed
│ │ │ + -- 2.56026s elapsed
│ │ │  
│ │ │  o4 = Tally{cokernel {1} | x   -2z -3y -y  0   0   | => 1}
│ │ │                      {1} | 3z  -y  0   x   2z  -2y |
│ │ │                      {1} | -3y x   z   -3z 0   0   |
│ │ │                      {1} | 0   0   -3y y   x   -3z |
│ │ │                      {1} | -2z 2y  x   0   -z  3y  |
│ │ │                      {1} | 0   0   -z  -3z -2y x   |
│ │ │ @@ -32,15 +32,15 @@
│ │ │                      {1} | 0  0   -2z z   3y  x  |
│ │ │              1
│ │ │             R  => 1
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 : elapsedTime tallySummands summands changeBaseField(GF(7, 2), F)
│ │ │ - -- 11.5598s elapsed
│ │ │ + -- 8.36651s elapsed
│ │ │  
│ │ │  o5 = Tally{cokernel {1} | (-2a+1)y x  z       | => 1      }
│ │ │                      {1} | x        az (-a-3)y |
│ │ │                      {1} | (2a-2)z  3y x       |
│ │ │             cokernel {1} | (-3a-2)y x        2z      | => 1
│ │ │                      {1} | x        (-2a+2)z (2a-1)y |
│ │ │                      {1} | 3az      -y       x       |
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/example-output/_is__Indecomposable.out
│ │ │ @@ -54,14 +54,14 @@
│ │ │  
│ │ │                              19
│ │ │  o8 : S-module, quotient of S
│ │ │  
│ │ │  i9 : assert(2 == rank FHM)
│ │ │  
│ │ │  i10 : assert elapsedTime isIndecomposable FHM
│ │ │ - -- 1.00223s elapsed
│ │ │ + -- .889618s elapsed
│ │ │  
│ │ │  i11 : assert({FHM} == summands FHM)
│ │ │  
│ │ │  i12 : assert FHM.cache.isIndecomposable
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/html/_is__Indecomposable.html
│ │ │ @@ -170,15 +170,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -124,15 +124,15 @@
│ │ │ │  0     0      0    0      0      0      0      -x_4   -x_3    x_3  x_2    x_1
│ │ │ │  -x_0   |
│ │ │ │  
│ │ │ │                              19
│ │ │ │  o8 : S-module, quotient of S
│ │ │ │  i9 : assert(2 == rank FHM)
│ │ │ │  i10 : assert elapsedTime isIndecomposable FHM
│ │ │ │ - -- 1.00223s elapsed
│ │ │ │ + -- .889618s elapsed
│ │ │ │  i11 : assert({FHM} == summands FHM)
│ │ │ │  i12 : assert FHM.cache.isIndecomposable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_r_e_c_t_S_u_m_m_a_n_d_s -- compute the direct summands of a module or coherent
│ │ │ │        sheaf
│ │ │ │      * _f_i_n_d_I_d_e_m_p_o_t_e_n_t_s -- construct idempotent endomorphisms
│ │ │ │  ********** WWaayyss ttoo uussee iissIInnddeeccoommppoossaabbllee:: **********
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/html/index.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  When applied to a graph, complementGraph returns the graph whose edge set is
│ │ │ │  the set of edges not in G. When applied to a hypergraph, the edge set is found
│ │ │ │  by taking the complement of each edge of H in the vertex set.
│ │ │ │  i1 : R = QQ[a,b,c,d,e];
│ │ │ │  i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle
│ │ │ │  i3 : complementGraph c5 -- the graph complement of the 5-cycle
│ │ │ │  
│ │ │ │ -o3 = Graph{"edges" => {{a, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
│ │ │ │ +o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}}
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : c5hypergraph = hyperGraph c5 -- the 5-cycle, but viewed as a hypergraph
│ │ │ │  
│ │ │ │  o4 = HyperGraph{"edges" => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_delete__Edges.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  i3 : T = {{a,b},{d,e}}
│ │ │ │  
│ │ │ │  o3 = {{a, b}, {d, e}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : gprime = deleteEdges (g,T)
│ │ │ │  
│ │ │ │ -o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │ │ +o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │ │                  "ring" => S
│ │ │ │                  "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o4 : HyperGraph
│ │ │ │  i5 : h = hyperGraph {a*b*c,c*d*e,a*e}
│ │ │ │  
│ │ │ │  o5 = HyperGraph{"edges" => {{a, b, c}, {a, e}, {c, d, e}}}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
│ │ │ @@ -93,28 +93,28 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,24 +25,24 @@
│ │ │ │  _T_i_m_e_L_i_m_i_t). The method will return null if it cannot find a hypergraph within
│ │ │ │  the branch and time limits.
│ │ │ │  i1 : R = QQ[x_1..x_5];
│ │ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │ │  
│ │ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ │ -                              2   4   5     1   5     1   2   3   4
│ │ │ │ +                              1   2   3     3   4     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 }}}
│ │ │ │ -                              1   2   4     1   5     2   3   4   5
│ │ │ │ +                              2   3   4     3   5     1   2   4   5
│ │ │ │                  "ring" => R
│ │ │ │                  "vertices" => {x , x , x , x , x }
│ │ │ │                                  1   2   3   4   5
│ │ │ │  
│ │ │ │  o4 : HyperGraph
│ │ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch
│ │ │ │  limit reached
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_spanning__Tree.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,15 +13,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns a breadth first spanning tree of a graph.
│ │ │ │  i1 : R = QQ[x_1..x_6];
│ │ │ │  i2 : C = cycle R; -- a 6-cycle
│ │ │ │  i3 : spanningTree C
│ │ │ │  
│ │ │ │  o3 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ │ -                         1   2     2   3     3   4     4   5     5   6
│ │ │ │ +                         1   2     3   4     4   5     1   6     5   6
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │ │                             1   2   3   4   5   6
│ │ │ │  
│ │ │ │  o3 : Graph
│ │ │ │  i4 : T = graph {x_1*x_2,x_2*x_3, x_1*x_4,x_1*x_5,x_5*x_6}; -- a tree (no
│ │ │ │  cycles)
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : G = graph {x_1*x_2,x_2*x_3,x_3*x_1,x_4*x_5,x_5*x_6,x_6*x_4}; -- two three
│ │ │ │  cycles
│ │ │ │  i7 : spanningTree G
│ │ │ │  
│ │ │ │  o7 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ │ -                         1   2     1   3     4   5     4   6
│ │ │ │ +                         1   3     2   3     4   6     5   6
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │ │                             1   2   3   4   5   6
│ │ │ │  
│ │ │ │  o7 : Graph
│ │ │ │  ********** WWaayyss ttoo uussee ssppaannnniinnggTTrreeee:: **********
│ │ │ │      * spanningTree(Graph)
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  emVyb0RpbVNvbHZlKElkZWFsKQ==
│ │ │  #:len=254
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh6ZXJvRGltU29sdmUsSWRlYWwpLCJ6ZXJvRGltU29s
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/example-output/___Eigen__Solver.out
│ │ │ @@ -15,14 +15,14 @@
│ │ │       a*b*e*f + a*d*e*f + c*d*e*f, a*b*c*d*e + a*b*c*d*f + a*b*c*e*f +
│ │ │       ------------------------------------------------------------------------
│ │ │       a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1)
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │  
│ │ │  i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .217098s elapsed
│ │ │ + -- .208344s elapsed
│ │ │  
│ │ │  i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/html/index.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,22 +29,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00258521s (cpu); 0.0025827s (thread); 0s (gc)
│ │ │ │ + -- used 0.00298032s (cpu); 0.00299111s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                        2    2             2           2
│ │ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.00151386s (cpu); 0.00151487s (thread); 0s (gc)
│ │ │ │ + -- used 0.00169817s (cpu); 0.0016979s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html
│ │ │ @@ -102,26 +102,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,22 +30,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00271762s (cpu); 0.00271441s (thread); 0s (gc)
│ │ │ │ + -- used 0.00265663s (cpu); 0.00265287s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                        2    2             2           2
│ │ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.00160798s (cpu); 0.00160903s (thread); 0s (gc)
│ │ │ │ + -- used 0.00149513s (cpu); 0.0014956s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.596064s (cpu); 0.508805s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.887842s (cpu); 0.609206s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T   T   T   T
│ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │  
│ │ │  o10 : R[T   ..T   ]
│ │ │           1,1   1,5
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -90,15 +90,15 @@
│ │ │ │  Note that the first return value of _g_e_t_B_o_u_n_d_a_r_y_P_r_e_i_m_a_g_e indicates that the
│ │ │ │  inputs are indeed boundaries, and the second value is the lift of the boundary
│ │ │ │  along the differential.
│ │ │ │  Given cycles z1,z2,z3 such that z1*z2 and z2*z3 are boundaries, the Massey
│ │ │ │  triple product of the homology classes represented by z1,z2 and z3 is the
│ │ │ │  homology class of lift12*z3 + z1*lift23. To see this, we compute and check:
│ │ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ │ -Finding easy relations           :  -- used 0.596064s (cpu); 0.508805s
│ │ │ │ +Finding easy relations           :  -- used 0.887842s (cpu); 0.609206s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2
│ │ │ │  o10 = x x y z T   T   T   T
│ │ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │ │  
│ │ │ │  o10 : R[T   ..T   ]
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html
│ │ │ @@ -124,15 +124,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
    
│ │ │                
    i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.174343s (cpu); 0.170772s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.183311s (cpu); 0.172616s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ -Finding easy relations           :  -- used 0.532566s (cpu); 0.419747s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.644486s (cpu); 0.539534s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
    
│ │ │                
    i9 : assert(2 == rank FHM)
    
│ │ │              
    
│ │ │                
    i10 : assert elapsedTime isIndecomposable FHM
│ │ │ - -- 1.00223s elapsed
    
│ │ │ + -- .889618s elapsed
│ │ │              
    
│ │ │                
    i11 : assert({FHM} == summands FHM)
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime tallySummands summands F
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │ - -- 3.35396s elapsed
│ │ │ + -- 2.56026s elapsed
│ │ │  
│ │ │  o4 = Tally{cokernel {1} | x   -2z -3y -y  0   0   | => 1}
│ │ │                      {1} | 3z  -y  0   x   2z  -2y |
│ │ │                      {1} | -3y x   z   -3z 0   0   |
│ │ │                      {1} | 0   0   -3y y   x   -3z |
│ │ │                      {1} | -2z 2y  x   0   -z  3y  |
│ │ │                      {1} | 0   0   -z  -3z -2y x   |
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime tallySummands summands changeBaseField(GF(7, 2), F)
│ │ │ - -- 11.5598s elapsed
│ │ │ + -- 8.36651s elapsed
│ │ │  
│ │ │  o5 = Tally{cokernel {1} | (-2a+1)y x  z       | => 1      }
│ │ │                      {1} | x        az (-a-3)y |
│ │ │                      {1} | (2a-2)z  3y x       |
│ │ │             cokernel {1} | (-3a-2)y x        2z      | => 1
│ │ │                      {1} | x        (-2a+2)z (2a-1)y |
│ │ │                      {1} | 3az      -y       x       |
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │  i1 : R = ZZ/7[x,y,z]/(x^3+y^3+z^3);
│ │ │ │  i2 : X = Proj R;
│ │ │ │  i3 : F = frobeniusPushforward(1, OO_X);
│ │ │ │  i4 : elapsedTime tallySummands summands F
│ │ │ │   -- try using changeBaseField with GF 49
│ │ │ │   -- try using changeBaseField with GF 49
│ │ │ │   -- try using changeBaseField with GF 49
│ │ │ │ - -- 3.35396s elapsed
│ │ │ │ + -- 2.56026s elapsed
│ │ │ │  
│ │ │ │  o4 = Tally{cokernel {1} | x   -2z -3y -y  0   0   | => 1}
│ │ │ │                      {1} | 3z  -y  0   x   2z  -2y |
│ │ │ │                      {1} | -3y x   z   -3z 0   0   |
│ │ │ │                      {1} | 0   0   -3y y   x   -3z |
│ │ │ │                      {1} | -2z 2y  x   0   -z  3y  |
│ │ │ │                      {1} | 0   0   -z  -3z -2y x   |
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │                      {1} | 3z -3y x   0   -2z -y |
│ │ │ │                      {1} | 0  0   -2z z   3y  x  |
│ │ │ │              1
│ │ │ │             R  => 1
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  i5 : elapsedTime tallySummands summands changeBaseField(GF(7, 2), F)
│ │ │ │ - -- 11.5598s elapsed
│ │ │ │ + -- 8.36651s elapsed
│ │ │ │  
│ │ │ │  o5 = Tally{cokernel {1} | (-2a+1)y x  z       | => 1      }
│ │ │ │                      {1} | x        az (-a-3)y |
│ │ │ │                      {1} | (2a-2)z  3y x       |
│ │ │ │             cokernel {1} | (-3a-2)y x        2z      | => 1
│ │ │ │                      {1} | x        (-2a+2)z (2a-1)y |
│ │ │ │                      {1} | 3az      -y       x       |
│ │ ├── ./usr/share/doc/Macaulay2/Dmodules/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  RG1vZHVsZXM=
│ │ │  #:len=758
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRC1tb2R1bGVzIHBhY2thZ2UgY29sbGVj
│ │ │  dGlvbiIsICJsaW5lbnVtIiA9PiAxNDIsICJmaWxlbmFtZSIgPT4gIkRtb2R1bGVzLm0yIiwgRGVz
│ │ ├── ./usr/share/doc/Macaulay2/EagonResolution/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  VHJhbnNwb3Nl
│ │ │  #:len=1380
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVHJhbnNwb3NlID0+IGZhbHNlLCBkZWZh
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│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Y29tcGxlbWVudEdyYXBo
│ │ │  #:len=2011
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY29tcGxlbWVudCBv
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│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_complement__Graph.out
│ │ │ @@ -2,15 +2,15 @@
│ │ │  
│ │ │  i1 : R = QQ[a,b,c,d,e];
│ │ │  
│ │ │  i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle
│ │ │  
│ │ │  i3 : complementGraph c5 -- the graph complement of the 5-cycle
│ │ │  
│ │ │ -o3 = Graph{"edges" => {{a, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
│ │ │ +o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : c5hypergraph = hyperGraph c5 -- the 5-cycle, but viewed as a hypergraph
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_delete__Edges.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │  
│ │ │  o3 = {{a, b}, {d, e}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : gprime = deleteEdges (g,T)
│ │ │  
│ │ │ -o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │ +o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │                  "ring" => S
│ │ │                  "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o4 : HyperGraph
│ │ │  
│ │ │  i5 : h = hyperGraph {a*b*c,c*d*e,a*e}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out
│ │ │ @@ -3,25 +3,25 @@
│ │ │  i1 : R = QQ[x_1..x_5];
│ │ │  
│ │ │  i2 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              2   4   5     1   5     1   2   3   4
│ │ │ +                              1   2   3     3   4     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 }}}
│ │ │ -                              1   2   4     1   5     2   3   4   5
│ │ │ +                              2   3   4     3   5     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
│ │ │  
│ │ │  i5 : randomHyperGraph(R,{4,4,2,2}) -- impossible, returns null when time/branch limit reached
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_spanning__Tree.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : R = QQ[x_1..x_6];
│ │ │  
│ │ │  i2 : C = cycle R; -- a 6-cycle
│ │ │  
│ │ │  i3 : spanningTree C
│ │ │  
│ │ │  o3 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     2   3     3   4     4   5     5   6
│ │ │ +                         1   2     3   4     4   5     1   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : T = graph {x_1*x_2,x_2*x_3, x_1*x_4,x_1*x_5,x_5*x_6}; -- a tree (no cycles)
│ │ │ @@ -21,15 +21,15 @@
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : G = graph {x_1*x_2,x_2*x_3,x_3*x_1,x_4*x_5,x_5*x_6,x_6*x_4}; -- two three cycles
│ │ │  
│ │ │  i7 : spanningTree G
│ │ │  
│ │ │  o7 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     1   3     4   5     4   6
│ │ │ +                         1   3     2   3     4   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o7 : Graph
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_complement__Graph.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │                
    i2 : c5 = graph {a*b,b*c,c*d,d*e,e*a}; -- graph of the 5-cycle
    
│ │ │              
    
│ │ │                
    i3 : complementGraph c5 -- the graph complement of the 5-cycle
│ │ │  
│ │ │ -o3 = Graph{"edges" => {{a, d}, {a, c}, {b, e}, {b, d}, {c, e}}}
│ │ │ +o3 = Graph{"edges" => {{a, c}, {b, e}, {b, d}, {c, e}, {a, d}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
    
│ │ │                
    i4 : gprime = deleteEdges (g,T)
│ │ │  
│ │ │ -o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │ +o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │                  "ring" => S
│ │ │                  "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              2   4   5     1   5     1   2   3   4
│ │ │ +                              1   2   3     3   4     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 }}}
│ │ │ -                              1   2   4     1   5     2   3   4   5
│ │ │ +                              2   3   4     3   5     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i3 : spanningTree C
│ │ │  
│ │ │  o3 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     2   3     3   4     4   5     5   6
│ │ │ +                         1   2     3   4     4   5     1   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o3 : Graph
    
│ │ │              
    
│ │ │                
    i7 : spanningTree G
│ │ │  
│ │ │  o7 = Graph{"edges" => {{x , x }, {x , x }, {x , x }, {x , x }}}
│ │ │ -                         1   2     1   3     4   5     4   6
│ │ │ +                         1   3     2   3     4   6     5   6
│ │ │             "ring" => R
│ │ │             "vertices" => {x , x , x , x , x , x }
│ │ │                             1   2   3   4   5   6
│ │ │  
│ │ │  o7 : Graph
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .217098s elapsed
    
│ │ │ + -- .208344s elapsed
│ │ │              
    
│ │ │                
    i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a*b*e*f + a*d*e*f + c*d*e*f, a*b*c*d*e + a*b*c*d*f + a*b*c*e*f +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a*b*d*e*f + a*c*d*e*f + b*c*d*e*f, a*b*c*d*e*f - 1)
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[a..f]
│ │ │ │  i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ │ - -- .217098s elapsed
│ │ │ │ + -- .208344s elapsed
│ │ │ │  i4 : #sols -- 156 solutions
│ │ │ │  
│ │ │ │  o4 = 156
│ │ │ │  The authors would like to acknowledge the June 2020 Macaulay2 workshop held
│ │ │ │  virtually at Warwick, where this package was first developed.
│ │ │ │  RReeffeerreenncceess:
│ │ │ │      * [1] Sturmfels, Bernd. Solving systems of polynomial equations. No. 97.
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZWxpbWluYXRlKElkZWFsLExpc3Qp
│ │ │  #:len=200
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgxLCAidW5kb2N1bWVudGVkIiA9PiB0
│ │ │  cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhlbGltaW5h
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -17,23 +17,23 @@
│ │ │  
│ │ │        2
│ │ │  o3 = x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00258521s (cpu); 0.0025827s (thread); 0s (gc)
│ │ │ + -- used 0.00298032s (cpu); 0.00299111s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00151386s (cpu); 0.00151487s (thread); 0s (gc)
│ │ │ + -- used 0.00169817s (cpu); 0.0016979s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out
│ │ │ @@ -17,23 +17,23 @@
│ │ │  
│ │ │        2
│ │ │  o3 = x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00271762s (cpu); 0.00271441s (thread); 0s (gc)
│ │ │ + -- used 0.00265663s (cpu); 0.00265287s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00160798s (cpu); 0.00160903s (thread); 0s (gc)
│ │ │ + -- used 0.00149513s (cpu); 0.0014956s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │  
│ │ │        8    5
│ │ │  o3 = x  + x  + x*c + d
│ │ │  
│ │ │  o3 : R
│ │ │  
│ │ │  i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.71167s (cpu); 1.39647s (thread); 0s (gc)
│ │ │ + -- used 1.35185s (cpu); 1.1365s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -73,15 +73,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0168256s (cpu); 0.0168298s (thread); 0s (gc)
│ │ │ + -- used 0.0143901s (cpu); 0.0143906s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/example-output/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  
│ │ │        8    5
│ │ │  o4 = x  + x  + x*c + d
│ │ │  
│ │ │  o4 : R
│ │ │  
│ │ │  i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.84957s (cpu); 1.47049s (thread); 0s (gc)
│ │ │ + -- used 1.35535s (cpu); 1.14189s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -75,15 +75,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │                 3          4        4
│ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0175768s (cpu); 0.017584s (thread); 0s (gc)
│ │ │ + -- used 0.0127674s (cpu); 0.0127667s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -108,26 +108,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00258521s (cpu); 0.0025827s (thread); 0s (gc)
│ │ │ + -- used 0.00298032s (cpu); 0.00299111s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00151386s (cpu); 0.00151487s (thread); 0s (gc)
│ │ │ + -- used 0.00169817s (cpu); 0.0016979s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00271762s (cpu); 0.00271441s (thread); 0s (gc)
│ │ │ + -- used 0.00265663s (cpu); 0.00265287s (thread); 0s (gc)
│ │ │  
│ │ │                        2    2             2           2
│ │ │  o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.00160798s (cpu); 0.00160903s (thread); 0s (gc)
│ │ │ + -- used 0.00149513s (cpu); 0.0014956s (thread); 0s (gc)
│ │ │  
│ │ │                          2    2             2           2
│ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.71167s (cpu); 1.39647s (thread); 0s (gc)
│ │ │ + -- used 1.35185s (cpu); 1.1365s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -169,15 +169,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0168256s (cpu); 0.0168298s (thread); 0s (gc)
│ │ │ + -- used 0.0143901s (cpu); 0.0143906s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  i3 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o3 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.71167s (cpu); 1.39647s (thread); 0s (gc)
│ │ │ │ + -- used 1.35185s (cpu); 1.1365s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -90,15 +90,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.0168256s (cpu); 0.0168298s (thread); 0s (gc)
│ │ │ │ + -- used 0.0143901s (cpu); 0.0143906s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : R
    
│ │ │              
    
│ │ │                
    i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.84957s (cpu); 1.47049s (thread); 0s (gc)
│ │ │ + -- used 1.35535s (cpu); 1.14189s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,15 +168,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0175768s (cpu); 0.017584s (thread); 0s (gc)
│ │ │ + -- used 0.0127674s (cpu); 0.0127667s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  i4 : g = x^8+x^5+c*x+d
│ │ │ │  
│ │ │ │        8    5
│ │ │ │  o4 = x  + x  + x*c + d
│ │ │ │  
│ │ │ │  o4 : R
│ │ │ │  i5 : time eliminate(ideal(f,g),x)
│ │ │ │ - -- used 1.84957s (cpu); 1.47049s (thread); 0s (gc)
│ │ │ │ + -- used 1.35535s (cpu); 1.14189s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -85,15 +85,15 @@
│ │ │ │       + 792a*b c*d - 1512a*b*c d + 648a*c d - 360a b*d  + 648a c*d  - 504b d
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                 3          4        4
│ │ │ │       - 216b*c*d  + 2052a*d  - 1944d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : time ideal resultant(f,g,x)
│ │ │ │ - -- used 0.0175768s (cpu); 0.017584s (thread); 0s (gc)
│ │ │ │ + -- used 0.0127674s (cpu); 0.0127667s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/EliminationMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  bWF4Q29sKE1hdHJpeCk=
│ │ │  #:len=257
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTExNSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWF4Q29sLE1hdHJpeCksIm1heENvbChNYXRyaXgp
│ │ ├── ./usr/share/doc/Macaulay2/EliminationTemplates/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  Y29weVRlbXBsYXRl
│ │ │  #:len=2172
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29waWVzIEVsaW1pbmF0aW9uVGVtcGxh
│ │ │  dGUgb2JqZWN0IiwgImxpbmVudW0iID0+IDk4NywgSW5wdXRzID0+IHtTUEFOe1RUeyJFIn0sIiwg
│ │ ├── ./usr/share/doc/Macaulay2/EllipticCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  RWxsUG9pbnRXID09IEVsbFBvaW50Vw==
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzQ3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wgPT0sRWxsUG9pbnRXLEVsbFBvaW50Vyks
│ │ ├── ./usr/share/doc/Macaulay2/EllipticIntegrals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  RWxsaXB0aWNJbnRlZ3JhbHM=
│ │ │  #:len=584
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoaXMg
│ │ │  cGFja2FnZSBwcm92aWRlcyBzb21lIGZ1bmN0aW9ucyBmb3IgY29tcHV0aW5nIGVsbGlwdGljIGlu
│ │ ├── ./usr/share/doc/Macaulay2/EngineTests/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  RW5naW5lVGVzdHM=
│ │ │  #:len=311
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSB0ZXN0IHN1aXRlIGZvciB0aGUgTWFj
│ │ │  YXVsYXkyIGVuZ2luZSIsICJsaW5lbnVtIiA9PiA0NzksICJmaWxlbmFtZSIgPT4gIkVuZ2luZVRl
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  RW51bWVyYXRpb25DdXJ2ZXM=
│ │ │  #:len=747
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRW51bWVyYXRpb24gb2YgcmF0aW9uYWwg
│ │ │  Y3VydmVzIHZpYSB0b3J1cyBhY3Rpb25zIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7IiJ9
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1331975673177
│ │ │  
│ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.031399s (cpu); 0.0313974s (thread); 0s (gc)
│ │ │ + -- used 0.0229211s (cpu); 0.0229215s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out
│ │ │ @@ -37,83 +37,83 @@
│ │ │  i6 : rationalCurve(2) - rationalCurve(1)/8
│ │ │  
│ │ │  o6 = 609250
│ │ │  
│ │ │  o6 : QQ
│ │ │  
│ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.353246s (cpu); 0.284105s (thread); 0s (gc)
│ │ │ + -- used 0.309067s (cpu); 0.250172s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.23024s (cpu); 0.163316s (thread); 0s (gc)
│ │ │ + -- used 0.11173s (cpu); 0.111737s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 6.4737s (cpu); 5.50773s (thread); 0s (gc)
│ │ │ + -- used 4.32446s (cpu); 3.74181s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.232795s (cpu); 0.175565s (thread); 0s (gc)
│ │ │ + -- used 0.195826s (cpu); 0.137673s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 6.08562s (cpu); 5.19424s (thread); 0s (gc)
│ │ │ + -- used 4.47141s (cpu); 3.89414s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 1.85501s (cpu); 1.49923s (thread); 0s (gc)
│ │ │ + -- used 1.40822s (cpu); 1.2115s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 8.7345s (cpu); 6.35557s (thread); 0s (gc)
│ │ │ + -- used 6.16973s (cpu); 4.89866s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 2.27006s (cpu); 1.97354s (thread); 0s (gc)
│ │ │ + -- used 1.41272s (cpu); 1.21421s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 8.47703s (cpu); 6.3453s (thread); 0s (gc)
│ │ │ + -- used 6.16676s (cpu); 4.88578s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 9.22095s (cpu); 6.77006s (thread); 0s (gc)
│ │ │ + -- used 6.29049s (cpu); 5.05962s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
    Computes the number of lines on a general hypersurface of degree 2n - 3 in \mathbb P^n.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the number of lines on a general hypersurface of degree
│ │ │ │              2n - 3 in \mathbb P^n
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Computes the number of lines on a general hypersurface of degree 2n - 3 in
│ │ │ │  \mathbb P^n.
│ │ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ │ - -- used 0.031399s (cpu); 0.0313974s (thread); 0s (gc)
│ │ │ │ + -- used 0.0229211s (cpu); 0.0229215s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       289139638632755625, 520764738758073845321}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html
│ │ │ @@ -157,15 +157,15 @@
│ │ │            
    The numbers of conics on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.031399s (cpu); 0.0313974s (thread); 0s (gc)
│ │ │ + -- used 0.0229211s (cpu); 0.0229215s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ - -- used 0.353246s (cpu); 0.284105s (thread); 0s (gc)
│ │ │ + -- used 0.309067s (cpu); 0.250172s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ @@ -173,27 +173,27 @@
│ │ │            
    For rational curves of degree 3:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -203,15 +203,15 @@
│ │ │            
    The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i8 : time rationalCurve(3)
│ │ │ - -- used 0.23024s (cpu); 0.163316s (thread); 0s (gc)
│ │ │ + -- used 0.11173s (cpu); 0.111737s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    
│ │ │              
    
│ │ │                
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 6.4737s (cpu); 5.50773s (thread); 0s (gc)
│ │ │ + -- used 4.32446s (cpu); 3.74181s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.232795s (cpu); 0.175565s (thread); 0s (gc)
│ │ │ + -- used 0.195826s (cpu); 0.137673s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
    
│ │ │              
    
│ │ │ @@ -219,15 +219,15 @@
│ │ │            
    The numbers of rational curves of degree 3 on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 6.08562s (cpu); 5.19424s (thread); 0s (gc)
│ │ │ + -- used 4.47141s (cpu); 3.89414s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │ @@ -235,27 +235,27 @@
│ │ │            
    For rational curves of degree 4:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time rationalCurve(4)
│ │ │ - -- used 1.85501s (cpu); 1.49923s (thread); 0s (gc)
│ │ │ + -- used 1.40822s (cpu); 1.2115s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │              
    
│ │ │                
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 8.7345s (cpu); 6.35557s (thread); 0s (gc)
│ │ │ + -- used 6.16973s (cpu); 4.89866s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
    
│ │ │              
    
│ │ │ @@ -263,15 +263,15 @@
│ │ │            
    The number of rational curves of degree 4 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 2.27006s (cpu); 1.97354s (thread); 0s (gc)
│ │ │ + -- used 1.41272s (cpu); 1.21421s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
    
│ │ │              
    
│ │ │ @@ -279,25 +279,25 @@
│ │ │            
    The numbers of rational curves of degree 4 on general complete intersections of types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 8.47703s (cpu); 6.3453s (thread); 0s (gc)
│ │ │ + -- used 6.16676s (cpu); 4.88578s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    
│ │ │              
    
│ │ │                
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 9.22095s (cpu); 6.77006s (thread); 0s (gc)
│ │ │ + -- used 6.29049s (cpu); 5.05962s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,85 +59,85 @@
│ │ │ │  
│ │ │ │  o6 = 609250
│ │ │ │  
│ │ │ │  o6 : QQ
│ │ │ │  The numbers of conics on general complete intersection Calabi-Yau threefolds
│ │ │ │  can be computed as follows:
│ │ │ │  i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8
│ │ │ │ - -- used 0.353246s (cpu); 0.284105s (thread); 0s (gc)
│ │ │ │ + -- used 0.309067s (cpu); 0.250172s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  For rational curves of degree 3:
│ │ │ │  i8 : time rationalCurve(3)
│ │ │ │ - -- used 0.23024s (cpu); 0.163316s (thread); 0s (gc)
│ │ │ │ + -- used 0.11173s (cpu); 0.111737s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       8564575000
│ │ │ │  o8 = ----------
│ │ │ │           27
│ │ │ │  
│ │ │ │  o8 : QQ
│ │ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ │ - -- used 6.4737s (cpu); 5.50773s (thread); 0s (gc)
│ │ │ │ + -- used 4.32446s (cpu); 3.74181s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        8564575000  422690816           4834592  11239424
│ │ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │ │            27          27                 3        27
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  The number of rational curves of degree 3 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ │ - -- used 0.232795s (cpu); 0.175565s (thread); 0s (gc)
│ │ │ │ + -- used 0.195826s (cpu); 0.137673s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 317206375
│ │ │ │  
│ │ │ │  o10 : QQ
│ │ │ │  The numbers of rational curves of degree 3 on general complete intersection
│ │ │ │  Calabi-Yau threefolds can be computed as follows:
│ │ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ │ - -- used 6.08562s (cpu); 5.19424s (thread); 0s (gc)
│ │ │ │ + -- used 4.47141s (cpu); 3.89414s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  For rational curves of degree 4:
│ │ │ │  i12 : time rationalCurve(4)
│ │ │ │ - -- used 1.85501s (cpu); 1.49923s (thread); 0s (gc)
│ │ │ │ + -- used 1.40822s (cpu); 1.2115s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 8.7345s (cpu); 6.35557s (thread); 0s (gc)
│ │ │ │ + -- used 6.16973s (cpu); 4.89866s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = 3883914084
│ │ │ │  
│ │ │ │  o13 : QQ
│ │ │ │  The number of rational curves of degree 4 on a general quintic threefold can be
│ │ │ │  computed as follows:
│ │ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ │ - -- used 2.27006s (cpu); 1.97354s (thread); 0s (gc)
│ │ │ │ + -- used 1.41272s (cpu); 1.21421s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = 242467530000
│ │ │ │  
│ │ │ │  o14 : QQ
│ │ │ │  The numbers of rational curves of degree 4 on general complete intersections of
│ │ │ │  types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
│ │ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ │ - -- used 8.47703s (cpu); 6.3453s (thread); 0s (gc)
│ │ │ │ + -- used 6.16676s (cpu); 4.88578s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3883902528
│ │ │ │  
│ │ │ │  o15 : QQ
│ │ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ │ - -- used 9.22095s (cpu); 6.77006s (thread); 0s (gc)
│ │ │ │ + -- used 6.29049s (cpu); 5.05962s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 1139448384
│ │ │ │  
│ │ │ │  o16 : QQ
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_e_C_o_v_e_r -- Multiple coverings of rational curves on Calabi-Yau
│ │ │ │        threefolds
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  YnVpbGRFTW9ub21pYWxNYXAoUmluZyxSaW5nLExpc3Qp
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIzNiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYnVpbGRFTW9ub21pYWxNYXAsUmluZyxSaW5nLExp
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out
│ │ │ @@ -10,34 +10,34 @@
│ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │                    1   1 0   1 0   0
│ │ │  
│ │ │  o3 : RingMap R <-- S
│ │ │  
│ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .0022445 seconds
│ │ │ -     -- used .0006884 seconds
│ │ │ +     -- used .001823 seconds
│ │ │ +     -- used .000595522 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00370187 seconds
│ │ │ -     -- used .00494948 seconds
│ │ │ +     -- used .00341028 seconds
│ │ │ +     -- used .0045019 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00769783 seconds
│ │ │ -     -- used .118616 seconds
│ │ │ +     -- used .00801113 seconds
│ │ │ +     -- used .117262 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0180006 seconds
│ │ │ -     -- used .319378 seconds
│ │ │ +     -- used .0177469 seconds
│ │ │ +     -- used .289903 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .039338 seconds
│ │ │ -     -- used 1.12239 seconds
│ │ │ +     -- used .0370655 seconds
│ │ │ +     -- used .993025 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html
│ │ │ @@ -106,34 +106,34 @@
│ │ │  o3 : RingMap R <-- S
    
│ │ │              
    
│ │ │                
    i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .0022445 seconds
│ │ │ -     -- used .0006884 seconds
│ │ │ +     -- used .001823 seconds
│ │ │ +     -- used .000595522 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00370187 seconds
│ │ │ -     -- used .00494948 seconds
│ │ │ +     -- used .00341028 seconds
│ │ │ +     -- used .0045019 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00769783 seconds
│ │ │ -     -- used .118616 seconds
│ │ │ +     -- used .00801113 seconds
│ │ │ +     -- used .117262 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0180006 seconds
│ │ │ -     -- used .319378 seconds
│ │ │ +     -- used .0177469 seconds
│ │ │ +     -- used .289903 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .039338 seconds
│ │ │ -     -- used 1.12239 seconds
│ │ │ +     -- used .0370655 seconds
│ │ │ +     -- used .993025 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,34 +33,34 @@
│ │ │ │                    2               2
│ │ │ │  o3 = map (R, S, {x , x x , x x , x })
│ │ │ │                    1   1 0   1 0   0
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- S
│ │ │ │  i4 : G = egbToric(m, OutFile=>stdio)
│ │ │ │  3
│ │ │ │ -     -- used .0022445 seconds
│ │ │ │ -     -- used .0006884 seconds
│ │ │ │ +     -- used .001823 seconds
│ │ │ │ +     -- used .000595522 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .00370187 seconds
│ │ │ │ -     -- used .00494948 seconds
│ │ │ │ +     -- used .00341028 seconds
│ │ │ │ +     -- used .0045019 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00769783 seconds
│ │ │ │ -     -- used .118616 seconds
│ │ │ │ +     -- used .00801113 seconds
│ │ │ │ +     -- used .117262 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0180006 seconds
│ │ │ │ -     -- used .319378 seconds
│ │ │ │ +     -- used .0177469 seconds
│ │ │ │ +     -- used .289903 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .039338 seconds
│ │ │ │ -     -- used 1.12239 seconds
│ │ │ │ +     -- used .0370655 seconds
│ │ │ │ +     -- used .993025 seconds
│ │ │ │  (49, 217)
│ │ │ │  
│ │ │ │                                     2
│ │ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/EuclideanDistanceDegree/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=40
│ │ │  bnVtZXJpY1dlaWdodEVERGVncmVlKC4uLixTdWJtb2RlbD0+Li4uKQ==
│ │ │  #:len=341
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDkxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tudW1lcmljV2VpZ2h0RUREZWdyZWUsU3VibW9kZWxd
│ │ ├── ./usr/share/doc/Macaulay2/ExampleSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  aGVhcnQ=
│ │ │  #:len=1516
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gOC1kaW1lbnNpb25hbCBlY29ub21p
│ │ │  Y3MgcHJvYmxlbSIsICJsaW5lbnVtIiA9PiA0OCwgSW5wdXRzID0+IHtTUEFOe1RUeyJrayJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/ExteriorExtensions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  YmFzZXM=
│ │ │  #:len=602
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2l2ZXMgYSBsaXN0IG9mIHRoZSBiYXNl
│ │ │  cyBvZiB0aGUgZGlmZmVyZW50IGdyYWRlZCBwaWVjZXMgb2YgdGhlIEV4dGVyaW9yRXh0ZW5zaW9u
│ │ ├── ./usr/share/doc/Macaulay2/ExteriorIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  c29sdmVNYWNhdWxheUV4cGFuc2lvbihMaXN0KQ==
│ │ │  #:len=301
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDMwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzb2x2ZU1hY2F1bGF5RXhwYW5zaW9uLExpc3QpLCJz
│ │ ├── ./usr/share/doc/Macaulay2/ExteriorModules/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  YWxtb3N0U3RhYmxlTW9kdWxlKE1vZHVsZSk=
│ │ │  #:len=292
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbG1vc3RTdGFibGVNb2R1bGUsTW9kdWxlKSwiYWxt
│ │ ├── ./usr/share/doc/Macaulay2/FGLM/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  ZmdsbShJZGVhbCxSaW5nKQ==
│ │ │  #:len=211
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzk2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmZ2xtLElkZWFsLFJpbmcpLCJmZ2xtKElkZWFsLFJp
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  cmVjdXJzaXZlTWlub3JzKFpaLE1hdHJpeCk=
│ │ │  #:len=272
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA3NCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVjdXJzaXZlTWlub3JzLFpaLE1hdHJpeCksInJl
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out
│ │ │ @@ -414,50 +414,50 @@
│ │ │                 3 2 4     3 6
│ │ │  o27 = ideal(12x x x  - 4x x )
│ │ │                 3 7 9     3 9
│ │ │  
│ │ │  o27 : Ideal of S
│ │ │  
│ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.0399774s (cpu); 0.0434466s (thread); 0s (gc)
│ │ │ + -- used 0.0520263s (cpu); 0.0480562s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.221823s (cpu); 0.174555s (thread); 0s (gc)
│ │ │ + -- used 0.233507s (cpu); 0.16315s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.358134s (cpu); 0.30248s (thread); 0s (gc)
│ │ │ + -- used 0.340606s (cpu); 0.270316s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.169855s (cpu); 0.172074s (thread); 0s (gc)
│ │ │ + -- used 0.165225s (cpu); 0.166123s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.2399s (cpu); 0.172785s (thread); 0s (gc)
│ │ │ + -- used 0.254745s (cpu); 0.167261s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.286617s (cpu); 0.214112s (thread); 0s (gc)
│ │ │ + -- used 0.265883s (cpu); 0.198058s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.135747s (cpu); 0.13629s (thread); 0s (gc)
│ │ │ + -- used 0.117473s (cpu); 0.120261s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 12.1654s (cpu); 9.70719s (thread); 0s (gc)
│ │ │ + -- used 11.7235s (cpu); 9.06138s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
│ │ │  
│ │ │  i36 : peek StrategyDefault
│ │ │  
│ │ │  o36 = OptionTable{GRevLexLargest => 0      }
│ │ │                    GRevLexSmallest => 16
│ │ │ @@ -488,15 +488,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.415617s (cpu); 0.354159s (thread); 0s (gc)
│ │ │ + -- used 0.397956s (cpu); 0.330573s (thread); 0s (gc)
│ │ │  chooseGoodMinors: found =20, attempted = 22
│ │ │  
│ │ │  o37 : Ideal of S
│ │ │  
│ │ │  i38 : peek StrategyDefaultNonRandom
│ │ │  
│ │ │  o38 = OptionTable{GRevLexLargest => 0      }
│ │ │ @@ -536,15 +536,15 @@
│ │ │  i41 : ptsStratGeometric = new OptionTable from (options chooseGoodMinors)#PointOptions;
│ │ │  
│ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │  
│ │ │  o42 = true
│ │ │  
│ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.731279s (cpu); 0.584895s (thread); 0s (gc)
│ │ │ + -- used 0.723933s (cpu); 0.577184s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
│ │ │  
│ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │  
│ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │                    DimensionFunction => dim
│ │ │ @@ -559,49 +559,49 @@
│ │ │  o44 : OptionTable
│ │ │  
│ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │  
│ │ │  o45 = false
│ │ │  
│ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.628132s (cpu); 0.473813s (thread); 0s (gc)
│ │ │ + -- used 0.672201s (cpu); 0.502326s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 2.16725s (cpu); 1.89894s (thread); 0s (gc)
│ │ │ + -- used 2.22179s (cpu); 1.91452s (thread); 0s (gc)
│ │ │  
│ │ │  o47 = true
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.2841s (cpu); 1.07067s (thread); 0s (gc)
│ │ │ + -- used 1.23396s (cpu); 1.01536s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.82661s (cpu); 2.5335s (thread); 0s (gc)
│ │ │ + -- used 2.84798s (cpu); 2.66102s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.82185s (cpu); 2.12806s (thread); 0s (gc)
│ │ │ + -- used 2.54176s (cpu); 1.97367s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.282679s (cpu); 0.282995s (thread); 0s (gc)
│ │ │ + -- used 0.285139s (cpu); 0.288722s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.96165s (cpu); 2.31539s (thread); 0s (gc)
│ │ │ + -- used 2.93292s (cpu); 2.21578s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.46682s (cpu); 2.82813s (thread); 0s (gc)
│ │ │ + -- used 3.27032s (cpu); 2.66421s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 63.8248s (cpu); 51.3047s (thread); 0s (gc)
│ │ │ + -- used 51.6339s (cpu); 43.9397s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 2.98865s (cpu); 2.35025s (thread); 0s (gc)
│ │ │ + -- used 2.89079s (cpu); 2.22088s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
│ │ │  
│ │ │  i56 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  o2 : Ideal of S
│ │ │  
│ │ │  i3 : dim (S/J)
│ │ │  
│ │ │  o3 = 4
│ │ │  
│ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 3.94924s (cpu); 2.5242s (thread); 0s (gc)
│ │ │ + -- used 3.10506s (cpu); 1.82235s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 14.4632s (cpu); 9.06112s (thread); 0s (gc)
│ │ │ + -- used 12.865s (cpu); 7.84076s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time regularInCodimension(1, S/J, Verbose=>true)
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -127,15 +127,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 3.66634s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ + -- used 3.58253s (cpu); 2.06505s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 452.908 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -175,15 +175,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 0.31421s (cpu); 0.194894s (thread); 0s (gc)
│ │ │ + -- used 0.266814s (cpu); 0.143501s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -197,15 +197,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.265969s (cpu); 0.170823s (thread); 0s (gc)
│ │ │ + -- used 0.267699s (cpu); 0.150712s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -219,15 +219,15 @@
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.436288s (cpu); 0.287639s (thread); 0s (gc)
│ │ │ + -- used 0.441888s (cpu); 0.245203s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 6, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -262,15 +262,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 1.22231s (cpu); 0.809378s (thread); 0s (gc)
│ │ │ + -- used 1.37352s (cpu); 0.833362s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 5, and computed = 5
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -313,15 +313,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.07283s (cpu); 0.688614s (thread); 0s (gc)
│ │ │ + -- used 0.970902s (cpu); 0.562956s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 15, and computed = 14
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5509279875405941999
│ │ │  
│ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │  
│ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 4.1651s elapsed
│ │ │ + -- 2.87644s elapsed
│ │ │  
│ │ │  o2 = true
│ │ │  
│ │ │  i3 : peek StrategyDefault
│ │ │  
│ │ │  o3 = OptionTable{GRevLexLargest => 0      }
│ │ │                   GRevLexSmallest => 16
│ │ │ @@ -16,12 +16,12 @@
│ │ │                   LexSmallest => 16
│ │ │                   LexSmallestTerm => 16
│ │ │                   Points => 0
│ │ │                   Random => 16
│ │ │                   RandomNonzero => 16
│ │ │  
│ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.67457s elapsed
│ │ │ + -- 1.17576s elapsed
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out
│ │ │ @@ -16,29 +16,29 @@
│ │ │  i5 : r = rank myDiff;
│ │ │  
│ │ │  i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom);
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.000947066s (cpu); 0.00263955s (thread); 0s (gc)
│ │ │ + -- used 0.000282021s (cpu); 0.00256493s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : I = ideal(x_2^8*x_10^3-3*x_1*x_2^7*x_10^2*x_11+3*x_1^2*x_2^6*x_10*x_11^2-x_1^3*x_2^5*x_11^3,x_5^5*x_6^3*x_11^3-3*x_5^6*x_6^2*x_11^2*x_12+3*x_5^7*x_6*x_11*x_12^2-x_5^8*x_12^3,x_1^5*x_2^3*x_4^3-3*x_1^6*x_2^2*x_4^2*x_5+3*x_1^7*x_2*x_4*x_5^2-x_1^8*x_5^3,x_6^8*x_11^3-3*x_5*x_6^7*x_11^2*x_12+3*x_5^2*x_6^6*x_11*x_12^2-x_5^3*x_6^5*x_12^3,x_8^3*x_10^8-3*x_7*x_8^2*x_10^7*x_11+3*x_7^2*x_8*x_10^6*x_11^2-x_7^3*x_10^5*x_11^3,x_2^8*x_4^3-3*x_1*x_2^7*x_4^2*x_5+3*x_1^2*x_2^6*x_4*x_5^2-x_1^3*x_2^5*x_5^3,-x_6^3*x_11^8+3*x_5*x_6^2*x_11^7*x_12-3*x_5^2*x_6*x_11^6*x_12^2+x_5^3*x_11^5*x_12^3,-x_6^3*x_7^3*x_9^5+3*x_4*x_6^2*x_7^2*x_9^6-3*x_4^2*x_6*x_7*x_9^7+x_4^3*x_9^8,x_8^8*x_10^3-3*x_7*x_8^7*x_10^2*x_11+3*x_7^2*x_8^6*x_10*x_11^2-x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);
│ │ │  
│ │ │                 ZZ
│ │ │  o8 : Ideal of ---[x  , x , x , x , x  , x , x , x  , x , x , x , x ]
│ │ │                127  11   8   1   9   12   6   5   10   2   4   3   7
│ │ │  
│ │ │  i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.00400074s (cpu); 0.00251447s (thread); 0s (gc)
│ │ │ + -- used 0.00076956s (cpu); 0.0024593s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.5373e-05s (cpu); 0.0022963s (thread); 0s (gc)
│ │ │ + -- used 0.000839978s (cpu); 0.00238827s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out
│ │ │ @@ -7,17 +7,17 @@
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : pdim(module I)
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.296971s (cpu); 0.159933s (thread); 0s (gc)
│ │ │ + -- used 0.312724s (cpu); 0.1765s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00988389s (cpu); 0.0120097s (thread); 0s (gc)
│ │ │ + -- used 0.00884699s (cpu); 0.0117094s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │  
│ │ │               6      7
│ │ │  o2 : Matrix R  <-- R
│ │ │  
│ │ │  i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.555663s (cpu); 0.481264s (thread); 0s (gc)
│ │ │ + -- used 0.459231s (cpu); 0.407503s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.57748s (cpu); 1.37879s (thread); 0s (gc)
│ │ │ + -- used 1.15613s (cpu); 1.04316s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : I1 == I2
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out
│ │ │ @@ -19,44 +19,44 @@
│ │ │  i6 : S = T/I;
│ │ │  
│ │ │  i7 : dim S
│ │ │  
│ │ │  o7 = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.921914s (cpu); 0.617916s (thread); 0s (gc)
│ │ │ + -- used 0.994644s (cpu); 0.583465s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 10.1044s (cpu); 6.75793s (thread); 0s (gc)
│ │ │ + -- used 8.06537s (cpu); 5.03693s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : R = QQ[c, f, g, h]/ideal(g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
│ │ │  
│ │ │  i11 : dim(R)
│ │ │  
│ │ │  o11 = 2
│ │ │  
│ │ │  i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0279591s (cpu); 0.0261627s (thread); 0s (gc)
│ │ │ + -- used 0.0160178s (cpu); 0.0165786s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.559038s (cpu); 0.316642s (thread); 0s (gc)
│ │ │ + -- used 0.487301s (cpu); 0.26534s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.441343s (cpu); 0.303101s (thread); 0s (gc)
│ │ │ + -- used 0.308915s (cpu); 0.200732s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.424787s (cpu); 0.217971s (thread); 0s (gc)
│ │ │ + -- used 0.362009s (cpu); 0.188854s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ @@ -381,15 +381,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 9.54962s (cpu); 6.2698s (thread); 0s (gc)
│ │ │ + -- used 8.40753s (cpu); 5.29969s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 327.599 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -462,15 +462,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.69942s (cpu); 1.13047s (thread); 0s (gc)
│ │ │ + -- used 1.5566s (cpu); 1.0524s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -486,68 +486,68 @@
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 30, and computed = 25
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 30, and computed = 25.  singular locus dimension appears to be = 1
│ │ │  
│ │ │  i18 : time regularInCodimension(2, S, VerifyNonRegular=>true)
│ │ │ - -- used 1.63296s (cpu); 0.972645s (thread); 0s (gc)
│ │ │ + -- used 1.53931s (cpu); 0.909701s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = false
│ │ │  
│ │ │  i19 : StrategyCurrent#Random = 0;
│ │ │  
│ │ │  i20 : StrategyCurrent#LexSmallest = 100;
│ │ │  
│ │ │  i21 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.597363s (cpu); 0.322568s (thread); 0s (gc)
│ │ │ + -- used 0.547244s (cpu); 0.301736s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.706068s (cpu); 0.366625s (thread); 0s (gc)
│ │ │ + -- used 0.62286s (cpu); 0.29794s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.49826s (cpu); 0.35781s (thread); 0s (gc)
│ │ │ + -- used 0.385356s (cpu); 0.255903s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.532771s (cpu); 0.31747s (thread); 0s (gc)
│ │ │ + -- used 0.44728s (cpu); 0.252185s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = true
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i27 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.58881s (cpu); 2.13991s (thread); 0s (gc)
│ │ │ + -- used 3.06187s (cpu); 1.72571s (thread); 0s (gc)
│ │ │  
│ │ │  i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.21572s (cpu); 1.86136s (thread); 0s (gc)
│ │ │ + -- used 2.96033s (cpu); 1.67439s (thread); 0s (gc)
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.328318s (cpu); 0.225783s (thread); 0s (gc)
│ │ │ + -- used 0.314649s (cpu); 0.19483s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.566594s (cpu); 0.371181s (thread); 0s (gc)
│ │ │ + -- used 0.478476s (cpu); 0.31264s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 2.2427s (cpu); 1.62597s (thread); 0s (gc)
│ │ │ + -- used 1.77716s (cpu); 1.23856s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
│ │ │  
│ │ │  i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.78805s (cpu); 1.22068s (thread); 0s (gc)
│ │ │ + -- used 1.51546s (cpu); 1.04747s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = true
│ │ │  
│ │ │  i34 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
│ │ │ @@ -577,71 +577,71 @@
│ │ │          
    
│ │ │            
    Here the $1$ passed to the function says how many minors to compute.  For instance, let's compute 8 minors for each of these strategies and see if that was enough to verify that the ring is regular in codimension 1.  In other words, if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/J$ has dimension 3).
    
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│ │ │          
    
│ │ │                
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.0399774s (cpu); 0.0434466s (thread); 0s (gc)
│ │ │ + -- used 0.0520263s (cpu); 0.0480562s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    
│ │ │              
    
│ │ │                
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.221823s (cpu); 0.174555s (thread); 0s (gc)
│ │ │ + -- used 0.233507s (cpu); 0.16315s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    
│ │ │              
    
│ │ │                
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.358134s (cpu); 0.30248s (thread); 0s (gc)
│ │ │ + -- used 0.340606s (cpu); 0.270316s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    
│ │ │              
    
│ │ │                
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.169855s (cpu); 0.172074s (thread); 0s (gc)
│ │ │ + -- used 0.165225s (cpu); 0.166123s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    
│ │ │              
    
│ │ │                
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.2399s (cpu); 0.172785s (thread); 0s (gc)
│ │ │ + -- used 0.254745s (cpu); 0.167261s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    
│ │ │              
    
│ │ │                
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.286617s (cpu); 0.214112s (thread); 0s (gc)
│ │ │ + -- used 0.265883s (cpu); 0.198058s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    
│ │ │              
    
│ │ │                
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.135747s (cpu); 0.13629s (thread); 0s (gc)
│ │ │ + -- used 0.117473s (cpu); 0.120261s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    
│ │ │              
    
│ │ │                
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 12.1654s (cpu); 9.70719s (thread); 0s (gc)
│ │ │ + -- used 11.7235s (cpu); 9.06138s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Indeed, in this example, even computing determinants of 1,000 random submatrices is not typically enough to verify that $V(J)$ is regular in codimension 1.  On the other hand, Points is almost always quite effective at finding valuable submatrices, but can be quite slow.  In this particular example, we can see that LexSmallestTerm also performs very well (and does it quickly). Since different strategies work better or worse on different examples, the default strategy actually mixes and matches various strategies.  The default strategy, which we now elucidate,
    
│ │ │ @@ -688,15 +688,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.415617s (cpu); 0.354159s (thread); 0s (gc)
│ │ │ + -- used 0.397956s (cpu); 0.330573s (thread); 0s (gc)
│ │ │  chooseGoodMinors: found =20, attempted = 22
│ │ │  
│ │ │  o37 : Ideal of S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -779,15 +779,15 @@
│ │ │  
│ │ │  o42 = true
    
│ │ │              
    
│ │ │                
    i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.731279s (cpu); 0.584895s (thread); 0s (gc)
│ │ │ + -- used 0.723933s (cpu); 0.577184s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
    
│ │ │              
    
│ │ │                
    i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │ @@ -811,15 +811,15 @@
│ │ │  
│ │ │  o45 = false
    
│ │ │              
    
│ │ │                
    i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.628132s (cpu); 0.473813s (thread); 0s (gc)
│ │ │ + -- used 0.672201s (cpu); 0.502326s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Other options may also be passed to the RandomPoints package via the PointOptions option.
    
│ │ │ @@ -827,71 +827,71 @@
│ │ │          
    
│ │ │            
    regularInCodimension:  It is reasonable to think that you should find a few minors (with one strategy or another), and see if perhaps the minors you have computed so far are enough to verify our ring is regular in codimension 1.  This is exactly what regularInCodimension does.  One can control at a fine level how frequently new minors are computed, and how frequently the dimension of what we have computed so far is checked, by the option codimCheckFunction.  For more on that, see RegularInCodimensionTutorial and regularInCodimension.  Let us finish running regularInCodimension on our example with several different strategies.
    
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│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 2.16725s (cpu); 1.89894s (thread); 0s (gc)
│ │ │ + -- used 2.22179s (cpu); 1.91452s (thread); 0s (gc)
│ │ │  
│ │ │  o47 = true
    
│ │ │              
    
│ │ │                
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.2841s (cpu); 1.07067s (thread); 0s (gc)
│ │ │ + -- used 1.23396s (cpu); 1.01536s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    
│ │ │              
    
│ │ │                
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.82661s (cpu); 2.5335s (thread); 0s (gc)
    
│ │ │ + -- used 2.84798s (cpu); 2.66102s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.82185s (cpu); 2.12806s (thread); 0s (gc)
    
│ │ │ + -- used 2.54176s (cpu); 1.97367s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.282679s (cpu); 0.282995s (thread); 0s (gc)
│ │ │ + -- used 0.285139s (cpu); 0.288722s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    
│ │ │              
    
│ │ │                
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.96165s (cpu); 2.31539s (thread); 0s (gc)
    
│ │ │ + -- used 2.93292s (cpu); 2.21578s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.46682s (cpu); 2.82813s (thread); 0s (gc)
    
│ │ │ + -- used 3.27032s (cpu); 2.66421s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 63.8248s (cpu); 51.3047s (thread); 0s (gc)
│ │ │ + -- used 51.6339s (cpu); 43.9397s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    
│ │ │              
    
│ │ │                
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 2.98865s (cpu); 2.35025s (thread); 0s (gc)
│ │ │ + -- used 2.89079s (cpu); 2.22088s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If regularInCodimension outputs nothing, then it couldn't verify that the ring was regular in that codimension.  We set MaxMinors => 100 to keep it from running too long with an ineffective strategy.  Again, even though GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular example, in others they perform better than other strategies.  Note similar considerations also apply to projDim.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -438,44 +438,44 @@
│ │ │ │  o27 : Ideal of S
│ │ │ │  Here the $1$ passed to the function says how many minors to compute. For
│ │ │ │  instance, let's compute 8 minors for each of these strategies and see if that
│ │ │ │  was enough to verify that the ring is regular in codimension 1. In other words,
│ │ │ │  if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/
│ │ │ │  J$ has dimension 3).
│ │ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ │ - -- used 0.0399774s (cpu); 0.0434466s (thread); 0s (gc)
│ │ │ │ + -- used 0.0520263s (cpu); 0.0480562s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 2
│ │ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ │ - -- used 0.221823s (cpu); 0.174555s (thread); 0s (gc)
│ │ │ │ + -- used 0.233507s (cpu); 0.16315s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = 3
│ │ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ │ - -- used 0.358134s (cpu); 0.30248s (thread); 0s (gc)
│ │ │ │ + -- used 0.340606s (cpu); 0.270316s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 1
│ │ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ │ - -- used 0.169855s (cpu); 0.172074s (thread); 0s (gc)
│ │ │ │ + -- used 0.165225s (cpu); 0.166123s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 2
│ │ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ │ - -- used 0.2399s (cpu); 0.172785s (thread); 0s (gc)
│ │ │ │ + -- used 0.254745s (cpu); 0.167261s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = 3
│ │ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J,
│ │ │ │  Strategy=>GRevLexSmallestTerm))
│ │ │ │ - -- used 0.286617s (cpu); 0.214112s (thread); 0s (gc)
│ │ │ │ + -- used 0.265883s (cpu); 0.198058s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = 3
│ │ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ │ - -- used 0.135747s (cpu); 0.13629s (thread); 0s (gc)
│ │ │ │ + -- used 0.117473s (cpu); 0.120261s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o34 = 3
│ │ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ │ - -- used 12.1654s (cpu); 9.70719s (thread); 0s (gc)
│ │ │ │ + -- used 11.7235s (cpu); 9.06138s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o35 = 1
│ │ │ │  Indeed, in this example, even computing determinants of 1,000 random
│ │ │ │  submatrices is not typically enough to verify that $V(J)$ is regular in
│ │ │ │  codimension 1. On the other hand, Points is almost always quite effective at
│ │ │ │  finding valuable submatrices, but can be quite slow. In this particular
│ │ │ │  example, we can see that LexSmallestTerm also performs very well (and does it
│ │ │ │ @@ -518,15 +518,15 @@
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │ - -- used 0.415617s (cpu); 0.354159s (thread); 0s (gc)
│ │ │ │ + -- used 0.397956s (cpu); 0.330573s (thread); 0s (gc)
│ │ │ │  chooseGoodMinors: found =20, attempted = 22
│ │ │ │  
│ │ │ │  o37 : Ideal of S
│ │ │ │  we can see different minors being chosen via different strategies.
│ │ │ │  Note, if one asks chooseGoodMinors for more than one minor, then any time a
│ │ │ │  Points strategy is selected, the point is found on $J$ plus the ideal of all
│ │ │ │  minors computed thus far.
│ │ │ │ @@ -587,15 +587,15 @@
│ │ │ │  i41 : ptsStratGeometric = new OptionTable from (options
│ │ │ │  chooseGoodMinors)#PointOptions;
│ │ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │ │  
│ │ │ │  o42 = true
│ │ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratGeometric))
│ │ │ │ - -- used 0.731279s (cpu); 0.584895s (thread); 0s (gc)
│ │ │ │ + -- used 0.723933s (cpu); 0.577184s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o43 = 2
│ │ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that
│ │ │ │  value
│ │ │ │  
│ │ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │ │                    DimensionFunction => dim
│ │ │ │ @@ -609,60 +609,60 @@
│ │ │ │  
│ │ │ │  o44 : OptionTable
│ │ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │ │  
│ │ │ │  o45 = false
│ │ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratRational))
│ │ │ │ - -- used 0.628132s (cpu); 0.473813s (thread); 0s (gc)
│ │ │ │ + -- used 0.672201s (cpu); 0.502326s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = 2
│ │ │ │  Other options may also be passed to the _R_a_n_d_o_m_P_o_i_n_t_s package via the
│ │ │ │  _P_o_i_n_t_O_p_t_i_o_n_s option.
│ │ │ │  rreegguullaarrIInnCCooddiimmeennssiioonn:: It is reasonable to think that you should find a few
│ │ │ │  minors (with one strategy or another), and see if perhaps the minors you have
│ │ │ │  computed so far are enough to verify our ring is regular in codimension 1. This
│ │ │ │  is exactly what regularInCodimension does. One can control at a fine level how
│ │ │ │  frequently new minors are computed, and how frequently the dimension of what we
│ │ │ │  have computed so far is checked, by the option codimCheckFunction. For more on
│ │ │ │  that, see _R_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n_T_u_t_o_r_i_a_l and _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n. Let us finish
│ │ │ │  running regularInCodimension on our example with several different strategies.
│ │ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefault)
│ │ │ │ - -- used 2.16725s (cpu); 1.89894s (thread); 0s (gc)
│ │ │ │ + -- used 2.22179s (cpu); 1.91452s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o47 = true
│ │ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultNonRandom)
│ │ │ │ - -- used 1.2841s (cpu); 1.07067s (thread); 0s (gc)
│ │ │ │ + -- used 1.23396s (cpu); 1.01536s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o48 = true
│ │ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ │ - -- used 2.82661s (cpu); 2.5335s (thread); 0s (gc)
│ │ │ │ + -- used 2.84798s (cpu); 2.66102s (thread); 0s (gc)
│ │ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallest)
│ │ │ │ - -- used 2.82185s (cpu); 2.12806s (thread); 0s (gc)
│ │ │ │ + -- used 2.54176s (cpu); 1.97367s (thread); 0s (gc)
│ │ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallestTerm)
│ │ │ │ - -- used 0.282679s (cpu); 0.282995s (thread); 0s (gc)
│ │ │ │ + -- used 0.285139s (cpu); 0.288722s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = true
│ │ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallest)
│ │ │ │ - -- used 2.96165s (cpu); 2.31539s (thread); 0s (gc)
│ │ │ │ + -- used 2.93292s (cpu); 2.21578s (thread); 0s (gc)
│ │ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallestTerm)
│ │ │ │ - -- used 3.46682s (cpu); 2.82813s (thread); 0s (gc)
│ │ │ │ + -- used 3.27032s (cpu); 2.66421s (thread); 0s (gc)
│ │ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ │ - -- used 63.8248s (cpu); 51.3047s (thread); 0s (gc)
│ │ │ │ + -- used 51.6339s (cpu); 43.9397s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o54 = true
│ │ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultWithPoints)
│ │ │ │ - -- used 2.98865s (cpu); 2.35025s (thread); 0s (gc)
│ │ │ │ + -- used 2.89079s (cpu); 2.22088s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o55 = true
│ │ │ │  If regularInCodimension outputs nothing, then it couldn't verify that the ring
│ │ │ │  was regular in that codimension. We set MaxMinors => 100 to keep it from
│ │ │ │  running too long with an ineffective strategy. Again, even though
│ │ │ │  GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular
│ │ │ │  example, in others they perform better than other strategies. Note similar
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html
│ │ │ @@ -86,23 +86,23 @@
│ │ │          
    
│ │ │            
    It is the cone over $P^2 \times E$ where $E$ is an elliptic curve.  We have embedded it with a Segre embedding inside $P^8$.  In particular, this example is even regular in codimension 3.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 3.94924s (cpu); 2.5242s (thread); 0s (gc)
│ │ │ + -- used 3.10506s (cpu); 1.82235s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │                
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 14.4632s (cpu); 9.06112s (thread); 0s (gc)
│ │ │ + -- used 12.865s (cpu); 7.84076s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the ideal made up of a small number of minors of the Jacobian matrix. In this example, instead of computing all relevant 1465128 minors to compute the singular locus, and then trying to compute the dimension of the ideal they generate, we instead compute a few of them.  regularInCodimension returns true if it verified that the ring is regular in codim 1 or 2 (respectively) and null if not.  Because of the randomness that exists in terms of selecting minors, the execution time can actually vary quite a bit.   Let's take a look at what is occurring by using the Verbose option.  We go through the output and explain what each line is telling us.
    
│ │ │ @@ -217,15 +217,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 3.66634s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ + -- used 3.58253s (cpu); 2.06505s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 452.908 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -273,15 +273,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 0.31421s (cpu); 0.194894s (thread); 0s (gc)
│ │ │ + -- used 0.266814s (cpu); 0.143501s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -306,15 +306,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.265969s (cpu); 0.170823s (thread); 0s (gc)
│ │ │ + -- used 0.267699s (cpu); 0.150712s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -339,15 +339,15 @@
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.436288s (cpu); 0.287639s (thread); 0s (gc)
│ │ │ + -- used 0.441888s (cpu); 0.245203s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 10 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 6, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -393,15 +393,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 1.22231s (cpu); 0.809378s (thread); 0s (gc)
│ │ │ + -- used 1.37352s (cpu); 0.833362s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 2, and computed = 2
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 5, and computed = 5
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -452,15 +452,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.07283s (cpu); 0.688614s (thread); 0s (gc)
│ │ │ + -- used 0.970902s (cpu); 0.562956s (thread); 0s (gc)
│ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5 minors, we will compute up to 25 of them.
│ │ │  regularInCodimension: About to enter loop
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 15, and computed = 14
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  i3 : dim (S/J)
│ │ │ │  
│ │ │ │  o3 = 4
│ │ │ │  It is the cone over $P^2 \times E$ where $E$ is an elliptic curve. We have
│ │ │ │  embedded it with a Segre embedding inside $P^8$. In particular, this example is
│ │ │ │  even regular in codimension 3.
│ │ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ │ - -- used 3.94924s (cpu); 2.5242s (thread); 0s (gc)
│ │ │ │ + -- used 3.10506s (cpu); 1.82235s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 14.4632s (cpu); 9.06112s (thread); 0s (gc)
│ │ │ │ + -- used 12.865s (cpu); 7.84076s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the
│ │ │ │  ideal made up of a small number of minors of the Jacobian matrix. In this
│ │ │ │  example, instead of computing all relevant 1465128 minors to compute the
│ │ │ │  singular locus, and then trying to compute the dimension of the ideal they
│ │ │ │  generate, we instead compute a few of them. regularInCodimension returns true
│ │ │ │ @@ -152,15 +152,15 @@
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │ - -- used 3.66634s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ │ + -- used 3.58253s (cpu); 2.06505s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 452.908 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 6
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -213,15 +213,15 @@
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │ - -- used 0.31421s (cpu); 0.194894s (thread); 0s (gc)
│ │ │ │ + -- used 0.266814s (cpu); 0.143501s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 7
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -244,15 +244,15 @@
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │ - -- used 0.265969s (cpu); 0.170823s (thread); 0s (gc)
│ │ │ │ + -- used 0.267699s (cpu); 0.150712s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 7
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -282,15 +282,15 @@
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │ - -- used 0.436288s (cpu); 0.287639s (thread); 0s (gc)
│ │ │ │ + -- used 0.441888s (cpu); 0.245203s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 10 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 3, and computed = 3
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -343,15 +343,15 @@
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │ - -- used 1.22231s (cpu); 0.809378s (thread); 0s (gc)
│ │ │ │ + -- used 1.37352s (cpu); 0.833362s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 2, and computed = 2
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ │ @@ -410,15 +410,15 @@
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │ - -- used 1.07283s (cpu); 0.688614s (thread); 0s (gc)
│ │ │ │ + -- used 0.970902s (cpu); 0.562956s (thread); 0s (gc)
│ │ │ │  regularInCodimension: ring dimension =4, there are 1465128 possible 5 by 5
│ │ │ │  minors, we will compute up to 25 of them.
│ │ │ │  regularInCodimension: About to enter loop
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 7
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │              
    		
│ │ │                
    i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 4.1651s elapsed
│ │ │ + -- 2.87644s elapsed
│ │ │  
│ │ │  o2 = true
    
│ │ │              
    
│ │ │  In this particular example, on one machine, we list average time to completion of each of the above strategies after 100 runs.        
      
│ │ │            
    • StrategyDefault: 1.65 seconds
      • 
│ │ │ @@ -127,15 +127,15 @@
│ │ │            
    • StrategyPoints: choose all submatrices via Points.
      • 
│ │ │            
    • StrategyDefaultWithPoints: like StrategyDefault but replaces the Random and RandomNonZero submatrices as with matrices chosen as in Points.
      • 
│ │ │          
    
│ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported.  It begins as the default strategy, but the user can modify it.

    Using a single heuristic  Alternatively, if the user only wants to use say LexSmallestTerm they can set, Strategy to point to that symbol, instead of a creating a custom strategy HashTable.  For example:         
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.67457s elapsed
│ │ │ + -- 1.17576s elapsed
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-
│ │ │ │  b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-
│ │ │ │  e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-
│ │ │ │  a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-
│ │ │ │  g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-
│ │ │ │  h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
│ │ │ │  i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ │ - -- 4.1651s elapsed
│ │ │ │ + -- 2.87644s elapsed
│ │ │ │  
│ │ │ │  o2 = true
│ │ │ │  In this particular example, on one machine, we list average time to completion
│ │ │ │  of each of the above strategies after 100 runs.
│ │ │ │      * StrategyDefault: 1.65 seconds
│ │ │ │      * StrategyRandom: 8.32 seconds
│ │ │ │      * StrategyDefaultNonRandom: 0.99 seconds
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported. It
│ │ │ │  begins as the default strategy, but the user can modify it.
│ │ │ │  
│ │ │ │  UUssiinngg aa ssiinnggllee hheeuurriissttiicc Alternatively, if the user only wants to use say
│ │ │ │  LexSmallestTerm they can set, Strategy to point to that symbol, instead of a
│ │ │ │  creating a custom strategy HashTable. For example:
│ │ │ │  i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ │ - -- 1.67457s elapsed
│ │ │ │ + -- 1.17576s elapsed
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _S_t_r_a_t_e_g_y_D_e_f_a_u_l_t is an _o_p_t_i_o_n_ _t_a_b_l_e.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/FastMinors.m2:2037:0.
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html
│ │ │ @@ -119,15 +119,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.000947066s (cpu); 0.00263955s (thread); 0s (gc)
│ │ │ + -- used 0.000282021s (cpu); 0.00256493s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │ @@ -141,24 +141,24 @@ │ │ │ o8 : Ideal of ---[x , x , x , x , x , x , x , x , x , x , x , x ] │ │ │ 127 11 8 1 9 12 6 5 10 2 4 3 7 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.00400074s (cpu); 0.00251447s (thread); 0s (gc)
│ │ │ + -- used 0.00076956s (cpu); 0.0024593s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.5373e-05s (cpu); 0.0022963s (thread); 0s (gc)
│ │ │ + -- used 0.000839978s (cpu); 0.00238827s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Notice in the first case the function returned null, because the depth of search was not high enough. It only computed codim 5 times. The second returned true, but it did so as soon as the answer was found (and before we hit the PairLimit limit).

    │ │ │ ├── html2text {} │ │ │ │ @@ -38,15 +38,15 @@ │ │ │ │ 30 12 │ │ │ │ o4 : Matrix R <-- R │ │ │ │ i5 : r = rank myDiff; │ │ │ │ i6 : J = chooseGoodMinors(15, r, myDiff, Strategy=>StrategyDefaultNonRandom); │ │ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ i7 : time isCodimAtLeast(3, J) │ │ │ │ - -- used 0.000947066s (cpu); 0.00263955s (thread); 0s (gc) │ │ │ │ + -- used 0.000282021s (cpu); 0.00256493s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = true │ │ │ │ The function works by computing gb(I, PairLimit=>f(i)) for successive values of │ │ │ │ i. Here f(i) is a function that takes t, some approximation of the base degree │ │ │ │ value of the polynomial ring (for example, in a standard graded polynomial │ │ │ │ ring, this is probably expected to be \{1\}). And i is a counting variable. You │ │ │ │ can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=> │ │ │ │ @@ -72,20 +72,20 @@ │ │ │ │ x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3- │ │ │ │ 3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3); │ │ │ │ │ │ │ │ ZZ │ │ │ │ o8 : Ideal of ---[x , x , x , x , x , x , x , x , x , x , x , x ] │ │ │ │ 127 11 8 1 9 12 6 5 10 2 4 3 7 │ │ │ │ i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true) │ │ │ │ - -- used 0.00400074s (cpu); 0.00251447s (thread); 0s (gc) │ │ │ │ + -- used 0.00076956s (cpu); 0.0024593s (thread); 0s (gc) │ │ │ │ isCodimAtLeast: Computing codim of monomials based on ideal generators. │ │ │ │ │ │ │ │ o9 = true │ │ │ │ i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false) │ │ │ │ - -- used 6.5373e-05s (cpu); 0.0022963s (thread); 0s (gc) │ │ │ │ + -- used 0.000839978s (cpu); 0.00238827s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = true │ │ │ │ Notice in the first case the function returned null, because the depth of │ │ │ │ search was not high enough. It only computed codim 5 times. The second returned │ │ │ │ true, but it did so as soon as the answer was found (and before we hit the │ │ │ │ PairLimit limit). │ │ │ │ ********** WWaayyss ttoo uussee iissCCooddiimmAAttLLeeaasstt:: ********** │ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ @@ -104,23 +104,23 @@ │ │ │ │ │ │ o3 = 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.296971s (cpu); 0.159933s (thread); 0s (gc)
│ │ │ + -- used 0.312724s (cpu); 0.1765s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00988389s (cpu); 0.0120097s (thread); 0s (gc)
│ │ │ + -- used 0.00884699s (cpu); 0.0117094s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    The option MaxMinors can be used to control how many minors are computed at each step. If this is not specified, the number of minors is a function of the dimension $d$ of the polynomial ring and the possible minors $c$. Specifically it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors => ZZ to specify that a fixed integer is used for each step. Alternatively, the user can control the number of minors computed at each step by setting the option MaxMinors => List. In this case, the list specifies how many minors to be computed at each step, (working backwards). Finally, you can also set MaxMinors to be a custom function of the dimension $d$ of the polynomial ring and the maximum number of minors.

    │ │ │ ├── html2text {} │ │ │ │ @@ -44,19 +44,19 @@ │ │ │ │ i2 : I = ideal((x^3+y)^2, (x^2+y^2)^2, (x+y^3)^2, (x*y)^2); │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : pdim(module I) │ │ │ │ │ │ │ │ o3 = 2 │ │ │ │ i4 : time projDim(module I, Strategy=>StrategyRandom) │ │ │ │ - -- used 0.296971s (cpu); 0.159933s (thread); 0s (gc) │ │ │ │ + -- used 0.312724s (cpu); 0.1765s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 1 │ │ │ │ i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1) │ │ │ │ - -- used 0.00988389s (cpu); 0.0120097s (thread); 0s (gc) │ │ │ │ + -- used 0.00884699s (cpu); 0.0117094s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 1 │ │ │ │ The option MaxMinors can be used to control how many minors are computed at │ │ │ │ each step. If this is not specified, the number of minors is a function of the │ │ │ │ dimension $d$ of the polynomial ring and the possible minors $c$. Specifically │ │ │ │ it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors │ │ │ │ => ZZ to specify that a fixed integer is used for each step. Alternatively, the │ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html │ │ │ @@ -97,23 +97,23 @@ │ │ │ 6 7 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.555663s (cpu); 0.481264s (thread); 0s (gc)
│ │ │ + -- used 0.459231s (cpu); 0.407503s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.57748s (cpu); 1.37879s (thread); 0s (gc)
│ │ │ + -- used 1.15613s (cpu); 1.04316s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : I1 == I2
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  strategy for minors
│ │ │ │  i1 : R = QQ[x,y];
│ │ │ │  i2 : M = random(R^{5,5,5,5,5,5}, R^7);
│ │ │ │  
│ │ │ │               6      7
│ │ │ │  o2 : Matrix R  <-- R
│ │ │ │  i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ │ - -- used 0.555663s (cpu); 0.481264s (thread); 0s (gc)
│ │ │ │ + -- used 0.459231s (cpu); 0.407503s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ │ - -- used 1.57748s (cpu); 1.37879s (thread); 0s (gc)
│ │ │ │ + -- used 1.15613s (cpu); 1.04316s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : I1 == I2
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r_s -- ideal generated by minors
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html
│ │ │ @@ -139,23 +139,23 @@
│ │ │  
│ │ │  o7 = 3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.921914s (cpu); 0.617916s (thread); 0s (gc)
│ │ │ + -- used 0.994644s (cpu); 0.583465s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 10.1044s (cpu); 6.75793s (thread); 0s (gc)
    │ │ │ + -- used 8.06537s (cpu); 5.03693s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │
    │ │ │
    │ │ │ @@ -173,39 +173,39 @@ │ │ │ │ │ │ o11 = 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0279591s (cpu); 0.0261627s (thread); 0s (gc)
│ │ │ + -- used 0.0160178s (cpu); 0.0165786s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.559038s (cpu); 0.316642s (thread); 0s (gc)
│ │ │ + -- used 0.487301s (cpu); 0.26534s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.441343s (cpu); 0.303101s (thread); 0s (gc)
│ │ │ + -- used 0.308915s (cpu); 0.200732s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.424787s (cpu); 0.217971s (thread); 0s (gc)
│ │ │ + -- used 0.362009s (cpu); 0.188854s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    The function works by choosing interesting looking submatrices, computing their determinants, and periodically (based on a logarithmic growth setting), computing the dimension of a subideal of the Jacobian. The option Verbose can be used to see this in action.

    │ │ │ @@ -538,15 +538,15 @@ │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ -- internalChooseMinor: Choosing LexSmallestTerm │ │ │ -- internalChooseMinor: Choosing GRevLexSmallest │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ - -- used 9.54962s (cpu); 6.2698s (thread); 0s (gc) │ │ │ + -- used 8.40753s (cpu); 5.29969s (thread); 0s (gc) │ │ │ regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 327.599 of them. │ │ │ regularInCodimension: About to enter loop │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices considered: 9, and computed = 9 │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ regularInCodimension: partial singular locus dimension computed, = 1 │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices considered: 11, and computed = 10 │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ @@ -627,15 +627,15 @@ │ │ │ -- internalChooseMinor: Choosing Random │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ -- internalChooseMinor: Choosing GRevLexSmallest │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ -- internalChooseMinor: Choosing LexSmallestTerm │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ -- internalChooseMinor: Choosing RandomNonZero │ │ │ - -- used 1.69942s (cpu); 1.13047s (thread); 0s (gc) │ │ │ + -- used 1.5566s (cpu); 1.0524s (thread); 0s (gc) │ │ │ regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 minors, we will compute up to 30 of them. │ │ │ regularInCodimension: About to enter loop │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices considered: 9, and computed = 9 │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ regularInCodimension: partial singular locus dimension computed, = 1 │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices considered: 11, and computed = 11 │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ @@ -659,15 +659,15 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i18 : time regularInCodimension(2, S, VerifyNonRegular=>true)
│ │ │ - -- used 1.63296s (cpu); 0.972645s (thread); 0s (gc)
│ │ │ + -- used 1.53931s (cpu); 0.909701s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = false
    │ │ │
    │ │ │
    │ │ │

    This function has many options which allow you to fine tune the strategy used to find interesting minors. You can pass it a HashTable specifying the strategy via the option Strategy. See LexSmallest for how to construct this HashTable. The default strategy is StrategyDefault, which seems to work well on the examples we have explored. However, caution must be exercised, because, even in the examples above, certain strategies work well while others do not. In the Abelian surface example, LexSmallest works very well, while LexSmallestTerm does not even typically correctly identify the ring as nonsingular (this is because there are a small number of entries with nonzero constant terms, which are selected repeatedly). However, in our first example, the LexSmallestTerm is much faster, and Random does not perform well at all.

    │ │ │ @@ -687,39 +687,39 @@ │ │ │ │ │ │ 
    i21 : StrategyCurrent#LexSmallestTerm = 0;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.597363s (cpu); 0.322568s (thread); 0s (gc)
│ │ │ + -- used 0.547244s (cpu); 0.301736s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.706068s (cpu); 0.366625s (thread); 0s (gc)
│ │ │ + -- used 0.62286s (cpu); 0.29794s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.49826s (cpu); 0.35781s (thread); 0s (gc)
│ │ │ + -- used 0.385356s (cpu); 0.255903s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.532771s (cpu); 0.31747s (thread); 0s (gc)
│ │ │ + -- used 0.44728s (cpu); 0.252185s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i26 : StrategyCurrent#LexSmallest = 0;
    │ │ │ @@ -729,51 +729,51 @@ │ │ │ │ │ │ 
    i27 : StrategyCurrent#LexSmallestTerm = 100;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.58881s (cpu); 2.13991s (thread); 0s (gc)
    │ │ │ + -- used 3.06187s (cpu); 1.72571s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.21572s (cpu); 1.86136s (thread); 0s (gc)
    │ │ │ + -- used 2.96033s (cpu); 1.67439s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.328318s (cpu); 0.225783s (thread); 0s (gc)
│ │ │ + -- used 0.314649s (cpu); 0.19483s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.566594s (cpu); 0.371181s (thread); 0s (gc)
│ │ │ + -- used 0.478476s (cpu); 0.31264s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 2.2427s (cpu); 1.62597s (thread); 0s (gc)
│ │ │ + -- used 1.77716s (cpu); 1.23856s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.78805s (cpu); 1.22068s (thread); 0s (gc)
│ │ │ + -- used 1.51546s (cpu); 1.04747s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    The minimum number of minors computed before checking the codimension can also be controlled by an option MinMinorsFunction. This is should be a function of a single variable, the number of minors computed. Finally, via the option CodimCheckFunction, you can pass the regularInCodimension a function which controls how frequently the codimension of the partial Jacobian ideal is computed. By default this is the floor of 1.3^k. Finally, passing the option Modulus => p will do the computation after changing the coefficient ring to ZZ/p.

    │ │ │ ├── html2text {} │ │ │ │ @@ -77,19 +77,19 @@ │ │ │ │ │ │ │ │ o5 : Ideal of T │ │ │ │ i6 : S = T/I; │ │ │ │ i7 : dim S │ │ │ │ │ │ │ │ o7 = 3 │ │ │ │ i8 : time regularInCodimension(1, S) │ │ │ │ - -- used 0.921914s (cpu); 0.617916s (thread); 0s (gc) │ │ │ │ + -- used 0.994644s (cpu); 0.583465s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = true │ │ │ │ i9 : time regularInCodimension(2, S) │ │ │ │ - -- used 10.1044s (cpu); 6.75793s (thread); 0s (gc) │ │ │ │ + -- used 8.06537s (cpu); 5.03693s (thread); 0s (gc) │ │ │ │ There are numerous examples where regularInCodimension is several orders of │ │ │ │ magnitude faster that calls of dim singularLocus. │ │ │ │ The following is a (pruned) affine chart on an Abelian surface obtained as a │ │ │ │ product of two elliptic curves. It is nonsingular, as our function verifies. If │ │ │ │ one does not prune it, then the dim singularLocus call takes an enormous amount │ │ │ │ of time, otherwise the running times of dim singularLocus and our function are │ │ │ │ frequently about the same. │ │ │ │ @@ -97,27 +97,27 @@ │ │ │ │ (g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3- │ │ │ │ f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h- │ │ │ │ c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c); │ │ │ │ i11 : dim(R) │ │ │ │ │ │ │ │ o11 = 2 │ │ │ │ i12 : time (dim singularLocus (R)) │ │ │ │ - -- used 0.0279591s (cpu); 0.0261627s (thread); 0s (gc) │ │ │ │ + -- used 0.0160178s (cpu); 0.0165786s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 = -1 │ │ │ │ i13 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.559038s (cpu); 0.316642s (thread); 0s (gc) │ │ │ │ + -- used 0.487301s (cpu); 0.26534s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 = true │ │ │ │ i14 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.441343s (cpu); 0.303101s (thread); 0s (gc) │ │ │ │ + -- used 0.308915s (cpu); 0.200732s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.424787s (cpu); 0.217971s (thread); 0s (gc) │ │ │ │ + -- used 0.362009s (cpu); 0.188854s (thread); 0s (gc) │ │ │ │ │ │ │ │ o15 = true │ │ │ │ The function works by choosing interesting looking submatrices, computing their │ │ │ │ determinants, and periodically (based on a logarithmic growth setting), │ │ │ │ computing the dimension of a subideal of the Jacobian. The option Verbose can │ │ │ │ be used to see this in action. │ │ │ │ i16 : time regularInCodimension(2, S, Verbose=>true) │ │ │ │ @@ -445,15 +445,15 @@ │ │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ │ -- internalChooseMinor: Choosing LexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallest │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ - -- used 9.54962s (cpu); 6.2698s (thread); 0s (gc) │ │ │ │ + -- used 8.40753s (cpu); 5.29969s (thread); 0s (gc) │ │ │ │ regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 │ │ │ │ minors, we will compute up to 327.599 of them. │ │ │ │ regularInCodimension: About to enter loop │ │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices │ │ │ │ considered: 9, and computed = 9 │ │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ │ regularInCodimension: partial singular locus dimension computed, = 1 │ │ │ │ @@ -550,15 +550,15 @@ │ │ │ │ -- internalChooseMinor: Choosing Random │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallest │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing LexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing RandomNonZero │ │ │ │ - -- used 1.69942s (cpu); 1.13047s (thread); 0s (gc) │ │ │ │ + -- used 1.5566s (cpu); 1.0524s (thread); 0s (gc) │ │ │ │ regularInCodimension: ring dimension =3, there are 17325 possible 4 by 4 │ │ │ │ minors, we will compute up to 30 of them. │ │ │ │ regularInCodimension: About to enter loop │ │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices │ │ │ │ considered: 9, and computed = 9 │ │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ │ regularInCodimension: partial singular locus dimension computed, = 1 │ │ │ │ @@ -589,15 +589,15 @@ │ │ │ │ that when the set where the minors computed so far has codimension n, then it │ │ │ │ evaluates the matrix at the generic point of a minimal prime of that set. If │ │ │ │ that evaluated Jacobian matrix has too low of a rank, then one has verified │ │ │ │ that variety is not regular in codimemsion n. We consider the same example as │ │ │ │ above, but notice now the function returns false instead of true. This │ │ │ │ sometimes can be slower and sometimes can be faster. │ │ │ │ i18 : time regularInCodimension(2, S, VerifyNonRegular=>true) │ │ │ │ - -- used 1.63296s (cpu); 0.972645s (thread); 0s (gc) │ │ │ │ + -- used 1.53931s (cpu); 0.909701s (thread); 0s (gc) │ │ │ │ │ │ │ │ o18 = false │ │ │ │ This function has many options which allow you to fine tune the strategy used │ │ │ │ to find interesting minors. You can pass it a HashTable specifying the strategy │ │ │ │ via the option Strategy. See _L_e_x_S_m_a_l_l_e_s_t for how to construct this HashTable. │ │ │ │ The default strategy is StrategyDefault, which seems to work well on the │ │ │ │ examples we have explored. However, caution must be exercised, because, even in │ │ │ │ @@ -607,49 +607,49 @@ │ │ │ │ because there are a small number of entries with nonzero constant terms, which │ │ │ │ are selected repeatedly). However, in our first example, the LexSmallestTerm is │ │ │ │ much faster, and Random does not perform well at all. │ │ │ │ i19 : StrategyCurrent#Random = 0; │ │ │ │ i20 : StrategyCurrent#LexSmallest = 100; │ │ │ │ i21 : StrategyCurrent#LexSmallestTerm = 0; │ │ │ │ i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.597363s (cpu); 0.322568s (thread); 0s (gc) │ │ │ │ + -- used 0.547244s (cpu); 0.301736s (thread); 0s (gc) │ │ │ │ │ │ │ │ o22 = true │ │ │ │ i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.706068s (cpu); 0.366625s (thread); 0s (gc) │ │ │ │ + -- used 0.62286s (cpu); 0.29794s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = true │ │ │ │ i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.49826s (cpu); 0.35781s (thread); 0s (gc) │ │ │ │ + -- used 0.385356s (cpu); 0.255903s (thread); 0s (gc) │ │ │ │ │ │ │ │ o24 = true │ │ │ │ i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.532771s (cpu); 0.31747s (thread); 0s (gc) │ │ │ │ + -- used 0.44728s (cpu); 0.252185s (thread); 0s (gc) │ │ │ │ │ │ │ │ o25 = true │ │ │ │ i26 : StrategyCurrent#LexSmallest = 0; │ │ │ │ i27 : StrategyCurrent#LexSmallestTerm = 100; │ │ │ │ i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 3.58881s (cpu); 2.13991s (thread); 0s (gc) │ │ │ │ + -- used 3.06187s (cpu); 1.72571s (thread); 0s (gc) │ │ │ │ i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 3.21572s (cpu); 1.86136s (thread); 0s (gc) │ │ │ │ + -- used 2.96033s (cpu); 1.67439s (thread); 0s (gc) │ │ │ │ i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.328318s (cpu); 0.225783s (thread); 0s (gc) │ │ │ │ + -- used 0.314649s (cpu); 0.19483s (thread); 0s (gc) │ │ │ │ │ │ │ │ o30 = true │ │ │ │ i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.566594s (cpu); 0.371181s (thread); 0s (gc) │ │ │ │ + -- used 0.478476s (cpu); 0.31264s (thread); 0s (gc) │ │ │ │ │ │ │ │ o31 = true │ │ │ │ i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 2.2427s (cpu); 1.62597s (thread); 0s (gc) │ │ │ │ + -- used 1.77716s (cpu); 1.23856s (thread); 0s (gc) │ │ │ │ │ │ │ │ o32 = true │ │ │ │ i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 1.78805s (cpu); 1.22068s (thread); 0s (gc) │ │ │ │ + -- used 1.51546s (cpu); 1.04747s (thread); 0s (gc) │ │ │ │ │ │ │ │ o33 = true │ │ │ │ The minimum number of minors computed before checking the codimension can also │ │ │ │ be controlled by an option MinMinorsFunction. This is should be a function of a │ │ │ │ single variable, the number of minors computed. Finally, via the option │ │ │ │ CodimCheckFunction, you can pass the regularInCodimension a function which │ │ │ │ controls how frequently the codimension of the partial Jacobian ideal is │ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ bmV4dERlZ3JlZShNYXRyaXgsWlosUmluZyk= │ │ │ #:len=288 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzY0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhuZXh0RGVncmVlLE1hdHJpeCxaWixSaW5nKSwibmV4 │ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out │ │ │ @@ -81,23 +81,23 @@ │ │ │ │ │ │ i14 : K3=nextDegree(gens ker Q2,2,S); │ │ │ │ │ │ 8 8 │ │ │ o14 : Matrix R <-- R │ │ │ │ │ │ i15 : time I=co1Fitting(K3) │ │ │ - -- used 0.00253889s (cpu); 0.00253616s (thread); 0s (gc) │ │ │ + -- used 0.0023063s (cpu); 0.0023044s (thread); 0s (gc) │ │ │ │ │ │ o15 = ideal (a a + a - a , a a - a , a a + a - a , a a - a ) │ │ │ 9 11 5 12 3 11 6 9 10 4 11 3 10 5 │ │ │ │ │ │ o15 : Ideal of R │ │ │ │ │ │ i16 : time J=fittingIdeal(2-1,coker K3); │ │ │ - -- used 0.00662056s (cpu); 0.00662794s (thread); 0s (gc) │ │ │ + -- used 0.00541265s (cpu); 0.00541325s (thread); 0s (gc) │ │ │ │ │ │ o16 : Ideal of R │ │ │ │ │ │ i17 : I==J │ │ │ │ │ │ o17 = true │ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ @@ -207,26 +207,26 @@ │ │ │ 8 8 │ │ │ o14 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00253889s (cpu); 0.00253616s (thread); 0s (gc)
│ │ │ + -- used 0.0023063s (cpu); 0.0023044s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │  
│ │ │  o15 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ - -- used 0.00662056s (cpu); 0.00662794s (thread); 0s (gc)
│ │ │ + -- used 0.00541265s (cpu); 0.00541325s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : I==J
│ │ │ ├── html2text {}
│ │ │ │ @@ -95,22 +95,22 @@
│ │ │ │                2      6
│ │ │ │  o13 : Matrix R  <-- R
│ │ │ │  i14 : K3=nextDegree(gens ker Q2,2,S);
│ │ │ │  
│ │ │ │                8      8
│ │ │ │  o14 : Matrix R  <-- R
│ │ │ │  i15 : time I=co1Fitting(K3)
│ │ │ │ - -- used 0.00253889s (cpu); 0.00253616s (thread); 0s (gc)
│ │ │ │ + -- used 0.0023063s (cpu); 0.0023044s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = ideal (a a   + a  - a  , a a   - a , a a   + a  - a  , a a   - a )
│ │ │ │                9 11    5    12   3 11    6   9 10    4    11   3 10    5
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : time J=fittingIdeal(2-1,coker K3);
│ │ │ │ - -- used 0.00662056s (cpu); 0.00662794s (thread); 0s (gc)
│ │ │ │ + -- used 0.00541265s (cpu); 0.00541325s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : I==J
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  Note that our method is a bit faster for this small example, and for rank 2
│ │ │ │  quotients of S^3=\mathbb{Z}[x,y]^3 the time difference is massive.
│ │ ├── ./usr/share/doc/Macaulay2/FirstPackage/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  Rmlyc3RQYWNrYWdl
│ │ │  #:len=509
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZXhhbXBsZSBNYWNhdWxheTIgcGFj
│ │ │  a2FnZSIsICJsaW5lbnVtIiA9PiA1MywgImZpbGVuYW1lIiA9PiAiRmlyc3RQYWNrYWdlLm0yIiwg
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  dmFsdWUodWludDE2KQ==
│ │ │  #:len=273
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTcxOSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodmFsdWUsdWludDE2KSwidmFsdWUodWludDE2KSIs
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Object.out
│ │ │ @@ -4,19 +4,19 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9050}
│ │ │ +o2 = int32{Address => 0x7f8573add510}
│ │ │  
│ │ │  i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9050
│ │ │ +o3 = 0x7f8573add510
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : class x
│ │ │  
│ │ │  o4 = int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  
│ │ │  o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog}
│ │ │  
│ │ │  o2 : ForeignObject of type char**
│ │ │  
│ │ │  i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7f40f2217e90, 0x7f40f2217e80, 0x7f40f2217e70}
│ │ │ +o3 = {0x7f8573b2ab10, 0x7f8573b2ab00, 0x7f8573b2aaf0}
│ │ │  
│ │ │  o3 : ForeignObject of type void**
│ │ │  
│ │ │  i4 : x = charstarstar {"foo", "bar", "baz"}
│ │ │  
│ │ │  o4 = {foo, bar, baz}
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type_sp__Visible__List.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {foo, bar}
│ │ │  
│ │ │  o1 : ForeignObject of type char**
│ │ │  
│ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7f40f2217d00, 0x7f40f2217cf0, 0x7f40f2217ce0}
│ │ │ +o2 = {0x7f8573b2a960, 0x7f8573b2a950, 0x7f8573b2a940}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
│ │ │  
│ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │  
│ │ │  o3 = int32[2]*
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Type_sp__Pointer.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1730835169888399450
│ │ │  
│ │ │  i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7f40f26a3da0
│ │ │ +o1 = 0x7f8573fe0980
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f40f26a3da0
│ │ │ +o2 = 0x7f8573fe0980
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Pointer.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d97f0
│ │ │ +o2 = 0x7f8573b0ff00
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 : int ptr
│ │ │  
│ │ │  o3 = 5
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp_st_spvoidstar.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 1731230829183683930
│ │ │  
│ │ │  i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7f40f21d91f0
│ │ │ +o1 = 0x7f8573b0fad0
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : int * ptr
│ │ │  
│ │ │  o2 = 5
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = myunion
│ │ │  
│ │ │  o1 : ForeignUnionType
│ │ │  
│ │ │  i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.91261e-310}
│ │ │ +o2 = HashTable{"bar" => 6.92726e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
│ │ │  
│ │ │  i3 : myunion pi
│ │ │  
│ │ │  o3 = HashTable{"bar" => 3.14159   }
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Pointer.out
│ │ │ @@ -4,28 +4,28 @@
│ │ │  
│ │ │  o1 = 20
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9960}
│ │ │ +o2 = int32{Address => 0x7f8573b0fc40}
│ │ │  
│ │ │  i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9960
│ │ │ +o3 = 0x7f8573b0fc40
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f40f21d9965
│ │ │ +o4 = 0x7f8573b0fc45
│ │ │  
│ │ │  o4 : Pointer
│ │ │  
│ │ │  i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f40f21d995d
│ │ │ +o5 = 0x7f8573b0fc3d
│ │ │  
│ │ │  o5 : Pointer
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Shared__Library.out
│ │ │ @@ -4,10 +4,10 @@
│ │ │  
│ │ │  o1 = mpfr
│ │ │  
│ │ │  o1 : SharedLibrary
│ │ │  
│ │ │  i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7f40f9b43ac0, mpfr}
│ │ │ +o2 = SharedLibrary{0x7f8588859ac0, mpfr}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/__st_spvoidstar_sp_eq_sp__Thing.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = 5
│ │ │  
│ │ │  o1 : ForeignObject of type int32
│ │ │  
│ │ │  i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d9220
│ │ │ +o2 = 0x7f8573b0f750
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 : *ptr = int 6
│ │ │  
│ │ │  o3 = 6
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_address.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 1730181884377373595
│ │ │  
│ │ │  i1 : address int
│ │ │  
│ │ │ -o1 = 0x562fbd063080
│ │ │ +o1 = 0x55c678e32080
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7f40f21d95f0
│ │ │ +o2 = 0x7f8573b0fde0
│ │ │  
│ │ │  o2 : Pointer
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Function.out
│ │ │ @@ -78,14 +78,14 @@
│ │ │  
│ │ │  o16 = free
│ │ │  
│ │ │  o16 : ForeignFunction
│ │ │  
│ │ │  i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f0558070f60
│ │ │ +o17 = 0x7fb098070f60
│ │ │  
│ │ │  o17 : ForeignObject of type void*
│ │ │  
│ │ │  i18 : registerFinalizer(x, free)
│ │ │  
│ │ │  i19 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_get__Memory.out
│ │ │ @@ -1,21 +1,21 @@
│ │ │  -- -*- M2-comint -*- hash: 10647988412767280310
│ │ │  
│ │ │  i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7f40f50ca360
│ │ │ +o1 = 0x7f85740b6880
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f40f21d9ae0
│ │ │ +o2 = 0x7f8573b0fac0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f40f21d9a00
│ │ │ +o3 = 0x7f8573b0f9d0
│ │ │  
│ │ │  o3 : ForeignObject of type void*
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.out
│ │ │ @@ -17,18 +17,18 @@
│ │ │  o3 = finalizer
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
│ │ │  
│ │ │  i5 : collectGarbage()
│ │ │ -freeing memory at 0x7f40dc0848b0
│ │ │ -freeing memory at 0x7f40dc084870
│ │ │ -freeing memory at 0x7f40dc084830
│ │ │ -freeing memory at 0x7f40dc0841d0
│ │ │ -freeing memory at 0x7f40dc0841f0
│ │ │ -freeing memory at 0x7f40dc0848d0
│ │ │ -freeing memory at 0x7f40dc084810
│ │ │ -freeing memory at 0x7f40dc084850
│ │ │ -freeing memory at 0x7f40dc084890
│ │ │ +freeing memory at 0x7f855c0841f0
│ │ │ +freeing memory at 0x7f855c084810
│ │ │ +freeing memory at 0x7f855c084830
│ │ │ +freeing memory at 0x7f855c084850
│ │ │ +freeing memory at 0x7f855c0848b0
│ │ │ +freeing memory at 0x7f855c084890
│ │ │ +freeing memory at 0x7f855c0848d0
│ │ │ +freeing memory at 0x7f855c084870
│ │ │ +freeing memory at 0x7f855c0841d0
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_value_lp__Foreign__Object_rp.out
│ │ │ @@ -20,21 +20,21 @@
│ │ │  
│ │ │  o4 = 5
│ │ │  
│ │ │  o4 : RR (of precision 53)
│ │ │  
│ │ │  i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7f40f21fd990
│ │ │ +o5 = 0x7f8573b0f200
│ │ │  
│ │ │  o5 : ForeignObject of type void*
│ │ │  
│ │ │  i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f40f21fd990
│ │ │ +o6 = 0x7f8573b0f200
│ │ │  
│ │ │  o6 : Pointer
│ │ │  
│ │ │  i7 : x = charstar "Hello, world!"
│ │ │  
│ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html
│ │ │ @@ -69,27 +69,27 @@
│ │ │  o1 : ForeignObject of type int32
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9050}
    │ │ │ +o2 = int32{Address => 0x7f8573add510} │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    To get this, use address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9050
│ │ │ +o3 = 0x7f8573add510
│ │ │  
│ │ │  o3 : Pointer
    │ │ │
    │ │ │
    │ │ │

    Use class to determine the type of the object.

    │ │ │ ├── html2text {} │ │ │ │ @@ -10,19 +10,19 @@ │ │ │ │ i1 : x = int 5 │ │ │ │ │ │ │ │ o1 = 5 │ │ │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ i2 : peek x │ │ │ │ │ │ │ │ -o2 = int32{Address => 0x7f40f21d9050} │ │ │ │ +o2 = int32{Address => 0x7f8573add510} │ │ │ │ To get this, use _a_d_d_r_e_s_s. │ │ │ │ i3 : address x │ │ │ │ │ │ │ │ -o3 = 0x7f40f21d9050 │ │ │ │ +o3 = 0x7f8573add510 │ │ │ │ │ │ │ │ o3 : Pointer │ │ │ │ Use _c_l_a_s_s to determine the type of the object. │ │ │ │ i4 : class x │ │ │ │ │ │ │ │ o4 = int32 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type.html │ │ │ @@ -79,15 +79,15 @@ │ │ │ o2 : ForeignObject of type char** │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7f40f2217e90, 0x7f40f2217e80, 0x7f40f2217e70}
│ │ │ +o3 = {0x7f8573b2ab10, 0x7f8573b2ab00, 0x7f8573b2aaf0}
│ │ │  
│ │ │  o3 : ForeignObject of type void**
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Foreign pointer arrays may be subscripted using _.

    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ "lazy", "dog"} │ │ │ │ │ │ │ │ o2 = {the, quick, brown, fox, jumps, over, the, lazy, dog} │ │ │ │ │ │ │ │ o2 : ForeignObject of type char** │ │ │ │ i3 : voidstarstar {address int 0, address int 1, address int 2} │ │ │ │ │ │ │ │ -o3 = {0x7f40f2217e90, 0x7f40f2217e80, 0x7f40f2217e70} │ │ │ │ +o3 = {0x7f8573b2ab10, 0x7f8573b2ab00, 0x7f8573b2aaf0} │ │ │ │ │ │ │ │ o3 : ForeignObject of type void** │ │ │ │ Foreign pointer arrays may be subscripted using __. │ │ │ │ i4 : x = charstarstar {"foo", "bar", "baz"} │ │ │ │ │ │ │ │ o4 = {foo, bar, baz} │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type_sp__Visible__List.html │ │ │ @@ -87,15 +87,15 @@ │ │ │ o1 : ForeignObject of type char** │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7f40f2217d00, 0x7f40f2217cf0, 0x7f40f2217ce0}
│ │ │ +o2 = {0x7f8573b2a960, 0x7f8573b2a950, 0x7f8573b2a940}
│ │ │  
│ │ │  o2 : ForeignObject of type void**
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : charstarstar {"foo", "bar"}
│ │ │ │  
│ │ │ │  o1 = {foo, bar}
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type char**
│ │ │ │  i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │ │  
│ │ │ │ -o2 = {0x7f40f2217d00, 0x7f40f2217cf0, 0x7f40f2217ce0}
│ │ │ │ +o2 = {0x7f8573b2a960, 0x7f8573b2a950, 0x7f8573b2a940}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void**
│ │ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ │  
│ │ │ │  o3 = int32[2]*
│ │ │ │  
│ │ │ │  o3 : ForeignPointerArrayType
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Type_sp__Pointer.html
│ │ │ @@ -78,24 +78,24 @@
│ │ │            
    To cast a Macaulay2 pointer to a foreign object with a pointer type, give the type followed by the pointer.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7f40f26a3da0
│ │ │ +o1 = 0x7f8573fe0980
│ │ │  
│ │ │  o1 : Pointer
    │ │ │
    │ │ │ 
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f40f26a3da0
│ │ │ +o2 = 0x7f8573fe0980
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -15,20 +15,20 @@ │ │ │ │ * Outputs: │ │ │ │ o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ To cast a Macaulay2 pointer to a foreign object with a pointer type, give the │ │ │ │ type followed by the pointer. │ │ │ │ i1 : ptr = address int 0 │ │ │ │ │ │ │ │ -o1 = 0x7f40f26a3da0 │ │ │ │ +o1 = 0x7f8573fe0980 │ │ │ │ │ │ │ │ o1 : Pointer │ │ │ │ i2 : voidstar ptr │ │ │ │ │ │ │ │ -o2 = 0x7f40f26a3da0 │ │ │ │ +o2 = 0x7f8573fe0980 │ │ │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _F_o_r_e_i_g_n_P_o_i_n_t_e_r_T_y_p_e_ _P_o_i_n_t_e_r -- cast a Macaulay2 pointer to a foreign │ │ │ │ pointer │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Pointer.html │ │ │ @@ -87,15 +87,15 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d97f0
│ │ │ +o2 = 0x7f8573b0ff00
│ │ │  
│ │ │  o2 : Pointer
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : int ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7f40f21d97f0
│ │ │ │ +o2 = 0x7f8573b0ff00
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : int ptr
│ │ │ │  
│ │ │ │  o3 = 5
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │            
    This is syntactic sugar for T value ptr (see ForeignType Pointer) for dereferencing pointers.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7f40f21d91f0
│ │ │ +o1 = 0x7f8573b0fad0
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    │ │ │
    │ │ │ 
    i2 : int * ptr
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _f_o_r_e_i_g_n_ _o_b_j_e_c_t, of type T;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This is syntactic sugar for T value ptr (see _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r) for
│ │ │ │  dereferencing pointers.
│ │ │ │  i1 : ptr = voidstar address int 5
│ │ │ │  
│ │ │ │ -o1 = 0x7f40f21d91f0
│ │ │ │ +o1 = 0x7f8573b0fad0
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type void*
│ │ │ │  i2 : int * ptr
│ │ │ │  
│ │ │ │  o2 = 5
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  o1 : ForeignUnionType
    │ │ │
    │ │ │ 
    i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.91261e-310}
│ │ │ +o2 = HashTable{"bar" => 6.92726e-310}
│ │ │                 "foo" => 27
│ │ │  
│ │ │  o2 : ForeignObject of type myunion
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -20,15 +20,15 @@ │ │ │ │ i1 : myunion = foreignUnionType("myunion", {"foo" => int, "bar" => double}) │ │ │ │ │ │ │ │ o1 = myunion │ │ │ │ │ │ │ │ o1 : ForeignUnionType │ │ │ │ i2 : myunion 27 │ │ │ │ │ │ │ │ -o2 = HashTable{"bar" => 6.91261e-310} │ │ │ │ +o2 = HashTable{"bar" => 6.92726e-310} │ │ │ │ "foo" => 27 │ │ │ │ │ │ │ │ o2 : ForeignObject of type myunion │ │ │ │ i3 : myunion pi │ │ │ │ │ │ │ │ o3 = HashTable{"bar" => 3.14159 } │ │ │ │ "foo" => 1413754136 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Pointer.html │ │ │ @@ -69,50 +69,50 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │
    │ │ │ 
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9960}
    │ │ │ +o2 = int32{Address => 0x7f8573b0fc40} │ │ │
    │ │ │
    │ │ │

    These pointers can be accessed using address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9960
│ │ │ +o3 = 0x7f8573b0fc40
│ │ │  
│ │ │  o3 : Pointer
    │ │ │
    │ │ │
    │ │ │

    Simple arithmetic can be performed on pointers.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f40f21d9965
│ │ │ +o4 = 0x7f8573b0fc45
│ │ │  
│ │ │  o4 : Pointer
    │ │ │
    │ │ │ 
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f40f21d995d
│ │ │ +o5 = 0x7f8573b0fc3d
│ │ │  
│ │ │  o5 : Pointer
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -10,30 +10,30 @@ │ │ │ │ i1 : x = int 20 │ │ │ │ │ │ │ │ o1 = 20 │ │ │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ i2 : peek x │ │ │ │ │ │ │ │ -o2 = int32{Address => 0x7f40f21d9960} │ │ │ │ +o2 = int32{Address => 0x7f8573b0fc40} │ │ │ │ These pointers can be accessed using _a_d_d_r_e_s_s. │ │ │ │ i3 : ptr = address x │ │ │ │ │ │ │ │ -o3 = 0x7f40f21d9960 │ │ │ │ +o3 = 0x7f8573b0fc40 │ │ │ │ │ │ │ │ o3 : Pointer │ │ │ │ Simple arithmetic can be performed on pointers. │ │ │ │ i4 : ptr + 5 │ │ │ │ │ │ │ │ -o4 = 0x7f40f21d9965 │ │ │ │ +o4 = 0x7f8573b0fc45 │ │ │ │ │ │ │ │ o4 : Pointer │ │ │ │ i5 : ptr - 3 │ │ │ │ │ │ │ │ -o5 = 0x7f40f21d995d │ │ │ │ +o5 = 0x7f8573b0fc3d │ │ │ │ │ │ │ │ o5 : Pointer │ │ │ │ ******** MMeennuu ******** │ │ │ │ * _n_u_l_l_P_o_i_n_t_e_r -- the null pointer │ │ │ │ * _a_d_d_r_e_s_s -- pointer to type or object │ │ │ │ * _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r -- dereference a pointer │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa ppooiinntteerr:: ********** │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html │ │ │ @@ -69,15 +69,15 @@ │ │ │ o1 : SharedLibrary │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7f40f9b43ac0, mpfr}
    │ │ │ +o2 = SharedLibrary{0x7f8588859ac0, mpfr} │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Menu

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -12,15 +12,15 @@ │ │ │ │ i1 : mpfr = openSharedLibrary "mpfr" │ │ │ │ │ │ │ │ o1 = mpfr │ │ │ │ │ │ │ │ o1 : SharedLibrary │ │ │ │ i2 : peek mpfr │ │ │ │ │ │ │ │ -o2 = SharedLibrary{0x7f40f9b43ac0, mpfr} │ │ │ │ +o2 = SharedLibrary{0x7f8588859ac0, mpfr} │ │ │ │ ******** MMeennuu ******** │ │ │ │ * _o_p_e_n_S_h_a_r_e_d_L_i_b_r_a_r_y -- open a shared library │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa sshhaarreedd lliibbrraarryy:: ********** │ │ │ │ * _o_p_e_n_S_h_a_r_e_d_L_i_b_r_a_r_y -- open a shared library │ │ │ │ ********** MMeetthhooddss tthhaatt uussee aa sshhaarreedd lliibbrraarryy:: ********** │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) -- see │ │ │ │ _f_o_r_e_i_g_n_F_u_n_c_t_i_o_n -- construct a foreign function │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/__st_spvoidstar_sp_eq_sp__Thing.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d9220
│ │ │ +o2 = 0x7f8573b0f750
│ │ │  
│ │ │  o2 : Pointer
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : *ptr = int 6
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7f40f21d9220
│ │ │ │ +o2 = 0x7f8573b0f750
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : *ptr = int 6
│ │ │ │  
│ │ │ │  o3 = 6
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_address.html
│ │ │ @@ -76,29 +76,29 @@
│ │ │            
      If x is a foreign type, then this returns the address to the ffi_type struct used by libffi to identify the type.
      
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : address int
│ │ │  
│ │ │ -o1 = 0x562fbd063080
│ │ │ +o1 = 0x55c678e32080
│ │ │  
│ │ │  o1 : Pointer
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7f40f21d95f0
│ │ │ +o2 = 0x7f8573b0fde0
│ │ │  
│ │ │  o2 : Pointer
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -11,22 +11,22 @@ │ │ │ │ * Outputs: │ │ │ │ o a _p_o_i_n_t_e_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ If x is a foreign type, then this returns the address to the ffi_type struct │ │ │ │ used by libffi to identify the type. │ │ │ │ i1 : address int │ │ │ │ │ │ │ │ -o1 = 0x562fbd063080 │ │ │ │ +o1 = 0x55c678e32080 │ │ │ │ │ │ │ │ o1 : Pointer │ │ │ │ If x is a foreign object, then this returns the address to the object. It │ │ │ │ behaves like the & "address-of" operator in C. │ │ │ │ i2 : address int 5 │ │ │ │ │ │ │ │ -o2 = 0x7f40f21d95f0 │ │ │ │ +o2 = 0x7f8573b0fde0 │ │ │ │ │ │ │ │ o2 : Pointer │ │ │ │ ********** WWaayyss ttoo uussee aaddddrreessss:: ********** │ │ │ │ * address(ForeignObject) │ │ │ │ * address(ForeignType) │ │ │ │ * address(Nothing) (missing documentation) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_foreign__Function.html │ │ │ @@ -237,15 +237,15 @@ │ │ │ o16 : ForeignFunction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f0558070f60
│ │ │ +o17 = 0x7fb098070f60
│ │ │  
│ │ │  o17 : ForeignObject of type void*
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : registerFinalizer(x, free)
    │ │ │ ├── html2text {} │ │ │ │ @@ -95,15 +95,15 @@ │ │ │ │ i16 : free = foreignFunction("free", void, voidstar) │ │ │ │ │ │ │ │ o16 = free │ │ │ │ │ │ │ │ o16 : ForeignFunction │ │ │ │ i17 : x = malloc 8 │ │ │ │ │ │ │ │ -o17 = 0x7f0558070f60 │ │ │ │ +o17 = 0x7fb098070f60 │ │ │ │ │ │ │ │ o17 : ForeignObject of type void* │ │ │ │ i18 : registerFinalizer(x, free) │ │ │ │ ********** WWaayyss ttoo uussee ffoorreeiiggnnFFuunnccttiioonn:: ********** │ │ │ │ * foreignFunction(Pointer,String,ForeignType,VisibleList) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,VisibleList) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_get__Memory.html │ │ │ @@ -82,43 +82,43 @@ │ │ │

    Allocate n bytes of memory using the GC garbage collector.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7f40f50ca360
│ │ │ +o1 = 0x7f85740b6880
│ │ │  
│ │ │  o1 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f40f21d9ae0
│ │ │ +o2 = 0x7f8573b0fac0
│ │ │  
│ │ │  o2 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f40f21d9a00
│ │ │ +o3 = 0x7f8573b0f9d0
│ │ │  
│ │ │  o3 : ForeignObject of type void*
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -14,30 +14,30 @@ │ │ │ │ o Atomic => ..., default value false │ │ │ │ * Outputs: │ │ │ │ o an instance of the type _v_o_i_d_s_t_a_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Allocate n bytes of memory using the _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r. │ │ │ │ i1 : ptr = getMemory 8 │ │ │ │ │ │ │ │ -o1 = 0x7f40f50ca360 │ │ │ │ +o1 = 0x7f85740b6880 │ │ │ │ │ │ │ │ o1 : ForeignObject of type void* │ │ │ │ If the memory will not contain any pointers, then set the Atomic option to │ │ │ │ _t_r_u_e. │ │ │ │ i2 : ptr = getMemory(8, Atomic => true) │ │ │ │ │ │ │ │ -o2 = 0x7f40f21d9ae0 │ │ │ │ +o2 = 0x7f8573b0fac0 │ │ │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ Alternatively, a foreign object type T may be specified. In this case, the │ │ │ │ number of bytes and whether the Atomic option should be set will be determined │ │ │ │ automatically. │ │ │ │ i3 : ptr = getMemory int │ │ │ │ │ │ │ │ -o3 = 0x7f40f21d9a00 │ │ │ │ +o3 = 0x7f8573b0f9d0 │ │ │ │ │ │ │ │ o3 : ForeignObject of type void* │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_g_i_s_t_e_r_F_i_n_a_l_i_z_e_r_(_F_o_r_e_i_g_n_O_b_j_e_c_t_,_F_u_n_c_t_i_o_n_) -- register a finalizer for a │ │ │ │ foreign object │ │ │ │ ********** WWaayyss ttoo uussee ggeettMMeemmoorryy:: ********** │ │ │ │ * getMemory(ForeignType) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.html │ │ │ @@ -105,23 +105,23 @@ │ │ │ │ │ │ 
    i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : collectGarbage()
│ │ │ -freeing memory at 0x7f40dc0848b0
│ │ │ -freeing memory at 0x7f40dc084870
│ │ │ -freeing memory at 0x7f40dc084830
│ │ │ -freeing memory at 0x7f40dc0841d0
│ │ │ -freeing memory at 0x7f40dc0841f0
│ │ │ -freeing memory at 0x7f40dc0848d0
│ │ │ -freeing memory at 0x7f40dc084810
│ │ │ -freeing memory at 0x7f40dc084850
│ │ │ -freeing memory at 0x7f40dc084890
    │ │ │ +freeing memory at 0x7f855c0841f0 │ │ │ +freeing memory at 0x7f855c084810 │ │ │ +freeing memory at 0x7f855c084830 │ │ │ +freeing memory at 0x7f855c084850 │ │ │ +freeing memory at 0x7f855c0848b0 │ │ │ +freeing memory at 0x7f855c084890 │ │ │ +freeing memory at 0x7f855c0848d0 │ │ │ +freeing memory at 0x7f855c084870 │ │ │ +freeing memory at 0x7f855c0841d0 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -31,23 +31,23 @@ │ │ │ │ i3 : finalizer = x -> (print("freeing memory at " | net x); free x) │ │ │ │ │ │ │ │ o3 = finalizer │ │ │ │ │ │ │ │ o3 : FunctionClosure │ │ │ │ i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer)) │ │ │ │ i5 : collectGarbage() │ │ │ │ -freeing memory at 0x7f40dc0848b0 │ │ │ │ -freeing memory at 0x7f40dc084870 │ │ │ │ -freeing memory at 0x7f40dc084830 │ │ │ │ -freeing memory at 0x7f40dc0841d0 │ │ │ │ -freeing memory at 0x7f40dc0841f0 │ │ │ │ -freeing memory at 0x7f40dc0848d0 │ │ │ │ -freeing memory at 0x7f40dc084810 │ │ │ │ -freeing memory at 0x7f40dc084850 │ │ │ │ -freeing memory at 0x7f40dc084890 │ │ │ │ +freeing memory at 0x7f855c0841f0 │ │ │ │ +freeing memory at 0x7f855c084810 │ │ │ │ +freeing memory at 0x7f855c084830 │ │ │ │ +freeing memory at 0x7f855c084850 │ │ │ │ +freeing memory at 0x7f855c0848b0 │ │ │ │ +freeing memory at 0x7f855c084890 │ │ │ │ +freeing memory at 0x7f855c0848d0 │ │ │ │ +freeing memory at 0x7f855c084870 │ │ │ │ +freeing memory at 0x7f855c0841d0 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_t_M_e_m_o_r_y -- allocate memory using the garbage collector │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _r_e_g_i_s_t_e_r_F_i_n_a_l_i_z_e_r_(_F_o_r_e_i_g_n_O_b_j_e_c_t_,_F_u_n_c_t_i_o_n_) -- register a finalizer for a │ │ │ │ foreign object │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_value_lp__Foreign__Object_rp.html │ │ │ @@ -121,24 +121,24 @@ │ │ │

      Foreign pointer objects are converted to Pointer objects.

      │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7f40f21fd990
│ │ │ +o5 = 0x7f8573b0f200
│ │ │  
│ │ │  o5 : ForeignObject of type void*
    │ │ │
    │ │ │ 
    i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f40f21fd990
│ │ │ +o6 = 0x7f8573b0f200
│ │ │  
│ │ │  o6 : Pointer
    │ │ │
    │ │ │
    │ │ │

    Foreign string objects are converted to strings.

    │ │ │ ├── html2text {} │ │ │ │ @@ -34,20 +34,20 @@ │ │ │ │ │ │ │ │ o4 = 5 │ │ │ │ │ │ │ │ o4 : RR (of precision 53) │ │ │ │ Foreign pointer objects are converted to _P_o_i_n_t_e_r objects. │ │ │ │ i5 : x = voidstar address int 5 │ │ │ │ │ │ │ │ -o5 = 0x7f40f21fd990 │ │ │ │ +o5 = 0x7f8573b0f200 │ │ │ │ │ │ │ │ o5 : ForeignObject of type void* │ │ │ │ i6 : value x │ │ │ │ │ │ │ │ -o6 = 0x7f40f21fd990 │ │ │ │ +o6 = 0x7f8573b0f200 │ │ │ │ │ │ │ │ o6 : Pointer │ │ │ │ Foreign string objects are converted to strings. │ │ │ │ i7 : x = charstar "Hello, world!" │ │ │ │ │ │ │ │ o7 = Hello, world! │ │ ├── ./usr/share/doc/Macaulay2/FormalGroupLaws/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ c2VyaWVzKFJpbmdFbGVtZW50LFpaKQ== │ │ │ #:len=1234 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29uc3RydWN0aW5nIGEgZm9ybWFsIHNl │ │ │ cmllcyIsICJsaW5lbnVtIiA9PiAzNzIsIElucHV0cyA9PiB7U1BBTntUVHsicyJ9LCIsICIsU1BB │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=13 │ │ │ dG9yaWNHcm9lYm5lcg== │ │ │ #:len=2522 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2FsY3VsYXRlcyBhIEdyb2VibmVyIGJh │ │ │ c2lzIG9mIHRoZSB0b3JpYyBpZGVhbCBJX0EsIGdpdmVuIEE7IGludm9rZXMgXCJncm9lYm5lclwi │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/example-output/_put__Matrix.out │ │ │ @@ -6,27 +6,27 @@ │ │ │ | 1 2 3 4 | │ │ │ │ │ │ 2 4 │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i2 : s = temporaryFileName() │ │ │ │ │ │ -o2 = /tmp/M2-15423-0/0 │ │ │ +o2 = /tmp/M2-19303-0/0 │ │ │ │ │ │ i3 : F = openOut(s) │ │ │ │ │ │ -o3 = /tmp/M2-15423-0/0 │ │ │ +o3 = /tmp/M2-19303-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : putMatrix(F,A) │ │ │ │ │ │ i5 : close(F) │ │ │ │ │ │ -o5 = /tmp/M2-15423-0/0 │ │ │ +o5 = /tmp/M2-19303-0/0 │ │ │ │ │ │ o5 : File │ │ │ │ │ │ i6 : getMatrix(s) │ │ │ │ │ │ o6 = | 1 1 1 1 | │ │ │ | 1 2 3 4 | │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/html/_put__Matrix.html │ │ │ @@ -84,36 +84,36 @@ │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15423-0/0
    │ │ │ +o2 = /tmp/M2-19303-0/0 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15423-0/0
│ │ │ +o3 = /tmp/M2-19303-0/0
│ │ │  
│ │ │  o3 : File
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : putMatrix(F,A)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15423-0/0
│ │ │ +o5 = /tmp/M2-19303-0/0
│ │ │  
│ │ │  o5 : File
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : getMatrix(s)
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,24 +16,24 @@
│ │ │ │  o1 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ │ │  
│ │ │ │                2       4
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  i2 : s = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-15423-0/0
│ │ │ │ +o2 = /tmp/M2-19303-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-15423-0/0
│ │ │ │ +o3 = /tmp/M2-19303-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-15423-0/0
│ │ │ │ +o5 = /tmp/M2-19303-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : getMatrix(s)
│ │ │ │  
│ │ │ │  o6 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FourierMotzkin/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  Zm91cmllck1vdHpraW4=
│ │ │  #:len=3387
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW50ZXJjaGFuZ2UgaW5lcXVhbGl0eS9n
│ │ │  ZW5lcmF0b3IgcmVwcmVzZW50YXRpb24gb2YgYSBwb2x5aGVkcmFsIGNvbmUiLCAibGluZW51bSIg
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=32
│ │ │  aXNGUFQoLi4uLFFHb3JlbnN0ZWluSW5kZXg9Pi4uLik=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTI0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tpc0ZQVCxRR29yZW5zdGVpbkluZGV4XSwiaXNGUFQo
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out
│ │ │ @@ -155,31 +155,31 @@
│ │ │  i26 : numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true) -- FinalAttempt improves the estimate slightly
│ │ │  
│ │ │  o26 = {.142067, .144}
│ │ │  
│ │ │  o26 : List
│ │ │  
│ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 1.56927s (cpu); 1.12322s (thread); 0s (gc)
│ │ │ + -- used 1.64145s (cpu); 1.13654s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.0262s (cpu); 0.753238s (thread); 0s (gc)
│ │ │ + -- used 0.898357s (cpu); 0.697521s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 0.740369s (cpu); 0.533854s (thread); 0s (gc)
│ │ │ + -- used 0.778547s (cpu); 0.572475s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out
│ │ │ @@ -43,34 +43,34 @@
│ │ │  o12 = 220
│ │ │  
│ │ │  i13 : R = ZZ/17[x,y,z];
│ │ │  
│ │ │  i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
│ │ │  
│ │ │  i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00433815s (cpu); 0.00433416s (thread); 0s (gc)
│ │ │ + -- used 0.00426976s (cpu); 0.00426659s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.293694s (cpu); 0.229445s (thread); 0s (gc)
│ │ │ + -- used 0.321186s (cpu); 0.24938s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
│ │ │  
│ │ │  i17 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i18 : f = x^3 + y^3 + z^3 + x*y*z;
│ │ │  
│ │ │  i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.252728s (cpu); 0.176544s (thread); 0s (gc)
│ │ │ + -- used 0.274383s (cpu); 0.192829s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.81045s (cpu); 1.35141s (thread); 0s (gc)
│ │ │ + -- used 1.31549s (cpu); 1.02569s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
│ │ │  
│ │ │  i21 : R = ZZ/3[x,y];
│ │ │  
│ │ │  i22 : M = ideal(x, y);
│ │ │  
│ │ │ @@ -85,34 +85,34 @@
│ │ │  o24 = 8
│ │ │  
│ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │  
│ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │  
│ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 0.778907s (cpu); 0.635009s (thread); 0s (gc)
│ │ │ + -- used 0.69176s (cpu); 0.557563s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.08232s (cpu); 0.827197s (thread); 0s (gc)
│ │ │ + -- used 1.09201s (cpu); 0.823976s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
│ │ │  
│ │ │  i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
│ │ │  
│ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 2.16055s (cpu); 1.60855s (thread); 0s (gc)
│ │ │ + -- used 1.46509s (cpu); 1.1744s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.636757s (cpu); 0.499453s (thread); 0s (gc)
│ │ │ + -- used 0.492991s (cpu); 0.422329s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
│ │ │  
│ │ │  i32 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html
│ │ │ @@ -368,37 +368,37 @@
│ │ │          
    
│ │ │            
    The computations performed when FinalAttempt is set to true are often slow, and often fail to improve the estimate, and for this reason, this option should be used sparingly. It is often more effective to increase the values of Attempts or DepthOfSearch, instead.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -228,29 +228,29 @@
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  The computations performed when FinalAttempt is set to true are often slow, and
│ │ │ │  often fail to improve the estimate, and for this reason, this option should be
│ │ │ │  used sparingly. It is often more effective to increase the values of Attempts
│ │ │ │  or DepthOfSearch, instead.
│ │ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ │ - -- used 1.56927s (cpu); 1.12322s (thread); 0s (gc)
│ │ │ │ + -- used 1.64145s (cpu); 1.13654s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {.142067, .144}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ │ - -- used 1.0262s (cpu); 0.753238s (thread); 0s (gc)
│ │ │ │ + -- used 0.898357s (cpu); 0.697521s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o28 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o28 : QQ
│ │ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ │ - -- used 0.740369s (cpu); 0.533854s (thread); 0s (gc)
│ │ │ │ + -- used 0.778547s (cpu); 0.572475s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o29 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o29 : QQ
│ │ │ │  As seen in several examples above, when the exact answer is not found, a list
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html
│ │ │ @@ -197,23 +197,23 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 1.56927s (cpu); 1.12322s (thread); 0s (gc)
│ │ │ + -- used 1.64145s (cpu); 1.13654s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.0262s (cpu); 0.753238s (thread); 0s (gc)
│ │ │ + -- used 0.898357s (cpu); 0.697521s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    
│ │ │              
    
│ │ │                
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 0.740369s (cpu); 0.533854s (thread); 0s (gc)
│ │ │ + -- used 0.778547s (cpu); 0.572475s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o29 = -
│ │ │        7
│ │ │  
│ │ │  o29 : QQ
    
│ │ │                  		
│ │ │                
    i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
    
│ │ │              
    
│ │ │                
    i15 : time frobeniusNu(3, f)
│ │ │ - -- used 0.00433815s (cpu); 0.00433416s (thread); 0s (gc)
│ │ │ + -- used 0.00426976s (cpu); 0.00426659s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    
│ │ │              
    
│ │ │                
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.293694s (cpu); 0.229445s (thread); 0s (gc)
│ │ │ + -- used 0.321186s (cpu); 0.24938s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The valid values for the option ContainmentTest are FrobeniusPower, FrobeniusRoot, and StandardPower. The default value of this option depends on what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to StandardPower if frobeniusNu is passed an ideal $I$. We describe the consequences of setting ContainmentTest to each of these values below.
    
│ │ │ @@ -230,23 +230,23 @@
│ │ │              
│ │ │                
    i18 : f = x^3 + y^3 + z^3 + x*y*z;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.252728s (cpu); 0.176544s (thread); 0s (gc)
│ │ │ + -- used 0.274383s (cpu); 0.192829s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.81045s (cpu); 1.35141s (thread); 0s (gc)
│ │ │ + -- used 1.31549s (cpu); 1.02569s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Finally, when ContainmentTest is set to FrobeniusPower, then instead of producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as defined in the paper Frobenius Powers by Hernández, Teixeira, and Witt, which can be computed with the function frobeniusPower, from the TestIdeals package. In particular, frobeniusNu(e,I,J) and frobeniusNu(e,I,J,ContainmentTest=>FrobeniusPower) need not agree. However, they will agree when $I$ is a principal ideal.
    
│ │ │ @@ -292,46 +292,46 @@
│ │ │              
│ │ │                
    i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 0.778907s (cpu); 0.635009s (thread); 0s (gc)
│ │ │ + -- used 0.69176s (cpu); 0.557563s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.08232s (cpu); 0.827197s (thread); 0s (gc)
│ │ │ + -- used 1.09201s (cpu); 0.823976s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : M = ideal(x, y, z);
│ │ │  
│ │ │  o29 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ - -- used 2.16055s (cpu); 1.60855s (thread); 0s (gc)
│ │ │ + -- used 1.46509s (cpu); 1.1744s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.636757s (cpu); 0.499453s (thread); 0s (gc)
│ │ │ + -- used 0.492991s (cpu); 0.422329s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    The option AtOrigin (default value true) can be turned off to tell frobeniusNu to effectively do the computation over all possible maximal ideals $J$ and take the minimum.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  algorithms, namely diagonal polynomials, binomials, forms in two variables, and
│ │ │ │  polynomials whose factors are in simple normal crossing. This feature can be
│ │ │ │  disabled by setting the option UseSpecialAlgorithms (default value true) to
│ │ │ │  false.
│ │ │ │  i13 : R = ZZ/17[x,y,z];
│ │ │ │  i14 : f = x^3 + y^4 + z^5; -- a diagonal polynomial
│ │ │ │  i15 : time frobeniusNu(3, f)
│ │ │ │ - -- used 0.00433815s (cpu); 0.00433416s (thread); 0s (gc)
│ │ │ │ + -- used 0.00426976s (cpu); 0.00426659s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3756
│ │ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ │ - -- used 0.293694s (cpu); 0.229445s (thread); 0s (gc)
│ │ │ │ + -- used 0.321186s (cpu); 0.24938s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 3756
│ │ │ │  The valid values for the option ContainmentTest are FrobeniusPower,
│ │ │ │  FrobeniusRoot, and StandardPower. The default value of this option depends on
│ │ │ │  what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to
│ │ │ │  FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to
│ │ │ │  StandardPower if frobeniusNu is passed an ideal $I$. We describe the
│ │ │ │ @@ -133,19 +133,19 @@
│ │ │ │  is contained in $J$. The output is unaffected, but this option often speeds up
│ │ │ │  computations, specially when a polynomial or principal ideal is passed as the
│ │ │ │  second argument.
│ │ │ │  i17 : R = ZZ/5[x,y,z];
│ │ │ │  i18 : f = x^3 + y^3 + z^3 + x*y*z;
│ │ │ │  i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by
│ │ │ │  default
│ │ │ │ - -- used 0.252728s (cpu); 0.176544s (thread); 0s (gc)
│ │ │ │ + -- used 0.274383s (cpu); 0.192829s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = 499
│ │ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ │ - -- used 1.81045s (cpu); 1.35141s (thread); 0s (gc)
│ │ │ │ + -- used 1.31549s (cpu); 1.02569s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = 499
│ │ │ │  Finally, when ContainmentTest is set to FrobeniusPower, then instead of
│ │ │ │  producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead
│ │ │ │  outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius
│ │ │ │  power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the
│ │ │ │  $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as
│ │ │ │ @@ -167,31 +167,31 @@
│ │ │ │  The function frobeniusNu works by searching through the list of potential
│ │ │ │  integers $n$ and checking containments of $I^n$ in the specified Frobenius
│ │ │ │  power of $J$. The way this search is approached is specified by the option
│ │ │ │  Search, which can be set to Binary (the default value) or Linear.
│ │ │ │  i25 : R = ZZ/5[x,y,z];
│ │ │ │  i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
│ │ │ │  i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ │ - -- used 0.778907s (cpu); 0.635009s (thread); 0s (gc)
│ │ │ │ + -- used 0.69176s (cpu); 0.557563s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = 1124
│ │ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ │ - -- used 1.08232s (cpu); 0.827197s (thread); 0s (gc)
│ │ │ │ + -- used 1.09201s (cpu); 0.823976s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 1124
│ │ │ │  i29 : M = ideal(x, y, z);
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : time frobeniusNu(2, M, M^2) -- uses binary search (default)
│ │ │ │ - -- used 2.16055s (cpu); 1.60855s (thread); 0s (gc)
│ │ │ │ + -- used 1.46509s (cpu); 1.1744s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 97
│ │ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets
│ │ │ │  luckier
│ │ │ │ - -- used 0.636757s (cpu); 0.499453s (thread); 0s (gc)
│ │ │ │ + -- used 0.492991s (cpu); 0.422329s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 97
│ │ │ │  The option AtOrigin (default value true) can be turned off to tell frobeniusNu
│ │ │ │  to effectively do the computation over all possible maximal ideals $J$ and take
│ │ │ │  the minimum.
│ │ │ │  i32 : R = ZZ/7[x,y];
│ │ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/FunctionFieldDesingularization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  YXJjcw==
│ │ │  #:len=3089
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJpbnRzIG5vZGUgbGFiZWxzIGZvciB0
│ │ │  aGUgZGVzaW5ndWxhcml6YXRpb24gdHJlZSIsICJsaW5lbnVtIiA9PiA2NTIsIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  UlJFRk1ldGhvZA==
│ │ │  #:len=209
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjEzMCwgInVuZG9jdW1lbnRlZCIgPT4g
│ │ │  dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiUlJFRk1l
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out
│ │ │ @@ -208,21 +208,21 @@
│ │ │        | 3/7 5/4 3/7  10   |
│ │ │        | 6/7 2/9 5    3/2  |
│ │ │  
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
│ │ │  
│ │ │  i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.68501s (cpu); 0.426849s (thread); 0s (gc)
│ │ │ + -- used 6.68513s (cpu); 1.12502s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
│ │ │  
│ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 2.0599s (cpu); 1.05036s (thread); 0s (gc)
│ │ │ + -- used 3.08113s (cpu); 0.992653s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
│ │ │  
│ │ │  i29 :
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html
│ │ │ @@ -391,25 +391,25 @@
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.68501s (cpu); 0.426849s (thread); 0s (gc)
│ │ │ + -- used 6.68513s (cpu); 1.12502s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 2.0599s (cpu); 1.05036s (thread); 0s (gc)
│ │ │ + -- used 3.08113s (cpu); 0.992653s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -241,21 +241,21 @@ │ │ │ │ o26 = | 7 6 3/10 10/9 | │ │ │ │ | 3/7 5/4 3/7 10 | │ │ │ │ | 6/7 2/9 5 3/2 | │ │ │ │ │ │ │ │ 3 4 │ │ │ │ o26 : Matrix QQ <-- QQ │ │ │ │ i27 : time C = orbitClosure(X,Mat) │ │ │ │ - -- used 0.68501s (cpu); 0.426849s (thread); 0s (gc) │ │ │ │ + -- used 6.68513s (cpu); 1.12502s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o27 : KClass │ │ │ │ i28 : time C = orbitClosure(X,Mat, RREFMethod => true) │ │ │ │ - -- used 2.0599s (cpu); 1.05036s (thread); 0s (gc) │ │ │ │ + -- used 3.08113s (cpu); 0.992653s (thread); 0s (gc) │ │ │ │ │ │ │ │ o28 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o28 : KClass │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_a_l_i_z_e_d_F_l_a_g_V_a_r_i_e_t_y -- makes a generalized flag variety as a GKM │ │ │ │ variety │ │ ├── ./usr/share/doc/Macaulay2/GameTheory/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ cHJvYmFiaWxpdHlSaW5nKExpc3Qp │ │ │ #:len=262 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU0Niwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocHJvYmFiaWxpdHlSaW5nLExpc3QpLCJwcm9iYWJp │ │ ├── ./usr/share/doc/Macaulay2/GenericInitialIdeal/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ Z2luKElkZWFsKQ== │ │ │ #:len=242 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTczLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnaW4sSWRlYWwpLCJnaW4oSWRlYWwpIiwiR2VuZXJp │ │ ├── ./usr/share/doc/Macaulay2/GeometricDecomposability/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ SXNJZGVhbFVubWl4ZWQ= │ │ │ #:len=1529 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3BlY2lmeSB3aGV0aGVyIGFuIGlkZWFs │ │ │ IGlzIHVubWl4ZWQiLCAibGluZW51bSIgPT4gMTkwNCwgU2VlQWxzbyA9PiBESVZ7SEVBREVSMnsi │ │ ├── ./usr/share/doc/Macaulay2/GradedLieAlgebras/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ WlogXyBFeHRBbGdlYnJh │ │ │ #:len=932 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2V0IHRoZSB6ZXJvIGVsZW1lbnQiLCBE │ │ │ ZXNjcmlwdGlvbiA9PiAoRElWe0hFQURFUjJ7IlN5bm9wc2lzIn0sVUx7TEl7REx7ImNsYXNzIiA9 │ │ ├── ./usr/share/doc/Macaulay2/GraphicalModels/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ VmFyaWFibGVOYW1l │ │ │ #:len=616 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl │ │ │ IGluZGV0ZXJtaW5hdGUgbmFtZSBpbiBtYXJrb3ZSaW5nIiwgImxpbmVudW0iID0+IDE4ODgsIFNl │ │ ├── ./usr/share/doc/Macaulay2/GraphicalModelsMLE/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ c29sdmVyTUxFKExpc3QsR3JhcGgp │ │ │ #:len=274 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk1OCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc29sdmVyTUxFLExpc3QsR3JhcGgpLCJzb2x2ZXJN │ │ ├── ./usr/share/doc/Macaulay2/Graphics/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ cGljdHVyZVpvbmUoU3BoZXJlKQ== │ │ │ #:len=747 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZCB0aGUgem9uZSB0aGF0IGNvbnRh │ │ │ aW5zIHRoZSBzcGhlcmUiLCAibGluZW51bSIgPT4gMTExNywgSW5wdXRzID0+IHtTUEFOe1RUeyJz │ │ ├── ./usr/share/doc/Macaulay2/Graphs/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=25 │ │ │ cmV2ZXJzZUJyZWFkdGhGaXJzdFNlYXJjaA== │ │ │ #:len=1565 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicnVucyBhIHJldmVyc2UgYnJlYWR0aCBm │ │ │ aXJzdCBzZWFyY2ggb24gdGhlIGRpZ3JhcGggc3RhcnRpbmcgYXQgYSBzcGVjaWZpZWQgbm9kZSIs │ │ ├── ./usr/share/doc/Macaulay2/Graphs/example-output/_new__Digraph.out │ │ │ @@ -32,12 +32,12 @@ │ │ │ 5 => {6} │ │ │ 6 => {} │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ i3 : keys H │ │ │ │ │ │ -o3 = {digraph, map, newDigraph} │ │ │ +o3 = {map, newDigraph, digraph} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Graphs/html/_new__Digraph.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : keys H
│ │ │  
│ │ │ -o3 = {digraph, map, newDigraph}
│ │ │ +o3 = {map, newDigraph, digraph}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 4 => {} │ │ │ │ 5 => {6} │ │ │ │ 6 => {} │ │ │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ i3 : keys H │ │ │ │ │ │ │ │ -o3 = {digraph, map, newDigraph} │ │ │ │ +o3 = {map, newDigraph, digraph} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_o_p_S_o_r_t -- topologically sort the vertices of a digraph │ │ │ │ * _S_o_r_t_e_d_D_i_g_r_a_p_h -- hashtable used in topSort │ │ │ │ * _t_o_p_o_l_o_g_i_c_a_l_S_o_r_t -- outputs a list of vertices in a topologically sorted │ │ │ │ order of a DAG. │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ ZmluZFdlaWdodENvbnN0cmFpbnRz │ │ │ #:len=2630 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyBhIG1hdHJpeCBvZiB3ZWln │ │ │ aHQgY29uc3RyYWludHMiLCAibGluZW51bSIgPT4gNjkxLCBJbnB1dHMgPT4ge1NQQU57VFR7Ik0i │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/___Groebner__Strata.out │ │ │ @@ -243,67 +243,72 @@ │ │ │ | 2 2 2 2 2 2 2 2 2 2 2 2 | │ │ │ |t - t + t t t - t t + t t t + t t t - t t t t + 25t t + 4t t t - 2t t t t - 2t t t t - 2t t t t + t t t t + t t t t - 3t t + 3t t t t - 26t t t | │ │ │ | 6 23 16 20 22 14 22 23 22 13 23 16 19 16 22 13 19 16 19 23 20 21 20 22 13 21 16 20 19 21 23 13 19 21 22 13 19 21 16 13 19 21 20 21 20 13 19 21 13 19 21| │ │ │ +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i12 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o12 = | -26 -24 6 39 17 28 -46 -12 11 -29 -10 -48 -36 -30 39 -29 -8 -22 -10 │ │ │ +o12 = | 37 -2 -19 -4 -13 20 50 -34 -12 -8 23 38 -36 -10 32 19 -22 -29 19 11 │ │ │ ----------------------------------------------------------------------- │ │ │ - 11 19 19 24 -29 | │ │ │ + -30 -29 24 -29 | │ │ │ │ │ │ 1 24 │ │ │ o12 : Matrix kk <-- kk │ │ │ │ │ │ i13 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o13 = | -30 -41 -39 30 -13 32 17 -12 29 25 -49 11 43 21 -42 -4 19 -47 39 -38 │ │ │ +o13 = | -4 -6 -29 43 12 -8 -47 31 -19 -3 -17 -2 -49 21 11 3 -47 19 -16 -38 39 │ │ │ ----------------------------------------------------------------------- │ │ │ - -16 0 -24 34 | │ │ │ + 0 -24 34 | │ │ │ │ │ │ 1 24 │ │ │ o13 : Matrix kk <-- kk │ │ │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o14 = ideal (a + 11b*c + 39c - 29a*d + 17b*d - 24c*d - 26d , a*b + 39b*c - │ │ │ + 2 2 2 │ │ │ +o14 = ideal (a - 12b*c - 4c - 8a*d - 13b*d - 2c*d + 37d , a*b + 32b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 10c - 29a*d - 48b*d + 28c*d + 6d , b + 19b*c - 8c + 19a*d - 22b*d - │ │ │ + 2 2 2 2 │ │ │ + 23c + 19a*d + 38b*d + 20c*d - 19d , b - 30b*c - 22c - 29a*d - 29b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 36c*d - 46d , a*c + 24b*c - 10c - 29a*d + 11b*d - 30c*d - 12d ) │ │ │ + 2 2 2 │ │ │ + - 36c*d + 50d , a*c + 24b*c + 19c - 29a*d + 11b*d - 10c*d - 34d ) │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ │ │ i15 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o15 = ideal (a + 29b*c + 30c + 25a*d - 13b*d - 41c*d - 30d , a*b - 42b*c - │ │ │ + 2 2 2 │ │ │ +o15 = ideal (a - 19b*c + 43c - 3a*d + 12b*d - 6c*d - 4d , a*b + 11b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 49c - 4a*d + 11b*d + 32c*d - 39d , b - 16b*c + 19c - 47b*d + 43c*d + │ │ │ + 2 2 2 2 │ │ │ + 17c + 3a*d - 2b*d - 8c*d - 29d , b + 39b*c - 47c + 19b*d - 49c*d - │ │ │ ----------------------------------------------------------------------- │ │ │ 2 2 2 │ │ │ - 17d , a*c - 24b*c + 39c + 34a*d - 38b*d + 21c*d - 12d ) │ │ │ + 47d , a*c - 24b*c - 16c + 34a*d - 38b*d + 21c*d + 31d ) │ │ │ │ │ │ o15 : Ideal of S │ │ │ │ │ │ i16 : decompose F1 │ │ │ │ │ │ - 2 2 2 │ │ │ -o16 = {ideal (a + 24b - 10c - 17d, b + 19b*c - 8c + 27b*d - 48c*d - 26d ), │ │ │ + 2 2 2 │ │ │ +o16 = {ideal (a + 24b + 19c + 36d, b - 30b*c - 22c - 40b*d + 10c*d - 17d ), │ │ │ ----------------------------------------------------------------------- │ │ │ - ideal (c - 29d, b + 23d, a - 14d)} │ │ │ + ideal (c - 29d, b + 30d, a - 38d)} │ │ │ │ │ │ o16 : List │ │ │ │ │ │ i17 : decompose F2 │ │ │ │ │ │ -o17 = {ideal (b - 42c - 18d, a + 41c + 33d), ideal (b + 26c - 29d, a - 44c - │ │ │ + 2 2 2 │ │ │ +o17 = {ideal (a*c - 24b*c - 16c + 34a*d - 38b*d + 21c*d + 31d , b + 39b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 9d)} │ │ │ + 2 2 2 │ │ │ + 47c + 19b*d - 49c*d - 47d , a*b + 11b*c - 17c + 3a*d - 2b*d - 8c*d - │ │ │ + ----------------------------------------------------------------------- │ │ │ + 2 2 2 2 │ │ │ + 29d , a - 19b*c + 43c - 3a*d + 12b*d - 6c*d - 4d )} │ │ │ │ │ │ o17 : List │ │ │ │ │ │ i18 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_nonminimal__Maps.out │ │ │ @@ -103,46 +103,46 @@ │ │ │ │ │ │ i13 : #compsJ │ │ │ │ │ │ o13 = 2 │ │ │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o14 = | -48 -21 -28 -32 16 31 15 -12 14 5 -13 29 -40 -22 2 30 25 -10 3 -1 3 │ │ │ +o14 = | -38 27 -21 -39 -14 16 35 -2 9 -17 -14 49 -46 -10 31 22 1 19 1 -18 18 │ │ │ ----------------------------------------------------------------------- │ │ │ - -30 19 -16 -24 -29 -29 24 -29 -24 -36 21 -8 -38 19 39 | │ │ │ + 24 -30 -24 12 -29 -36 39 19 -8 21 -16 -29 -22 -29 -38 | │ │ │ │ │ │ 1 36 │ │ │ o14 : Matrix kk <-- kk │ │ │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o15 = | 6 29 -50 41 50 -19 -15 -34 -28 -44 21 -45 -23 -13 -23 25 31 -18 45 22 │ │ │ +o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16 │ │ │ ----------------------------------------------------------------------- │ │ │ - -40 17 -47 38 -25 -43 19 26 -47 -15 34 16 0 -28 -39 2 | │ │ │ + -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 | │ │ │ │ │ │ 1 36 │ │ │ o15 : Matrix kk <-- kk │ │ │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o16 = ideal (a + 2b*c + 5c + 29a*d + 31b*d - 28c*d - 48d , a*b - 24b*c + │ │ │ + 2 2 2 │ │ │ +o16 = ideal (a + 31b*c - 17c + 49a*d + 16b*d - 21c*d - 38d , a*b + 12b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 3c + 3a*d - 40b*d + 14c*d - 21d , a*c + 21b*c - 29c + 24a*d - b*d - │ │ │ + 2 2 2 │ │ │ + c + 18a*d - 46b*d + 9c*d + 27d , a*c - 16b*c - 29c + 39a*d - 18b*d - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 2 │ │ │ - 22c*d + 16d , b + 19b*c - 29c - 8a*d - 30b*d + 30c*d + 15d , b*c - │ │ │ + 2 2 2 2 2 │ │ │ + 10c*d - 14d , b - 29b*c + 19c - 29a*d + 24b*d + 22c*d + 35d , b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 3 3 2 │ │ │ - 24b*c*d + 19c d - 29a*d + 25b*d - 13c*d - 32d , c + 39b*c*d - 36c d │ │ │ + 2 2 2 2 3 3 2 │ │ │ + 8b*c*d - 30c d - 36a*d + b*d - 14c*d - 39d , c - 38b*c*d + 21c d - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 3 │ │ │ - - 38a*d - 16b*d - 10c*d - 12d ) │ │ │ + 2 2 2 3 │ │ │ + 22a*d - 24b*d + 19c*d - 2d ) │ │ │ │ │ │ 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 - 23b*c - 44c - 45a*d - 19b*d - 50c*d + 6d , a*b - 25b*c + │ │ │ + 2 2 2 │ │ │ +o18 = ideal (a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 45c - 40a*d - 23b*d - 28c*d + 29d , a*c + 16b*c - 43c + 26a*d + 22b*d │ │ │ + 2 2 2 │ │ │ + 45c - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c + 33a*d + 16b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 2 │ │ │ - - 13c*d + 50d , b - 39b*c - 47c + 17b*d + 25c*d - 15d , b*c - │ │ │ + 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 3 3 2 │ │ │ - 15b*c*d - 47c d + 19a*d + 31b*d + 21c*d + 41d , c + 2b*c*d + 34c 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 3 │ │ │ - - 28a*d + 38b*d - 18c*d - 34d ) │ │ │ + 2 2 3 │ │ │ + - 47b*d - 39c*d + 40d ) │ │ │ │ │ │ o18 : Ideal of S │ │ │ │ │ │ i19 : betti res F2 │ │ │ │ │ │ 0 1 2 3 │ │ │ o19 = total: 1 6 8 3 │ │ │ @@ -179,27 +179,24 @@ │ │ │ 1: . 4 4 1 │ │ │ 2: . 2 4 2 │ │ │ │ │ │ o19 : BettiTally │ │ │ │ │ │ i20 : netList decompose F1 │ │ │ │ │ │ - +-------------------------------------------------------+ │ │ │ -o20 = |ideal (c - 49d, b - 6d, a - 15d) | │ │ │ - +-------------------------------------------------------+ │ │ │ - |ideal (c + 24d, b + 37d, a + 35d) | │ │ │ - +-------------------------------------------------------+ │ │ │ - | 2 2 | │ │ │ - |ideal (b - 18c - 32d, a + 46c + 33d, c - 26c*d - 15d )| │ │ │ - +-------------------------------------------------------+ │ │ │ - | 2 2 | │ │ │ - |ideal (b - 37c - 27d, a + 41c + 29d, c + 15c*d + 16d )| │ │ │ - +-------------------------------------------------------+ │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ +o20 = |ideal (c - d, b + 48d, a + 42d) | │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ + |ideal (c + 39d, b - 23d, a + 34d) | │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 16b - 29c + 10d, b*c - 38c + 23b*d + 43c*d - 16d , b + 28c + 25b*d + 24c*d - 38d )| │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ 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 + 16b*c - 43c + 26a*d + 22b*d - 13c*d + 50d , b - 39b*c - 47c + 17b*d + 25c*d - 15d , a*b - 25b*c + 45c - 40a*d - 23b*d - 28c*d + 29d , a - 23b*c - 44c - 45a*d - 19b*d - 50c*d + 6d , c + 2b*c*d + 34c d - 28a*d + 38b*d - 18c*d - 34d , b*c - 15b*c*d - 47c d + 19a*d + 31b*d + 21c*d + 41d )| │ │ │ - +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 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 )| │ │ │ + +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/example-output/_random__Point__On__Rational__Variety_lp__Ideal_rp.out │ │ │ @@ -12,15 +12,15 @@ │ │ │ 2 │ │ │ c*e + b*f, - c*d + b*e, - c*e + b*f, - e + d*f) │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ i4 : pt = randomPointOnRationalVariety I │ │ │ │ │ │ -o4 = | -25 20 -30 -16 24 -36 | │ │ │ +o4 = | 1 49 24 -23 -36 -30 | │ │ │ │ │ │ 1 6 │ │ │ o4 : Matrix kk <-- kk │ │ │ │ │ │ i5 : sub(I, pt) == 0 │ │ │ │ │ │ o5 = true │ │ │ @@ -200,67 +200,72 @@ │ │ │ │ │ │ o12 = {11, 8} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ -o13 = | -46 -11 28 49 -48 -26 38 -46 -45 19 -32 17 -10 -29 17 -8 -29 -9 -22 │ │ │ +o13 = | -42 12 -9 -46 -17 -41 -23 -6 12 -38 21 -11 19 -29 -3 -22 -16 -13 -8 │ │ │ ----------------------------------------------------------------------- │ │ │ - -24 -29 -38 19 -16 | │ │ │ + 19 -10 -29 -29 -24 | │ │ │ │ │ │ 1 24 │ │ │ o13 : Matrix kk <-- kk │ │ │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ - 2 2 2 │ │ │ -o14 = ideal (a - 45b*c + 49c + 19a*d - 48b*d - 11c*d - 46d , a*b + 17b*c - │ │ │ + 2 2 2 │ │ │ +o14 = ideal (a + 12b*c - 46c - 38a*d - 17b*d + 12c*d - 42d , a*b - 3b*c + │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 32c - 8a*d + 17b*d - 26c*d + 28d , a*c - 29b*c - 29c - 38a*d - 9b*d - │ │ │ + 2 2 2 │ │ │ + 21c - 22a*d - 11b*d - 41c*d - 9d , a*c - 10b*c - 16c - 29a*d - 13b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - 10c*d + 38d , b + 19b*c - 22c - 16a*d - 24b*d - 29c*d - 46d ) │ │ │ + 2 2 2 2 │ │ │ + + 19c*d - 23d , b - 29b*c - 8c - 24a*d + 19b*d - 29c*d - 6d ) │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ - 2 2 2 │ │ │ -o15 = {ideal (a - 29b - 29c - d, b + 19b*c - 22c + 17b*d + 12c*d + 39d ), │ │ │ + 2 2 2 │ │ │ +o15 = {ideal (a - 10b - 16c - 41d, b - 29b*c - 8c - 19b*d - 9c*d + 20d ), │ │ │ ----------------------------------------------------------------------- │ │ │ - ideal (c - 38d, b - 49d, a + 41d)} │ │ │ + ideal (c - 29d, b + 16d, a - 20d)} │ │ │ │ │ │ o15 : List │ │ │ │ │ │ i16 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ -o16 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39 │ │ │ +o16 = | -6 -13 21 -15 -31 6 -9 -1 50 -39 -18 21 -39 31 -31 44 -18 -13 -47 21 │ │ │ ----------------------------------------------------------------------- │ │ │ - -13 -18 34 0 | │ │ │ + 34 19 39 0 | │ │ │ │ │ │ 1 24 │ │ │ o16 : Matrix kk <-- kk │ │ │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ - 2 2 2 │ │ │ -o17 = ideal (a - 7b*c + 30c - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c - │ │ │ + 2 2 2 │ │ │ +o17 = ideal (a + 50b*c - 15c - 39a*d - 31b*d - 13c*d - 6d , a*b - 31b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 │ │ │ - 5c + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c - 18a*d + 21b*d │ │ │ + 2 2 2 │ │ │ + 18c + 44a*d + 21b*d + 6c*d + 21d , a*c + 34b*c - 18c + 19a*d - 13b*d │ │ │ ----------------------------------------------------------------------- │ │ │ - 2 2 2 2 │ │ │ - + 19c*d - 20d , b + 34b*c - 47c - 39b*d + 20c*d - 35d ) │ │ │ + 2 2 2 2 │ │ │ + - 39c*d - 9d , b + 39b*c - 47c + 21b*d + 31c*d - d ) │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ -o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c - │ │ │ + 2 2 2 │ │ │ +o18 = {ideal (a*c + 34b*c - 18c + 19a*d - 13b*d - 39c*d - 9d , b + 39b*c - │ │ │ ----------------------------------------------------------------------- │ │ │ - 5d)} │ │ │ + 2 2 2 │ │ │ + 47c + 21b*d + 31c*d - d , a*b - 31b*c - 18c + 44a*d + 21b*d + 6c*d + │ │ │ + ----------------------------------------------------------------------- │ │ │ + 2 2 2 2 │ │ │ + 21d , a + 50b*c - 15c - 39a*d - 31b*d - 13c*d - 6d )} │ │ │ │ │ │ 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 = || 42 -50 -50 38 -39 6 -1 47 49 -18 18 -28 -47 19 48 34 -13 11 -16 -38 39 21 -43 -39 | | │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + || -44 -31 -9 -21 -42 -36 -47 -20 38 -34 -3 -43 22 16 -35 2 -48 32 -28 -15 -47 38 -47 45 || │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + || 45 -29 -1 -4 42 -35 4 -13 18 -17 1 21 39 -23 50 15 -11 -11 19 47 -16 7 48 43 | | │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + || 35 -44 -33 -8 21 -2 -44 -20 19 -28 19 27 11 40 34 33 1 -14 35 36 11 -38 -3 46 | | │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + || 33 47 46 16 -22 -25 -44 -36 -30 -37 30 -25 -47 29 -41 2 -13 -41 -47 22 -23 -7 -10 15 | | │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + || 43 1 -41 23 -42 -14 37 -50 -32 -20 -5 -49 -9 32 -18 -22 24 43 -18 30 39 27 -30 -32 | | │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + || 12 8 26 15 22 12 0 -5 6 17 -21 -18 -33 -49 -19 33 -20 0 -15 -48 39 0 44 -19 | | │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + || -48 -27 -8 -33 -35 -16 -31 -44 -46 -49 -21 -3 -26 13 -40 4 -11 -48 36 -39 9 -39 -8 22 || │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + || 34 29 30 -1 34 -22 32 -6 39 -28 -15 37 41 -30 47 -22 -6 -7 -8 43 36 -3 35 16 | | │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + || 10 -5 -38 -25 -21 21 14 42 19 -41 8 -35 25 -31 29 3 -49 38 -35 -9 6 40 -13 -2 | | │ │ │ + +-----------------------------------------------------------------------------------------+ │ │ │ + │ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) │ │ │ + │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ -o13 = || -6 -26 42 -37 -23 28 29 -18 -3 -16 5 23 34 19 -32 -13 -38 15 -18 21 -39 -47 39 -43 | | │ │ │ +o14 = || 1 4 44 1 40 20 29 -26 42 -16 -4 50 -40 48 23 -34 -35 37 30 4 -47 27 -31 0 | | │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ - || 14 -34 -33 9 27 11 32 41 4 -28 15 -36 2 16 -21 -48 -15 -20 -34 38 45 22 -47 -47 | | │ │ │ + || -29 -11 2 28 -12 -21 -30 20 -1 -42 27 -45 -48 -14 -3 -45 -37 30 -31 -39 -48 -29 47 0 || │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ - || 32 35 11 -9 -27 15 0 -12 7 19 24 7 15 -23 28 -11 47 -40 -17 7 43 39 -16 48 | | │ │ │ + || -20 14 48 3 14 33 18 -46 24 45 -1 -29 1 31 -44 -2 -22 40 28 -49 -18 46 10 0 | | │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ - || -49 -8 -5 -45 -36 -5 47 -21 -34 35 -25 32 33 40 35 1 36 1 -28 -38 46 11 11 -3 | | │ │ │ + || 21 25 -23 -45 -48 -40 8 -7 -6 -1 14 3 -13 -44 -44 21 -41 3 30 7 13 -17 8 0 | | │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ - || 35 14 -9 -22 -14 19 -10 48 23 -47 50 9 2 29 -37 -13 22 2 -37 -7 15 -47 -23 -10 | | │ │ │ + || -15 26 -17 12 0 -5 35 -11 0 -31 -39 40 49 -21 -42 28 42 -18 -29 8 30 -46 23 0 | | │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ - || 35 -49 -37 32 -48 42 -10 -49 42 -18 -34 25 -22 32 -35 24 30 -15 -20 27 -32 -9 39 -30 || │ │ │ + || -17 38 -49 -9 2 -1 -9 2 3 49 33 49 -46 25 1 0 -18 12 15 -28 18 -16 27 0 | | │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ - || -29 -40 30 -5 36 -41 -24 1 34 -15 20 -33 33 -49 -14 -20 -48 21 17 0 -19 -33 39 44 | | │ │ │ + || -50 3 -12 -21 -10 4 36 -14 19 -9 43 -48 44 -35 16 6 20 -39 23 -21 -37 -23 19 0 | | │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ - || -28 27 38 13 36 38 0 35 12 36 -33 -24 4 13 -35 -11 -39 34 -49 -39 22 -26 9 -8 | | │ │ │ + || 36 -17 -27 37 -4 12 42 -16 -15 12 -3 -14 -28 38 -24 18 -9 6 -47 0 -28 47 -33 0 | | │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ - || 8 -20 1 -28 27 -39 40 0 -38 -8 44 -44 -22 -30 -28 -6 43 50 -28 -3 16 41 36 35 | | │ │ │ + || 43 44 31 14 -9 -4 -4 -40 -48 45 47 17 -37 29 -9 10 -28 -33 -29 28 26 5 42 0 | | │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ - || -12 -11 18 -43 28 44 44 5 -12 -35 19 -50 3 -31 3 -49 -9 -50 -41 40 -2 25 6 -13 | | │ │ │ + || 17 -47 -23 8 -18 -45 -25 2 44 41 -19 -37 5 16 2 38 -13 -20 30 44 4 22 -29 0 | | │ │ │ +----------------------------------------------------------------------------------------+ │ │ │ │ │ │ -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 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 || │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 4 0 | | │ │ │ - +-------------------------------------------------------------------------------------+ │ │ │ - │ │ │ i15 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_nonminimal__Maps.html │ │ │ @@ -237,52 +237,52 @@ │ │ │ o13 = 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o14 = | -48 -21 -28 -32 16 31 15 -12 14 5 -13 29 -40 -22 2 30 25 -10 3 -1 3
│ │ │ +o14 = | -38 27 -21 -39 -14 16 35 -2 9 -17 -14 49 -46 -10 31 22 1 19 1 -18 18
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -30 19 -16 -24 -29 -29 24 -29 -24 -36 21 -8 -38 19 39 |
│ │ │ +      24 -30 -24 12 -29 -36 39 19 -8 21 -16 -29 -22 -29 -38 |
│ │ │  
│ │ │                 1       36
│ │ │  o14 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o15 = | 6 29 -50 41 50 -19 -15 -34 -28 -44 21 -45 -23 -13 -23 25 31 -18 45 22
│ │ │ +o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -40 17 -47 38 -25 -43 19 26 -47 -15 34 16 0 -28 -39 2 |
│ │ │ +      -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 |
│ │ │  
│ │ │                 1       36
│ │ │  o15 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2            2                              2               
│ │ │ -o16 = ideal (a  + 2b*c + 5c  + 29a*d + 31b*d - 28c*d - 48d , a*b - 24b*c +
│ │ │ +              2              2                              2               
│ │ │ +o16 = ideal (a  + 31b*c - 17c  + 49a*d + 16b*d - 21c*d - 38d , a*b + 12b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                             2                   2                
│ │ │ -      3c  + 3a*d - 40b*d + 14c*d - 21d , a*c + 21b*c - 29c  + 24a*d - b*d -
│ │ │ +       2                             2                   2                  
│ │ │ +      c  + 18a*d - 46b*d + 9c*d + 27d , a*c - 16b*c - 29c  + 39a*d - 18b*d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                             2     2  
│ │ │ -      22c*d + 16d , b  + 19b*c - 29c  - 8a*d - 30b*d + 30c*d + 15d , b*c  -
│ │ │ +                 2   2              2                              2     2  
│ │ │ +      10c*d - 14d , b  - 29b*c + 19c  - 29a*d + 24b*d + 22c*d + 35d , b*c  -
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2         2        2        2      3   3                2 
│ │ │ -      24b*c*d + 19c d - 29a*d  + 25b*d  - 13c*d  - 32d , c  + 39b*c*d - 36c d
│ │ │ +                  2         2      2        2      3   3                2   
│ │ │ +      8b*c*d - 30c d - 36a*d  + b*d  - 14c*d  - 39d , c  - 38b*c*d + 21c d -
│ │ │        -----------------------------------------------------------------------
│ │ │ -             2        2        2      3
│ │ │ -      - 38a*d  - 16b*d  - 10c*d  - 12d )
│ │ │ +           2        2        2     3
│ │ │ +      22a*d  - 24b*d  + 19c*d  - 2d )
│ │ │  
│ │ │  o16 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : betti res F1
│ │ │ @@ -296,28 +296,28 @@
│ │ │  o17 : BettiTally
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                             2               
│ │ │ -o18 = ideal (a  - 23b*c - 44c  - 45a*d - 19b*d - 50c*d + 6d , a*b - 25b*c +
│ │ │ +              2              2                              2               
│ │ │ +o18 = ideal (a  - 35b*c + 22c  - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                              2                   2                
│ │ │ -      45c  - 40a*d - 23b*d - 28c*d + 29d , a*c + 16b*c - 43c  + 26a*d + 22b*d
│ │ │ +         2                              2                  2                
│ │ │ +      45c  - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c  + 33a*d + 16b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2     2  
│ │ │ -      - 13c*d + 50d , b  - 39b*c - 47c  + 17b*d + 25c*d - 15d , b*c  -
│ │ │ +                   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      3   3               2 
│ │ │ -      15b*c*d - 47c d + 19a*d  + 31b*d  + 21c*d  + 41d , c  + 2b*c*d + 34c 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      3
│ │ │ -      - 28a*d  + 38b*d  - 18c*d  - 34d )
│ │ │ +             2        2      3
│ │ │ +      - 47b*d  - 39c*d  + 40d )
│ │ │  
│ │ │  o18 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i19 : betti res F2
│ │ │ @@ -336,35 +336,32 @@
│ │ │            
    What are the ideals F1 and F2?
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i20 : netList decompose F1
│ │ │  
│ │ │ -      +-------------------------------------------------------+
│ │ │ -o20 = |ideal (c - 49d, b - 6d, a - 15d)                       |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |ideal (c + 24d, b + 37d, a + 35d)                      |
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 18c - 32d, a + 46c + 33d, c  - 26c*d - 15d )|
│ │ │ -      +-------------------------------------------------------+
│ │ │ -      |                                      2              2 |
│ │ │ -      |ideal (b - 37c - 27d, a + 41c + 29d, c  + 15c*d + 16d )|
│ │ │ -      +-------------------------------------------------------+
    │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ +o20 = |ideal (c - d, b + 48d, a + 42d) | │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ + |ideal (c + 39d, b - 23d, a + 34d) | │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │ + | 2 2 2 2 2 | │ │ │ + |ideal (a - 16b - 29c + 10d, b*c - 38c + 23b*d + 43c*d - 16d , b + 28c + 25b*d + 24c*d - 38d )| │ │ │ + +------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │ 
    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 + 16b*c - 43c  + 26a*d + 22b*d - 13c*d + 50d , b  - 39b*c - 47c  + 17b*d + 25c*d - 15d , a*b - 25b*c + 45c  - 40a*d - 23b*d - 28c*d + 29d , a  - 23b*c - 44c  - 45a*d - 19b*d - 50c*d + 6d , c  + 2b*c*d + 34c d - 28a*d  + 38b*d  - 18c*d  - 34d , b*c  - 15b*c*d - 47c d + 19a*d  + 31b*d  + 21c*d  + 41d )|
│ │ │ -      +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
    │ │ │ + +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │ + | 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 )| │ │ │ + +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ │ │ │
    │ │ │
    │ │ │

    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 +170,122 @@ │ │ │ │ 32 13 21 33 19 31 │ │ │ │ i12 : compsJ = decompose J; │ │ │ │ i13 : #compsJ │ │ │ │ │ │ │ │ o13 = 2 │ │ │ │ i14 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o14 = | -48 -21 -28 -32 16 31 15 -12 14 5 -13 29 -40 -22 2 30 25 -10 3 -1 3 │ │ │ │ +o14 = | -38 27 -21 -39 -14 16 35 -2 9 -17 -14 49 -46 -10 31 22 1 19 1 -18 18 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -30 19 -16 -24 -29 -29 24 -29 -24 -36 21 -8 -38 19 39 | │ │ │ │ + 24 -30 -24 12 -29 -36 39 19 -8 21 -16 -29 -22 -29 -38 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o14 : Matrix kk <-- kk │ │ │ │ i15 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o15 = | 6 29 -50 41 50 -19 -15 -34 -28 -44 21 -45 -23 -13 -23 25 31 -18 45 22 │ │ │ │ +o15 = | 30 10 43 20 -39 23 -30 40 -34 22 46 -25 21 -18 -35 -1 21 -39 -45 16 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -40 17 -47 38 -25 -43 19 26 -47 -15 34 16 0 -28 -39 2 | │ │ │ │ + -35 -5 19 -47 -20 -13 34 33 -28 -43 22 2 0 -15 -47 38 | │ │ │ │ │ │ │ │ 1 36 │ │ │ │ o15 : Matrix kk <-- kk │ │ │ │ i16 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o16 = ideal (a + 2b*c + 5c + 29a*d + 31b*d - 28c*d - 48d , a*b - 24b*c + │ │ │ │ + 2 2 2 │ │ │ │ +o16 = ideal (a + 31b*c - 17c + 49a*d + 16b*d - 21c*d - 38d , a*b + 12b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 3c + 3a*d - 40b*d + 14c*d - 21d , a*c + 21b*c - 29c + 24a*d - b*d - │ │ │ │ + 2 2 2 │ │ │ │ + c + 18a*d - 46b*d + 9c*d + 27d , a*c - 16b*c - 29c + 39a*d - 18b*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 2 │ │ │ │ - 22c*d + 16d , b + 19b*c - 29c - 8a*d - 30b*d + 30c*d + 15d , b*c - │ │ │ │ + 2 2 2 2 2 │ │ │ │ + 10c*d - 14d , b - 29b*c + 19c - 29a*d + 24b*d + 22c*d + 35d , b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 3 3 2 │ │ │ │ - 24b*c*d + 19c d - 29a*d + 25b*d - 13c*d - 32d , c + 39b*c*d - 36c d │ │ │ │ + 2 2 2 2 3 3 2 │ │ │ │ + 8b*c*d - 30c d - 36a*d + b*d - 14c*d - 39d , c - 38b*c*d + 21c d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 3 │ │ │ │ - - 38a*d - 16b*d - 10c*d - 12d ) │ │ │ │ + 2 2 2 3 │ │ │ │ + 22a*d - 24b*d + 19c*d - 2d ) │ │ │ │ │ │ │ │ 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 - 23b*c - 44c - 45a*d - 19b*d - 50c*d + 6d , a*b - 25b*c + │ │ │ │ + 2 2 2 │ │ │ │ +o18 = ideal (a - 35b*c + 22c - 25a*d + 23b*d + 43c*d + 30d , a*b - 20b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 45c - 40a*d - 23b*d - 28c*d + 29d , a*c + 16b*c - 43c + 26a*d + 22b*d │ │ │ │ + 2 2 2 │ │ │ │ + 45c - 35a*d + 21b*d - 34c*d + 10d , a*c + 2b*c - 13c + 33a*d + 16b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 2 │ │ │ │ - - 13c*d + 50d , b - 39b*c - 47c + 17b*d + 25c*d - 15d , b*c - │ │ │ │ + 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 3 3 2 │ │ │ │ - 15b*c*d - 47c d + 19a*d + 31b*d + 21c*d + 41d , c + 2b*c*d + 34c 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 3 │ │ │ │ - - 28a*d + 38b*d - 18c*d - 34d ) │ │ │ │ + 2 2 3 │ │ │ │ + - 47b*d - 39c*d + 40d ) │ │ │ │ │ │ │ │ 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 - 49d, b - 6d, a - 15d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - |ideal (c + 24d, b + 37d, a + 35d) | │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - | 2 2 | │ │ │ │ - |ideal (b - 18c - 32d, a + 46c + 33d, c - 26c*d - 15d )| │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ - | 2 2 | │ │ │ │ - |ideal (b - 37c - 27d, a + 41c + 29d, c + 15c*d + 16d )| │ │ │ │ - +-------------------------------------------------------+ │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +------------------------+ │ │ │ │ +o20 = |ideal (c - d, b + 48d, a + 42d) │ │ │ │ +| │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +------------------------+ │ │ │ │ + |ideal (c + 39d, b - 23d, a + 34d) │ │ │ │ +| │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +------------------------+ │ │ │ │ + | 2 2 2 2 │ │ │ │ +2 | │ │ │ │ + |ideal (a - 16b - 29c + 10d, b*c - 38c + 23b*d + 43c*d - 16d , b + 28c │ │ │ │ ++ 25b*d + 24c*d - 38d )| │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +------------------------+ │ │ │ │ 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 + 16b*c - 43c + 26a*d + 22b*d - 13c*d + 50d , b - 39b*c - │ │ │ │ -47c + 17b*d + 25c*d - 15d , a*b - 25b*c + 45c - 40a*d - 23b*d - 28c*d + 29d , │ │ │ │ -a - 23b*c - 44c - 45a*d - 19b*d - 50c*d + 6d , c + 2b*c*d + 34c d - 28a*d + │ │ │ │ -38b*d - 18c*d - 34d , b*c - 15b*c*d - 47c d + 19a*d + 31b*d + 21c*d + 41d │ │ │ │ +-+ │ │ │ │ + | 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 │ │ │ │ )| │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----+ │ │ │ │ +-+ │ │ │ │ We can determine what these represent. One should be a set of 6 points, where 5 │ │ │ │ lie on a plane. The other should be 6 points with 3 points on one line, and the │ │ │ │ other 3 points on a skew line. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_P_o_i_n_t_O_n_R_a_t_i_o_n_a_l_V_a_r_i_e_t_y -- find a random point on a variety that can │ │ │ │ be detected to be rational │ │ │ │ ********** WWaayyss ttoo uussee nnoonnmmiinniimmaallMMaappss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/GroebnerStrata/html/_random__Point__On__Rational__Variety_lp__Ideal_rp.html │ │ │ @@ -101,15 +101,15 @@ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : pt = randomPointOnRationalVariety I
│ │ │  
│ │ │ -o4 = | -25 20 -30 -16 24 -36 |
│ │ │ +o4 = | 1 49 24 -23 -36 -30 |
│ │ │  
│ │ │                1       6
│ │ │  o4 : Matrix kk  <-- kk
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -323,90 +323,95 @@ │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i13 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o13 = | -46 -11 28 49 -48 -26 38 -46 -45 19 -32 17 -10 -29 17 -8 -29 -9 -22
│ │ │ +o13 = | -42 12 -9 -46 -17 -41 -23 -6 12 -38 21 -11 19 -29 -3 -22 -16 -13 -8
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -24 -29 -38 19 -16 |
│ │ │ +      19 -10 -29 -29 -24 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o14 = ideal (a  - 45b*c + 49c  + 19a*d - 48b*d - 11c*d - 46d , a*b + 17b*c -
│ │ │ +              2              2                              2              
│ │ │ +o14 = ideal (a  + 12b*c - 46c  - 38a*d - 17b*d + 12c*d - 42d , a*b - 3b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2                   2                 
│ │ │ -      32c  - 8a*d + 17b*d - 26c*d + 28d , a*c - 29b*c - 29c  - 38a*d - 9b*d -
│ │ │ +         2                             2                   2                
│ │ │ +      21c  - 22a*d - 11b*d - 41c*d - 9d , a*c - 10b*c - 16c  - 29a*d - 13b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2   2              2                              2
│ │ │ -      10c*d + 38d , b  + 19b*c - 22c  - 16a*d - 24b*d - 29c*d - 46d )
│ │ │ +                   2   2             2                             2
│ │ │ +      + 19c*d - 23d , b  - 29b*c - 8c  - 24a*d + 19b*d - 29c*d - 6d )
│ │ │  
│ │ │  o14 : Ideal of S
    │ │ │
    │ │ │ 
    i15 : decompose F1
│ │ │  
│ │ │ -                                  2              2                      2
│ │ │ -o15 = {ideal (a - 29b - 29c - d, b  + 19b*c - 22c  + 17b*d + 12c*d + 39d ),
│ │ │ +                                    2             2                     2
│ │ │ +o15 = {ideal (a - 10b - 16c - 41d, b  - 29b*c - 8c  - 19b*d - 9c*d + 20d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 38d, b - 49d, a + 41d)}
│ │ │ +      ideal (c - 29d, b + 16d, a - 20d)}
│ │ │  
│ │ │  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 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39
│ │ │ +o16 = | -6 -13 21 -15 -31 6 -9 -1 50 -39 -18 21 -39 31 -31 44 -18 -13 -47 21
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -13 -18 34 0 |
│ │ │ +      34 19 39 0 |
│ │ │  
│ │ │                 1       24
│ │ │  o16 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i17 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2             2                             2               
│ │ │ -o17 = ideal (a  - 7b*c + 30c  - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c -
│ │ │ +              2              2                             2               
│ │ │ +o17 = ideal (a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d , a*b - 31b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -        2                              2                   2                
│ │ │ -      5c  + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c  - 18a*d + 21b*d
│ │ │ +         2                             2                   2                
│ │ │ +      18c  + 44a*d + 21b*d + 6c*d + 21d , a*c + 34b*c - 18c  + 19a*d - 13b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                   2   2              2                      2
│ │ │ -      + 19c*d - 20d , b  + 34b*c - 47c  - 39b*d + 20c*d - 35d )
│ │ │ +                  2   2              2                    2
│ │ │ +      - 39c*d - 9d , b  + 39b*c - 47c  + 21b*d + 31c*d - d )
│ │ │  
│ │ │  o17 : Ideal of S
    │ │ │
    │ │ │ 
    i18 : decompose F2
│ │ │  
│ │ │ -o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c -
│ │ │ +                               2                             2   2          
│ │ │ +o18 = {ideal (a*c + 34b*c - 18c  + 19a*d - 13b*d - 39c*d - 9d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      5d)}
│ │ │ +         2                    2                   2                         
│ │ │ +      47c  + 21b*d + 31c*d - d , a*b - 31b*c - 18c  + 44a*d + 21b*d + 6c*d +
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         2   2              2                             2
│ │ │ +      21d , a  + 50b*c - 15c  - 39a*d - 31b*d - 13c*d - 6d )}
│ │ │  
│ │ │  o18 : List
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │ ├── html2text {} │ │ │ │ @@ -28,15 +28,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 │ │ │ │ c*e + b*f, - c*d + b*e, - c*e + b*f, - e + d*f) │ │ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ i4 : pt = randomPointOnRationalVariety I │ │ │ │ │ │ │ │ -o4 = | -25 20 -30 -16 24 -36 | │ │ │ │ +o4 = | 1 49 24 -23 -36 -30 | │ │ │ │ │ │ │ │ 1 6 │ │ │ │ o4 : Matrix kk <-- kk │ │ │ │ i5 : sub(I, pt) == 0 │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : S = kk[a..d]; │ │ │ │ @@ -212,67 +212,72 @@ │ │ │ │ │ │ │ │ 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 = | -46 -11 28 49 -48 -26 38 -46 -45 19 -32 17 -10 -29 17 -8 -29 -9 -22 │ │ │ │ +o13 = | -42 12 -9 -46 -17 -41 -23 -6 12 -38 21 -11 19 -29 -3 -22 -16 -13 -8 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -24 -29 -38 19 -16 | │ │ │ │ + 19 -10 -29 -29 -24 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o13 : Matrix kk <-- kk │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ -o14 = ideal (a - 45b*c + 49c + 19a*d - 48b*d - 11c*d - 46d , a*b + 17b*c - │ │ │ │ +o14 = ideal (a + 12b*c - 46c - 38a*d - 17b*d + 12c*d - 42d , a*b - 3b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 │ │ │ │ - 32c - 8a*d + 17b*d - 26c*d + 28d , a*c - 29b*c - 29c - 38a*d - 9b*d - │ │ │ │ + 21c - 22a*d - 11b*d - 41c*d - 9d , a*c - 10b*c - 16c - 29a*d - 13b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 10c*d + 38d , b + 19b*c - 22c - 16a*d - 24b*d - 29c*d - 46d ) │ │ │ │ + 2 2 2 2 │ │ │ │ + + 19c*d - 23d , b - 29b*c - 8c - 24a*d + 19b*d - 29c*d - 6d ) │ │ │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : decompose F1 │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o15 = {ideal (a - 29b - 29c - d, b + 19b*c - 22c + 17b*d + 12c*d + 39d ), │ │ │ │ + 2 2 2 │ │ │ │ +o15 = {ideal (a - 10b - 16c - 41d, b - 29b*c - 8c - 19b*d - 9c*d + 20d ), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - ideal (c - 38d, b - 49d, a + 41d)} │ │ │ │ + ideal (c - 29d, b + 16d, a - 20d)} │ │ │ │ │ │ │ │ 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 = | -8 28 25 30 -12 -42 -20 -35 -7 -37 -5 45 19 20 -24 34 39 21 -47 -39 │ │ │ │ +o16 = | -6 -13 21 -15 -31 6 -9 -1 50 -39 -18 21 -39 31 -31 44 -18 -13 -47 21 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -13 -18 34 0 | │ │ │ │ + 34 19 39 0 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o16 : Matrix kk <-- kk │ │ │ │ i17 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o17 = ideal (a - 7b*c + 30c - 37a*d - 12b*d + 28c*d - 8d , a*b - 24b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o17 = ideal (a + 50b*c - 15c - 39a*d - 31b*d - 13c*d - 6d , a*b - 31b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 5c + 34a*d + 45b*d - 42c*d + 25d , a*c - 13b*c + 39c - 18a*d + 21b*d │ │ │ │ + 2 2 2 │ │ │ │ + 18c + 44a*d + 21b*d + 6c*d + 21d , a*c + 34b*c - 18c + 19a*d - 13b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - + 19c*d - 20d , b + 34b*c - 47c - 39b*d + 20c*d - 35d ) │ │ │ │ + 2 2 2 2 │ │ │ │ + - 39c*d - 9d , b + 39b*c - 47c + 21b*d + 31c*d - d ) │ │ │ │ │ │ │ │ o17 : Ideal of S │ │ │ │ i18 : decompose F2 │ │ │ │ │ │ │ │ -o18 = {ideal (b - 12c + 48d, a - 16c + d), ideal (b + 46c + 14d, a + 31c - │ │ │ │ + 2 2 2 │ │ │ │ +o18 = {ideal (a*c + 34b*c - 18c + 19a*d - 13b*d - 39c*d - 9d , b + 39b*c - │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 2 2 2 │ │ │ │ + 47c + 21b*d + 31c*d - d , a*b - 31b*c - 18c + 44a*d + 21b*d + 6c*d + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 5d)} │ │ │ │ + 2 2 2 2 │ │ │ │ + 21d , a + 50b*c - 15c - 39a*d - 31b*d - 13c*d - 6d )} │ │ │ │ │ │ │ │ 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 │ │ │ @@ -191,64 +191,64 @@ │ │ │

    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 = || 42 -50 -50 38 -39 6 -1 47 49 -18 18 -28 -47 19 48 34 -13 11 -16 -38 39 21 -43 -39 |    |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || -44 -31 -9 -21 -42 -36 -47 -20 38 -34 -3 -43 22 16 -35 2 -48 32 -28 -15 -47 38 -47 45 ||
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 45 -29 -1 -4 42 -35 4 -13 18 -17 1 21 39 -23 50 15 -11 -11 19 47 -16 7 48 43 |         |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 35 -44 -33 -8 21 -2 -44 -20 19 -28 19 27 11 40 34 33 1 -14 35 36 11 -38 -3 46 |        |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 33 47 46 16 -22 -25 -44 -36 -30 -37 30 -25 -47 29 -41 2 -13 -41 -47 22 -23 -7 -10 15 | |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 43 1 -41 23 -42 -14 37 -50 -32 -20 -5 -49 -9 32 -18 -22 24 43 -18 30 39 27 -30 -32 |   |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 12 8 26 15 22 12 0 -5 6 17 -21 -18 -33 -49 -19 33 -20 0 -15 -48 39 0 44 -19 |          |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || -48 -27 -8 -33 -35 -16 -31 -44 -46 -49 -21 -3 -26 13 -40 4 -11 -48 36 -39 9 -39 -8 22 ||
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 34 29 30 -1 34 -22 32 -6 39 -28 -15 37 41 -30 47 -22 -6 -7 -8 43 36 -3 35 16 |         |
│ │ │ +      +-----------------------------------------------------------------------------------------+
│ │ │ +      || 10 -5 -38 -25 -21 21 14 42 19 -41 8 -35 25 -31 29 3 -49 38 -35 -9 6 40 -13 -2 |        |
│ │ │ +      +-----------------------------------------------------------------------------------------+
    │ │ │ +
    │ │ │ + 
    i14 : netList randomPointsOnRationalVariety(compsJ_1, 10)
│ │ │ +
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -o13 = || -6 -26 42 -37 -23 28 29 -18 -3 -16 5 23 34 19 -32 -13 -38 15 -18 21 -39 -47 39 -43 |  |
│ │ │ +o14 = || 1 4 44 1 40 20 29 -26 42 -16 -4 50 -40 48 23 -34 -35 37 30 4 -47 27 -31 0 |           |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 14 -34 -33 9 27 11 32 41 4 -28 15 -36 2 16 -21 -48 -15 -20 -34 38 45 22 -47 -47 |     |
│ │ │ +      || -29 -11 2 28 -12 -21 -30 20 -1 -42 27 -45 -48 -14 -3 -45 -37 30 -31 -39 -48 -29 47 0 ||
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 32 35 11 -9 -27 15 0 -12 7 19 24 7 15 -23 28 -11 47 -40 -17 7 43 39 -16 48 |          |
│ │ │ +      || -20 14 48 3 14 33 18 -46 24 45 -1 -29 1 31 -44 -2 -22 40 28 -49 -18 46 10 0 |         |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -49 -8 -5 -45 -36 -5 47 -21 -34 35 -25 32 33 40 35 1 36 1 -28 -38 46 11 11 -3 |       |
│ │ │ +      || 21 25 -23 -45 -48 -40 8 -7 -6 -1 14 3 -13 -44 -44 21 -41 3 30 7 13 -17 8 0 |          |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 35 14 -9 -22 -14 19 -10 48 23 -47 50 9 2 29 -37 -13 22 2 -37 -7 15 -47 -23 -10 |      |
│ │ │ +      || -15 26 -17 12 0 -5 35 -11 0 -31 -39 40 49 -21 -42 28 42 -18 -29 8 30 -46 23 0 |       |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 35 -49 -37 32 -48 42 -10 -49 42 -18 -34 25 -22 32 -35 24 30 -15 -20 27 -32 -9 39 -30 ||
│ │ │ +      || -17 38 -49 -9 2 -1 -9 2 3 49 33 49 -46 25 1 0 -18 12 15 -28 18 -16 27 0 |             |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -29 -40 30 -5 36 -41 -24 1 34 -15 20 -33 33 -49 -14 -20 -48 21 17 0 -19 -33 39 44 |   |
│ │ │ +      || -50 3 -12 -21 -10 4 36 -14 19 -9 43 -48 44 -35 16 6 20 -39 23 -21 -37 -23 19 0 |      |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -28 27 38 13 36 38 0 35 12 36 -33 -24 4 13 -35 -11 -39 34 -49 -39 22 -26 9 -8 |       |
│ │ │ +      || 36 -17 -27 37 -4 12 42 -16 -15 12 -3 -14 -28 38 -24 18 -9 6 -47 0 -28 47 -33 0 |      |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || 8 -20 1 -28 27 -39 40 0 -38 -8 44 -44 -22 -30 -28 -6 43 50 -28 -3 16 41 36 35 |       |
│ │ │ +      || 43 44 31 14 -9 -4 -4 -40 -48 45 47 17 -37 29 -9 10 -28 -33 -29 28 26 5 42 0 |         |
│ │ │        +----------------------------------------------------------------------------------------+
│ │ │ -      || -12 -11 18 -43 28 44 44 5 -12 -35 19 -50 3 -31 3 -49 -9 -50 -41 40 -2 25 6 -13 |      |
│ │ │ +      || 17 -47 -23 8 -18 -45 -25 2 44 41 -19 -37 5 16 2 38 -13 -20 30 44 4 22 -29 0 |         |
│ │ │        +----------------------------------------------------------------------------------------+
    │ │ │
    │ │ │ - 
    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 |   |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 -48 30 -48 0 ||
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 40 -18 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 3 13 0 |     |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 -18 30 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 12 18 0 |     |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 -37 0 |      |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 6 -28 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 -33 26 0 |    |
│ │ │ -      +-------------------------------------------------------------------------------------+
│ │ │ -      || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 -20 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 = || -6 -26 42 -37 -23 28 29 -18 -3 -16 5 23 34 19 -32 -13 -38 15 -18 21 - │ │ │ │ -39 -47 39 -43 | | │ │ │ │ +-----------------+ │ │ │ │ +o13 = || 42 -50 -50 38 -39 6 -1 47 49 -18 18 -28 -47 19 48 34 -13 11 -16 -38 39 │ │ │ │ +21 -43 -39 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +-----------------+ │ │ │ │ + || -44 -31 -9 -21 -42 -36 -47 -20 38 -34 -3 -43 22 16 -35 2 -48 32 -28 - │ │ │ │ +15 -47 38 -47 45 || │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +-----------------+ │ │ │ │ + || 45 -29 -1 -4 42 -35 4 -13 18 -17 1 21 39 -23 50 15 -11 -11 19 47 -16 7 │ │ │ │ +48 43 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +-----------------+ │ │ │ │ + || 35 -44 -33 -8 21 -2 -44 -20 19 -28 19 27 11 40 34 33 1 -14 35 36 11 - │ │ │ │ +38 -3 46 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +-----------------+ │ │ │ │ + || 33 47 46 16 -22 -25 -44 -36 -30 -37 30 -25 -47 29 -41 2 -13 -41 -47 22 │ │ │ │ +-23 -7 -10 15 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +-----------------+ │ │ │ │ + || 43 1 -41 23 -42 -14 37 -50 -32 -20 -5 -49 -9 32 -18 -22 24 43 -18 30 │ │ │ │ +39 27 -30 -32 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +-----------------+ │ │ │ │ + || 12 8 26 15 22 12 0 -5 6 17 -21 -18 -33 -49 -19 33 -20 0 -15 -48 39 0 │ │ │ │ +44 -19 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +-----------------+ │ │ │ │ + || -48 -27 -8 -33 -35 -16 -31 -44 -46 -49 -21 -3 -26 13 -40 4 -11 -48 36 │ │ │ │ +-39 9 -39 -8 22 || │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +-----------------+ │ │ │ │ + || 34 29 30 -1 34 -22 32 -6 39 -28 -15 37 41 -30 47 -22 -6 -7 -8 43 36 - │ │ │ │ +3 35 16 | | │ │ │ │ + +------------------------------------------------------------------------ │ │ │ │ +-----------------+ │ │ │ │ + || 10 -5 -38 -25 -21 21 14 42 19 -41 8 -35 25 -31 29 3 -49 38 -35 -9 6 40 │ │ │ │ +-13 -2 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ -----------------+ │ │ │ │ - || 14 -34 -33 9 27 11 32 41 4 -28 15 -36 2 16 -21 -48 -15 -20 -34 38 45 │ │ │ │ -22 -47 -47 | | │ │ │ │ +-----------------+ │ │ │ │ +i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) │ │ │ │ + │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------+ │ │ │ │ - || 32 35 11 -9 -27 15 0 -12 7 19 24 7 15 -23 28 -11 47 -40 -17 7 43 39 - │ │ │ │ -16 48 | | │ │ │ │ +o14 = || 1 4 44 1 40 20 29 -26 42 -16 -4 50 -40 48 23 -34 -35 37 30 4 -47 27 - │ │ │ │ +31 0 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------+ │ │ │ │ - || -49 -8 -5 -45 -36 -5 47 -21 -34 35 -25 32 33 40 35 1 36 1 -28 -38 46 │ │ │ │ -11 11 -3 | | │ │ │ │ + || -29 -11 2 28 -12 -21 -30 20 -1 -42 27 -45 -48 -14 -3 -45 -37 30 -31 - │ │ │ │ +39 -48 -29 47 0 || │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------+ │ │ │ │ - || 35 14 -9 -22 -14 19 -10 48 23 -47 50 9 2 29 -37 -13 22 2 -37 -7 15 -47 │ │ │ │ --23 -10 | | │ │ │ │ + || -20 14 48 3 14 33 18 -46 24 45 -1 -29 1 31 -44 -2 -22 40 28 -49 -18 46 │ │ │ │ +10 0 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------+ │ │ │ │ - || 35 -49 -37 32 -48 42 -10 -49 42 -18 -34 25 -22 32 -35 24 30 -15 -20 27 │ │ │ │ --32 -9 39 -30 || │ │ │ │ + || 21 25 -23 -45 -48 -40 8 -7 -6 -1 14 3 -13 -44 -44 21 -41 3 30 7 13 -17 │ │ │ │ +8 0 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------+ │ │ │ │ - || -29 -40 30 -5 36 -41 -24 1 34 -15 20 -33 33 -49 -14 -20 -48 21 17 0 - │ │ │ │ -19 -33 39 44 | | │ │ │ │ + || -15 26 -17 12 0 -5 35 -11 0 -31 -39 40 49 -21 -42 28 42 -18 -29 8 30 - │ │ │ │ +46 23 0 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------+ │ │ │ │ - || -28 27 38 13 36 38 0 35 12 36 -33 -24 4 13 -35 -11 -39 34 -49 -39 22 - │ │ │ │ -26 9 -8 | | │ │ │ │ + || -17 38 -49 -9 2 -1 -9 2 3 49 33 49 -46 25 1 0 -18 12 15 -28 18 -16 27 │ │ │ │ +0 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------+ │ │ │ │ - || 8 -20 1 -28 27 -39 40 0 -38 -8 44 -44 -22 -30 -28 -6 43 50 -28 -3 16 │ │ │ │ -41 36 35 | | │ │ │ │ + || -50 3 -12 -21 -10 4 36 -14 19 -9 43 -48 44 -35 16 6 20 -39 23 -21 -37 │ │ │ │ +-23 19 0 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------+ │ │ │ │ - || -12 -11 18 -43 28 44 44 5 -12 -35 19 -50 3 -31 3 -49 -9 -50 -41 40 - │ │ │ │ -2 25 6 -13 | | │ │ │ │ + || 36 -17 -27 37 -4 12 42 -16 -15 12 -3 -14 -28 38 -24 18 -9 6 -47 0 -28 │ │ │ │ +47 -33 0 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ ----------------+ │ │ │ │ -i14 : netList randomPointsOnRationalVariety(compsJ_1, 10) │ │ │ │ - │ │ │ │ + || 43 44 31 14 -9 -4 -4 -40 -48 45 47 17 -37 29 -9 10 -28 -33 -29 28 26 5 │ │ │ │ +42 0 | | │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ -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 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || -1 -5 -10 -10 -11 42 6 46 -4 47 42 -40 47 -27 -20 49 -39 -31 -37 -29 - │ │ │ │ -48 30 -48 0 || │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 29 18 20 1 18 26 -31 -45 -21 10 22 -30 10 32 -31 -21 -49 28 -22 46 1 │ │ │ │ -40 -18 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || -17 3 17 -9 -36 -45 49 30 -45 24 -28 41 8 -4 -26 -28 7 30 -41 -17 -13 │ │ │ │ -3 13 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 37 33 -47 -20 -49 45 29 19 41 13 -38 44 23 40 -48 45 8 -29 42 -46 49 - │ │ │ │ -18 30 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || -9 -3 -26 13 35 49 -8 49 -40 13 -20 9 27 5 -8 -15 -28 15 -18 -16 -46 │ │ │ │ -12 18 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 28 32 0 0 -17 -44 25 42 7 -35 29 -17 19 8 -9 -26 -21 23 20 -23 44 -39 │ │ │ │ --37 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || -30 -29 27 14 17 39 33 15 -35 50 -50 45 -33 13 24 -44 0 -47 -9 47 -28 │ │ │ │ -6 -28 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 7 -12 42 -29 30 1 3 -28 -7 36 -26 -40 42 38 -20 -23 28 -29 -28 5 -37 - │ │ │ │ -33 26 0 | | │ │ │ │ - +------------------------------------------------------------------------ │ │ │ │ --------------+ │ │ │ │ - || 28 -10 13 -39 -20 11 13 -13 -37 8 -36 -29 -29 17 24 -50 44 30 -13 22 5 │ │ │ │ --20 4 0 | | │ │ │ │ +----------------+ │ │ │ │ + || 17 -47 -23 8 -18 -45 -25 2 44 41 -19 -37 5 16 2 38 -13 -20 30 44 4 22 │ │ │ │ +-29 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/GroebnerStrata/html/index.html │ │ │ @@ -355,85 +355,90 @@ │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i12 : pt1 = randomPointOnRationalVariety compsJ_0
│ │ │  
│ │ │ -o12 = | -26 -24 6 39 17 28 -46 -12 11 -29 -10 -48 -36 -30 39 -29 -8 -22 -10
│ │ │ +o12 = | 37 -2 -19 -4 -13 20 50 -34 -12 -8 23 38 -36 -10 32 19 -22 -29 19 11
│ │ │        -----------------------------------------------------------------------
│ │ │ -      11 19 19 24 -29 |
│ │ │ +      -30 -29 24 -29 |
│ │ │  
│ │ │                 1       24
│ │ │  o12 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i13 : pt2 = randomPointOnRationalVariety compsJ_1
│ │ │  
│ │ │ -o13 = | -30 -41 -39 30 -13 32 17 -12 29 25 -49 11 43 21 -42 -4 19 -47 39 -38
│ │ │ +o13 = | -4 -6 -29 43 12 -8 -47 31 -19 -3 -17 -2 -49 21 11 3 -47 19 -16 -38 39
│ │ │        -----------------------------------------------------------------------
│ │ │ -      -16 0 -24 34 |
│ │ │ +      0 -24 34 |
│ │ │  
│ │ │                 1       24
│ │ │  o13 : Matrix kk  <-- kk
    │ │ │
    │ │ │ 
    i14 : F1 = sub(F, (vars S)|pt1)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o14 = ideal (a  + 11b*c + 39c  - 29a*d + 17b*d - 24c*d - 26d , a*b + 39b*c -
│ │ │ +              2             2                            2               
│ │ │ +o14 = ideal (a  - 12b*c - 4c  - 8a*d - 13b*d - 2c*d + 37d , a*b + 32b*c +
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2             2                  
│ │ │ -      10c  - 29a*d - 48b*d + 28c*d + 6d , b  + 19b*c - 8c  + 19a*d - 22b*d -
│ │ │ +         2                              2   2              2                
│ │ │ +      23c  + 19a*d + 38b*d + 20c*d - 19d , b  - 30b*c - 22c  - 29a*d - 29b*d
│ │ │        -----------------------------------------------------------------------
│ │ │ -                 2                   2                              2
│ │ │ -      36c*d - 46d , a*c + 24b*c - 10c  - 29a*d + 11b*d - 30c*d - 12d )
│ │ │ +                   2                   2                              2
│ │ │ +      - 36c*d + 50d , a*c + 24b*c + 19c  - 29a*d + 11b*d - 10c*d - 34d )
│ │ │  
│ │ │  o14 : Ideal of S
    │ │ │
    │ │ │ 
    i15 : F2 = sub(F, (vars S)|pt2)
│ │ │  
│ │ │ -              2              2                              2               
│ │ │ -o15 = ideal (a  + 29b*c + 30c  + 25a*d - 13b*d - 41c*d - 30d , a*b - 42b*c -
│ │ │ +              2              2                           2               
│ │ │ +o15 = ideal (a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d , a*b + 11b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -         2                             2   2              2                  
│ │ │ -      49c  - 4a*d + 11b*d + 32c*d - 39d , b  - 16b*c + 19c  - 47b*d + 43c*d +
│ │ │ +         2                           2   2              2                  
│ │ │ +      17c  + 3a*d - 2b*d - 8c*d - 29d , b  + 39b*c - 47c  + 19b*d - 49c*d -
│ │ │        -----------------------------------------------------------------------
│ │ │           2                   2                              2
│ │ │ -      17d , a*c - 24b*c + 39c  + 34a*d - 38b*d + 21c*d - 12d )
│ │ │ +      47d , a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d )
│ │ │  
│ │ │  o15 : Ideal of S
    │ │ │
    │ │ │ 
    i16 : decompose F1
│ │ │  
│ │ │ -                                    2             2                      2
│ │ │ -o16 = {ideal (a + 24b - 10c - 17d, b  + 19b*c - 8c  + 27b*d - 48c*d - 26d ),
│ │ │ +                                    2              2                      2
│ │ │ +o16 = {ideal (a + 24b + 19c + 36d, b  - 30b*c - 22c  - 40b*d + 10c*d - 17d ),
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ideal (c - 29d, b + 23d, a - 14d)}
│ │ │ +      ideal (c - 29d, b + 30d, a - 38d)}
│ │ │  
│ │ │  o16 : List
    │ │ │
    │ │ │ 
    i17 : decompose F2
│ │ │  
│ │ │ -o17 = {ideal (b - 42c - 18d, a + 41c + 33d), ideal (b + 26c - 29d, a - 44c -
│ │ │ +                               2                              2   2          
│ │ │ +o17 = {ideal (a*c - 24b*c - 16c  + 34a*d - 38b*d + 21c*d + 31d , b  + 39b*c -
│ │ │        -----------------------------------------------------------------------
│ │ │ -      9d)}
│ │ │ +         2                      2                   2                       
│ │ │ +      47c  + 19b*d - 49c*d - 47d , a*b + 11b*c - 17c  + 3a*d - 2b*d - 8c*d -
│ │ │ +      -----------------------------------------------------------------------
│ │ │ +         2   2              2                           2
│ │ │ +      29d , a  - 19b*c + 43c  - 3a*d + 12b*d - 6c*d - 4d )}
│ │ │  
│ │ │  o17 : List
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │ ├── html2text {} │ │ │ │ @@ -424,65 +424,70 @@ │ │ │ │ -----------------------------------------------------------+ │ │ │ │ This tells us that there are 2 components (at least over the given field). │ │ │ │ Their dimensions are 11, 8. │ │ │ │ We can find random points on each component, since these components are │ │ │ │ rational. │ │ │ │ i12 : pt1 = randomPointOnRationalVariety compsJ_0 │ │ │ │ │ │ │ │ -o12 = | -26 -24 6 39 17 28 -46 -12 11 -29 -10 -48 -36 -30 39 -29 -8 -22 -10 │ │ │ │ +o12 = | 37 -2 -19 -4 -13 20 50 -34 -12 -8 23 38 -36 -10 32 19 -22 -29 19 11 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 11 19 19 24 -29 | │ │ │ │ + -30 -29 24 -29 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o12 : Matrix kk <-- kk │ │ │ │ i13 : pt2 = randomPointOnRationalVariety compsJ_1 │ │ │ │ │ │ │ │ -o13 = | -30 -41 -39 30 -13 32 17 -12 29 25 -49 11 43 21 -42 -4 19 -47 39 -38 │ │ │ │ +o13 = | -4 -6 -29 43 12 -8 -47 31 -19 -3 -17 -2 -49 21 11 3 -47 19 -16 -38 39 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - -16 0 -24 34 | │ │ │ │ + 0 -24 34 | │ │ │ │ │ │ │ │ 1 24 │ │ │ │ o13 : Matrix kk <-- kk │ │ │ │ i14 : F1 = sub(F, (vars S)|pt1) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o14 = ideal (a + 11b*c + 39c - 29a*d + 17b*d - 24c*d - 26d , a*b + 39b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o14 = ideal (a - 12b*c - 4c - 8a*d - 13b*d - 2c*d + 37d , a*b + 32b*c + │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 10c - 29a*d - 48b*d + 28c*d + 6d , b + 19b*c - 8c + 19a*d - 22b*d - │ │ │ │ + 2 2 2 2 │ │ │ │ + 23c + 19a*d + 38b*d + 20c*d - 19d , b - 30b*c - 22c - 29a*d - 29b*d │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 │ │ │ │ - 36c*d - 46d , a*c + 24b*c - 10c - 29a*d + 11b*d - 30c*d - 12d ) │ │ │ │ + 2 2 2 │ │ │ │ + - 36c*d + 50d , a*c + 24b*c + 19c - 29a*d + 11b*d - 10c*d - 34d ) │ │ │ │ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : F2 = sub(F, (vars S)|pt2) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o15 = ideal (a + 29b*c + 30c + 25a*d - 13b*d - 41c*d - 30d , a*b - 42b*c - │ │ │ │ + 2 2 2 │ │ │ │ +o15 = ideal (a - 19b*c + 43c - 3a*d + 12b*d - 6c*d - 4d , a*b + 11b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 2 2 2 2 │ │ │ │ - 49c - 4a*d + 11b*d + 32c*d - 39d , b - 16b*c + 19c - 47b*d + 43c*d + │ │ │ │ + 2 2 2 2 │ │ │ │ + 17c + 3a*d - 2b*d - 8c*d - 29d , b + 39b*c - 47c + 19b*d - 49c*d - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 2 2 │ │ │ │ - 17d , a*c - 24b*c + 39c + 34a*d - 38b*d + 21c*d - 12d ) │ │ │ │ + 47d , a*c - 24b*c - 16c + 34a*d - 38b*d + 21c*d + 31d ) │ │ │ │ │ │ │ │ o15 : Ideal of S │ │ │ │ i16 : decompose F1 │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o16 = {ideal (a + 24b - 10c - 17d, b + 19b*c - 8c + 27b*d - 48c*d - 26d ), │ │ │ │ + 2 2 2 │ │ │ │ +o16 = {ideal (a + 24b + 19c + 36d, b - 30b*c - 22c - 40b*d + 10c*d - 17d ), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - ideal (c - 29d, b + 23d, a - 14d)} │ │ │ │ + ideal (c - 29d, b + 30d, a - 38d)} │ │ │ │ │ │ │ │ o16 : List │ │ │ │ i17 : decompose F2 │ │ │ │ │ │ │ │ -o17 = {ideal (b - 42c - 18d, a + 41c + 33d), ideal (b + 26c - 29d, a - 44c - │ │ │ │ + 2 2 2 │ │ │ │ +o17 = {ideal (a*c - 24b*c - 16c + 34a*d - 38b*d + 21c*d + 31d , b + 39b*c - │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 9d)} │ │ │ │ + 2 2 2 │ │ │ │ + 47c + 19b*d - 49c*d - 47d , a*b + 11b*c - 17c + 3a*d - 2b*d - 8c*d - │ │ │ │ + ----------------------------------------------------------------------- │ │ │ │ + 2 2 2 2 │ │ │ │ + 29d , a - 19b*c + 43c - 3a*d + 12b*d - 6c*d - 4d )} │ │ │ │ │ │ │ │ o17 : List │ │ │ │ Note, the general element of one component is a plane conic union a point. (The │ │ │ │ dimension of the locus of all such is: (choice of plane) + (choice of degree 2 │ │ │ │ in plane) + choice of point. This is 3 + 5 + 3 = 11. │ │ │ │ The other component consists of two skew lines. This has dimension (choice of │ │ │ │ line) + (choice of line). This is 4 + 4 = 8. Also notice that the 2 skew lines │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ c2V0V2Fsa1RyYWNlKFpaKQ== │ │ │ #:len=251 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTYxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzZXRXYWxrVHJhY2UsWlopLCJzZXRXYWxrVHJhY2Uo │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out │ │ │ @@ -11,21 +11,21 @@ │ │ │ i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}]; │ │ │ │ │ │ i4 : I2 = sub(I1, R2); │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ │ │ i5 : elapsedTime gb I2 │ │ │ - -- 3.51847s elapsed │ │ │ + -- 1.78781s elapsed │ │ │ │ │ │ o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16] │ │ │ │ │ │ o5 : GroebnerBasis │ │ │ │ │ │ i6 : elapsedTime groebnerWalk(gb I1, R2) │ │ │ - -- 3.08455s elapsed │ │ │ + -- 1.46346s elapsed │ │ │ │ │ │ o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0] │ │ │ │ │ │ o6 : GroebnerBasis │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html │ │ │ @@ -97,30 +97,30 @@ │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime gb I2
│ │ │ - -- 3.51847s elapsed
│ │ │ + -- 1.78781s elapsed
│ │ │  
│ │ │  o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16]
│ │ │  
│ │ │  o5 : GroebnerBasis
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime groebnerWalk(gb I1, R2)
│ │ │ - -- 3.08455s elapsed
│ │ │ + -- 1.46346s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]
│ │ │  
│ │ │  o6 : GroebnerBasis
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -38,23 +38,23 @@ │ │ │ │ using a different weight vector and then graded reverse lexicographic we could │ │ │ │ substitute and compute directly, │ │ │ │ i3 : R2 = QQ[x,y,z,u,v, MonomialOrder=>Weights=>{0,0,0,1,1}]; │ │ │ │ i4 : I2 = sub(I1, R2); │ │ │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ i5 : elapsedTime gb I2 │ │ │ │ - -- 3.51847s elapsed │ │ │ │ + -- 1.78781s elapsed │ │ │ │ │ │ │ │ o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16] │ │ │ │ │ │ │ │ o5 : GroebnerBasis │ │ │ │ but it is faster to compute directly in the first order and then use the │ │ │ │ Groebner walk. │ │ │ │ i6 : elapsedTime groebnerWalk(gb I1, R2) │ │ │ │ - -- 3.08455s elapsed │ │ │ │ + -- 1.46346s elapsed │ │ │ │ │ │ │ │ o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0] │ │ │ │ │ │ │ │ o6 : GroebnerBasis │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The target ring must be the same ring as the ring of the starting ideal, except │ │ │ │ with different monomial order. The ring must be a polynomial ring over a field. │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=23 │ │ │ aWRlYWxPZlByb2plY3RpdmVQb2ludHM= │ │ │ #:len=1240 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIGlkZWFsIG9mIHNl │ │ │ dCBvZiBwb2ludHMiLCAibGluZW51bSIgPT4gNDI0LCBJbnB1dHMgPT4ge1NQQU57VFR7IkwifSwi │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_hadamard__Power_lp__List_cm__Z__Z_rp.out │ │ │ @@ -6,22 +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, 4, 8}, Point{1, 0, 16}, Point{1, 0, 1}, Point{1, 1, -}, │ │ │ - 8 │ │ │ + │ │ │ +o2 = {Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16}, Point{1, 0, 1}, │ │ │ + │ │ │ ------------------------------------------------------------------------ │ │ │ - 1 │ │ │ - Point{1, 0, 2}, Point{1, 0, -}, Point{1, 2, 1}, Point{1, 0, 4}, Point{1, │ │ │ - 2 │ │ │ + 1 1 │ │ │ + Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1, 2, 1}, Point{1, │ │ │ + 8 2 │ │ │ ------------------------------------------------------------------------ │ │ │ - 1 │ │ │ - 0, -}, Point{1, 8, 64}} │ │ │ - 4 │ │ │ + 1 │ │ │ + 0, 4}, Point{1, 0, -}} │ │ │ + 4 │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_ideal__Of__Projective__Points.out │ │ │ @@ -30,17 +30,17 @@ │ │ │ 2 2 │ │ │ + x*y - 6x*z ) │ │ │ │ │ │ o4 : Ideal of S │ │ │ │ │ │ i5 : X2 = hadamardPower(X,2) │ │ │ │ │ │ -o5 = {Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, Point{0, 1, 1}, │ │ │ +o5 = {Point{0, 1, 0}, Point{0, 2, -1}, Point{0, 1, 1}, Point{1, 1, 0}, │ │ │ ------------------------------------------------------------------------ │ │ │ - Point{1, 1, 0}, Point{1, 2, 0}} │ │ │ + Point{1, 2, 0}, Point{1, 4, 1}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : I2 == idealOfProjectivePoints(X2,S) │ │ │ │ │ │ o6 = true │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Power_lp__List_cm__Z__Z_rp.html │ │ │ @@ -89,25 +89,25 @@ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : hadamardPower(L,3)
│ │ │  
│ │ │ -                                                                   1  
│ │ │ -o2 = {Point{1, 4, 8}, Point{1, 0, 16}, Point{1, 0, 1}, Point{1, 1, -},
│ │ │ -                                                                   8  
│ │ │ +                                                                       
│ │ │ +o2 = {Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16}, Point{1, 0, 1},
│ │ │ +                                                                       
│ │ │       ------------------------------------------------------------------------
│ │ │ -                                 1                                           
│ │ │ -     Point{1, 0, 2}, Point{1, 0, -}, Point{1, 2, 1}, Point{1, 0, 4}, Point{1,
│ │ │ -                                 2                                           
│ │ │ +                 1                               1                           
│ │ │ +     Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1, 2, 1}, Point{1,
│ │ │ +                 8                               2                           
│ │ │       ------------------------------------------------------------------------
│ │ │ -        1
│ │ │ -     0, -}, Point{1, 8, 64}}
│ │ │ -        4
│ │ │ +                        1
│ │ │ +     0, 4}, Point{1, 0, -}}
│ │ │ +                        4
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -21,25 +21,25 @@ │ │ │ │ 1 │ │ │ │ o1 = {Point{1, 1, -}, Point{1, 0, 1}, Point{1, 2, 4}} │ │ │ │ 2 │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : hadamardPower(L,3) │ │ │ │ │ │ │ │ - 1 │ │ │ │ -o2 = {Point{1, 4, 8}, Point{1, 0, 16}, Point{1, 0, 1}, Point{1, 1, -}, │ │ │ │ - 8 │ │ │ │ + │ │ │ │ +o2 = {Point{1, 8, 64}, Point{1, 4, 8}, Point{1, 0, 16}, Point{1, 0, 1}, │ │ │ │ + │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 1 │ │ │ │ - Point{1, 0, 2}, Point{1, 0, -}, Point{1, 2, 1}, Point{1, 0, 4}, Point{1, │ │ │ │ - 2 │ │ │ │ + 1 1 │ │ │ │ + Point{1, 1, -}, Point{1, 0, 2}, Point{1, 0, -}, Point{1, 2, 1}, Point{1, │ │ │ │ + 8 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 1 │ │ │ │ - 0, -}, Point{1, 8, 64}} │ │ │ │ - 4 │ │ │ │ + 1 │ │ │ │ + 0, 4}, Point{1, 0, -}} │ │ │ │ + 4 │ │ │ │ │ │ │ │ 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/_ideal__Of__Projective__Points.html │ │ │ @@ -121,17 +121,17 @@ │ │ │ o4 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : X2 = hadamardPower(X,2)
│ │ │  
│ │ │ -o5 = {Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, Point{0, 1, 1},
│ │ │ +o5 = {Point{0, 1, 0}, Point{0, 2, -1}, Point{0, 1, 1}, Point{1, 1, 0},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Point{1, 1, 0}, Point{1, 2, 0}}
│ │ │ +     Point{1, 2, 0}, Point{1, 4, 1}}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : I2 == idealOfProjectivePoints(X2,S)
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,17 +39,17 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            2       2
│ │ │ │       + x*y  - 6x*z )
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : X2 = hadamardPower(X,2)
│ │ │ │  
│ │ │ │ -o5 = {Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, Point{0, 1, 1},
│ │ │ │ +o5 = {Point{0, 1, 0}, Point{0, 2, -1}, Point{0, 1, 1}, Point{1, 1, 0},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Point{1, 1, 0}, Point{1, 2, 0}}
│ │ │ │ +     Point{1, 2, 0}, Point{1, 4, 1}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : I2 == idealOfProjectivePoints(X2,S)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  ********** WWaayyss ttoo uussee iiddeeaallOOffPPrroojjeeccttiivveePPooiinnttss:: **********
│ │ │ │      * idealOfProjectivePoints(List,Ring)
│ │ ├── ./usr/share/doc/Macaulay2/HigherCIOperators/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  Y2lPcGVyYXRvclJlc29sdXRpb24=
│ │ │  #:len=2606
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiXCJsaWZ0IHJlc29sdXRpb24gZnJvbSBj
│ │ │  b21wbGV0ZSBpbnRlcnNlY3Rpb24gdXNpbmcgaGlnaGVyIGNpLW9wZXJhdG9yc1wiIiwgImxpbmVu
│ │ ├── ./usr/share/doc/Macaulay2/HighestWeights/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  R3JvdXBBY3Rpbmc=
│ │ │  #:len=621
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3RvcmVzIHRoZSBEeW5raW4gdHlwZSBv
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│ │ ├── ./usr/share/doc/Macaulay2/HodgeIntegrals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  a2FwcGE=
│ │ │  #:len=1396
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│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  ZXVsZXJPcGVyYXRvcnM=
│ │ │  #:len=1959
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXVsZXIgT3BlcmF0b3JzIiwgImxpbmVu
│ │ │  dW0iID0+IDE0MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJBIn0sIiwgIixTUEFOeyJhICIsVE8ye25l
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000004839s elapsed
│ │ │ - -- .000004879s elapsed
│ │ │ - -- .00000513s elapsed
│ │ │ - -- .0000051s elapsed
│ │ │ - -- .000003987s elapsed
│ │ │ + -- .000004451s elapsed
│ │ │ + -- .000004284s elapsed
│ │ │ + -- .000004476s elapsed
│ │ │ + -- .000005696s elapsed
│ │ │ + -- .000005507s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3, t_2*t_4);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : solveFrobeniusIdeal I
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000006092s elapsed
│ │ │ + -- .000005573s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX 
│ │ │ @@ -26,15 +26,15 @@
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │  
│ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000007755s elapsed
│ │ │ + -- .000005307s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html
│ │ │ @@ -139,19 +139,19 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000004839s elapsed
│ │ │ - -- .000004879s elapsed
│ │ │ - -- .00000513s elapsed
│ │ │ - -- .0000051s elapsed
│ │ │ - -- .000003987s elapsed
│ │ │ + -- .000004451s elapsed
│ │ │ + -- .000004284s elapsed
│ │ │ + -- .000004476s elapsed
│ │ │ + -- .000005696s elapsed
│ │ │ + -- .000005507s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,19 +57,19 @@
│ │ │ │  o5 = {9, 1, 99999, 9999999, 3, 999}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : netList cssLeadTerm(Hbeta, w)
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000004839s elapsed
│ │ │ │ - -- .000004879s elapsed
│ │ │ │ - -- .00000513s elapsed
│ │ │ │ - -- .0000051s elapsed
│ │ │ │ - -- .000003987s elapsed
│ │ │ │ + -- .000004451s elapsed
│ │ │ │ + -- .000004284s elapsed
│ │ │ │ + -- .000004476s elapsed
│ │ │ │ + -- .000005696s elapsed
│ │ │ │ + -- .000005507s elapsed
│ │ │ │  
│ │ │ │       +----------------------------------------------------+
│ │ │ │       |   1 5   5 5                                        |
│ │ │ │       | - - - - - -                                        |
│ │ │ │       |   2 2   2 2                                        |
│ │ │ │  o6 = |x   x x   x                                         |
│ │ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : solveFrobeniusIdeal I
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000006092s elapsed
│ │ │ + -- .000005573s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX 
│ │ │ @@ -120,15 +120,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : solveFrobeniusIdeal(I, W)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000007755s elapsed
│ │ │ + -- .000005307s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  t_2*t_4);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : solveFrobeniusIdeal I
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000006092s elapsed
│ │ │ │ + -- .000005573s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1             1    2   1             1             1             1    2
│ │ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000007755s elapsed
│ │ │ │ + -- .000005307s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1             1    2   1             1             1             1    2
│ │ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  YWxsZ2VucyhER0FsZ2VicmEsWlop
│ │ │  #:len=269
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbGxnZW5zLERHQWxnZWJyYSxaWiksImFsbGdlbnMo
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/example-output/_bracket.out
│ │ │ @@ -88,106 +88,106 @@
│ │ │  
│ │ │  o13 = 600
│ │ │  
│ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │  
│ │ │  i15 : H'
│ │ │  
│ │ │ -o15 = {({T   , T   }, - T   T    - T   T    + y*T    + z*T   ), ({T   ,
│ │ │ -          1,4   2,3      1,2 2,2    1,4 2,3      3,2      3,4      1,3 
│ │ │ +o15 = {({T   , T   }, - T   T    + y*T   ), ({T   , T   }, T   T    -
│ │ │ +          1,5   2,5      1,5 2,5      3,8      1,4   2,1    1,4 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    - z*T    + y*T   ), ({T   , T   }, - T   T    -
│ │ │ -       2,4    1,3 2,4      3,5      3,7      1,3   2,1      1,3 2,1  
│ │ │ +      T   T    + x*T    ), ({T   , T   }, - T   T    - T   T    + x*T   ),
│ │ │ +       1,1 2,5      3,10      1,4   2,2      1,1 2,1    1,4 2,2      3,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ -       1,5 2,2    1,1 2,3      3,2      3,4      1,1   2,2      1,1 2,2  
│ │ │ +      ({T   , T   }, T   T    + T   T    + T   T    + y*T    - z*T   ),
│ │ │ +         1,2   2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   },
│ │ │ -         3,3      1,5   2,4      1,2 2,3    1,5 2,4      3,5      1,3   2,5  
│ │ │ +      ({T   , T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   },
│ │ │ +         1,5   2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3   2,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ -         1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,2      1,2 2,2  
│ │ │ +      T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   , T   },
│ │ │ +       1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3   2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T    + z*T   ), ({T   , T   }, - T   T    + T   T    -
│ │ │ -       1,4 2,3      3,2      3,4      1,4   2,3      1,4 2,3    1,3 2,5  
│ │ │ +      T   T    + y*T    - z*T   ), ({T   , T   }, - T   T    - T   T    +
│ │ │ +       1,3 2,1      3,1      3,2      1,5   2,2      1,5 2,2    1,4 2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ),
│ │ │ -         3,8      3,9      1,2   2,5      1,5 2,3    1,2 2,5      3,9  
│ │ │ +      z*T    + z*T    ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ +         3,2      3,10      1,3   2,2    1,4 2,1    1,3 2,2      3,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ -         1,4   2,5      1,4 2,5      3,8      1,5   2,3      1,5 2,3  
│ │ │ +      y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, - T   T   
│ │ │ +         3,3      1,2   2,4      1,2 2,4      3,6      1,1   2,4      1,5 2,1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T   ), ({T   , T   }, T   T    + x*T    - z*T   ), ({T   ,
│ │ │ -       1,2 2,5      3,9      1,3   2,3    1,3 2,3      3,5      3,7      1,1 
│ │ │ +      - T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    + z*T   ),
│ │ │ +         1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2      3,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, - T   T    - T   T    - T   T    + z*T    + x*T   ), ({T   ,
│ │ │ -       2,3      1,3 2,1    1,5 2,2    1,1 2,3      3,2      3,4      1,4 
│ │ │ +      ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   }, -
│ │ │ +         1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   }, -
│ │ │ -       2,4    1,4 2,4    1,5 2,5      3,7      3,9      1,2   2,3    
│ │ │ +      T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ), ({T   ,
│ │ │ +       1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2      1,5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    - T   T    -
│ │ │ -       1,2 2,3    1,5 2,4      3,5      1,5   2,2      1,3 2,1    1,5 2,2  
│ │ │ +      T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +       2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + z*T    + x*T   ), ({T   , T   }, T   T    + T   T    +
│ │ │ -       1,1 2,3      3,2      3,4      1,4   2,4    1,2 2,1    1,3 2,3  
│ │ │ +      T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ +       1,5 2,3    1,3 2,4      3,5      3,6      1,4   2,3      1,2 2,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T    - z*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ -       1,4 2,4      3,4      3,7      1,3   2,5    1,5 2,1    1,3 2,5  
│ │ │ +      T   T    + y*T    + z*T   ), ({T   , T   }, T   T    - z*T    +
│ │ │ +       1,4 2,3      3,2      3,4      1,3   2,4    1,3 2,4      3,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T    ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ -         3,8      3,10      1,5   2,1    1,5 2,1    1,3 2,5      3,8  
│ │ │ +      y*T   ), ({T   , T   }, - T   T    - T   T    - T   T    + z*T    +
│ │ │ +         3,7      1,3   2,1      1,3 2,1    1,5 2,2    1,1 2,3      3,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T    ), ({T   , T   }, - T   T    - T   T    + z*T    + z*T    ),
│ │ │ -         3,10      1,4   2,5      1,5 2,2    1,4 2,5      3,2      3,10  
│ │ │ +      x*T   ), ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T   
│ │ │ +         3,4      1,1   2,2      1,1 2,2      3,3      1,5   2,4      1,2 2,3
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    - T   T    + x*T   ), ({T   , T   }, T   T   
│ │ │ -         1,1   2,1      1,1 2,1    1,4 2,2      3,1      1,3   2,4    1,5 2,3
│ │ │ +      - T   T    + y*T   ), ({T   , T   }, - T   T    + T   T    - z*T    +
│ │ │ +         1,5 2,4      3,5      1,3   2,5      1,4 2,3    1,3 2,5      3,8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      + T   T    - z*T    + x*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ -         1,3 2,4      3,5      3,6      1,4   2,1    1,4 2,1    1,3 2,2  
│ │ │ +      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T    + z*T   ),
│ │ │ +         3,9      1,2   2,2      1,2 2,2    1,4 2,3      3,2      3,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   },
│ │ │ -         3,1      3,3      1,5   2,5      1,5 2,5      3,8      1,4   2,1  
│ │ │ +      ({T   , T   }, - T   T    + T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +         1,4   2,3      1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + x*T    ), ({T   , T   }, - T   T    - T   T    +
│ │ │ -       1,4 2,1    1,1 2,5      3,10      1,4   2,2      1,1 2,1    1,4 2,2  
│ │ │ +      - T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    + x*T   ),
│ │ │ +         1,5 2,3    1,2 2,5      3,9      1,4   2,5      1,4 2,5      3,8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   ), ({T   , T   }, T   T    + T   T    + T   T    + y*T    -
│ │ │ -         3,1      1,2   2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4  
│ │ │ +      ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   }, T   T   
│ │ │ +         1,5   2,3      1,5 2,3    1,2 2,5      3,9      1,3   2,3    1,3 2,3
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    - z*T    + z*T   ), ({T   ,
│ │ │ -         3,7      1,5   2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3 
│ │ │ +      + x*T    - z*T   ), ({T   , T   }, - T   T    - T   T    - T   T    +
│ │ │ +           3,5      3,7      1,1   2,3      1,3 2,1    1,5 2,2    1,1 2,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   ,
│ │ │ -       2,3    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3 
│ │ │ +      z*T    + x*T   ), ({T   , T   }, T   T    - T   T    - z*T    +
│ │ │ +         3,2      3,4      1,4   2,4    1,4 2,4    1,5 2,5      3,7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    + y*T    - z*T   ), ({T   , T   }, - T   T    -
│ │ │ -       2,1    1,3 2,1      3,1      3,2      1,5   2,2      1,5 2,2  
│ │ │ +      z*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   },
│ │ │ +         3,9      1,2   2,3      1,2 2,3    1,5 2,4      3,5      1,5   2,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + z*T    + z*T    ), ({T   , T   }, T   T    + T   T    -
│ │ │ -       1,4 2,5      3,2      3,10      1,3   2,2    1,4 2,1    1,3 2,2  
│ │ │ +      - T   T    - T   T    - T   T    + z*T    + x*T   ), ({T   , T   },
│ │ │ +         1,3 2,1    1,5 2,2    1,1 2,3      3,2      3,4      1,4   2,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, -
│ │ │ -         3,1      3,3      1,2   2,4      1,2 2,4      3,6      1,1   2,4    
│ │ │ +      T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   , T   },
│ │ │ +       1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3   2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ -       1,5 2,1    1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2  
│ │ │ +      T   T    + T   T    - z*T    + y*T    ), ({T   , T   }, T   T    +
│ │ │ +       1,5 2,1    1,3 2,5      3,8      3,10      1,5   2,1    1,5 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   },
│ │ │ -         3,3      1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4  
│ │ │ +      T   T    - z*T    + y*T    ), ({T   , T   }, - T   T    - T   T    +
│ │ │ +       1,3 2,5      3,8      3,10      1,4   2,5      1,5 2,2    1,4 2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ),
│ │ │ -         1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2  
│ │ │ +      z*T    + z*T    ), ({T   , T   }, - T   T    - T   T    + x*T   ),
│ │ │ +         3,2      3,10      1,1   2,1      1,1 2,1    1,4 2,2      3,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ -         1,5   2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3  
│ │ │ +      ({T   , T   }, T   T    + T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +         1,3   2,4    1,5 2,3    1,3 2,4      3,5      3,6      1,4   2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + T   T    - z*T    + x*T   )}
│ │ │ -       1,5 2,3    1,3 2,4      3,5      3,6
│ │ │ +      T   T    + T   T    - z*T    + y*T   )}
│ │ │ +       1,4 2,1    1,3 2,2      3,1      3,3
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : H#(H'_0)
│ │ │  
│ │ │  o16 = -1
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/html/_bracket.html
│ │ │ @@ -223,106 +223,106 @@
│ │ │                
    i14 : H' = select(keys H, k->H#k != 0);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : H'
│ │ │  
│ │ │ -o15 = {({T   , T   }, - T   T    - T   T    + y*T    + z*T   ), ({T   ,
│ │ │ -          1,4   2,3      1,2 2,2    1,4 2,3      3,2      3,4      1,3 
│ │ │ +o15 = {({T   , T   }, - T   T    + y*T   ), ({T   , T   }, T   T    -
│ │ │ +          1,5   2,5      1,5 2,5      3,8      1,4   2,1    1,4 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    - z*T    + y*T   ), ({T   , T   }, - T   T    -
│ │ │ -       2,4    1,3 2,4      3,5      3,7      1,3   2,1      1,3 2,1  
│ │ │ +      T   T    + x*T    ), ({T   , T   }, - T   T    - T   T    + x*T   ),
│ │ │ +       1,1 2,5      3,10      1,4   2,2      1,1 2,1    1,4 2,2      3,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ -       1,5 2,2    1,1 2,3      3,2      3,4      1,1   2,2      1,1 2,2  
│ │ │ +      ({T   , T   }, T   T    + T   T    + T   T    + y*T    - z*T   ),
│ │ │ +         1,2   2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   },
│ │ │ -         3,3      1,5   2,4      1,2 2,3    1,5 2,4      3,5      1,3   2,5  
│ │ │ +      ({T   , T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   },
│ │ │ +         1,5   2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3   2,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ -         1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,2      1,2 2,2  
│ │ │ +      T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   , T   },
│ │ │ +       1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3   2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T    + z*T   ), ({T   , T   }, - T   T    + T   T    -
│ │ │ -       1,4 2,3      3,2      3,4      1,4   2,3      1,4 2,3    1,3 2,5  
│ │ │ +      T   T    + y*T    - z*T   ), ({T   , T   }, - T   T    - T   T    +
│ │ │ +       1,3 2,1      3,1      3,2      1,5   2,2      1,5 2,2    1,4 2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ),
│ │ │ -         3,8      3,9      1,2   2,5      1,5 2,3    1,2 2,5      3,9  
│ │ │ +      z*T    + z*T    ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ +         3,2      3,10      1,3   2,2    1,4 2,1    1,3 2,2      3,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ -         1,4   2,5      1,4 2,5      3,8      1,5   2,3      1,5 2,3  
│ │ │ +      y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, - T   T   
│ │ │ +         3,3      1,2   2,4      1,2 2,4      3,6      1,1   2,4      1,5 2,1
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T   ), ({T   , T   }, T   T    + x*T    - z*T   ), ({T   ,
│ │ │ -       1,2 2,5      3,9      1,3   2,3    1,3 2,3      3,5      3,7      1,1 
│ │ │ +      - T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    + z*T   ),
│ │ │ +         1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2      3,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, - T   T    - T   T    - T   T    + z*T    + x*T   ), ({T   ,
│ │ │ -       2,3      1,3 2,1    1,5 2,2    1,1 2,3      3,2      3,4      1,4 
│ │ │ +      ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   }, -
│ │ │ +         1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4    
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   }, -
│ │ │ -       2,4    1,4 2,4    1,5 2,5      3,7      3,9      1,2   2,3    
│ │ │ +      T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ), ({T   ,
│ │ │ +       1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2      1,5 
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    - T   T    -
│ │ │ -       1,2 2,3    1,5 2,4      3,5      1,5   2,2      1,3 2,1    1,5 2,2  
│ │ │ +      T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +       2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + z*T    + x*T   ), ({T   , T   }, T   T    + T   T    +
│ │ │ -       1,1 2,3      3,2      3,4      1,4   2,4    1,2 2,1    1,3 2,3  
│ │ │ +      T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ +       1,5 2,3    1,3 2,4      3,5      3,6      1,4   2,3      1,2 2,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + y*T    - z*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ -       1,4 2,4      3,4      3,7      1,3   2,5    1,5 2,1    1,3 2,5  
│ │ │ +      T   T    + y*T    + z*T   ), ({T   , T   }, T   T    - z*T    +
│ │ │ +       1,4 2,3      3,2      3,4      1,3   2,4    1,3 2,4      3,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T    ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ -         3,8      3,10      1,5   2,1    1,5 2,1    1,3 2,5      3,8  
│ │ │ +      y*T   ), ({T   , T   }, - T   T    - T   T    - T   T    + z*T    +
│ │ │ +         3,7      1,3   2,1      1,3 2,1    1,5 2,2    1,1 2,3      3,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      y*T    ), ({T   , T   }, - T   T    - T   T    + z*T    + z*T    ),
│ │ │ -         3,10      1,4   2,5      1,5 2,2    1,4 2,5      3,2      3,10  
│ │ │ +      x*T   ), ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T   
│ │ │ +         3,4      1,1   2,2      1,1 2,2      3,3      1,5   2,4      1,2 2,3
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    - T   T    + x*T   ), ({T   , T   }, T   T   
│ │ │ -         1,1   2,1      1,1 2,1    1,4 2,2      3,1      1,3   2,4    1,5 2,3
│ │ │ +      - T   T    + y*T   ), ({T   , T   }, - T   T    + T   T    - z*T    +
│ │ │ +         1,5 2,4      3,5      1,3   2,5      1,4 2,3    1,3 2,5      3,8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      + T   T    - z*T    + x*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ -         1,3 2,4      3,5      3,6      1,4   2,1    1,4 2,1    1,3 2,2  
│ │ │ +      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T    + z*T   ),
│ │ │ +         3,9      1,2   2,2      1,2 2,2    1,4 2,3      3,2      3,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   },
│ │ │ -         3,1      3,3      1,5   2,5      1,5 2,5      3,8      1,4   2,1  
│ │ │ +      ({T   , T   }, - T   T    + T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +         1,4   2,3      1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    + x*T    ), ({T   , T   }, - T   T    - T   T    +
│ │ │ -       1,4 2,1    1,1 2,5      3,10      1,4   2,2      1,1 2,1    1,4 2,2  
│ │ │ +      - T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    + x*T   ),
│ │ │ +         1,5 2,3    1,2 2,5      3,9      1,4   2,5      1,4 2,5      3,8  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      x*T   ), ({T   , T   }, T   T    + T   T    + T   T    + y*T    -
│ │ │ -         3,1      1,2   2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4  
│ │ │ +      ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   }, T   T   
│ │ │ +         1,5   2,3      1,5 2,3    1,2 2,5      3,9      1,3   2,3    1,3 2,3
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    - z*T    + z*T   ), ({T   ,
│ │ │ -         3,7      1,5   2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3 
│ │ │ +      + x*T    - z*T   ), ({T   , T   }, - T   T    - T   T    - T   T    +
│ │ │ +           3,5      3,7      1,1   2,3      1,3 2,1    1,5 2,2    1,1 2,3  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   ,
│ │ │ -       2,3    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3 
│ │ │ +      z*T    + x*T   ), ({T   , T   }, T   T    - T   T    - z*T    +
│ │ │ +         3,2      3,4      1,4   2,4    1,4 2,4    1,5 2,5      3,7  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   }, T   T    + y*T    - z*T   ), ({T   , T   }, - T   T    -
│ │ │ -       2,1    1,3 2,1      3,1      3,2      1,5   2,2      1,5 2,2  
│ │ │ +      z*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   },
│ │ │ +         3,9      1,2   2,3      1,2 2,3    1,5 2,4      3,5      1,5   2,2  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + z*T    + z*T    ), ({T   , T   }, T   T    + T   T    -
│ │ │ -       1,4 2,5      3,2      3,10      1,3   2,2    1,4 2,1    1,3 2,2  
│ │ │ +      - T   T    - T   T    - T   T    + z*T    + x*T   ), ({T   , T   },
│ │ │ +         1,3 2,1    1,5 2,2    1,1 2,3      3,2      3,4      1,4   2,4  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, -
│ │ │ -         3,1      3,3      1,2   2,4      1,2 2,4      3,6      1,1   2,4    
│ │ │ +      T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   , T   },
│ │ │ +       1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3   2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    - T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ -       1,5 2,1    1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2  
│ │ │ +      T   T    + T   T    - z*T    + y*T    ), ({T   , T   }, T   T    +
│ │ │ +       1,5 2,1    1,3 2,5      3,8      3,10      1,5   2,1    1,5 2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   },
│ │ │ -         3,3      1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4  
│ │ │ +      T   T    - z*T    + y*T    ), ({T   , T   }, - T   T    - T   T    +
│ │ │ +       1,3 2,5      3,8      3,10      1,4   2,5      1,5 2,2    1,4 2,5  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      - T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ),
│ │ │ -         1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2  
│ │ │ +      z*T    + z*T    ), ({T   , T   }, - T   T    - T   T    + x*T   ),
│ │ │ +         3,2      3,10      1,1   2,1      1,1 2,1    1,4 2,2      3,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      ({T   , T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ -         1,5   2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3  
│ │ │ +      ({T   , T   }, T   T    + T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ +         1,3   2,4    1,5 2,3    1,3 2,4      3,5      3,6      1,4   2,1  
│ │ │        -----------------------------------------------------------------------
│ │ │ -      T   T    + T   T    - z*T    + x*T   )}
│ │ │ -       1,5 2,3    1,3 2,4      3,5      3,6
│ │ │ +      T   T    + T   T    - z*T    + y*T   )}
│ │ │ +       1,4 2,1    1,3 2,2      3,1      3,3
│ │ │  
│ │ │  o15 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : H#(H'_0)
│ │ │ ├── html2text {}
│ │ │ │ @@ -119,106 +119,106 @@
│ │ │ │  i12 : H = bracket(A,2,3);
│ │ │ │  i13 : #keys H
│ │ │ │  
│ │ │ │  o13 = 600
│ │ │ │  i14 : H' = select(keys H, k->H#k != 0);
│ │ │ │  i15 : H'
│ │ │ │  
│ │ │ │ -o15 = {({T   , T   }, - T   T    - T   T    + y*T    + z*T   ), ({T   ,
│ │ │ │ -          1,4   2,3      1,2 2,2    1,4 2,3      3,2      3,4      1,3
│ │ │ │ +o15 = {({T   , T   }, - T   T    + y*T   ), ({T   , T   }, T   T    -
│ │ │ │ +          1,5   2,5      1,5 2,5      3,8      1,4   2,1    1,4 2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, T   T    - z*T    + y*T   ), ({T   , T   }, - T   T    -
│ │ │ │ -       2,4    1,3 2,4      3,5      3,7      1,3   2,1      1,3 2,1
│ │ │ │ +      T   T    + x*T    ), ({T   , T   }, - T   T    - T   T    + x*T   ),
│ │ │ │ +       1,1 2,5      3,10      1,4   2,2      1,1 2,1    1,4 2,2      3,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    - T   T    + z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ │ -       1,5 2,2    1,1 2,3      3,2      3,4      1,1   2,2      1,1 2,2
│ │ │ │ +      ({T   , T   }, T   T    + T   T    + T   T    + y*T    - z*T   ),
│ │ │ │ +         1,2   2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   },
│ │ │ │ -         3,3      1,5   2,4      1,2 2,3    1,5 2,4      3,5      1,3   2,5
│ │ │ │ +      ({T   , T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   },
│ │ │ │ +         1,5   2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3   2,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      - T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ │ -         1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,2      1,2 2,2
│ │ │ │ +      T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   , T   },
│ │ │ │ +       1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3   2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + y*T    + z*T   ), ({T   , T   }, - T   T    + T   T    -
│ │ │ │ -       1,4 2,3      3,2      3,4      1,4   2,3      1,4 2,3    1,3 2,5
│ │ │ │ +      T   T    + y*T    - z*T   ), ({T   , T   }, - T   T    - T   T    +
│ │ │ │ +       1,3 2,1      3,1      3,2      1,5   2,2      1,5 2,2    1,4 2,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T    + x*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ),
│ │ │ │ -         3,8      3,9      1,2   2,5      1,5 2,3    1,2 2,5      3,9
│ │ │ │ +      z*T    + z*T    ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ │ +         3,2      3,10      1,3   2,2    1,4 2,1    1,3 2,2      3,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ │ -         1,4   2,5      1,4 2,5      3,8      1,5   2,3      1,5 2,3
│ │ │ │ +      y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, - T   T
│ │ │ │ +         3,3      1,2   2,4      1,2 2,4      3,6      1,1   2,4      1,5 2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + y*T   ), ({T   , T   }, T   T    + x*T    - z*T   ), ({T   ,
│ │ │ │ -       1,2 2,5      3,9      1,3   2,3    1,3 2,3      3,5      3,7      1,1
│ │ │ │ +      - T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    + z*T   ),
│ │ │ │ +         1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2      3,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, - T   T    - T   T    - T   T    + z*T    + x*T   ), ({T   ,
│ │ │ │ -       2,3      1,3 2,1    1,5 2,2    1,1 2,3      3,2      3,4      1,4
│ │ │ │ +      ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   }, -
│ │ │ │ +         1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, T   T    - T   T    - z*T    + z*T   ), ({T   , T   }, -
│ │ │ │ -       2,4    1,4 2,4    1,5 2,5      3,7      3,9      1,2   2,3
│ │ │ │ +      T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ), ({T   ,
│ │ │ │ +       1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2      1,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    - T   T    -
│ │ │ │ -       1,2 2,3    1,5 2,4      3,5      1,5   2,2      1,3 2,1    1,5 2,2
│ │ │ │ +      T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ │ +       2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + z*T    + x*T   ), ({T   , T   }, T   T    + T   T    +
│ │ │ │ -       1,1 2,3      3,2      3,4      1,4   2,4    1,2 2,1    1,3 2,3
│ │ │ │ +      T   T    + T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    -
│ │ │ │ +       1,5 2,3    1,3 2,4      3,5      3,6      1,4   2,3      1,2 2,2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + y*T    - z*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ │ -       1,4 2,4      3,4      3,7      1,3   2,5    1,5 2,1    1,3 2,5
│ │ │ │ +      T   T    + y*T    + z*T   ), ({T   , T   }, T   T    - z*T    +
│ │ │ │ +       1,4 2,3      3,2      3,4      1,3   2,4    1,3 2,4      3,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T    + y*T    ), ({T   , T   }, T   T    + T   T    - z*T    +
│ │ │ │ -         3,8      3,10      1,5   2,1    1,5 2,1    1,3 2,5      3,8
│ │ │ │ +      y*T   ), ({T   , T   }, - T   T    - T   T    - T   T    + z*T    +
│ │ │ │ +         3,7      1,3   2,1      1,3 2,1    1,5 2,2    1,1 2,3      3,2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      y*T    ), ({T   , T   }, - T   T    - T   T    + z*T    + z*T    ),
│ │ │ │ -         3,10      1,4   2,5      1,5 2,2    1,4 2,5      3,2      3,10
│ │ │ │ +      x*T   ), ({T   , T   }, - T   T    + x*T   ), ({T   , T   }, - T   T
│ │ │ │ +         3,4      1,1   2,2      1,1 2,2      3,3      1,5   2,4      1,2 2,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T   , T   }, - T   T    - T   T    + x*T   ), ({T   , T   }, T   T
│ │ │ │ -         1,1   2,1      1,1 2,1    1,4 2,2      3,1      1,3   2,4    1,5 2,3
│ │ │ │ +      - T   T    + y*T   ), ({T   , T   }, - T   T    + T   T    - z*T    +
│ │ │ │ +         1,5 2,4      3,5      1,3   2,5      1,4 2,3    1,3 2,5      3,8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      + T   T    - z*T    + x*T   ), ({T   , T   }, T   T    + T   T    -
│ │ │ │ -         1,3 2,4      3,5      3,6      1,4   2,1    1,4 2,1    1,3 2,2
│ │ │ │ +      x*T   ), ({T   , T   }, - T   T    - T   T    + y*T    + z*T   ),
│ │ │ │ +         3,9      1,2   2,2      1,2 2,2    1,4 2,3      3,2      3,4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   },
│ │ │ │ -         3,1      3,3      1,5   2,5      1,5 2,5      3,8      1,4   2,1
│ │ │ │ +      ({T   , T   }, - T   T    + T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ │ +         1,4   2,3      1,4 2,3    1,3 2,5      3,8      3,9      1,2   2,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    - T   T    + x*T    ), ({T   , T   }, - T   T    - T   T    +
│ │ │ │ -       1,4 2,1    1,1 2,5      3,10      1,4   2,2      1,1 2,1    1,4 2,2
│ │ │ │ +      - T   T    - T   T    + y*T   ), ({T   , T   }, - T   T    + x*T   ),
│ │ │ │ +         1,5 2,3    1,2 2,5      3,9      1,4   2,5      1,4 2,5      3,8
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      x*T   ), ({T   , T   }, T   T    + T   T    + T   T    + y*T    -
│ │ │ │ -         3,1      1,2   2,1    1,2 2,1    1,3 2,3    1,4 2,4      3,4
│ │ │ │ +      ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   }, T   T
│ │ │ │ +         1,5   2,3      1,5 2,3    1,2 2,5      3,9      1,3   2,3    1,3 2,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    - z*T    + z*T   ), ({T   ,
│ │ │ │ -         3,7      1,5   2,5    1,4 2,4    1,5 2,5      3,7      3,9      1,3
│ │ │ │ +      + x*T    - z*T   ), ({T   , T   }, - T   T    - T   T    - T   T    +
│ │ │ │ +           3,5      3,7      1,1   2,3      1,3 2,1    1,5 2,2    1,1 2,3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   ,
│ │ │ │ -       2,3    1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3
│ │ │ │ +      z*T    + x*T   ), ({T   , T   }, T   T    - T   T    - z*T    +
│ │ │ │ +         3,2      3,4      1,4   2,4    1,4 2,4    1,5 2,5      3,7
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   }, T   T    + y*T    - z*T   ), ({T   , T   }, - T   T    -
│ │ │ │ -       2,1    1,3 2,1      3,1      3,2      1,5   2,2      1,5 2,2
│ │ │ │ +      z*T   ), ({T   , T   }, - T   T    - T   T    + y*T   ), ({T   , T   },
│ │ │ │ +         3,9      1,2   2,3      1,2 2,3    1,5 2,4      3,5      1,5   2,2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + z*T    + z*T    ), ({T   , T   }, T   T    + T   T    -
│ │ │ │ -       1,4 2,5      3,2      3,10      1,3   2,2    1,4 2,1    1,3 2,2
│ │ │ │ +      - T   T    - T   T    - T   T    + z*T    + x*T   ), ({T   , T   },
│ │ │ │ +         1,3 2,1    1,5 2,2    1,1 2,3      3,2      3,4      1,4   2,4
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T    + y*T   ), ({T   , T   }, - T   T    + y*T   ), ({T   , T   }, -
│ │ │ │ -         3,1      3,3      1,2   2,4      1,2 2,4      3,6      1,1   2,4
│ │ │ │ +      T   T    + T   T    + T   T    + y*T    - z*T   ), ({T   , T   },
│ │ │ │ +       1,2 2,1    1,3 2,3    1,4 2,4      3,4      3,7      1,3   2,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    - T   T    - z*T    + x*T   ), ({T   , T   }, - T   T    +
│ │ │ │ -       1,5 2,1    1,1 2,4      3,4      3,7      1,4   2,2      1,4 2,2
│ │ │ │ +      T   T    + T   T    - z*T    + y*T    ), ({T   , T   }, T   T    +
│ │ │ │ +       1,5 2,1    1,3 2,5      3,8      3,10      1,5   2,1    1,5 2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      z*T   ), ({T   , T   }, T   T    - T   T    + x*T    ), ({T   , T   },
│ │ │ │ -         3,3      1,1   2,5    1,4 2,1    1,1 2,5      3,10      1,5   2,4
│ │ │ │ +      T   T    - z*T    + y*T    ), ({T   , T   }, - T   T    - T   T    +
│ │ │ │ +       1,3 2,5      3,8      3,10      1,4   2,5      1,5 2,2    1,4 2,5
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      - T   T    + z*T   ), ({T   , T   }, T   T    - z*T    + x*T   ),
│ │ │ │ -         1,5 2,4      3,6      1,3   2,2    1,3 2,2      3,1      3,2
│ │ │ │ +      z*T    + z*T    ), ({T   , T   }, - T   T    - T   T    + x*T   ),
│ │ │ │ +         3,2      3,10      1,1   2,1      1,1 2,1    1,4 2,2      3,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      ({T   , T   }, - T   T    - T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ │ -         1,5   2,1      1,5 2,1    1,1 2,4      3,4      3,7      1,5   2,3
│ │ │ │ +      ({T   , T   }, T   T    + T   T    - z*T    + x*T   ), ({T   , T   },
│ │ │ │ +         1,3   2,4    1,5 2,3    1,3 2,4      3,5      3,6      1,4   2,1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ -      T   T    + T   T    - z*T    + x*T   )}
│ │ │ │ -       1,5 2,3    1,3 2,4      3,5      3,6
│ │ │ │ +      T   T    + T   T    - z*T    + y*T   )}
│ │ │ │ +       1,4 2,1    1,3 2,2      3,1      3,3
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : H#(H'_0)
│ │ │ │  
│ │ │ │  o16 = -1
│ │ │ │  
│ │ │ │  o16 : S[T   ..T   , T   ..T    , T   ..T    , T   ..T    ]
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  cmVzdHJpY3Rpb24oQXJyYW5nZW1lbnQsTGlzdCk=
│ │ │  #:len=343
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc3OCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVzdHJpY3Rpb24sQXJyYW5nZW1lbnQsTGlzdCks
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_cone_lp__Arrangement_cm__Ring__Element_rp.out
│ │ │ @@ -44,15 +44,15 @@
│ │ │  
│ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │  
│ │ │  o13 : Hyperplane Arrangement 
│ │ │  
│ │ │  i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {y, x, x - y}
│ │ │ +o14 = {x - y, y, x}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
│ │ │  
│ │ │  i15 : assert isCentral cA''
│ │ │  
│ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.out
│ │ │ @@ -10,35 +10,35 @@
│ │ │  
│ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │  
│ │ │  o2 : Hyperplane Arrangement 
│ │ │  
│ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : restriction(A,1)
│ │ │  
│ │ │  o4 = {x, z, x, x - z}
│ │ │  
│ │ │  o4 : Hyperplane Arrangement 
│ │ │  
│ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {x - z, z, x}
│ │ │ +o5 = {z, x, x - z}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │  
│ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : prune image der(A,m)
│ │ │  
│ │ │        3
│ │ │  o9 = R
│ │ │ @@ -59,16 +59,16 @@
│ │ │  
│ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │  
│ │ │  i12 : A = arrangement "bracelet";
│ │ │  
│ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         2   3    4   3   4   2    3    4   2    4
│ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │  
│ │ │  o13 : Sequence
│ │ │  
│ │ │  i14 : C = restriction(A,0)
│ │ │  
│ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_trim_lp__Arrangement_rp.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  o2 = {x, x, 0, y, y, y, x + y, x + y, x + y, x + y, x + y}
│ │ │  
│ │ │  o2 : Hyperplane Arrangement 
│ │ │  
│ │ │  i3 : A' = trim A
│ │ │  
│ │ │ -o3 = {x + y, y, x}
│ │ │ +o3 = {y, x, x + y}
│ │ │  
│ │ │  o3 : Hyperplane Arrangement 
│ │ │  
│ │ │  i4 : assert(ring A' === R)
│ │ │  
│ │ │  i5 : assert(trim A' == A')
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_type__B_lp__Z__Z_cm__Ring_rp.out
│ │ │ @@ -33,16 +33,16 @@
│ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │         1   1    2   1    2   2
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
│ │ │  
│ │ │  i7 : ring A3
│ │ │  
│ │ │       ZZ
│ │ │  o7 = --[x ..x ]
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_cone_lp__Arrangement_cm__Ring__Element_rp.html
│ │ │ @@ -172,15 +172,15 @@
│ │ │  o13 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {y, x, x - y}
│ │ │ +o14 = {x - y, y, x}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : assert isCentral cA''
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -60,15 +60,15 @@
│ │ │ │  i13 : cone(A, x)
│ │ │ │  
│ │ │ │  o13 = {x, y, x - y, 0, - x + y, x}
│ │ │ │  
│ │ │ │  o13 : Hyperplane Arrangement
│ │ │ │  i14 : cA'' = trim cone(A, x)
│ │ │ │  
│ │ │ │ -o14 = {y, x, x - y}
│ │ │ │ +o14 = {x - y, y, x}
│ │ │ │  
│ │ │ │  o14 : Hyperplane Arrangement
│ │ │ │  i15 : assert isCentral cA''
│ │ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ │ │  When the second input is a _S_y_m_b_o_l, this method creates a new ring from the
│ │ │ │  underlying ring of $A$ by adjoining the symbol as a variable and constructs the
│ │ │ │  cone in this new ring.
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o2 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : restriction(A,1)
│ │ │ @@ -118,15 +118,15 @@
│ │ │  o4 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {x - z, z, x}
│ │ │ +o5 = {z, x, x - z}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$, where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the Euler restriction is free as well, and the modules of logarithmic derivations form a short exact sequence.  See the paper of Abe, Terao and Wakefield for details.
    
│ │ │ @@ -137,15 +137,15 @@
│ │ │                
    i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : prune image der(A,m)
│ │ │ @@ -186,16 +186,16 @@
│ │ │                
    i12 : A = arrangement "bracelet";
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         2   3    4   3   4   2    3    4   2    4
│ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │  
│ │ │  o13 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : C = restriction(A,0)
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,36 +33,36 @@
│ │ │ │  i2 : A = arrangement {x,y,z,x-y,x-z}
│ │ │ │  
│ │ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │ │  
│ │ │ │  o2 : Hyperplane Arrangement
│ │ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │ │  
│ │ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : restriction(A,1)
│ │ │ │  
│ │ │ │  o4 = {x, z, x, x - z}
│ │ │ │  
│ │ │ │  o4 : Hyperplane Arrangement
│ │ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │ │  
│ │ │ │ -o5 = {x - z, z, x}
│ │ │ │ +o5 = {z, x, x - z}
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$,
│ │ │ │  where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the
│ │ │ │  Euler restriction is free as well, and the modules of _l_o_g_a_r_i_t_h_m_i_c_ _d_e_r_i_v_a_t_i_o_n_s
│ │ │ │  form a short exact sequence. See the paper of Abe, Terao and Wakefield for
│ │ │ │  details.
│ │ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │ │  
│ │ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : prune image der(A,m)
│ │ │ │  
│ │ │ │        3
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │ @@ -80,16 +80,16 @@
│ │ │ │  
│ │ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │ │  It may be the case that the Euler restriction is free, while the naive
│ │ │ │  restriction is not:
│ │ │ │  i12 : A = arrangement "bracelet";
│ │ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │ │  
│ │ │ │ -o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ -         2   3    4   3   4   2    3    4   2    4
│ │ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ │ │  i14 : C = restriction(A,0)
│ │ │ │  
│ │ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_trim_lp__Arrangement_rp.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : A' = trim A
│ │ │  
│ │ │ -o3 = {x + y, y, x}
│ │ │ +o3 = {y, x, x + y}
│ │ │  
│ │ │  o3 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : assert(ring A' === R)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : A = arrangement{x,x,0_R,y,y,y,x+y,x+y,x+y,x+y,x+y}
│ │ │ │  
│ │ │ │  o2 = {x, x, 0, y, y, y, x + y, x + y, x + y, x + y, x + y}
│ │ │ │  
│ │ │ │  o2 : Hyperplane Arrangement
│ │ │ │  i3 : A' = trim A
│ │ │ │  
│ │ │ │ -o3 = {x + y, y, x}
│ │ │ │ +o3 = {y, x, x + y}
│ │ │ │  
│ │ │ │  o3 : Hyperplane Arrangement
│ │ │ │  i4 : assert(ring A' === R)
│ │ │ │  i5 : assert(trim A' == A')
│ │ │ │  i6 : assert(trim A' == A')
│ │ │ │  Some natural operations produce non-simple hyperplane arrangements.
│ │ │ │  i7 : A'' = restriction(A, y)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_type__B_lp__Z__Z_cm__Ring_rp.html
│ │ │ @@ -130,16 +130,16 @@
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       2   1   1    2
│ │ │ +o6 = {x  + x , x , x }
│ │ │ +       1    2   2   1
│ │ │  
│ │ │  o6 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : ring A3
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,16 +51,16 @@
│ │ │ │  
│ │ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │ │         1   1    2   1    2   2
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  i6 : trim A3
│ │ │ │  
│ │ │ │ -o6 = {x , x , x  + x }
│ │ │ │ -       2   1   1    2
│ │ │ │ +o6 = {x  + x , x , x }
│ │ │ │ +       1    2   2   1
│ │ │ │  
│ │ │ │  o6 : Hyperplane Arrangement
│ │ │ │  i7 : ring A3
│ │ │ │  
│ │ │ │       ZZ
│ │ │ │  o7 = --[x ..x ]
│ │ │ │        2  1   2
│ │ ├── ./usr/share/doc/Macaulay2/IncidenceCorrespondenceCohomology/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  cmVjdXJzaXZlRGl2aWRlZENvaG9tb2xvZ3k=
│ │ │  #:len=3143
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgZGltZW5zaW9uIG9yIGNo
│ │ │  YXJhY3RlciBvZiBzaGVhZiBjb2hvbW9sb2d5IG9mIHR3aXN0cyBvZiBkaXZpZGVkIHBvd2VycyBv
│ │ ├── ./usr/share/doc/Macaulay2/IntegerProgramming/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:32 2026
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=36
│ │ │  YWRhcHRlZE1vbm9taWFsT3JkZXIoLi4uLEZpZWxkPT4uLi4p
│ │ │  #:len=286
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTI2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1thZGFwdGVkTW9ub21pYWxPcmRlcixGaWVsZF0sImFk
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  aW50ZWdyYWxDbG9zdXJlKC4uLixMaW1pdD0+Li4uKQ==
│ │ │  #:len=1120
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZG8gYSBwYXJ0aWFsIGludGVncmFsIGNs
│ │ │  b3N1cmUiLCAibGluZW51bSIgPT4gMTM5NywgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -16,15 +16,15 @@
│ │ │  i3 : R = S/f
│ │ │  
│ │ │  o3 = R
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : time R' = integralClosure R
│ │ │ - -- used 0.387896s (cpu); 0.303595s (thread); 0s (gc)
│ │ │ + -- used 0.386165s (cpu); 0.309925s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -83,15 +83,15 @@
│ │ │  i9 : R = S/f
│ │ │  
│ │ │  o9 = R
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │  
│ │ │  i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.508329s (cpu); 0.342291s (thread); 0s (gc)
│ │ │ + -- used 0.360586s (cpu); 0.287809s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
│ │ │  
│ │ │  i11 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -150,15 +150,15 @@
│ │ │  i15 : R = S/f
│ │ │  
│ │ │  o15 = R
│ │ │  
│ │ │  o15 : QuotientRing
│ │ │  
│ │ │  i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.507202s (cpu); 0.349308s (thread); 0s (gc)
│ │ │ + -- used 0.488122s (cpu); 0.332852s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -208,15 +208,15 @@
│ │ │  i20 : R = S/f
│ │ │  
│ │ │  o20 = R
│ │ │  
│ │ │  o20 : QuotientRing
│ │ │  
│ │ │  i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.400249s (cpu); 0.317688s (thread); 0s (gc)
│ │ │ + -- used 0.389919s (cpu); 0.315813s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
│ │ │  
│ │ │  i22 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -266,15 +266,15 @@
│ │ │  i25 : R = S/f
│ │ │  
│ │ │  o25 = R
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │  
│ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.79625s (cpu); 0.550685s (thread); 0s (gc)
│ │ │ + -- used 0.772773s (cpu); 0.540577s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
│ │ │  
│ │ │  i27 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -324,15 +324,15 @@
│ │ │  i30 : R = S/f
│ │ │  
│ │ │  o30 = R
│ │ │  
│ │ │  o30 : QuotientRing
│ │ │  
│ │ │  i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.508579s (cpu); 0.336293s (thread); 0s (gc)
│ │ │ + -- used 0.477415s (cpu); 0.326729s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
│ │ │  
│ │ │  i32 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -382,15 +382,15 @@
│ │ │  i35 : R = S/f
│ │ │  
│ │ │  o35 = R
│ │ │  
│ │ │  o35 : QuotientRing
│ │ │  
│ │ │  i36 : time R' = integralClosure R
│ │ │ - -- used 0.0428345s (cpu); 0.0428212s (thread); 0s (gc)
│ │ │ + -- used 0.0466942s (cpu); 0.046696s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
│ │ │  
│ │ │  i37 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -432,15 +432,15 @@
│ │ │  i40 : R = S/I
│ │ │  
│ │ │  o40 = R
│ │ │  
│ │ │  o40 : QuotientRing
│ │ │  
│ │ │  i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.167523s (cpu); 0.0828249s (thread); 0s (gc)
│ │ │ + -- used 0.15173s (cpu); 0.0714075s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
│ │ │  
│ │ │  i42 : icFractions R
│ │ │  
│ │ │ @@ -467,15 +467,15 @@
│ │ │  i45 : R = S/I
│ │ │  
│ │ │  o45 = R
│ │ │  
│ │ │  o45 : QuotientRing
│ │ │  
│ │ │  i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0639871s (cpu); 0.0639869s (thread); 0s (gc)
│ │ │ + -- used 0.0668493s (cpu); 0.0668461s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
│ │ │  
│ │ │  i47 : icFractions R
│ │ │  
│ │ │ @@ -501,15 +501,15 @@
│ │ │  i50 : R = S/I
│ │ │  
│ │ │  o50 = R
│ │ │  
│ │ │  o50 : QuotientRing
│ │ │  
│ │ │  i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0434306s (cpu); 0.0434315s (thread); 0s (gc)
│ │ │ + -- used 0.151552s (cpu); 0.0753283s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
│ │ │  
│ │ │  i52 : icFractions R
│ │ │  
│ │ │ @@ -536,15 +536,15 @@
│ │ │  i55 : R = S/I
│ │ │  
│ │ │  o55 = R
│ │ │  
│ │ │  o55 : QuotientRing
│ │ │  
│ │ │  i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.175803s (cpu); 0.0898169s (thread); 0s (gc)
│ │ │ + -- used 0.0457782s (cpu); 0.0457781s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
│ │ │  
│ │ │  i57 : icFractions R
│ │ │  
│ │ │ @@ -633,15 +633,15 @@
│ │ │  i66 : R = S/I
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │  
│ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0622734s (cpu); 0.0622745s (thread); 0s (gc)
│ │ │ + -- used 0.0659785s (cpu); 0.0659792s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
│ │ │  
│ │ │  i68 : icFractions R
│ │ │  
│ │ │ @@ -722,15 +722,15 @@
│ │ │  i77 : R = S/I
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │  
│ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.500553s (cpu); 0.333412s (thread); 0s (gc)
│ │ │ + -- used 0.474793s (cpu); 0.315983s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
│ │ │  
│ │ │  i79 : icFractions R
│ │ │  
│ │ │ @@ -750,15 +750,15 @@
│ │ │  i81 : R = S/sub(I,S)
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │  
│ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.543183s (cpu); 0.38548s (thread); 0s (gc)
│ │ │ + -- used 0.594658s (cpu); 0.365376s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -778,20 +778,20 @@
│ │ │  i85 : R = S/sub(I,S)
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000532358 sec #minors 4]
│ │ │ + [jacobian time .000558853 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .228455 sec  #fractions 6]
│ │ │ - [step 1:   time .246733 sec  #fractions 6]
│ │ │ - -- used 0.478873s (cpu); 0.304753s (thread); 0s (gc)
│ │ │ + [step 0:   time .106181 sec  #fractions 6]
│ │ │ + [step 1:   time .245944 sec  #fractions 6]
│ │ │ + -- used 0.355931s (cpu); 0.267095s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
│ │ │  
│ │ │  i87 : icFractions R
│ │ │  
│ │ │ @@ -811,20 +811,20 @@
│ │ │  i89 : R = S/sub(I,S)
│ │ │  
│ │ │  o89 = R
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │  
│ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000530094 sec #minors 4]
│ │ │ + [jacobian time .000524536 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .210442 sec  #fractions 6]
│ │ │ - [step 1:   time .269174 sec  #fractions 6]
│ │ │ - -- used 0.483329s (cpu); 0.298388s (thread); 0s (gc)
│ │ │ + [step 0:   time .213689 sec  #fractions 6]
│ │ │ + [step 1:   time .254215 sec  #fractions 6]
│ │ │ + -- used 0.471844s (cpu); 0.304723s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -844,20 +844,20 @@
│ │ │  i93 : R = S/sub(I,S)
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000774762 sec #minors 1]
│ │ │ + [jacobian time .000672198 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .271111 sec  #fractions 6]
│ │ │ - [step 1:   time .874897 sec  #fractions 6]
│ │ │ - -- used 1.15027s (cpu); 0.732279s (thread); 0s (gc)
│ │ │ + [step 0:   time .257929 sec  #fractions 6]
│ │ │ + [step 1:   time .679148 sec  #fractions 6]
│ │ │ + -- used 0.941001s (cpu); 0.600883s (thread); 0s (gc)
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
│ │ │  
│ │ │  i95 : icFractions R
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out
│ │ │ @@ -1,50 +1,50 @@
│ │ │  -- -*- M2-comint -*- hash: 13177954069434615273
│ │ │  
│ │ │  i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
│ │ │  
│ │ │  i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000522279 sec #minors 3]
│ │ │ + [jacobian time .000528384 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00233339 seconds
│ │ │ -      idlizer1:  .00747003 seconds
│ │ │ -      idlizer2:  .00844957 seconds
│ │ │ -      minpres:   .00811702 seconds
│ │ │ -  time .0371527 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00247504 seconds
│ │ │ +      idlizer1:  .00847245 seconds
│ │ │ +      idlizer2:  .010069 seconds
│ │ │ +      minpres:   .00983557 seconds
│ │ │ +  time .0423551 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00248298 seconds
│ │ │ -      idlizer1:  .0119562 seconds
│ │ │ -      idlizer2:  .0111 seconds
│ │ │ -      minpres:   .0117312 seconds
│ │ │ -  time .0484066 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00246977 seconds
│ │ │ +      idlizer1:  .0128471 seconds
│ │ │ +      idlizer2:  .0109258 seconds
│ │ │ +      minpres:   .013127 seconds
│ │ │ +  time .0514417 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00413852 seconds
│ │ │ -      idlizer1:  .0169512 seconds
│ │ │ -      idlizer2:  .0143372 seconds
│ │ │ -      minpres:   .0137525 seconds
│ │ │ -  time .203496 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00272689 seconds
│ │ │ +      idlizer1:  .011831 seconds
│ │ │ +      idlizer2:  .0111346 seconds
│ │ │ +      minpres:   .0104825 seconds
│ │ │ +  time .143873 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00279962 seconds
│ │ │ -      idlizer1:  .0141692 seconds
│ │ │ -      idlizer2:  .0166105 seconds
│ │ │ -      minpres:   .0188805 seconds
│ │ │ -  time .0680688 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00306728 seconds
│ │ │ +      idlizer1:  .0134167 seconds
│ │ │ +      idlizer2:  .0142861 seconds
│ │ │ +      minpres:   .018816 seconds
│ │ │ +  time .0641517 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .0025556 seconds
│ │ │ -      idlizer1:  .00935896 seconds
│ │ │ -      idlizer2:  .0181192 seconds
│ │ │ -      minpres:   .0119314 seconds
│ │ │ -  time .0555899 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00311785 seconds
│ │ │ +      idlizer1:  .00982251 seconds
│ │ │ +      idlizer2:  .0184608 seconds
│ │ │ +      minpres:   .0119702 seconds
│ │ │ +  time .0576614 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00261851 seconds
│ │ │ -      idlizer1:  .00849894 seconds
│ │ │ -  time .0184597 sec  #fractions 5]
│ │ │ - -- used 0.435062s (cpu); 0.361171s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .002594 seconds
│ │ │ +      idlizer1:  .00900577 seconds
│ │ │ +  time .0191255 sec  #fractions 5]
│ │ │ + -- used 0.382847s (cpu); 0.314165s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │  i3 : trim ideal R'
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out
│ │ │ @@ -13,26 +13,26 @@
│ │ │  
│ │ │                  2      2    2        2   2 2     2
│ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │  
│ │ │  o3 : Ideal of S
│ │ │  
│ │ │  i4 : time integralClosure J
│ │ │ - -- used 1.00348s (cpu); 0.739924s (thread); 0s (gc)
│ │ │ + -- used 1.15719s (cpu); 0.697718s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.733674s (cpu); 0.56695s (thread); 0s (gc)
│ │ │ + -- used 1.16334s (cpu); 0.547475s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time R' = integralClosure R
│ │ │ - -- used 0.387896s (cpu); 0.303595s (thread); 0s (gc)
│ │ │ + -- used 0.386165s (cpu); 0.309925s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -191,15 +191,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.508329s (cpu); 0.342291s (thread); 0s (gc)
│ │ │ + -- used 0.360586s (cpu); 0.287809s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -278,15 +278,15 @@
│ │ │  
│ │ │  o15 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.507202s (cpu); 0.349308s (thread); 0s (gc)
│ │ │ + -- used 0.488122s (cpu); 0.332852s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -353,15 +353,15 @@
│ │ │  
│ │ │  o20 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.400249s (cpu); 0.317688s (thread); 0s (gc)
│ │ │ + -- used 0.389919s (cpu); 0.315813s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -428,15 +428,15 @@ │ │ │ │ │ │ o25 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.79625s (cpu); 0.550685s (thread); 0s (gc)
│ │ │ + -- used 0.772773s (cpu); 0.540577s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -503,15 +503,15 @@ │ │ │ │ │ │ o30 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.508579s (cpu); 0.336293s (thread); 0s (gc)
│ │ │ + -- used 0.477415s (cpu); 0.326729s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -578,15 +578,15 @@ │ │ │ │ │ │ o35 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.0428345s (cpu); 0.0428212s (thread); 0s (gc)
│ │ │ + -- used 0.0466942s (cpu); 0.046696s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -648,15 +648,15 @@ │ │ │ │ │ │ o40 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.167523s (cpu); 0.0828249s (thread); 0s (gc)
│ │ │ + -- used 0.15173s (cpu); 0.0714075s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -700,15 +700,15 @@ │ │ │ │ │ │ o45 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0639871s (cpu); 0.0639869s (thread); 0s (gc)
│ │ │ + -- used 0.0668493s (cpu); 0.0668461s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -751,15 +751,15 @@ │ │ │ │ │ │ o50 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0434306s (cpu); 0.0434315s (thread); 0s (gc)
│ │ │ + -- used 0.151552s (cpu); 0.0753283s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -803,15 +803,15 @@ │ │ │ │ │ │ o55 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.175803s (cpu); 0.0898169s (thread); 0s (gc)
│ │ │ + -- used 0.0457782s (cpu); 0.0457781s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -938,15 +938,15 @@ │ │ │ │ │ │ o66 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0622734s (cpu); 0.0622745s (thread); 0s (gc)
│ │ │ + -- used 0.0659785s (cpu); 0.0659792s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1062,15 +1062,15 @@ │ │ │ │ │ │ o77 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.500553s (cpu); 0.333412s (thread); 0s (gc)
│ │ │ + -- used 0.474793s (cpu); 0.315983s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1104,15 +1104,15 @@ │ │ │ │ │ │ o81 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.543183s (cpu); 0.38548s (thread); 0s (gc)
│ │ │ + -- used 0.594658s (cpu); 0.365376s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1146,20 +1146,20 @@ │ │ │ │ │ │ o85 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000532358 sec #minors 4]
│ │ │ + [jacobian time .000558853 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .228455 sec  #fractions 6]
│ │ │ - [step 1:   time .246733 sec  #fractions 6]
│ │ │ - -- used 0.478873s (cpu); 0.304753s (thread); 0s (gc)
│ │ │ + [step 0:   time .106181 sec  #fractions 6]
│ │ │ + [step 1:   time .245944 sec  #fractions 6]
│ │ │ + -- used 0.355931s (cpu); 0.267095s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1193,20 +1193,20 @@ │ │ │ │ │ │ o89 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000530094 sec #minors 4]
│ │ │ + [jacobian time .000524536 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .210442 sec  #fractions 6]
│ │ │ - [step 1:   time .269174 sec  #fractions 6]
│ │ │ - -- used 0.483329s (cpu); 0.298388s (thread); 0s (gc)
│ │ │ + [step 0:   time .213689 sec  #fractions 6]
│ │ │ + [step 1:   time .254215 sec  #fractions 6]
│ │ │ + -- used 0.471844s (cpu); 0.304723s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1243,20 +1243,20 @@ │ │ │ │ │ │ o93 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000774762 sec #minors 1]
│ │ │ + [jacobian time .000672198 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .271111 sec  #fractions 6]
│ │ │ - [step 1:   time .874897 sec  #fractions 6]
│ │ │ - -- used 1.15027s (cpu); 0.732279s (thread); 0s (gc)
│ │ │ + [step 0:   time .257929 sec  #fractions 6]
│ │ │ + [step 1:   time .679148 sec  #fractions 6]
│ │ │ + -- used 0.941001s (cpu); 0.600883s (thread); 0s (gc)
│ │ │  
│ │ │  o94 = R'
│ │ │  
│ │ │  o94 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -48,15 +48,15 @@ │ │ │ │ o2 : Ideal of S │ │ │ │ i3 : R = S/f │ │ │ │ │ │ │ │ o3 = R │ │ │ │ │ │ │ │ o3 : QuotientRing │ │ │ │ i4 : time R' = integralClosure R │ │ │ │ - -- used 0.387896s (cpu); 0.303595s (thread); 0s (gc) │ │ │ │ + -- used 0.386165s (cpu); 0.309925s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = R' │ │ │ │ │ │ │ │ o4 : QuotientRing │ │ │ │ i5 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------+ │ │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ o8 : Ideal of S │ │ │ │ i9 : R = S/f │ │ │ │ │ │ │ │ o9 = R │ │ │ │ │ │ │ │ o9 : QuotientRing │ │ │ │ i10 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.508329s (cpu); 0.342291s (thread); 0s (gc) │ │ │ │ + -- used 0.360586s (cpu); 0.287809s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = R' │ │ │ │ │ │ │ │ o10 : QuotientRing │ │ │ │ i11 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -199,15 +199,15 @@ │ │ │ │ o14 : Ideal of S │ │ │ │ i15 : R = S/f │ │ │ │ │ │ │ │ o15 = R │ │ │ │ │ │ │ │ o15 : QuotientRing │ │ │ │ i16 : time R' = integralClosure(R, Strategy => AllCodimensions) │ │ │ │ - -- used 0.507202s (cpu); 0.349308s (thread); 0s (gc) │ │ │ │ + -- used 0.488122s (cpu); 0.332852s (thread); 0s (gc) │ │ │ │ │ │ │ │ o16 = R' │ │ │ │ │ │ │ │ o16 : QuotientRing │ │ │ │ i17 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -281,15 +281,15 @@ │ │ │ │ o19 : Ideal of S │ │ │ │ i20 : R = S/f │ │ │ │ │ │ │ │ o20 = R │ │ │ │ │ │ │ │ o20 : QuotientRing │ │ │ │ i21 : time R' = integralClosure(R, Strategy => SimplifyFractions) │ │ │ │ - -- used 0.400249s (cpu); 0.317688s (thread); 0s (gc) │ │ │ │ + -- used 0.389919s (cpu); 0.315813s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 = R' │ │ │ │ │ │ │ │ o21 : QuotientRing │ │ │ │ i22 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -363,15 +363,15 @@ │ │ │ │ o24 : Ideal of S │ │ │ │ i25 : R = S/f │ │ │ │ │ │ │ │ o25 = R │ │ │ │ │ │ │ │ o25 : QuotientRing │ │ │ │ i26 : time R' = integralClosure (R, Strategy => RadicalCodim1) │ │ │ │ - -- used 0.79625s (cpu); 0.550685s (thread); 0s (gc) │ │ │ │ + -- used 0.772773s (cpu); 0.540577s (thread); 0s (gc) │ │ │ │ │ │ │ │ o26 = R' │ │ │ │ │ │ │ │ o26 : QuotientRing │ │ │ │ i27 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -445,15 +445,15 @@ │ │ │ │ o29 : Ideal of S │ │ │ │ i30 : R = S/f │ │ │ │ │ │ │ │ o30 = R │ │ │ │ │ │ │ │ o30 : QuotientRing │ │ │ │ i31 : time R' = integralClosure (R, Strategy => Vasconcelos) │ │ │ │ - -- used 0.508579s (cpu); 0.336293s (thread); 0s (gc) │ │ │ │ + -- used 0.477415s (cpu); 0.326729s (thread); 0s (gc) │ │ │ │ │ │ │ │ o31 = R' │ │ │ │ │ │ │ │ o31 : QuotientRing │ │ │ │ i32 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -527,15 +527,15 @@ │ │ │ │ o34 : Ideal of S │ │ │ │ i35 : R = S/f │ │ │ │ │ │ │ │ o35 = R │ │ │ │ │ │ │ │ o35 : QuotientRing │ │ │ │ i36 : time R' = integralClosure R │ │ │ │ - -- used 0.0428345s (cpu); 0.0428212s (thread); 0s (gc) │ │ │ │ + -- used 0.0466942s (cpu); 0.046696s (thread); 0s (gc) │ │ │ │ │ │ │ │ o36 = R' │ │ │ │ │ │ │ │ o36 : QuotientRing │ │ │ │ i37 : netList (ideal R')_* │ │ │ │ │ │ │ │ +-----------+ │ │ │ │ @@ -573,15 +573,15 @@ │ │ │ │ o39 : Ideal of S │ │ │ │ i40 : R = S/I │ │ │ │ │ │ │ │ o40 = R │ │ │ │ │ │ │ │ o40 : QuotientRing │ │ │ │ i41 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.167523s (cpu); 0.0828249s (thread); 0s (gc) │ │ │ │ + -- used 0.15173s (cpu); 0.0714075s (thread); 0s (gc) │ │ │ │ │ │ │ │ o41 = R' │ │ │ │ │ │ │ │ o41 : QuotientRing │ │ │ │ i42 : icFractions R │ │ │ │ │ │ │ │ 2 │ │ │ │ @@ -603,15 +603,15 @@ │ │ │ │ o44 : Ideal of S │ │ │ │ i45 : R = S/I │ │ │ │ │ │ │ │ o45 = R │ │ │ │ │ │ │ │ o45 : QuotientRing │ │ │ │ i46 : time R' = integralClosure(R, Strategy => AllCodimensions) │ │ │ │ - -- used 0.0639871s (cpu); 0.0639869s (thread); 0s (gc) │ │ │ │ + -- used 0.0668493s (cpu); 0.0668461s (thread); 0s (gc) │ │ │ │ │ │ │ │ o46 = R' │ │ │ │ │ │ │ │ o46 : QuotientRing │ │ │ │ i47 : icFractions R │ │ │ │ │ │ │ │ b*d │ │ │ │ @@ -632,15 +632,15 @@ │ │ │ │ o49 : Ideal of S │ │ │ │ i50 : R = S/I │ │ │ │ │ │ │ │ o50 = R │ │ │ │ │ │ │ │ o50 : QuotientRing │ │ │ │ i51 : time R' = integralClosure (R, Strategy => RadicalCodim1) │ │ │ │ - -- used 0.0434306s (cpu); 0.0434315s (thread); 0s (gc) │ │ │ │ + -- used 0.151552s (cpu); 0.0753283s (thread); 0s (gc) │ │ │ │ │ │ │ │ o51 = R' │ │ │ │ │ │ │ │ o51 : QuotientRing │ │ │ │ i52 : icFractions R │ │ │ │ │ │ │ │ 2 │ │ │ │ @@ -662,15 +662,15 @@ │ │ │ │ o54 : Ideal of S │ │ │ │ i55 : R = S/I │ │ │ │ │ │ │ │ o55 = R │ │ │ │ │ │ │ │ o55 : QuotientRing │ │ │ │ i56 : time R' = integralClosure (R, Strategy => Vasconcelos) │ │ │ │ - -- used 0.175803s (cpu); 0.0898169s (thread); 0s (gc) │ │ │ │ + -- used 0.0457782s (cpu); 0.0457781s (thread); 0s (gc) │ │ │ │ │ │ │ │ o56 = R' │ │ │ │ │ │ │ │ o56 : QuotientRing │ │ │ │ i57 : icFractions R │ │ │ │ │ │ │ │ 2 │ │ │ │ @@ -755,15 +755,15 @@ │ │ │ │ o65 : BettiTally │ │ │ │ i66 : R = S/I │ │ │ │ │ │ │ │ o66 = R │ │ │ │ │ │ │ │ o66 : QuotientRing │ │ │ │ i67 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.0622734s (cpu); 0.0622745s (thread); 0s (gc) │ │ │ │ + -- used 0.0659785s (cpu); 0.0659792s (thread); 0s (gc) │ │ │ │ │ │ │ │ o67 = R' │ │ │ │ │ │ │ │ o67 : QuotientRing │ │ │ │ i68 : icFractions R │ │ │ │ │ │ │ │ 2 2 │ │ │ │ @@ -839,15 +839,15 @@ │ │ │ │ o76 : BettiTally │ │ │ │ i77 : R = S/I │ │ │ │ │ │ │ │ o77 = R │ │ │ │ │ │ │ │ o77 : QuotientRing │ │ │ │ i78 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.500553s (cpu); 0.333412s (thread); 0s (gc) │ │ │ │ + -- used 0.474793s (cpu); 0.315983s (thread); 0s (gc) │ │ │ │ │ │ │ │ o78 = R' │ │ │ │ │ │ │ │ o78 : QuotientRing │ │ │ │ i79 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -863,15 +863,15 @@ │ │ │ │ o80 : PolynomialRing │ │ │ │ i81 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o81 = R │ │ │ │ │ │ │ │ o81 : QuotientRing │ │ │ │ i82 : time R' = integralClosure(R, Strategy => AllCodimensions) │ │ │ │ - -- used 0.543183s (cpu); 0.38548s (thread); 0s (gc) │ │ │ │ + -- used 0.594658s (cpu); 0.365376s (thread); 0s (gc) │ │ │ │ │ │ │ │ o82 = R' │ │ │ │ │ │ │ │ o82 : QuotientRing │ │ │ │ i83 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -887,20 +887,20 @@ │ │ │ │ o84 : PolynomialRing │ │ │ │ i85 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o85 = R │ │ │ │ │ │ │ │ o85 : QuotientRing │ │ │ │ i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1) │ │ │ │ - [jacobian time .000532358 sec #minors 4] │ │ │ │ + [jacobian time .000558853 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .228455 sec #fractions 6] │ │ │ │ - [step 1: time .246733 sec #fractions 6] │ │ │ │ - -- used 0.478873s (cpu); 0.304753s (thread); 0s (gc) │ │ │ │ + [step 0: time .106181 sec #fractions 6] │ │ │ │ + [step 1: time .245944 sec #fractions 6] │ │ │ │ + -- used 0.355931s (cpu); 0.267095s (thread); 0s (gc) │ │ │ │ │ │ │ │ o86 = R' │ │ │ │ │ │ │ │ o86 : QuotientRing │ │ │ │ i87 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -916,20 +916,20 @@ │ │ │ │ o88 : PolynomialRing │ │ │ │ i89 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o89 = R │ │ │ │ │ │ │ │ o89 : QuotientRing │ │ │ │ i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1) │ │ │ │ - [jacobian time .000530094 sec #minors 4] │ │ │ │ + [jacobian time .000524536 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .210442 sec #fractions 6] │ │ │ │ - [step 1: time .269174 sec #fractions 6] │ │ │ │ - -- used 0.483329s (cpu); 0.298388s (thread); 0s (gc) │ │ │ │ + [step 0: time .213689 sec #fractions 6] │ │ │ │ + [step 1: time .254215 sec #fractions 6] │ │ │ │ + -- used 0.471844s (cpu); 0.304723s (thread); 0s (gc) │ │ │ │ │ │ │ │ o90 = R' │ │ │ │ │ │ │ │ o90 : QuotientRing │ │ │ │ i91 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -948,20 +948,20 @@ │ │ │ │ i93 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o93 = R │ │ │ │ │ │ │ │ o93 : QuotientRing │ │ │ │ i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, │ │ │ │ StartWithOneMinor}, Verbosity => 1) │ │ │ │ - [jacobian time .000774762 sec #minors 1] │ │ │ │ + [jacobian time .000672198 sec #minors 1] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .271111 sec #fractions 6] │ │ │ │ - [step 1: time .874897 sec #fractions 6] │ │ │ │ - -- used 1.15027s (cpu); 0.732279s (thread); 0s (gc) │ │ │ │ + [step 0: time .257929 sec #fractions 6] │ │ │ │ + [step 1: time .679148 sec #fractions 6] │ │ │ │ + -- used 0.941001s (cpu); 0.600883s (thread); 0s (gc) │ │ │ │ │ │ │ │ o94 = R' │ │ │ │ │ │ │ │ o94 : QuotientRing │ │ │ │ i95 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 2 3 2 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ @@ -76,52 +76,52 @@ │ │ │ │ │ │ 
    i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000522279 sec #minors 3]
│ │ │ + [jacobian time .000528384 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00233339 seconds
│ │ │ -      idlizer1:  .00747003 seconds
│ │ │ -      idlizer2:  .00844957 seconds
│ │ │ -      minpres:   .00811702 seconds
│ │ │ -  time .0371527 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00247504 seconds
│ │ │ +      idlizer1:  .00847245 seconds
│ │ │ +      idlizer2:  .010069 seconds
│ │ │ +      minpres:   .00983557 seconds
│ │ │ +  time .0423551 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00248298 seconds
│ │ │ -      idlizer1:  .0119562 seconds
│ │ │ -      idlizer2:  .0111 seconds
│ │ │ -      minpres:   .0117312 seconds
│ │ │ -  time .0484066 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00246977 seconds
│ │ │ +      idlizer1:  .0128471 seconds
│ │ │ +      idlizer2:  .0109258 seconds
│ │ │ +      minpres:   .013127 seconds
│ │ │ +  time .0514417 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00413852 seconds
│ │ │ -      idlizer1:  .0169512 seconds
│ │ │ -      idlizer2:  .0143372 seconds
│ │ │ -      minpres:   .0137525 seconds
│ │ │ -  time .203496 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00272689 seconds
│ │ │ +      idlizer1:  .011831 seconds
│ │ │ +      idlizer2:  .0111346 seconds
│ │ │ +      minpres:   .0104825 seconds
│ │ │ +  time .143873 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00279962 seconds
│ │ │ -      idlizer1:  .0141692 seconds
│ │ │ -      idlizer2:  .0166105 seconds
│ │ │ -      minpres:   .0188805 seconds
│ │ │ -  time .0680688 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00306728 seconds
│ │ │ +      idlizer1:  .0134167 seconds
│ │ │ +      idlizer2:  .0142861 seconds
│ │ │ +      minpres:   .018816 seconds
│ │ │ +  time .0641517 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .0025556 seconds
│ │ │ -      idlizer1:  .00935896 seconds
│ │ │ -      idlizer2:  .0181192 seconds
│ │ │ -      minpres:   .0119314 seconds
│ │ │ -  time .0555899 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00311785 seconds
│ │ │ +      idlizer1:  .00982251 seconds
│ │ │ +      idlizer2:  .0184608 seconds
│ │ │ +      minpres:   .0119702 seconds
│ │ │ +  time .0576614 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00261851 seconds
│ │ │ -      idlizer1:  .00849894 seconds
│ │ │ -  time .0184597 sec  #fractions 5]
│ │ │ - -- used 0.435062s (cpu); 0.361171s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .002594 seconds
│ │ │ +      idlizer1:  .00900577 seconds
│ │ │ +  time .0191255 sec  #fractions 5]
│ │ │ + -- used 0.382847s (cpu); 0.314165s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = R'
│ │ │  
│ │ │  o2 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -12,52 +12,52 @@ │ │ │ │ displayed. A value of 0 means: keep quiet. │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ When the computation takes a considerable time, this function can be used to │ │ │ │ decide if it will ever finish, or to get a feel for what is happening during │ │ │ │ the computation. │ │ │ │ i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3); │ │ │ │ i2 : time R' = integralClosure(R, Verbosity => 2) │ │ │ │ - [jacobian time .000522279 sec #minors 3] │ │ │ │ + [jacobian time .000528384 sec #minors 3] │ │ │ │ integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ [step 0: │ │ │ │ - radical (use minprimes) .00233339 seconds │ │ │ │ - idlizer1: .00747003 seconds │ │ │ │ - idlizer2: .00844957 seconds │ │ │ │ - minpres: .00811702 seconds │ │ │ │ - time .0371527 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00247504 seconds │ │ │ │ + idlizer1: .00847245 seconds │ │ │ │ + idlizer2: .010069 seconds │ │ │ │ + minpres: .00983557 seconds │ │ │ │ + time .0423551 sec #fractions 4] │ │ │ │ [step 1: │ │ │ │ - radical (use minprimes) .00248298 seconds │ │ │ │ - idlizer1: .0119562 seconds │ │ │ │ - idlizer2: .0111 seconds │ │ │ │ - minpres: .0117312 seconds │ │ │ │ - time .0484066 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00246977 seconds │ │ │ │ + idlizer1: .0128471 seconds │ │ │ │ + idlizer2: .0109258 seconds │ │ │ │ + minpres: .013127 seconds │ │ │ │ + time .0514417 sec #fractions 4] │ │ │ │ [step 2: │ │ │ │ - radical (use minprimes) .00413852 seconds │ │ │ │ - idlizer1: .0169512 seconds │ │ │ │ - idlizer2: .0143372 seconds │ │ │ │ - minpres: .0137525 seconds │ │ │ │ - time .203496 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00272689 seconds │ │ │ │ + idlizer1: .011831 seconds │ │ │ │ + idlizer2: .0111346 seconds │ │ │ │ + minpres: .0104825 seconds │ │ │ │ + time .143873 sec #fractions 5] │ │ │ │ [step 3: │ │ │ │ - radical (use minprimes) .00279962 seconds │ │ │ │ - idlizer1: .0141692 seconds │ │ │ │ - idlizer2: .0166105 seconds │ │ │ │ - minpres: .0188805 seconds │ │ │ │ - time .0680688 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00306728 seconds │ │ │ │ + idlizer1: .0134167 seconds │ │ │ │ + idlizer2: .0142861 seconds │ │ │ │ + minpres: .018816 seconds │ │ │ │ + time .0641517 sec #fractions 5] │ │ │ │ [step 4: │ │ │ │ - radical (use minprimes) .0025556 seconds │ │ │ │ - idlizer1: .00935896 seconds │ │ │ │ - idlizer2: .0181192 seconds │ │ │ │ - minpres: .0119314 seconds │ │ │ │ - time .0555899 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00311785 seconds │ │ │ │ + idlizer1: .00982251 seconds │ │ │ │ + idlizer2: .0184608 seconds │ │ │ │ + minpres: .0119702 seconds │ │ │ │ + time .0576614 sec #fractions 5] │ │ │ │ [step 5: │ │ │ │ - radical (use minprimes) .00261851 seconds │ │ │ │ - idlizer1: .00849894 seconds │ │ │ │ - time .0184597 sec #fractions 5] │ │ │ │ - -- used 0.435062s (cpu); 0.361171s (thread); 0s (gc) │ │ │ │ + radical (use minprimes) .002594 seconds │ │ │ │ + idlizer1: .00900577 seconds │ │ │ │ + time .0191255 sec #fractions 5] │ │ │ │ + -- used 0.382847s (cpu); 0.314165s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = R' │ │ │ │ │ │ │ │ o2 : QuotientRing │ │ │ │ i3 : trim ideal R' │ │ │ │ │ │ │ │ 3 2 2 2 4 4 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ @@ -114,29 +114,29 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time integralClosure J
│ │ │ - -- used 1.00348s (cpu); 0.739924s (thread); 0s (gc)
│ │ │ + -- used 1.15719s (cpu); 0.697718s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │  
│ │ │  o4 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ - -- used 0.733674s (cpu); 0.56695s (thread); 0s (gc)
│ │ │ + -- used 1.16334s (cpu); 0.547475s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,25 +46,25 @@
│ │ │ │  i3 : J = ideal jacobian ideal F
│ │ │ │  
│ │ │ │                  2      2    2        2   2 2     2
│ │ │ │  o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b  + 3c )
│ │ │ │  
│ │ │ │  o3 : Ideal of S
│ │ │ │  i4 : time integralClosure J
│ │ │ │ - -- used 1.00348s (cpu); 0.739924s (thread); 0s (gc)
│ │ │ │ + -- used 1.15719s (cpu); 0.697718s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o4 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
│ │ │ │ - -- used 0.733674s (cpu); 0.56695s (thread); 0s (gc)
│ │ │ │ + -- used 1.16334s (cpu); 0.547475s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  c2NocmVpZXJHcmFwaA==
│ │ │  #:len=1712
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NocmVpZXIgZ3JhcGggb2YgYSBmaW5p
│ │ │  dGUgZ3JvdXAiLCAibGluZW51bSIgPT4gMjYyLCBJbnB1dHMgPT4ge1NQQU57VFR7IkcifSwiLCAi
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o3 : DiagonalAction
│ │ │  
│ │ │  i4 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o4 = false
│ │ │  
│ │ │  i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00739572s elapsed
│ │ │ + -- .00299791s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (ζ ζ  + ζ   + ζ  )T + (ζ ζ  + ζ  + ζ  + ζ   + ζ  ζ   + ζ  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (ζ ζ  + ζ ζ  + ζ ζ  + ζ ζ   + 1 + ζ   + ζ  ζ  + ζ  ζ   + ζ  ζ   + ζ  )T 
│ │ │ @@ -51,10 +51,10 @@
│ │ │           0   1
│ │ │  
│ │ │  i6 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000676866s elapsed
│ │ │ + -- .000686775s elapsed
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hironaka__Decomposition.out
│ │ │ @@ -5,13 +5,13 @@
│ │ │  o1 = QQ[x..y] <- <| 0 -1 |>
│ │ │                    | 1 0  |
│ │ │  
│ │ │  o1 : FiniteGroupAction
│ │ │  
│ │ │  i2 : hironakaDecomposition C4
│ │ │  
│ │ │ -        2    2   2 2        3       3
│ │ │ -o2 = ({x  + y , x y }, {1, x y - x*y })
│ │ │ +        2    2     3       3        4    4
│ │ │ +o2 = ({x  + y , - x y + x*y }, {1, x  + y })
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out
│ │ │ @@ -23,23 +23,23 @@
│ │ │  o3 = QQ[x..z] <- <| 0 -1 0  |, | 0 -1 0 |>
│ │ │                    | 1 0  0  |  | 1 0  0 |
│ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.487583s (cpu); 0.324563s (thread); 0s (gc)
│ │ │ + -- used 0.442286s (cpu); 0.311121s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   4    4
│ │ │ -o4 = {z , x  + y , x  + y }
│ │ │ +       2   2    2   3       3
│ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.444139s (cpu); 0.375345s (thread); 0s (gc)
│ │ │ + -- used 0.374906s (cpu); 0.304636s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -90,23 +90,23 @@
│ │ │       ------------------------------------------------------------------------
│ │ │          2 6    8
│ │ │       90y z  + z }
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.137804s (cpu); 0.0658453s (thread); 0s (gc)
│ │ │ + -- used 0.115563s (cpu); 0.0472676s (thread); 0s (gc)
│ │ │  
│ │ │ -          3       3
│ │ │ -o6 = {1, x y - x*y }
│ │ │ +          4    4
│ │ │ +o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.41802s (cpu); 1.13291s (thread); 0s (gc)
│ │ │ + -- used 1.16427s (cpu); 0.892318s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .422203s elapsed
│ │ │ + -- .277118s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -32,15 +32,15 @@
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .339469s elapsed
│ │ │ + -- .25003s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .735714s elapsed
│ │ │ + -- .376134s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ - -- .126449s elapsed
│ │ │ + -- .0660564s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_primary__Invariants.out
│ │ │ @@ -16,16 +16,16 @@
│ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : primaryInvariants S3
│ │ │  
│ │ │ -                                   3    3    3
│ │ │ -o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │ +                  2    2    2
│ │ │ +o4 = {x + y + z, x  + y  + z , x*y*z}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : K=GF(101)
│ │ │  
│ │ │  o5 = K
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_primary__Invariants_lp..._cm__Dade_eq_gt..._rp.out
│ │ │ @@ -77,16 +77,16 @@
│ │ │                2 4              5              5          6
│ │ │       69234300y z  + 13824000x*z  + 13824000y*z  + 810000z }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : primaryInvariants(S3)
│ │ │  
│ │ │ -                  2    2    2
│ │ │ -o5 = {x + y + z, x  + y  + z , x*y*z}
│ │ │ +                                   3    3    3
│ │ │ +o5 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : K=GF(101)
│ │ │  
│ │ │  o6 = K
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.out
│ │ │ @@ -16,13 +16,13 @@
│ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : primaryInvariants(S3,DegreeVector=>{3,3,4})
│ │ │  
│ │ │ -       3    3    3   2       2    2     2       2      2   4    4    4
│ │ │ -o4 = {x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z , x  + y  + z }
│ │ │ +              3    3    3   2 2    2 2    2 2
│ │ │ +o4 = {x*y*z, x  + y  + z , x y  + x z  + y z }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00739572s elapsed
│ │ │ + -- .00299791s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (ζ ζ  + ζ   + ζ  )T + (ζ ζ  + ζ  + ζ  + ζ   + ζ  ζ   + ζ  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (ζ ζ  + ζ ζ  + ζ ζ  + ζ ζ   + 1 + ζ   + ζ  ζ  + ζ  ζ   + ζ  ζ   + ζ  )T 
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o6 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000676866s elapsed
    │ │ │ + -- .000686775s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    For the programmer

    │ │ │ ├── html2text {} │ │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ | 0 -1 1 | │ │ │ │ │ │ │ │ o3 : DiagonalAction │ │ │ │ i4 : T.cache.?equivariantHilbert │ │ │ │ │ │ │ │ o4 = false │ │ │ │ i5 : elapsedTime equivariantHilbertSeries(T, Order => 5) │ │ │ │ - -- .00739572s elapsed │ │ │ │ + -- .00299791s elapsed │ │ │ │ │ │ │ │ -1 -1 2 2 -2 -1 -1 -2 2 │ │ │ │ o5 = 1 + (ζ ζ + ζ + ζ )T + (ζ ζ + ζ + ζ + ζ + ζ ζ + ζ │ │ │ │ )T + │ │ │ │ 0 1 1 0 0 1 0 1 1 0 1 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 3 2 2 -1 -3 -1 -1 -2 -2 -1 -3 3 │ │ │ │ @@ -57,13 +57,13 @@ │ │ │ │ │ │ │ │ o5 : ZZ[ζ ..ζ ][T] │ │ │ │ 0 1 │ │ │ │ i6 : T.cache.?equivariantHilbert │ │ │ │ │ │ │ │ o6 = true │ │ │ │ i7 : elapsedTime equivariantHilbertSeries(T, Order => 5); │ │ │ │ - -- .000676866s elapsed │ │ │ │ + -- .000686775s elapsed │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _e_q_u_i_v_a_r_i_a_n_t_H_i_l_b_e_r_t is a _s_y_m_b_o_l. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/InvariantRing/AbelianGroupsDoc.m2:225:0. │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hironaka__Decomposition.html │ │ │ @@ -92,16 +92,16 @@ │ │ │ o1 : FiniteGroupAction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : hironakaDecomposition C4
│ │ │  
│ │ │ -        2    2   2 2        3       3
│ │ │ -o2 = ({x  + y , x y }, {1, x y - x*y })
│ │ │ +        2    2     3       3        4    4
│ │ │ +o2 = ({x  + y , - x y + x*y }, {1, x  + y })
│ │ │  
│ │ │  o2 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │

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

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

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.487583s (cpu); 0.324563s (thread); 0s (gc)
│ │ │ + -- used 0.442286s (cpu); 0.311121s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   4    4
│ │ │ -o4 = {z , x  + y , x  + y }
│ │ │ +       2   2    2   3       3
│ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.444139s (cpu); 0.375345s (thread); 0s (gc)
│ │ │ + -- used 0.374906s (cpu); 0.304636s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -173,26 +173,26 @@
│ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.137804s (cpu); 0.0658453s (thread); 0s (gc)
│ │ │ + -- used 0.115563s (cpu); 0.0472676s (thread); 0s (gc)
│ │ │  
│ │ │ -          3       3
│ │ │ -o6 = {1, x y - x*y }
│ │ │ +          4    4
│ │ │ +o6 = {1, x  + y }
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.41802s (cpu); 1.13291s (thread); 0s (gc)
│ │ │ + -- used 1.16427s (cpu); 0.892318s (thread); 0s (gc)
│ │ │  
│ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4  
│ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │       ------------------------------------------------------------------------
│ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x 
│ │ │       ------------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,22 +69,22 @@
│ │ │ │                    | 0 0  -1 |  | 0 0  1 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  The two algorithms used in _p_r_i_m_a_r_y_I_n_v_a_r_i_a_n_t_s are timed. One sees that the Dade
│ │ │ │  algorithm is faster, however the primary invariants output are all of degree 8
│ │ │ │  and have ugly coefficients.
│ │ │ │  i4 : time P1=primaryInvariants C4xC2
│ │ │ │ - -- used 0.487583s (cpu); 0.324563s (thread); 0s (gc)
│ │ │ │ + -- used 0.442286s (cpu); 0.311121s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -       2   2    2   4    4
│ │ │ │ -o4 = {z , x  + y , x  + y }
│ │ │ │ +       2   2    2   3       3
│ │ │ │ +o4 = {z , x  + y , x y - x*y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.444139s (cpu); 0.375345s (thread); 0s (gc)
│ │ │ │ + -- used 0.374906s (cpu); 0.304636s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                     8                 7                   6 2
│ │ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     5 3                  4 4                 3 5
│ │ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -138,22 +138,22 @@
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  The extra work done by the default algorithm to ensure an optimal hsop is
│ │ │ │  rewarded by needing to calculate a smaller collection of corresponding
│ │ │ │  secondary invariants. In fact, it has proved quicker overall to calculate the
│ │ │ │  invariant ring based on the optimal algorithm rather than the Dade algorithm.
│ │ │ │  i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ │ - -- used 0.137804s (cpu); 0.0658453s (thread); 0s (gc)
│ │ │ │ + -- used 0.115563s (cpu); 0.0472676s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -          3       3
│ │ │ │ -o6 = {1, x y - x*y }
│ │ │ │ +          4    4
│ │ │ │ +o6 = {1, x  + y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 1.41802s (cpu); 1.13291s (thread); 0s (gc)
│ │ │ │ + -- used 1.16427s (cpu); 0.892318s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4
│ │ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : elapsedTime invariants S4
│ │ │ - -- .422203s elapsed
│ │ │ + -- .277118s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -122,15 +122,15 @@
│ │ │
    		│ │ │ 
    i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .339469s elapsed
│ │ │ + -- .25003s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  o3 = R <- <| 0 1 0 0 |, | 0 0 0 1 |>
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .422203s elapsed
│ │ │ │ + -- .277118s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │ @@ -50,15 +50,15 @@
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │ │  
│ │ │ │  Warning: stopping condition not met!
│ │ │ │  Output may not generate the entire ring of invariants.
│ │ │ │  Increase value of DegreeBound.
│ │ │ │  
│ │ │ │ - -- .339469s elapsed
│ │ │ │ + -- .25003s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : elapsedTime invariants S4
│ │ │ - -- .735714s elapsed
│ │ │ + -- .376134s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ - -- .126449s elapsed
│ │ │ + -- .0660564s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,27 +26,27 @@
│ │ │ │  o3 = R <- <| 0 1 0 0 |, | 0 0 0 1 |>
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .735714s elapsed
│ │ │ │ + -- .376134s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │        4
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ │ - -- .126449s elapsed
│ │ │ │ + -- .0660564s elapsed
│ │ │ │  
│ │ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants.html
│ │ │ @@ -106,16 +106,16 @@
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : primaryInvariants S3
│ │ │  
│ │ │ -                                   3    3    3
│ │ │ -o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │ +                  2    2    2
│ │ │ +o4 = {x + y + z, x  + y  + z , x*y*z}
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │

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

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

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

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

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

    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -94,16 +94,16 @@ │ │ │ │ 69234300y z + 13824000x*z + 13824000y*z + 810000z } │ │ │ │ │ │ │ │ o4 : List │ │ │ │ Compare this result to the hsop output when Dade is left to its default value │ │ │ │ _f_a_l_s_e. │ │ │ │ i5 : primaryInvariants(S3) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o5 = {x + y + z, x + y + z , x*y*z} │ │ │ │ + 3 3 3 │ │ │ │ +o5 = {x + y + z, x*y + x*z + y*z, x + y + z } │ │ │ │ │ │ │ │ o5 : List │ │ │ │ Below, the invariant ring QQ[x,y,z]S3 is calculated with K being the field with │ │ │ │ 101 elements. │ │ │ │ i6 : K=GF(101) │ │ │ │ │ │ │ │ o6 = K │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants_lp..._cm__Degree__Vector_eq_gt..._rp.html │ │ │ @@ -102,16 +102,16 @@ │ │ │ o3 : FiniteGroupAction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : primaryInvariants(S3,DegreeVector=>{3,3,4})
│ │ │  
│ │ │ -       3    3    3   2       2    2     2       2      2   4    4    4
│ │ │ -o4 = {x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z , x  + y  + z }
│ │ │ +              3    3    3   2 2    2 2    2 2
│ │ │ +o4 = {x*y*z, x  + y  + z , x y  + x z  + y z }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -34,16 +34,16 @@ │ │ │ │ o3 = QQ[x..z] <- <| 0 1 0 |, | 0 1 0 |> │ │ │ │ | 0 0 1 | | 1 0 0 | │ │ │ │ | 1 0 0 | | 0 0 1 | │ │ │ │ │ │ │ │ o3 : FiniteGroupAction │ │ │ │ i4 : primaryInvariants(S3,DegreeVector=>{3,3,4}) │ │ │ │ │ │ │ │ - 3 3 3 2 2 2 2 2 2 4 4 4 │ │ │ │ -o4 = {x + y + z , x y + x*y + x z + y z + x*z + y*z , x + y + z } │ │ │ │ + 3 3 3 2 2 2 2 2 2 │ │ │ │ +o4 = {x*y*z, x + y + z , x y + x z + y z } │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Currently users can only use _p_r_i_m_a_r_y_I_n_v_a_r_i_a_n_t_s to calculate a hsop for the │ │ │ │ invariant ring over a finite field by using the Dade algorithm. Users should │ │ │ │ enter the finite field as a _G_a_l_o_i_s_F_i_e_l_d or a quotient field of the form _Z_Z/ │ │ │ │ p and are advised to ensure that the ground field has cardinality greater than │ │ ├── ./usr/share/doc/Macaulay2/InverseSystems/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ ZnJvbUR1YWwoTWF0cml4KQ== │ │ │ #:len=249 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzMyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmcm9tRHVhbCxNYXRyaXgpLCJmcm9tRHVhbChNYXRy │ │ ├── ./usr/share/doc/Macaulay2/InvolutiveBases/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=38 │ │ │ aW52UmVkdWNlKFJpbmdFbGVtZW50LEludm9sdXRpdmVCYXNpcyk= │ │ │ #:len=299 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzMywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaW52UmVkdWNlLFJpbmdFbGVtZW50LEludm9sdXRp │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ aXNJc29tb3JwaGljKE1hdHJpeCxNYXRyaXgp │ │ │ #:len=270 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTMxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0lzb21vcnBoaWMsTWF0cml4LE1hdHJpeCksImlz │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/example-output/_is__Isomorphic.out │ │ │ @@ -156,20 +156,20 @@ │ │ │ {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 x_1 x_0 | {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_0^2 -x_2 x_1 | │ │ │ {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 -x_2 x_1 -x_3 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_1^2 -x_3 x_2 | │ │ │ │ │ │ 40 │ │ │ o22 : S-module, subquotient of S │ │ │ │ │ │ i23 : elapsedTime isIsomorphic(T1, T2) │ │ │ - -- 1.69407s elapsed │ │ │ + -- 1.48977s elapsed │ │ │ │ │ │ o23 = true │ │ │ │ │ │ i24 : elapsedTime isomorphism(T1, T2) │ │ │ - -- .000029024s elapsed │ │ │ + -- .000021198s elapsed │ │ │ │ │ │ o24 = {-1} | 1 -3976 -13490 13495 -2886 2577 14757 -881 7677 │ │ │ {-1} | -2527 -13566 2778 -6934 -14806 4619 -13099 6022 -10907 │ │ │ {-1} | -15420 5642 1489 1354 4591 11881 -5253 7296 -1098 │ │ │ {-1} | 7909 -12428 -2260 -8465 12113 -6893 8411 4186 -9393 │ │ │ {-1} | -9615 2934 10440 5015 8145 -5585 1360 3295 12851 │ │ │ {-1} | -4881 -7984 12700 -10391 -10009 -14538 13207 262 -6500 │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/html/_is__Isomorphic.html │ │ │ @@ -333,23 +333,23 @@ │ │ │ 40 │ │ │ o22 : S-module, subquotient of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.69407s elapsed
│ │ │ + -- 1.48977s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .000029024s elapsed
│ │ │ + -- .000021198s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ │ ├── html2text {}
│ │ │ │ @@ -724,19 +724,19 @@
│ │ │ │  0      0    0   0      0    0   0      0    0   0      0    0   0      0    0
│ │ │ │  0      0    0   0      0    0   0      0    0   0      0    0   -x_1^2 -x_3 x_2
│ │ │ │  |
│ │ │ │  
│ │ │ │                                  40
│ │ │ │  o22 : S-module, subquotient of S
│ │ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ │ - -- 1.69407s elapsed
│ │ │ │ + -- 1.48977s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ │ - -- .000029024s elapsed
│ │ │ │ + -- .000021198s elapsed
│ │ │ │  
│ │ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677
│ │ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098
│ │ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393
│ │ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851
│ │ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ ├── ./usr/share/doc/Macaulay2/JSON/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  TmFtZVNlcGFyYXRvcg==
│ │ │  #:len=210
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzQzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJOYW1lU2VwYXJhdG9yIiwiTmFtZVNlcGFyYXRvciIs
│ │ ├── ./usr/share/doc/Macaulay2/JSON/example-output/_from__J__S__O__N.out
│ │ │ @@ -37,19 +37,19 @@
│ │ │  
│ │ │  o8 = {1, 2, 3}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-35451-0/0.json
│ │ │ +o9 = /tmp/M2-48810-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-35451-0/0.json
│ │ │ +o10 = /tmp/M2-48810-0/0.json
│ │ │  
│ │ │  o10 : File
│ │ │  
│ │ │  i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
│ │ │ @@ -170,22 +170,22 @@
│ │ │            
    The input may also be a file containing JSON data.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-35451-0/0.json
    │ │ │ +o9 = /tmp/M2-48810-0/0.json │ │ │
    │ │ │ 
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-35451-0/0.json
│ │ │ +o10 = /tmp/M2-48810-0/0.json
│ │ │  
│ │ │  o10 : File
    │ │ │
    │ │ │ 
    i11 : fromJSON openIn jsonFile
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,18 +49,18 @@
│ │ │ │  
│ │ │ │  o8 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  The input may also be a file containing JSON data.
│ │ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-35451-0/0.json
│ │ │ │ +o9 = /tmp/M2-48810-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-35451-0/0.json
│ │ │ │ +o10 = /tmp/M2-48810-0/0.json
│ │ │ │  
│ │ │ │  o10 : File
│ │ │ │  i11 : fromJSON openIn jsonFile
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o11 : List
│ │ ├── ./usr/share/doc/Macaulay2/JSONRPC/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  bWFrZVJlcXVlc3QoTGlzdCk=
│ │ │  #:len=236
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlUmVxdWVzdCxMaXN0KSwibWFrZVJlcXVlc3Qo
│ │ ├── ./usr/share/doc/Macaulay2/Jets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  amV0cyhaWixBZmZpbmVWYXJpZXR5KQ==
│ │ │  #:len=2469
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGpldHMgb2YgYW4gYWZmaW5lIHZh
│ │ │  cmlldHkiLCAibGluZW51bSIgPT4gMTcxMCwgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out
│ │ │ @@ -17,24 +17,24 @@
│ │ │  o3 = ideal (y0*z0*x2 + x0*z0*y2 + x0*y0*z2 + z0*x1*y1 + y0*x1*z1 + x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1 + x0*z0*y1 + x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o3 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .0023111s elapsed
│ │ │ + -- .00248559s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │  
│ │ │  i5 : elapsedTime radical J2I
│ │ │ - -- .424366s elapsed
│ │ │ + -- .21892s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out
│ │ │ @@ -33,15 +33,15 @@
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : I.cache.?jet
│ │ │  
│ │ │  o7 = false
│ │ │  
│ │ │  i8 : elapsedTime jets(3,I)
│ │ │ - -- .00766552s elapsed
│ │ │ + -- .00677761s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i9 : I.cache.?jet
│ │ │ @@ -53,23 +53,23 @@
│ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │                                 | 2x0x1-y1       |
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i11 : elapsedTime jets(3,I)
│ │ │ - -- .0025072s elapsed
│ │ │ + -- .00242244s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │  
│ │ │  i12 : elapsedTime jets(2,I)
│ │ │ - -- .00226291s elapsed
│ │ │ + -- .00221632s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │  
│ │ │  i13 : Q = R/I
│ │ │ @@ -148,15 +148,15 @@
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : f.cache.?jet
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : elapsedTime jets(3,f)
│ │ │ - -- .0113717s elapsed
│ │ │ + -- .0107398s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -173,15 +173,15 @@
│ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │                                 | t2 2t0t2+t1^2  |
│ │ │                                 | t1 2t0t1       |
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
│ │ │  
│ │ │  i27 : elapsedTime jets(2,f)
│ │ │ - -- .000732911s elapsed
│ │ │ + -- .000694758s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html
│ │ │ @@ -92,27 +92,27 @@
│ │ │          
    
│ │ │            
    However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is generated by the individual terms in the generators of the ideal of jets. This observation provides an alternative algorithm for computing radicals of jets of monomial ideals, which can be faster than the default radical computation in Macaulay2.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial
│ │ │ │  ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is
│ │ │ │  generated by the individual terms in the generators of the ideal of jets. This
│ │ │ │  observation provides an alternative algorithm for computing radicals of jets of
│ │ │ │  monomial ideals, which can be faster than the default radical computation in
│ │ │ │  Macaulay2.
│ │ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ │ - -- .0023111s elapsed
│ │ │ │ + -- .00248559s elapsed
│ │ │ │  
│ │ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │ │  
│ │ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  i5 : elapsedTime radical J2I
│ │ │ │ - -- .424366s elapsed
│ │ │ │ + -- .21892s elapsed
│ │ │ │  
│ │ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  For a monomial hypersurface, [GS06, Theorem 3.2] describes the minimal primes
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │  
│ │ │  o7 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -151,26 +151,26 @@
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -295,15 +295,15 @@
│ │ │  
│ │ │  o23 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .0023111s elapsed
│ │ │ + -- .00248559s elapsed
│ │ │  
│ │ │  o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0)
│ │ │  
│ │ │  o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime radical J2I
│ │ │ - -- .424366s elapsed
│ │ │ + -- .21892s elapsed
│ │ │  
│ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │       ------------------------------------------------------------------------
│ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │  
│ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime jets(3,I)
│ │ │ - -- .00766552s elapsed
│ │ │ + -- .00677761s elapsed
│ │ │  
│ │ │                                                    2                 2
│ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime jets(3,I)
│ │ │ - -- .0025072s elapsed
│ │ │ + -- .00242244s elapsed
│ │ │  
│ │ │                                                     2                 2
│ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
    
│ │ │              
    
│ │ │                
    i12 : elapsedTime jets(2,I)
│ │ │ - -- .00226291s elapsed
│ │ │ + -- .00221632s elapsed
│ │ │  
│ │ │                               2                 2
│ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime jets(3,f)
│ │ │ - -- .0113717s elapsed
│ │ │ + -- .0107398s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -329,15 +329,15 @@
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime jets(2,f)
│ │ │ - -- .000732911s elapsed
│ │ │ + -- .000694758s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,15 +41,15 @@
│ │ │ │  o6 = ideal(x  - y)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : I.cache.?jet
│ │ │ │  
│ │ │ │  o7 = false
│ │ │ │  i8 : elapsedTime jets(3,I)
│ │ │ │ - -- .00766552s elapsed
│ │ │ │ + -- .00677761s elapsed
│ │ │ │  
│ │ │ │                                                    2                 2
│ │ │ │  o8 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o8 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i9 : I.cache.?jet
│ │ │ │  
│ │ │ │ @@ -58,22 +58,22 @@
│ │ │ │  
│ │ │ │  o10 = CacheTable{jetsMatrix => | 2x0x3-y3+2x1x2 |}
│ │ │ │                                 | 2x0x2-y2+x1^2  |
│ │ │ │                                 | 2x0x1-y1       |
│ │ │ │                                 | x0^2-y0        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i11 : elapsedTime jets(3,I)
│ │ │ │ - -- .0025072s elapsed
│ │ │ │ + -- .00242244s elapsed
│ │ │ │  
│ │ │ │                                                     2                 2
│ │ │ │  o11 = ideal (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o11 : Ideal of QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ │  i12 : elapsedTime jets(2,I)
│ │ │ │ - -- .00226291s elapsed
│ │ │ │ + -- .00221632s elapsed
│ │ │ │  
│ │ │ │                               2                 2
│ │ │ │  o12 = ideal (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │ │  
│ │ │ │  o12 : Ideal of QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  For quotient rings, data is stored under *.jet. Each jets order gives rise to a
│ │ │ │  different quotient that is stored separately under *.jet.jetsRing (order zero
│ │ │ │ @@ -153,15 +153,15 @@
│ │ │ │  i22 : isWellDefined f
│ │ │ │  
│ │ │ │  o22 = true
│ │ │ │  i23 : f.cache.?jet
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : elapsedTime jets(3,f)
│ │ │ │ - -- .0113717s elapsed
│ │ │ │ + -- .0107398s elapsed
│ │ │ │  
│ │ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3,
│ │ │ │  y3]                                                      2                    2
│ │ │ │  o24 = map (QQ[t0][t1][t2][t3], ------------------------------------------------
│ │ │ │  ----------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                                                        2
│ │ │ │  2
│ │ │ │ @@ -183,15 +183,15 @@
│ │ │ │  
│ │ │ │  o26 = CacheTable{jetsMatrix => | t3 2t0t3+2t1t2 |}
│ │ │ │                                 | t2 2t0t2+t1^2  |
│ │ │ │                                 | t1 2t0t1       |
│ │ │ │                                 | t0 t0^2        |
│ │ │ │                   jetsMaxOrder => 3
│ │ │ │  i27 : elapsedTime jets(2,f)
│ │ │ │ - -- .000732911s elapsed
│ │ │ │ + -- .000694758s elapsed
│ │ │ │  
│ │ │ │                                     QQ[x0, y0][x1, y1][x2, y2]
│ │ │ │  2                    2
│ │ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2,
│ │ │ │  2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │ │                                              2                 2
│ │ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  Y2Fub25pY2FsSG9tb3RvcGllcyguLi4sRmluZUdyYWRpbmc9Pi4uLik=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU5Niwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY2Fub25pY2FsSG9tb3RvcGllcyxGaW5lR3JhZGlu
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │        32003  0   5   0   5         32003  0   5   0   5          32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5           32003  0   5   0   5          32003  0   5   0   5
│ │ │                                                                                                                                                                                                                                                                                           
│ │ │       0                            1                             2                              3                              4                              5                              6                              7                              8                             9
│ │ │  
│ │ │  o3 : Complex
│ │ │  
│ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0249269s elapsed
│ │ │ + -- .0248151s elapsed
│ │ │  
│ │ │  o5 = 350
│ │ │  
│ │ │  i6 : betti F_a, betti F
│ │ │  
│ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ @@ -46,19 +46,19 @@
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │  
│ │ │  o9 = 14
│ │ │  
│ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00229463s elapsed
│ │ │ - -- .00936212s elapsed
│ │ │ - -- .0239474s elapsed
│ │ │ - -- .011302s elapsed
│ │ │ - -- .00360161s elapsed
│ │ │ + -- .00257373s elapsed
│ │ │ + -- .00596806s elapsed
│ │ │ + -- .0237775s elapsed
│ │ │ + -- .0151649s elapsed
│ │ │ + -- .00386886s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out
│ │ │ @@ -3,20 +3,20 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00231006s elapsed
│ │ │ - -- .0217836s elapsed
│ │ │ - -- .0230895s elapsed
│ │ │ - -- .0101838s elapsed
│ │ │ - -- .00353337s elapsed
│ │ │ - -- .294917s elapsed
│ │ │ + -- .00234744s elapsed
│ │ │ + -- .00699052s elapsed
│ │ │ + -- .0237436s elapsed
│ │ │ + -- .0104529s elapsed
│ │ │ + -- .00386939s elapsed
│ │ │ + -- .308367s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -26,15 +26,15 @@
│ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .125196s elapsed
│ │ │ + -- .162155s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,22 +48,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o5 : BettiTally
│ │ │  
│ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00475633s elapsed
│ │ │ - -- .0968306s elapsed
│ │ │ - -- .111433s elapsed
│ │ │ - -- 1.47579s elapsed
│ │ │ - -- .551722s elapsed
│ │ │ - -- .0398804s elapsed
│ │ │ - -- .00727613s elapsed
│ │ │ - -- 6.05064s elapsed
│ │ │ + -- .00862631s elapsed
│ │ │ + -- .0319998s elapsed
│ │ │ + -- .0946381s elapsed
│ │ │ + -- .963039s elapsed
│ │ │ + -- .443734s elapsed
│ │ │ + -- .0404824s elapsed
│ │ │ + -- .00688091s elapsed
│ │ │ + -- 4.79371s elapsed
│ │ │  
│ │ │  i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out
│ │ │ @@ -3,19 +3,19 @@
│ │ │  i1 : a=5,b=5
│ │ │  
│ │ │  o1 = (5, 5)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00224642s elapsed
│ │ │ - -- .010053s elapsed
│ │ │ - -- .0365563s elapsed
│ │ │ - -- .0100498s elapsed
│ │ │ - -- .00353941s elapsed
│ │ │ + -- .0022132s elapsed
│ │ │ + -- .00589313s elapsed
│ │ │ + -- .0229214s elapsed
│ │ │ + -- .00965196s elapsed
│ │ │ + -- .00342279s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -48,15 +48,15 @@
│ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │  
│ │ │                ZZ
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
│ │ │  
│ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .196215s elapsed
│ │ │ + -- .163686s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -70,22 +70,22 @@
│ │ │           1: . . . . . . . . . .
│ │ │           2: . . . . . . . . . .
│ │ │           3: . . . . . . . . . .
│ │ │  
│ │ │  o6 : BettiTally
│ │ │  
│ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00490087s elapsed
│ │ │ - -- .0189491s elapsed
│ │ │ - -- .0966489s elapsed
│ │ │ - -- 1.19159s elapsed
│ │ │ - -- .524549s elapsed
│ │ │ - -- .0405632s elapsed
│ │ │ - -- .00680951s elapsed
│ │ │ - -- 5.37805s elapsed
│ │ │ + -- .00485784s elapsed
│ │ │ + -- .016181s elapsed
│ │ │ + -- .102199s elapsed
│ │ │ + -- .834527s elapsed
│ │ │ + -- .377223s elapsed
│ │ │ + -- .0364651s elapsed
│ │ │ + -- .00655898s elapsed
│ │ │ + -- 4.38601s elapsed
│ │ │  
│ │ │  i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │  
│ │ │  o8 : List
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out
│ │ │ @@ -3,82 +3,82 @@
│ │ │  i1 : a=4,b=4
│ │ │  
│ │ │  o1 = (4, 4)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : d=carpetDet(a,b)
│ │ │ - -- .00631474s elapsed
│ │ │ - -- .0125312s elapsed
│ │ │ + -- .00693904s elapsed
│ │ │ + -- .0129015s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000299471s elapsed
│ │ │ + -- .000271922s elapsed
│ │ │  1
│ │ │ - -- .000146984s elapsed
│ │ │ + -- .00017458s elapsed
│ │ │  1
│ │ │ - -- .00012821s elapsed
│ │ │ + -- .000164461s elapsed
│ │ │  1
│ │ │ - -- .000124793s elapsed
│ │ │ + -- .000172983s elapsed
│ │ │  1
│ │ │ - -- .000129423s elapsed
│ │ │ + -- .000169161s elapsed
│ │ │  2
│ │ │ - -- .000136796s elapsed
│ │ │ + -- .000167066s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000137637s elapsed
│ │ │ + -- .000168909s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000148369s elapsed
│ │ │ + -- .000181632s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000144691s elapsed
│ │ │ + -- .000217755s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000132719s elapsed
│ │ │ + -- .000168632s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000125866s elapsed
│ │ │ + -- .000163946s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132297s elapsed
│ │ │ + -- .000155117s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140242s elapsed
│ │ │ + -- .000179053s elapsed
│ │ │  2
│ │ │ - -- .000122429s elapsed
│ │ │ + -- .000143671s elapsed
│ │ │  2
│ │ │ - -- .000133088s elapsed
│ │ │ + -- .000180136s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120466s elapsed
│ │ │ + -- .000149416s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000136004s elapsed
│ │ │ + -- .000165456s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000131376s elapsed
│ │ │ + -- .000211582s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000208069s elapsed
│ │ │ + -- .000227564s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140834s elapsed
│ │ │ + -- .00019736s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000134101s elapsed
│ │ │ + -- .000181052s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000126856s elapsed
│ │ │ + -- .000153433s elapsed
│ │ │  2
│ │ │ - -- .000129172s elapsed
│ │ │ + -- .000170227s elapsed
│ │ │  1
│ │ │ - -- .000130714s elapsed
│ │ │ + -- .000160982s elapsed
│ │ │  1
│ │ │ - -- .000156823s elapsed
│ │ │ + -- .000173068s elapsed
│ │ │  1
│ │ │ - -- .000123991s elapsed
│ │ │ + -- .000169889s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
│ │ │  
│ │ │  i3 : factor d
│ │ │  
│ │ │        32 6
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out
│ │ │ @@ -3,17 +3,17 @@
│ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │  
│ │ │  o1 = (9, 7)
│ │ │  
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : computeBound 3
│ │ │ - -- .268927s elapsed
│ │ │ - -- .33298s elapsed
│ │ │ - -- .23855s elapsed
│ │ │ - -- .286255s elapsed
│ │ │ - -- .382732s elapsed
│ │ │ - -- .298869s elapsed
│ │ │ + -- .124649s elapsed
│ │ │ + -- .151499s elapsed
│ │ │ + -- .153721s elapsed
│ │ │ + -- .150302s elapsed
│ │ │ + -- .164229s elapsed
│ │ │ + -- .184608s elapsed
│ │ │  
│ │ │  o2 = 6
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out
│ │ │ @@ -9,19 +9,19 @@
│ │ │  i2 : e=(-1,5)
│ │ │  
│ │ │  o2 = (-1, 5)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .0030227s elapsed
│ │ │ - -- .00794913s elapsed
│ │ │ - -- .023886s elapsed
│ │ │ - -- .0114692s elapsed
│ │ │ - -- .00368452s elapsed
│ │ │ + -- .00235267s elapsed
│ │ │ + -- .00563066s elapsed
│ │ │ + -- .0319852s elapsed
│ │ │ + -- .0144942s elapsed
│ │ │ + -- .00362263s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -49,15 +49,15 @@
│ │ │  i4 : keys h
│ │ │  
│ │ │  o4 = {0, 2, 3, 5}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .147997s elapsed
│ │ │ + -- .184665s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -77,19 +77,19 @@
│ │ │  i7 : e=(-1,5^2)
│ │ │  
│ │ │  o7 = (-1, 25)
│ │ │  
│ │ │  o7 : Sequence
│ │ │  
│ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00447939s elapsed
│ │ │ - -- .0118282s elapsed
│ │ │ - -- .043588s elapsed
│ │ │ - -- .0167169s elapsed
│ │ │ - -- .00606616s elapsed
│ │ │ + -- .00247991s elapsed
│ │ │ + -- .00623003s elapsed
│ │ │ + -- .0245963s elapsed
│ │ │ + -- .00957759s elapsed
│ │ │ + -- .00369199s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out
│ │ │ @@ -1,172 +1,172 @@
│ │ │  -- -*- M2-comint -*- hash: 1729182891690704738
│ │ │  
│ │ │  i1 : a=4
│ │ │  
│ │ │  o1 = 4
│ │ │  
│ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0201004s elapsed
│ │ │ + -- .0203358s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000054162s elapsed
│ │ │ + -- .000051772s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000094566s elapsed
│ │ │ + -- .000320232s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000071484s elapsed
│ │ │ + -- .000079284s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000081081s elapsed
│ │ │ + -- .000080741s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000092442s elapsed
│ │ │ + -- .000089847s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000092503s elapsed
│ │ │ + -- .000089005s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024937s elapsed
│ │ │ + -- .000027777s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000097393s elapsed
│ │ │ + -- .000075537s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000083987s elapsed
│ │ │ + -- .000098361s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000091982s elapsed
│ │ │ + -- .000102344s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000094778s elapsed
│ │ │ + -- .000094893s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000082534s elapsed
│ │ │ + -- .000083678s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .000077535s elapsed
│ │ │ + -- .000078244s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000072095s elapsed
│ │ │ + -- .000074842s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000067977s elapsed
│ │ │ + -- .0000697s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000022111s elapsed
│ │ │ + -- .000024599s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000071755s elapsed
│ │ │ + -- .000070606s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000026329s elapsed
│ │ │ + -- .000030046s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  o3 : Complex
    
│ │ │              
    
│ │ │                
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0249269s elapsed
│ │ │ + -- .0248151s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │              
    
│ │ │                
    i6 : betti F_a, betti F
│ │ │ @@ -146,19 +146,19 @@
│ │ │  
│ │ │  o9 = 14
    
│ │ │              
    
│ │ │                
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00229463s elapsed
│ │ │ - -- .00936212s elapsed
│ │ │ - -- .0239474s elapsed
│ │ │ - -- .011302s elapsed
│ │ │ - -- .00360161s elapsed
│ │ │ + -- .00257373s elapsed
│ │ │ + -- .00596806s elapsed
│ │ │ + -- .0237775s elapsed
│ │ │ + -- .0151649s elapsed
│ │ │ + -- .00386886s elapsed
│ │ │  
│ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │       0                            1                             2
│ │ │ │  3                              4                              5
│ │ │ │  6                              7                              8
│ │ │ │  9
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  i4 : L = analyzeStrand(F,a); #L
│ │ │ │ - -- .0249269s elapsed
│ │ │ │ + -- .0248151s elapsed
│ │ │ │  
│ │ │ │  o5 = 350
│ │ │ │  i6 : betti F_a, betti F
│ │ │ │  
│ │ │ │                 0         0  1   2   3   4   5   6   7  8 9
│ │ │ │  o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7)
│ │ │ │            6: 350      0: 1  .   .   .   .   .   .   .  . .
│ │ │ │ @@ -72,19 +72,19 @@
│ │ │ │  o7 = 2   3
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ │ │  i8 : L3 = select(L,c->c%3==0); #L3
│ │ │ │  
│ │ │ │  o9 = 14
│ │ │ │  i10 : carpetBettiTable(a,b,3)
│ │ │ │ - -- .00229463s elapsed
│ │ │ │ - -- .00936212s elapsed
│ │ │ │ - -- .0239474s elapsed
│ │ │ │ - -- .011302s elapsed
│ │ │ │ - -- .00360161s elapsed
│ │ │ │ + -- .00257373s elapsed
│ │ │ │ + -- .00596806s elapsed
│ │ │ │ + -- .0237775s elapsed
│ │ │ │ + -- .0151649s elapsed
│ │ │ │ + -- .00386886s elapsed
│ │ │ │  
│ │ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html
│ │ │ @@ -88,20 +88,20 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00231006s elapsed
│ │ │ - -- .0217836s elapsed
│ │ │ - -- .0230895s elapsed
│ │ │ - -- .0101838s elapsed
│ │ │ - -- .00353337s elapsed
│ │ │ - -- .294917s elapsed
│ │ │ + -- .00234744s elapsed
│ │ │ + -- .00699052s elapsed
│ │ │ + -- .0237436s elapsed
│ │ │ + -- .0104529s elapsed
│ │ │ + -- .00386939s elapsed
│ │ │ + -- .308367s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -117,15 +117,15 @@
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .125196s elapsed
│ │ │ + -- .162155s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -145,22 +145,22 @@
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00475633s elapsed
│ │ │ - -- .0968306s elapsed
│ │ │ - -- .111433s elapsed
│ │ │ - -- 1.47579s elapsed
│ │ │ - -- .551722s elapsed
│ │ │ - -- .0398804s elapsed
│ │ │ - -- .00727613s elapsed
│ │ │ - -- 6.05064s elapsed
    
│ │ │ + -- .00862631s elapsed
│ │ │ + -- .0319998s elapsed
│ │ │ + -- .0946381s elapsed
│ │ │ + -- .963039s elapsed
│ │ │ + -- .443734s elapsed
│ │ │ + -- .0404824s elapsed
│ │ │ + -- .00688091s elapsed
│ │ │ + -- 4.79371s elapsed
│ │ │              
    
│ │ │                
    i7 : carpetBettiTable(h,7)
│ │ │  
│ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ │ - -- .00231006s elapsed
│ │ │ │ - -- .0217836s elapsed
│ │ │ │ - -- .0230895s elapsed
│ │ │ │ - -- .0101838s elapsed
│ │ │ │ - -- .00353337s elapsed
│ │ │ │ - -- .294917s elapsed
│ │ │ │ + -- .00234744s elapsed
│ │ │ │ + -- .00699052s elapsed
│ │ │ │ + -- .0237436s elapsed
│ │ │ │ + -- .0104529s elapsed
│ │ │ │ + -- .00386939s elapsed
│ │ │ │ + -- .308367s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │  o2 : BettiTally
│ │ │ │  i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i4 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .125196s elapsed
│ │ │ │ + -- .162155s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -66,22 +66,22 @@
│ │ │ │  o5 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00475633s elapsed
│ │ │ │ - -- .0968306s elapsed
│ │ │ │ - -- .111433s elapsed
│ │ │ │ - -- 1.47579s elapsed
│ │ │ │ - -- .551722s elapsed
│ │ │ │ - -- .0398804s elapsed
│ │ │ │ - -- .00727613s elapsed
│ │ │ │ - -- 6.05064s elapsed
│ │ │ │ + -- .00862631s elapsed
│ │ │ │ + -- .0319998s elapsed
│ │ │ │ + -- .0946381s elapsed
│ │ │ │ + -- .963039s elapsed
│ │ │ │ + -- .443734s elapsed
│ │ │ │ + -- .0404824s elapsed
│ │ │ │ + -- .00688091s elapsed
│ │ │ │ + -- 4.79371s elapsed
│ │ │ │  i7 : carpetBettiTable(h,7)
│ │ │ │  
│ │ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ │ │           2: .  .   .   .    .    . 1155 1408 891 320 55  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html
│ │ │ @@ -85,19 +85,19 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00224642s elapsed
│ │ │ - -- .010053s elapsed
│ │ │ - -- .0365563s elapsed
│ │ │ - -- .0100498s elapsed
│ │ │ - -- .00353941s elapsed
│ │ │ + -- .0022132s elapsed
│ │ │ + -- .00589313s elapsed
│ │ │ + -- .0229214s elapsed
│ │ │ + -- .00965196s elapsed
│ │ │ + -- .00342279s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -139,15 +139,15 @@
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .196215s elapsed
│ │ │ + -- .163686s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -167,22 +167,22 @@
│ │ │  
│ │ │  o6 : BettiTally
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00490087s elapsed
│ │ │ - -- .0189491s elapsed
│ │ │ - -- .0966489s elapsed
│ │ │ - -- 1.19159s elapsed
│ │ │ - -- .524549s elapsed
│ │ │ - -- .0405632s elapsed
│ │ │ - -- .00680951s elapsed
│ │ │ - -- 5.37805s elapsed
    
│ │ │ + -- .00485784s elapsed
│ │ │ + -- .016181s elapsed
│ │ │ + -- .102199s elapsed
│ │ │ + -- .834527s elapsed
│ │ │ + -- .377223s elapsed
│ │ │ + -- .0364651s elapsed
│ │ │ + -- .00655898s elapsed
│ │ │ + -- 4.38601s elapsed
│ │ │              
    
│ │ │                
    i8 : keys h
│ │ │  
│ │ │  o8 = {0, 2, 3, 5}
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,19 +21,19 @@
│ │ │ │  resulting data allow us to compute the Betti tables for arbitrary primes.
│ │ │ │  i1 : a=5,b=5
│ │ │ │  
│ │ │ │  o1 = (5, 5)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : h=carpetBettiTables(a,b)
│ │ │ │ - -- .00224642s elapsed
│ │ │ │ - -- .010053s elapsed
│ │ │ │ - -- .0365563s elapsed
│ │ │ │ - -- .0100498s elapsed
│ │ │ │ - -- .00353941s elapsed
│ │ │ │ + -- .0022132s elapsed
│ │ │ │ + -- .00589313s elapsed
│ │ │ │ + -- .0229214s elapsed
│ │ │ │ + -- .00965196s elapsed
│ │ │ │ + -- .00342279s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │ │                 3  0   5   0   5
│ │ │ │  i5 : elapsedTime T'=minimalBetti J
│ │ │ │ - -- .196215s elapsed
│ │ │ │ + -- .163686s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -83,22 +83,22 @@
│ │ │ │  o6 = total: . . . . . . . . . .
│ │ │ │           1: . . . . . . . . . .
│ │ │ │           2: . . . . . . . . . .
│ │ │ │           3: . . . . . . . . . .
│ │ │ │  
│ │ │ │  o6 : BettiTally
│ │ │ │  i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ │ - -- .00490087s elapsed
│ │ │ │ - -- .0189491s elapsed
│ │ │ │ - -- .0966489s elapsed
│ │ │ │ - -- 1.19159s elapsed
│ │ │ │ - -- .524549s elapsed
│ │ │ │ - -- .0405632s elapsed
│ │ │ │ - -- .00680951s elapsed
│ │ │ │ - -- 5.37805s elapsed
│ │ │ │ + -- .00485784s elapsed
│ │ │ │ + -- .016181s elapsed
│ │ │ │ + -- .102199s elapsed
│ │ │ │ + -- .834527s elapsed
│ │ │ │ + -- .377223s elapsed
│ │ │ │ + -- .0364651s elapsed
│ │ │ │ + -- .00655898s elapsed
│ │ │ │ + -- 4.38601s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -85,82 +85,82 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : d=carpetDet(a,b)
│ │ │ - -- .00631474s elapsed
│ │ │ - -- .0125312s elapsed
│ │ │ + -- .00693904s elapsed
│ │ │ + -- .0129015s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000299471s elapsed
│ │ │ + -- .000271922s elapsed
│ │ │  1
│ │ │ - -- .000146984s elapsed
│ │ │ + -- .00017458s elapsed
│ │ │  1
│ │ │ - -- .00012821s elapsed
│ │ │ + -- .000164461s elapsed
│ │ │  1
│ │ │ - -- .000124793s elapsed
│ │ │ + -- .000172983s elapsed
│ │ │  1
│ │ │ - -- .000129423s elapsed
│ │ │ + -- .000169161s elapsed
│ │ │  2
│ │ │ - -- .000136796s elapsed
│ │ │ + -- .000167066s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000137637s elapsed
│ │ │ + -- .000168909s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000148369s elapsed
│ │ │ + -- .000181632s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000144691s elapsed
│ │ │ + -- .000217755s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000132719s elapsed
│ │ │ + -- .000168632s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000125866s elapsed
│ │ │ + -- .000163946s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132297s elapsed
│ │ │ + -- .000155117s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140242s elapsed
│ │ │ + -- .000179053s elapsed
│ │ │  2
│ │ │ - -- .000122429s elapsed
│ │ │ + -- .000143671s elapsed
│ │ │  2
│ │ │ - -- .000133088s elapsed
│ │ │ + -- .000180136s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120466s elapsed
│ │ │ + -- .000149416s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000136004s elapsed
│ │ │ + -- .000165456s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000131376s elapsed
│ │ │ + -- .000211582s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000208069s elapsed
│ │ │ + -- .000227564s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140834s elapsed
│ │ │ + -- .00019736s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000134101s elapsed
│ │ │ + -- .000181052s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000126856s elapsed
│ │ │ + -- .000153433s elapsed
│ │ │  2
│ │ │ - -- .000129172s elapsed
│ │ │ + -- .000170227s elapsed
│ │ │  1
│ │ │ - -- .000130714s elapsed
│ │ │ + -- .000160982s elapsed
│ │ │  1
│ │ │ - -- .000156823s elapsed
│ │ │ + -- .000173068s elapsed
│ │ │  1
│ │ │ - -- .000123991s elapsed
│ │ │ + -- .000169889s elapsed
│ │ │  1
│ │ │  
│ │ │  o2 = 3131031158784
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,82 +19,82 @@
│ │ │ │  determinants and return their product.
│ │ │ │  i1 : a=4,b=4
│ │ │ │  
│ │ │ │  o1 = (4, 4)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : d=carpetDet(a,b)
│ │ │ │ - -- .00631474s elapsed
│ │ │ │ - -- .0125312s elapsed
│ │ │ │ + -- .00693904s elapsed
│ │ │ │ + -- .0129015s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │ - -- .000299471s elapsed
│ │ │ │ + -- .000271922s elapsed
│ │ │ │  1
│ │ │ │ - -- .000146984s elapsed
│ │ │ │ + -- .00017458s elapsed
│ │ │ │  1
│ │ │ │ - -- .00012821s elapsed
│ │ │ │ + -- .000164461s elapsed
│ │ │ │  1
│ │ │ │ - -- .000124793s elapsed
│ │ │ │ + -- .000172983s elapsed
│ │ │ │  1
│ │ │ │ - -- .000129423s elapsed
│ │ │ │ + -- .000169161s elapsed
│ │ │ │  2
│ │ │ │ - -- .000136796s elapsed
│ │ │ │ + -- .000167066s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000137637s elapsed
│ │ │ │ + -- .000168909s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000148369s elapsed
│ │ │ │ + -- .000181632s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000144691s elapsed
│ │ │ │ + -- .000217755s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000132719s elapsed
│ │ │ │ + -- .000168632s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000125866s elapsed
│ │ │ │ + -- .000163946s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000132297s elapsed
│ │ │ │ + -- .000155117s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000140242s elapsed
│ │ │ │ + -- .000179053s elapsed
│ │ │ │  2
│ │ │ │ - -- .000122429s elapsed
│ │ │ │ + -- .000143671s elapsed
│ │ │ │  2
│ │ │ │ - -- .000133088s elapsed
│ │ │ │ + -- .000180136s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000120466s elapsed
│ │ │ │ + -- .000149416s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000136004s elapsed
│ │ │ │ + -- .000165456s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000131376s elapsed
│ │ │ │ + -- .000211582s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000208069s elapsed
│ │ │ │ + -- .000227564s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000140834s elapsed
│ │ │ │ + -- .00019736s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000134101s elapsed
│ │ │ │ + -- .000181052s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000126856s elapsed
│ │ │ │ + -- .000153433s elapsed
│ │ │ │  2
│ │ │ │ - -- .000129172s elapsed
│ │ │ │ + -- .000170227s elapsed
│ │ │ │  1
│ │ │ │ - -- .000130714s elapsed
│ │ │ │ + -- .000160982s elapsed
│ │ │ │  1
│ │ │ │ - -- .000156823s elapsed
│ │ │ │ + -- .000173068s elapsed
│ │ │ │  1
│ │ │ │ - -- .000123991s elapsed
│ │ │ │ + -- .000169889s elapsed
│ │ │ │  1
│ │ │ │  
│ │ │ │  o2 = 3131031158784
│ │ │ │  i3 : factor d
│ │ │ │  
│ │ │ │        32 6
│ │ │ │  o3 = 2  3
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -90,20 +90,20 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
    
│ │ │                
    i2 : computeBound 3
│ │ │ - -- .268927s elapsed
│ │ │ - -- .33298s elapsed
│ │ │ - -- .23855s elapsed
│ │ │ - -- .286255s elapsed
│ │ │ - -- .382732s elapsed
│ │ │ - -- .298869s elapsed
│ │ │ + -- .124649s elapsed
│ │ │ + -- .151499s elapsed
│ │ │ + -- .153721s elapsed
│ │ │ + -- .150302s elapsed
│ │ │ + -- .164229s elapsed
│ │ │ + -- .184608s elapsed
│ │ │  
│ │ │  o2 = 6
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,20 +25,20 @@
│ │ │ │  classes mod k. We conjecture that c=k^2-k.
│ │ │ │  i1 : (a,b)=computeBound(6,4,3)
│ │ │ │  
│ │ │ │  o1 = (9, 7)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : computeBound 3
│ │ │ │ - -- .268927s elapsed
│ │ │ │ - -- .33298s elapsed
│ │ │ │ - -- .23855s elapsed
│ │ │ │ - -- .286255s elapsed
│ │ │ │ - -- .382732s elapsed
│ │ │ │ - -- .298869s elapsed
│ │ │ │ + -- .124649s elapsed
│ │ │ │ + -- .151499s elapsed
│ │ │ │ + -- .153721s elapsed
│ │ │ │ + -- .150302s elapsed
│ │ │ │ + -- .164229s elapsed
│ │ │ │ + -- .184608s elapsed
│ │ │ │  
│ │ │ │  o2 = 6
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_l_a_t_i_v_e_E_q_u_a_t_i_o_n_s -- compute the relative quadrics
│ │ │ │  ********** WWaayyss ttoo uussee ccoommppuutteeBBoouunndd:: **********
│ │ │ │      * computeBound(ZZ)
│ │ │ │      * computeBound(ZZ,ZZ,ZZ)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html
│ │ │ @@ -95,19 +95,19 @@
│ │ │  
│ │ │  o2 : Sequence
    │ │ │
    │ │ │ 
    i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .0030227s elapsed
│ │ │ - -- .00794913s elapsed
│ │ │ - -- .023886s elapsed
│ │ │ - -- .0114692s elapsed
│ │ │ - -- .00368452s elapsed
│ │ │ + -- .00235267s elapsed
│ │ │ + -- .00563066s elapsed
│ │ │ + -- .0319852s elapsed
│ │ │ + -- .0144942s elapsed
│ │ │ + -- .00362263s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -141,15 +141,15 @@
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .147997s elapsed
│ │ │ + -- .184665s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -183,19 +183,19 @@
│ │ │  
│ │ │  o7 : Sequence
    │ │ │
    │ │ │ 
    i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00447939s elapsed
│ │ │ - -- .0118282s elapsed
│ │ │ - -- .043588s elapsed
│ │ │ - -- .0167169s elapsed
│ │ │ - -- .00606616s elapsed
│ │ │ + -- .00247991s elapsed
│ │ │ + -- .00623003s elapsed
│ │ │ + -- .0245963s elapsed
│ │ │ + -- .00957759s elapsed
│ │ │ + -- .00369199s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,19 +27,19 @@
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : e=(-1,5)
│ │ │ │  
│ │ │ │  o2 = (-1, 5)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .0030227s elapsed
│ │ │ │ - -- .00794913s elapsed
│ │ │ │ - -- .023886s elapsed
│ │ │ │ - -- .0114692s elapsed
│ │ │ │ - -- .00368452s elapsed
│ │ │ │ + -- .00235267s elapsed
│ │ │ │ + -- .00563066s elapsed
│ │ │ │ + -- .0319852s elapsed
│ │ │ │ + -- .0144942s elapsed
│ │ │ │ + -- .00362263s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  o3 : HashTable
│ │ │ │  i4 : keys h
│ │ │ │  
│ │ │ │  o4 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ │ - -- .147997s elapsed
│ │ │ │ + -- .184665s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ │ @@ -94,19 +94,19 @@
│ │ │ │  these mistakes.
│ │ │ │  i7 : e=(-1,5^2)
│ │ │ │  
│ │ │ │  o7 = (-1, 25)
│ │ │ │  
│ │ │ │  o7 : Sequence
│ │ │ │  i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ │ - -- .00447939s elapsed
│ │ │ │ - -- .0118282s elapsed
│ │ │ │ - -- .043588s elapsed
│ │ │ │ - -- .0167169s elapsed
│ │ │ │ - -- .00606616s elapsed
│ │ │ │ + -- .00247991s elapsed
│ │ │ │ + -- .00623003s elapsed
│ │ │ │ + -- .0245963s elapsed
│ │ │ │ + -- .00957759s elapsed
│ │ │ │ + -- .00369199s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html
│ │ │ @@ -83,172 +83,172 @@
│ │ │  
│ │ │  o1 = 4
    │ │ │
    │ │ │ 
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0201004s elapsed
│ │ │ + -- .0203358s elapsed
│ │ │  (number of blocks= , 18)
│ │ │  (size of the matrices, Tally{1 => 4})
│ │ │                               2 => 6
│ │ │                               3 => 2
│ │ │                               4 => 6
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      7: 1 1
│ │ │ - -- .000054162s elapsed
│ │ │ + -- .000051772s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000094566s elapsed
│ │ │ + -- .000320232s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000071484s elapsed
│ │ │ + -- .000079284s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000081081s elapsed
│ │ │ + -- .000080741s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000092442s elapsed
│ │ │ + -- .000089847s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000092503s elapsed
│ │ │ + -- .000089005s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024937s elapsed
│ │ │ + -- .000027777s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000097393s elapsed
│ │ │ + -- .000075537s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000083987s elapsed
│ │ │ + -- .000098361s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000091982s elapsed
│ │ │ + -- .000102344s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000094778s elapsed
│ │ │ + -- .000094893s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000082534s elapsed
│ │ │ + -- .000083678s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .000077535s elapsed
│ │ │ + -- .000078244s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000072095s elapsed
│ │ │ + -- .000074842s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000067977s elapsed
│ │ │ + -- .0000697s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000022111s elapsed
│ │ │ + -- .000024599s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000071755s elapsed
│ │ │ + -- .000070606s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000026329s elapsed
│ │ │ + -- .000030046s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,172 +19,172 @@
│ │ │ │  grading. Viewed as a resolution over QQ(e_1,e_2), this resolution is non-
│ │ │ │  minimal and carries further gradings. We decompose the crucial map of the a-th
│ │ │ │  strand into blocks, compute their determinants, and factor the product.
│ │ │ │  i1 : a=4
│ │ │ │  
│ │ │ │  o1 = 4
│ │ │ │  i2 : (d1,d2)=resonanceDet(a)
│ │ │ │ - -- .0201004s elapsed
│ │ │ │ + -- .0203358s elapsed
│ │ │ │  (number of blocks= , 18)
│ │ │ │  (size of the matrices, Tally{1 => 4})
│ │ │ │                               2 => 6
│ │ │ │                               3 => 2
│ │ │ │                               4 => 6
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      7: 1 1
│ │ │ │ - -- .000054162s elapsed
│ │ │ │ + -- .000051772s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . 2
│ │ │ │ - -- .000094566s elapsed
│ │ │ │ + -- .000320232s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . .
│ │ │ │      9: . 2
│ │ │ │ - -- .000071484s elapsed
│ │ │ │ + -- .000079284s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │      7: 2 .
│ │ │ │      8: 1 .
│ │ │ │      9: . 1
│ │ │ │     10: . 2
│ │ │ │ - -- .000081081s elapsed
│ │ │ │ + -- .000080741s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (-3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      7: 1 .
│ │ │ │      8: 1 .
│ │ │ │      9: 2 2
│ │ │ │     10: . 1
│ │ │ │     11: . 1
│ │ │ │ - -- .000092442s elapsed
│ │ │ │ + -- .000089847s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      8: 1 .
│ │ │ │      9: 2 1
│ │ │ │     10: 1 2
│ │ │ │     11: . 1
│ │ │ │ - -- .000092503s elapsed
│ │ │ │ + -- .000089005s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      9: 1 1
│ │ │ │ - -- .000024937s elapsed
│ │ │ │ + -- .000027777s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      9: 1 1
│ │ │ │     10: 1 1
│ │ │ │ - -- .000097393s elapsed
│ │ │ │ + -- .000075537s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000083987s elapsed
│ │ │ │ + -- .000098361s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 2 1
│ │ │ │     11: 1 2
│ │ │ │     12: . 1
│ │ │ │ - -- .000091982s elapsed
│ │ │ │ + -- .000102344s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 1 .
│ │ │ │     11: 2 2
│ │ │ │     12: . 1
│ │ │ │     13: . 1
│ │ │ │ - -- .000094778s elapsed
│ │ │ │ + -- .000094893s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000082534s elapsed
│ │ │ │ + -- .000083678s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │     10: 2 .
│ │ │ │     11: 1 .
│ │ │ │     12: . 1
│ │ │ │     13: . 2
│ │ │ │ - -- .000077535s elapsed
│ │ │ │ + -- .000078244s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     10: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000072095s elapsed
│ │ │ │ + -- .000074842s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     11: 2 .
│ │ │ │     12: . .
│ │ │ │     13: . 2
│ │ │ │ - -- .000067977s elapsed
│ │ │ │ + -- .0000697s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000022111s elapsed
│ │ │ │ + -- .000024599s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     12: 2 .
│ │ │ │     13: . 2
│ │ │ │ - -- .000071755s elapsed
│ │ │ │ + -- .000070606s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     13: 1 1
│ │ │ │ - -- .000026329s elapsed
│ │ │ │ + -- .000030046s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │  
│ │ │ │         6      32    32
│ │ │ │  o2 = (3 , (e )  (e )  )
│ │ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Surfaces/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  cHJvamVjdChWaXNpYmxlTGlzdCxFbWJlZGRlZEszc3VyZmFjZSk=
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTQ4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/Kronecker/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  a3JvbmVja2VyTm9ybWFsRm9ybShNYXRyaXgp
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIwNiwgc3ltYm9sIERvY3VtZW50VGFn
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│ │ ├── ./usr/share/doc/Macaulay2/KustinMiller/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=37
│ │ │  a3VzdGluTWlsbGVyQ29tcGxleCguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=577
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3B0aW9uIHRvIHByaW50IGludGVybWVk
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│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  UmVhbFhE
│ │ │  #:len=499
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidXNlIGV4dGVuZGVkIGV4cG9uZW50IHJl
│ │ │  YWwgbnVtYmVycyIsIERlc2NyaXB0aW9uID0+IChUVHsiUmVhbFhEIn0sIiAtLSBhIHN0cmF0ZWd5
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/example-output/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -7,55 +7,55 @@
│ │ │  
│ │ │  i2 : m = syz m1;
│ │ │  
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i3 : time LLL m;
│ │ │ - -- used 0.00867687s (cpu); 0.00867114s (thread); 0s (gc)
│ │ │ + -- used 0.00836645s (cpu); 0.00836518s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0275296s (cpu); 0.0275336s (thread); 0s (gc)
│ │ │ + -- used 0.0226383s (cpu); 0.0226427s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.104159s (cpu); 0.104163s (thread); 0s (gc)
│ │ │ + -- used 0.0979495s (cpu); 0.0979375s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0107558s (cpu); 0.010756s (thread); 0s (gc)
│ │ │ + -- used 0.0121687s (cpu); 0.012015s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0473029s (cpu); 0.0473041s (thread); 0s (gc)
│ │ │ + -- used 0.0547823s (cpu); 0.0547867s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0557251s (cpu); 0.0557271s (thread); 0s (gc)
│ │ │ + -- used 0.0531711s (cpu); 0.0531759s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.317152s (cpu); 0.317156s (thread); 0s (gc)
│ │ │ + -- used 0.306457s (cpu); 0.306462s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11309s (cpu); 0.113092s (thread); 0s (gc)
│ │ │ + -- used 0.128516s (cpu); 0.12852s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -144,78 +144,78 @@
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i3 : time LLL m;
│ │ │ - -- used 0.00867687s (cpu); 0.00867114s (thread); 0s (gc)
│ │ │ + -- used 0.00836645s (cpu); 0.00836518s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0275296s (cpu); 0.0275336s (thread); 0s (gc)
│ │ │ + -- used 0.0226383s (cpu); 0.0226427s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.104159s (cpu); 0.104163s (thread); 0s (gc)
│ │ │ + -- used 0.0979495s (cpu); 0.0979375s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0107558s (cpu); 0.010756s (thread); 0s (gc)
│ │ │ + -- used 0.0121687s (cpu); 0.012015s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0473029s (cpu); 0.0473041s (thread); 0s (gc)
│ │ │ + -- used 0.0547823s (cpu); 0.0547867s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0557251s (cpu); 0.0557271s (thread); 0s (gc)
│ │ │ + -- used 0.0531711s (cpu); 0.0531759s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.317152s (cpu); 0.317156s (thread); 0s (gc)
│ │ │ + -- used 0.306457s (cpu); 0.306462s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │ 
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11309s (cpu); 0.113092s (thread); 0s (gc)
│ │ │ + -- used 0.128516s (cpu); 0.12852s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -115,50 +115,50 @@ │ │ │ │ 50 50 │ │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ i2 : m = syz m1; │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o2 : Matrix ZZ <-- ZZ │ │ │ │ i3 : time LLL m; │ │ │ │ - -- used 0.00867687s (cpu); 0.00867114s (thread); 0s (gc) │ │ │ │ + -- used 0.00836645s (cpu); 0.00836518s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o3 : Matrix ZZ <-- ZZ │ │ │ │ i4 : time LLL(m, Strategy=>CohenEngine); │ │ │ │ - -- used 0.0275296s (cpu); 0.0275336s (thread); 0s (gc) │ │ │ │ + -- used 0.0226383s (cpu); 0.0226427s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ i5 : time LLL(m, Strategy=>CohenTopLevel); │ │ │ │ - -- used 0.104159s (cpu); 0.104163s (thread); 0s (gc) │ │ │ │ + -- used 0.0979495s (cpu); 0.0979375s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o5 : Matrix ZZ <-- ZZ │ │ │ │ i6 : time LLL(m, Strategy=>{Givens,RealFP}); │ │ │ │ - -- used 0.0107558s (cpu); 0.010756s (thread); 0s (gc) │ │ │ │ + -- used 0.0121687s (cpu); 0.012015s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o6 : Matrix ZZ <-- ZZ │ │ │ │ i7 : time LLL(m, Strategy=>{Givens,RealQP}); │ │ │ │ - -- used 0.0473029s (cpu); 0.0473041s (thread); 0s (gc) │ │ │ │ + -- used 0.0547823s (cpu); 0.0547867s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o7 : Matrix ZZ <-- ZZ │ │ │ │ i8 : time LLL(m, Strategy=>{Givens,RealXD}); │ │ │ │ - -- used 0.0557251s (cpu); 0.0557271s (thread); 0s (gc) │ │ │ │ + -- used 0.0531711s (cpu); 0.0531759s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o8 : Matrix ZZ <-- ZZ │ │ │ │ i9 : time LLL(m, Strategy=>{Givens,RealRR}); │ │ │ │ - -- used 0.317152s (cpu); 0.317156s (thread); 0s (gc) │ │ │ │ + -- used 0.306457s (cpu); 0.306462s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o9 : Matrix ZZ <-- ZZ │ │ │ │ i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP}); │ │ │ │ - -- used 0.11309s (cpu); 0.113092s (thread); 0s (gc) │ │ │ │ + -- used 0.128516s (cpu); 0.12852s (thread); 0s (gc) │ │ │ │ │ │ │ │ 50 47 │ │ │ │ o10 : Matrix ZZ <-- ZZ │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For most of the options, the columns do not need to be linearly independent. │ │ │ │ The strategies CohenEngine and CohenTopLevel currently require the columns to │ │ │ │ be linearly independent. │ │ ├── ./usr/share/doc/Macaulay2/LanguageServer/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ c3RhcnQoTFNQU2VydmVyKQ== │ │ │ #:len=246 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjYsIHN5bWJvbCBEb2N1bWVudFRhZyA9 │ │ │ PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KHN0YXJ0LExTUFNlcnZlciksInN0YXJ0KExTUFNlcnZl │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=28 │ │ │ YXJlSXNvbW9ycGhpYyhNYXRyaXgsTWF0cml4KQ== │ │ │ #:len=289 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhcmVJc29tb3JwaGljLE1hdHJpeCxNYXRyaXgpLCJh │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out │ │ │ @@ -16,14 +16,14 @@ │ │ │ │ │ │ 3 8 │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i5 : P = convexHull(M); │ │ │ │ │ │ i6 : time areIsomorphic(P,P); │ │ │ - -- used 0.74582s (cpu); 0.506968s (thread); 0s (gc) │ │ │ + -- used 1.08722s (cpu); 0.495031s (thread); 0s (gc) │ │ │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ - -- used 0.361052s (cpu); 0.277771s (thread); 0s (gc) │ │ │ + -- used 0.383989s (cpu); 0.248529s (thread); 0s (gc) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html │ │ │ @@ -125,21 +125,21 @@ │ │ │ │ │ │ 
    i5 : P = convexHull(M);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.74582s (cpu); 0.506968s (thread); 0s (gc)
    │ │ │ + -- used 1.08722s (cpu); 0.495031s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.361052s (cpu); 0.277771s (thread); 0s (gc)
    │ │ │ + -- used 0.383989s (cpu); 0.248529s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use areIsomorphic:

    │ │ │ ├── html2text {} │ │ │ │ @@ -35,17 +35,17 @@ │ │ │ │ | 0 0 1 0 1 0 1 1 | │ │ │ │ | 0 0 0 1 0 1 1 1 | │ │ │ │ │ │ │ │ 3 8 │ │ │ │ o4 : Matrix ZZ <-- ZZ │ │ │ │ i5 : P = convexHull(M); │ │ │ │ i6 : time areIsomorphic(P,P); │ │ │ │ - -- used 0.74582s (cpu); 0.506968s (thread); 0s (gc) │ │ │ │ + -- used 1.08722s (cpu); 0.495031s (thread); 0s (gc) │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ │ - -- used 0.361052s (cpu); 0.277771s (thread); 0s (gc) │ │ │ │ + -- used 0.383989s (cpu); 0.248529s (thread); 0s (gc) │ │ │ │ ********** WWaayyss ttoo uussee aarreeIIssoommoorrpphhiicc:: ********** │ │ │ │ * areIsomorphic(Matrix,Matrix) │ │ │ │ * areIsomorphic(Polyhedron,Polyhedron) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _a_r_e_I_s_o_m_o_r_p_h_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/LexIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ TGV4SWRlYWxz │ │ │ #:len=470 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciB3b3JraW5nIHdp │ │ │ dGggbGV4IGlkZWFscyIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsiIixFTXsiTGV4 │ │ ├── ./usr/share/doc/Macaulay2/LieAlgebraRepresentations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=43 │ │ │ c3ltbWV0cmljUG93ZXIoWlosTGllQWxnZWJyYVJlcHJlc2VudGF0aW9uKQ== │ │ │ #:len=1614 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIGV4cGxpY2l0IGFj │ │ │ dGlvbiBvbiAkXFxvcGVyYXRvcm5hbWV7U3ltfV5kIFYkIGZvciBhICRcXG1hdGhmcmFre2d9JC1t │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ c3VwcG9ydE9mVG9yKENvbXBsZXgp │ │ │ #:len=277 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE0Mywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3VwcG9ydE9mVG9yLENvbXBsZXgpLCJzdXBwb3J0 │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ @@ -29,21 +29,21 @@ │ │ │ i5 : findRegion({{0,0},{4,4}},M,f) │ │ │ │ │ │ o5 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : elapsedTime findRegion({{0,0},{4,4}},M,f) │ │ │ - -- .134906s elapsed │ │ │ + -- .0703809s elapsed │ │ │ │ │ │ o6 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}}) │ │ │ - -- .0127315s elapsed │ │ │ + -- .0140789s elapsed │ │ │ │ │ │ o7 = {{1, 2}, {3, 1}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out │ │ │ @@ -30,21 +30,21 @@ │ │ │ i5 : apply(L, d -> isLinearComplex res prune truncate(d,M)) │ │ │ │ │ │ o5 = {true, true} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M) │ │ │ - -- 3.72413s elapsed │ │ │ + -- 2.84812s elapsed │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ - -- .115325s elapsed │ │ │ + -- .0527743s elapsed │ │ │ │ │ │ o7 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html │ │ │ @@ -130,25 +130,25 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ - -- .134906s elapsed
│ │ │ + -- .0703809s elapsed
│ │ │  
│ │ │  o6 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})
│ │ │ - -- .0127315s elapsed
│ │ │ + -- .0140789s elapsed
│ │ │  
│ │ │  o7 = {{1, 2}, {3, 1}}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,22 +48,22 @@ │ │ │ │ │ │ │ │ o5 : List │ │ │ │ If some degrees d are known to satisfy f(d,M), then they can be specified using │ │ │ │ the option Inner in order to expedite the computation. Similarly, degrees not │ │ │ │ above those given in Outer will be assumed not to satisfy f(d,M). If f takes │ │ │ │ options these can also be given to findRegion. │ │ │ │ i6 : elapsedTime findRegion({{0,0},{4,4}},M,f) │ │ │ │ - -- .134906s elapsed │ │ │ │ + -- .0703809s elapsed │ │ │ │ │ │ │ │ o6 = {{1, 2}, {3, 1}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{ │ │ │ │ {1,1}}) │ │ │ │ - -- .0127315s elapsed │ │ │ │ + -- .0140789s elapsed │ │ │ │ │ │ │ │ o7 = {{1, 2}, {3, 1}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** CCoonnttrriibbuuttoorrss ********** │ │ │ │ Mahrud Sayrafi contributed to the code for this function. │ │ │ │ ********** CCaavveeaatt ********** │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ @@ -128,25 +128,25 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ - -- 3.72413s elapsed
│ │ │ + -- 2.84812s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .115325s elapsed
│ │ │ + -- .0527743s elapsed
│ │ │  
│ │ │  o7 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,21 +48,21 @@ │ │ │ │ o5 = {true, true} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ The output is a list of the minimal multidegrees $d$ such that the sum of the │ │ │ │ positive coordinates of $b-d$ is at most $i$ for all degrees $b$ appearing in │ │ │ │ the i-th step of the resolution of $M$. │ │ │ │ i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M) │ │ │ │ - -- 3.72413s elapsed │ │ │ │ + -- 2.84812s elapsed │ │ │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ │ - -- .115325s elapsed │ │ │ │ + -- .0527743s elapsed │ │ │ │ │ │ │ │ o7 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ In general linearTruncationsBound will not find the minimal degrees where $M$ │ │ │ │ has a linear resolution but will be faster than repeatedly truncating $M$. │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ TG9jYWxSaW5ncw== │ │ │ #:len=5280 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTG9jYWxpemF0aW9ucyBvZiBwb2x5bm9t │ │ │ aWFsIHJpbmdzIGF0IHByaW1lIGlkZWFscyIsICJsaW5lbnVtIiA9PiA2MiwgImZpbGVuYW1lIiA9 │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ │ │ │ o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 | │ │ │ │ │ │ 1 │ │ │ o4 : RP-module, quotient of RP │ │ │ │ │ │ i5 : elapsedTime hilbertSamuelFunction(M, 0, 6) │ │ │ - -- .192462s elapsed │ │ │ + -- .169891s elapsed │ │ │ │ │ │ o5 = {1, 3, 6, 7, 6, 3, 1} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : oo//sum │ │ │ │ │ │ @@ -44,21 +44,21 @@ │ │ │ │ │ │ 2 3 │ │ │ o10 = ideal (x , y ) │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ │ │ i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds │ │ │ - -- .0128718s elapsed │ │ │ + -- .0142991s elapsed │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ o11 : List │ │ │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ - -- .376491s elapsed │ │ │ + -- .254115s elapsed │ │ │ │ │ │ o12 = {6, 12, 18, 24, 30, 36} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html │ │ │ @@ -116,15 +116,15 @@ │ │ │ 1 │ │ │ o4 : RP-module, quotient of RP │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .192462s elapsed
│ │ │ + -- .169891s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -168,25 +168,25 @@ │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0128718s elapsed
│ │ │ + -- .0142991s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .376491s elapsed
│ │ │ + -- .254115s elapsed
│ │ │  
│ │ │  o12 = {6, 12, 18, 24, 30, 36}
│ │ │  
│ │ │  o12 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -41,15 +41,15 @@ │ │ │ │ i4 : M = RP^1/I │ │ │ │ │ │ │ │ o4 = cokernel | x5+y3+z3 y5+x3+z3 z5+x3+y3 | │ │ │ │ │ │ │ │ 1 │ │ │ │ o4 : RP-module, quotient of RP │ │ │ │ i5 : elapsedTime hilbertSamuelFunction(M, 0, 6) │ │ │ │ - -- .192462s elapsed │ │ │ │ + -- .169891s elapsed │ │ │ │ │ │ │ │ o5 = {1, 3, 6, 7, 6, 3, 1} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : oo//sum │ │ │ │ │ │ │ │ o6 = 27 │ │ │ │ @@ -65,21 +65,21 @@ │ │ │ │ i10 : q = ideal"x2,y3" │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o10 = ideal (x , y ) │ │ │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds │ │ │ │ - -- .0128718s elapsed │ │ │ │ + -- .0142991s elapsed │ │ │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ │ │ o11 : List │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ │ - -- .376491s elapsed │ │ │ │ + -- .254115s elapsed │ │ │ │ │ │ │ │ o12 = {6, 12, 18, 24, 30, 36} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Hilbert-Samuel function with respect to a parameter ideal other than the │ │ │ │ maximal ideal can be slower. │ │ ├── ./usr/share/doc/Macaulay2/M0nbar/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ dGV4KEN1cnZlQ2xhc3NSZXByZXNlbnRhdGl2ZU0wbmJhcik= │ │ │ #:len=923 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydCB0byBUZVggZm9ybWF0Iiwg │ │ │ ImxpbmVudW0iID0+IDE1ODQsIElucHV0cyA9PiB7U1BBTntUVHsiQyJ9LCIsICIsU1BBTnsiYW4g │ │ ├── ./usr/share/doc/Macaulay2/MCMApproximations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ Y29BcHByb3hpbWF0aW9uKC4uLixUb3RhbD0+Li4uKQ== │ │ │ #:len=293 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tjb0FwcHJveGltYXRpb24sVG90YWxdLCJjb0FwcHJv │ │ ├── ./usr/share/doc/Macaulay2/MRDI/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ c2F2ZU1SREkoLi4uLEZpbGVOYW1lPT4uLi4p │ │ │ #:len=232 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTc3LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzYXZlTVJESSxGaWxlTmFtZV0sInNhdmVNUkRJKC4u │ │ ├── ./usr/share/doc/Macaulay2/MRDI/example-output/_load__M__R__D__I.out │ │ │ @@ -57,15 +57,15 @@ │ │ │ │ │ │ i9 : I === J │ │ │ │ │ │ o9 = true │ │ │ │ │ │ i10 : fn = temporaryFileName() | ".mrdi" │ │ │ │ │ │ -o10 = /tmp/M2-40008-0/0.mrdi │ │ │ +o10 = /tmp/M2-56066-0/0.mrdi │ │ │ │ │ │ i11 : saveMRDI(I, FileName => fn) │ │ │ │ │ │ o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name": │ │ │ "Ideal"}, "data": [[[["0", "0", "2", "0"], ["1", "1"]], [["0", "1", │ │ │ "0", "1"], ["-1", "1"]]], [[["0", "1", "1", "0"], ["1", "1"]], [["1", │ │ │ "0", "0", "1"], ["-1", "1"]]], [[["0", "2", "0", "0"], ["1", "1"]], │ │ ├── ./usr/share/doc/Macaulay2/MRDI/example-output/_save__M__R__D__I.out │ │ │ @@ -39,15 +39,15 @@ │ │ │ "1"]]]], "_ns": {"Macaulay2": ["https://macaulay2.com", "1.26.06"]}, │ │ │ "_refs": {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params": │ │ │ {"_type": "Ring", "data": "QQ"}, "name": "PolynomialRing"}, "data": │ │ │ {"variables": ["x", "y", "z"]}}}} │ │ │ │ │ │ i7 : fn = temporaryFileName() | ".mrdi" │ │ │ │ │ │ -o7 = /tmp/M2-40027-0/0.mrdi │ │ │ +o7 = /tmp/M2-56105-0/0.mrdi │ │ │ │ │ │ i8 : saveMRDI(f, FileName => fn) │ │ │ │ │ │ o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name": │ │ │ "RingElement"}, "data": [[["2", "0", "0"], ["1", "1"]], [["0", "1", │ │ │ "1"], ["1", "1"]], [["1", "0", "0"], ["-3", "1"]]], "_ns": {"Macaulay2": │ │ │ ["https://macaulay2.com", "1.26.06"]}, "_refs": │ │ │ @@ -82,16 +82,16 @@ │ │ │ {6 => ({Macaulay2, saveMRDI}, QuotientRing) } │ │ │ {7 => ({Macaulay2, saveMRDI}, GaloisField) } │ │ │ │ │ │ o12 : NumberedVerticalList │ │ │ │ │ │ i13 : methods {"Oscar", saveMRDI} │ │ │ │ │ │ -o13 = {0 => ({Oscar, saveMRDI}, RingElement) } │ │ │ - {1 => ({Oscar, saveMRDI}, Ring) } │ │ │ - {2 => ({Oscar, saveMRDI}, PolynomialRing)} │ │ │ - {3 => ({Oscar, saveMRDI}, QQ) } │ │ │ - {4 => ({Oscar, saveMRDI}, ZZ) } │ │ │ +o13 = {0 => ({Oscar, saveMRDI}, QQ) } │ │ │ + {1 => ({Oscar, saveMRDI}, ZZ) } │ │ │ + {2 => ({Oscar, saveMRDI}, RingElement) } │ │ │ + {3 => ({Oscar, saveMRDI}, Ring) } │ │ │ + {4 => ({Oscar, saveMRDI}, PolynomialRing)} │ │ │ │ │ │ o13 : NumberedVerticalList │ │ │ │ │ │ i14 : │ │ ├── ./usr/share/doc/Macaulay2/MRDI/html/_load__M__R__D__I.html │ │ │ @@ -176,15 +176,15 @@ │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i10 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o10 = /tmp/M2-40008-0/0.mrdi
    │ │ │ +o10 = /tmp/M2-56066-0/0.mrdi │ │ │
    │ │ │ 
    i11 : saveMRDI(I, FileName => fn)
│ │ │  
│ │ │  o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : I === J
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  Objects can be loaded from a file as well using _g_e_t.
│ │ │ │  i10 : fn = temporaryFileName() | ".mrdi"
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-40008-0/0.mrdi
│ │ │ │ +o10 = /tmp/M2-56066-0/0.mrdi
│ │ │ │  i11 : saveMRDI(I, FileName => fn)
│ │ │ │  
│ │ │ │  o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ │        "Ideal"}, "data": [[[["0", "0", "2", "0"], ["1", "1"]], [["0", "1",
│ │ │ │        "0", "1"], ["-1", "1"]]], [[["0", "1", "1", "0"], ["1", "1"]], [["1",
│ │ │ │        "0", "0", "1"], ["-1", "1"]]], [[["0", "2", "0", "0"], ["1", "1"]],
│ │ │ │        [["1", "0", "1", "0"], ["-1", "1"]]]], "_ns": {"Macaulay2":
│ │ ├── ./usr/share/doc/Macaulay2/MRDI/html/_save__M__R__D__I.html
│ │ │ @@ -160,15 +160,15 @@
│ │ │            
    The output can be written directly to a file using the FileName option.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o7 = /tmp/M2-40027-0/0.mrdi
    
│ │ │ +o7 = /tmp/M2-56105-0/0.mrdi
│ │ │              
    
│ │ │                
    i8 : saveMRDI(f, FileName => fn)
│ │ │  
│ │ │  o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ @@ -236,19 +236,19 @@
│ │ │  o12 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i13 : methods {"Oscar", saveMRDI}
│ │ │  
│ │ │ -o13 = {0 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ -      {1 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ -      {2 => ({Oscar, saveMRDI}, PolynomialRing)}
│ │ │ -      {3 => ({Oscar, saveMRDI}, QQ)            }
│ │ │ -      {4 => ({Oscar, saveMRDI}, ZZ)            }
│ │ │ +o13 = {0 => ({Oscar, saveMRDI}, QQ)            }
│ │ │ +      {1 => ({Oscar, saveMRDI}, ZZ)            }
│ │ │ +      {2 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ +      {3 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ +      {4 => ({Oscar, saveMRDI}, PolynomialRing)}
│ │ │  
│ │ │  o13 : NumberedVerticalList
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Additional types can be supported by calling addSaveMethod.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,15 +66,15 @@
│ │ │ │       "1"]]]], "_ns": {"Macaulay2": ["https://macaulay2.com", "1.26.06"]},
│ │ │ │       "_refs": {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params":
│ │ │ │       {"_type": "Ring", "data": "QQ"}, "name": "PolynomialRing"}, "data":
│ │ │ │       {"variables": ["x", "y", "z"]}}}}
│ │ │ │  The output can be written directly to a file using the _F_i_l_e_N_a_m_e option.
│ │ │ │  i7 : fn = temporaryFileName() | ".mrdi"
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-40027-0/0.mrdi
│ │ │ │ +o7 = /tmp/M2-56105-0/0.mrdi
│ │ │ │  i8 : saveMRDI(f, FileName => fn)
│ │ │ │  
│ │ │ │  o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ │       "RingElement"}, "data": [[["2", "0", "0"], ["1", "1"]], [["0", "1",
│ │ │ │       "1"], ["1", "1"]], [["1", "0", "0"], ["-3", "1"]]], "_ns": {"Macaulay2":
│ │ │ │       ["https://macaulay2.com", "1.26.06"]}, "_refs":
│ │ │ │       {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params": {"_type":
│ │ │ │ @@ -110,19 +110,19 @@
│ │ │ │        {5 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ │        {6 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ │        {7 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ │  
│ │ │ │  o12 : NumberedVerticalList
│ │ │ │  i13 : methods {"Oscar", saveMRDI}
│ │ │ │  
│ │ │ │ -o13 = {0 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ │ -      {1 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ │ -      {2 => ({Oscar, saveMRDI}, PolynomialRing)}
│ │ │ │ -      {3 => ({Oscar, saveMRDI}, QQ)            }
│ │ │ │ -      {4 => ({Oscar, saveMRDI}, ZZ)            }
│ │ │ │ +o13 = {0 => ({Oscar, saveMRDI}, QQ)            }
│ │ │ │ +      {1 => ({Oscar, saveMRDI}, ZZ)            }
│ │ │ │ +      {2 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ │ +      {3 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ │ +      {4 => ({Oscar, saveMRDI}, PolynomialRing)}
│ │ │ │  
│ │ │ │  o13 : NumberedVerticalList
│ │ │ │  Additional types can be supported by calling _a_d_d_S_a_v_e_M_e_t_h_o_d.
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Not all Macaulay2 types have save methods defined. Attempting to serialize an
│ │ │ │  unsupported type will produce an error. Quotient rings other than finite prime
│ │ │ │  fields are not yet supported.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  dW5pbnN0YWxsUGFja2FnZShTdHJpbmcp
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjg5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh1bmluc3RhbGxQYWNrYWdlLFN0cmluZyksInVuaW5z
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Command.out
│ │ │ @@ -5,12 +5,12 @@
│ │ │  i2 : f
│ │ │  
│ │ │  o2 = 1073741824
│ │ │  
│ │ │  i3 : (c = Command "date";)
│ │ │  
│ │ │  i4 : c
│ │ │ -Sat Oct 24 03:11:57 UTC 2026
│ │ │ +Thu Oct 29 12:05:54 UTC 2026
│ │ │  
│ │ │  o4 = 0
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Database.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 9579076464446459296
│ │ │  
│ │ │  i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-12569-0/0.dbm
│ │ │ +o1 = /tmp/M2-14359-0/0.dbm
│ │ │  
│ │ │  i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-12569-0/0.dbm
│ │ │ +o2 = /tmp/M2-14359-0/0.dbm
│ │ │  
│ │ │  o2 : Database
│ │ │  
│ │ │  i3 : x#"first" = "hi there"
│ │ │  
│ │ │  o3 = hi there
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 1731899428494721487
│ │ │  
│ │ │  i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 69379379194        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 69261986618        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 219217920
│ │ │ -               "numGCs" => 1481
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 254820352
│ │ │ +               "numGCs" => 1449
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
│ │ │  
│ │ │  i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 219217920
│ │ │ +o2 = 254820352
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out
│ │ │ @@ -40,20 +40,20 @@
│ │ │  o6 : PolynomialRing
│ │ │  
│ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00860229s (cpu); 0.00859565s (thread); 0s (gc)
│ │ │ + -- used 0.00436071s (cpu); 0.00435728s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0658401s (cpu); 0.0658504s (thread); 0s (gc)
│ │ │ + -- used 0.038211s (cpu); 0.0382163s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : numgens J
│ │ │  
│ │ │  o10 = 1067
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Ideal.out
│ │ │ @@ -77,15 +77,15 @@
│ │ │  o11 = monomialIdeal (a*b*c, a d, b*c*d)
│ │ │  
│ │ │  o11 : MonomialIdeal of R
│ │ │  
│ │ │  i12 : standardPairs I
│ │ │  
│ │ │                                                                             
│ │ │ -o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {a, c}}, {1,
│ │ │ +o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1,
│ │ │        -----------------------------------------------------------------------
│ │ │                             2
│ │ │        {b, a}}, {b, {c}}, {b , {c}}}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : independentSets I
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__List.out
│ │ │ @@ -37,19 +37,19 @@
│ │ │  i10 : s = new MutableList
│ │ │  
│ │ │  o10 = MutableList{}
│ │ │  
│ │ │  o10 : MutableList
│ │ │  
│ │ │  i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at each step
│ │ │ - -- .00605048s elapsed
│ │ │ + -- .00421461s elapsed
│ │ │  
│ │ │  i12 : t = new MutableList from 1000
│ │ │  
│ │ │  o12 = MutableList{...1000...}
│ │ │  
│ │ │  o12 : MutableList
│ │ │  
│ │ │  i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ - -- .000712171s elapsed
│ │ │ + -- .000272994s elapsed
│ │ │  
│ │ │  i14 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutex.out
│ │ │ @@ -8,60 +8,54 @@
│ │ │  
│ │ │  o2 = sayhello
│ │ │  
│ │ │  o2 : FunctionClosure
│ │ │  
│ │ │  i3 : T = apply(10, i -> schedule(() -> sayhello i))
│ │ │  
│ │ │ -o3 = {<>, <>, <>, <>, <>, <>, <>, <>, <>, <>, <>, <>, <>, <>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     result available, task done>>, <>,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     <>}
│ │ │ +     <>, <>, <>}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : while not all(T, isReady) do null
│ │ │  
│ │ │  i5 : stack sort lines msgs
│ │ │  
│ │ │  o5 = hello from thread #0
│ │ │ -     hello from thread #1
│ │ │       hello from thread #2
│ │ │       hello from thread #3
│ │ │       hello from thread #4
│ │ │       hello from thread #5
│ │ │ -     hello from thread #6
│ │ │       hello from thread #7
│ │ │  
│ │ │  i6 : m = new Mutex
│ │ │  
│ │ │  o6 = m
│ │ │  
│ │ │  o6 : Mutex
│ │ │  
│ │ │  i7 : msgs = ""
│ │ │  
│ │ │  o7 = 
│ │ │  
│ │ │  i8 : T = apply(10, i -> schedule(() -> (lock m; sayhello i; unlock m)))
│ │ │  
│ │ │ -o8 = {<>, <>, <>, <>, <>, <>, <>,
│ │ │ +     done>>, <>, <>, <>, <>, <>, <>, <>, <>, <>}
│ │ │ +     result available, task done>>, <>, <>}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : while not all(T, isReady) do null
│ │ │  
│ │ │  i10 : stack sort lines msgs
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out
│ │ │ @@ -3,13 +3,13 @@
│ │ │  i1 : M = random(RR^200, RR^200);
│ │ │  
│ │ │                  200         200
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
│ │ │  
│ │ │  i2 : time SVD(M);
│ │ │ - -- used 0.0228821s (cpu); 0.0228783s (thread); 0s (gc)
│ │ │ + -- used 0.0307446s (cpu); 0.0307456s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0228804s (cpu); 0.0228876s (thread); 0s (gc)
│ │ │ + -- used 0.0300902s (cpu); 0.0301002s (thread); 0s (gc)
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out
│ │ │ @@ -2,23 +2,23 @@
│ │ │  
│ │ │  i1 : A = set {1,2};
│ │ │  
│ │ │  i2 : R = QQ[a..d];
│ │ │  
│ │ │  i3 : B = set{a^2-b*c,b*d}
│ │ │  
│ │ │ -           2
│ │ │ -o3 = set {a  - b*c, b*d}
│ │ │ +                2
│ │ │ +o3 = set {b*d, a  - b*c}
│ │ │  
│ │ │  o3 : Set
│ │ │  
│ │ │  i4 : toList B
│ │ │  
│ │ │ -       2
│ │ │ -o4 = {a  - b*c, b*d}
│ │ │ +            2
│ │ │ +o4 = {b*d, a  - b*c}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : member(1,A)
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out
│ │ │ @@ -347,15 +347,15 @@
│ │ │                 | b e h k n q |
│ │ │                 | c f i l o r |
│ │ │  
│ │ │                               3
│ │ │  o58 : R-module, quotient of R
│ │ │  
│ │ │  i59 : time C = resolution M
│ │ │ - -- used 0.00458017s (cpu); 0.00457468s (thread); 0s (gc)
│ │ │ + -- used 0.00308399s (cpu); 0.00306297s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o59 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out
│ │ │ @@ -14,10 +14,10 @@
│ │ │  
│ │ │  i4 : peek read f
│ │ │  
│ │ │  o4 = "hi there"
│ │ │  
│ │ │  i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
│ │ │ +o5 = true
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out
│ │ │ @@ -1,9 +1,9 @@
│ │ │  -- -*- M2-comint -*- hash: 1330379359420
│ │ │  
│ │ │  i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0003750952360409743
│ │ │ +o1 = .0003064144001933974
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_caching_spcomputation_spresults.out
│ │ │ @@ -4,20 +4,20 @@
│ │ │  
│ │ │  i2 : R = QQ[x,y,z];
│ │ │  
│ │ │  i3 : M = coker vars R;
│ │ │  
│ │ │  i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .0035172s elapsed
│ │ │ + -- .00338879s elapsed
│ │ │  
│ │ │  o4 = 3
│ │ │  
│ │ │  i5 : elapsedTime pdim' M
│ │ │ - -- 9.22e-7s elapsed
│ │ │ + -- .000002474s elapsed
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : peek M.cache
│ │ │  
│ │ │  o6 = CacheTable{cache => MutableHashTable{}                                              }
│ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_can__Use__Hilbert__Hint.out
│ │ │ @@ -211,15 +211,15 @@
│ │ │        -----------------------------------------------------------------------
│ │ │             5
│ │ │        + 12w )
│ │ │  
│ │ │  o17 : Ideal of Rlex
│ │ │  
│ │ │  i18 : elapsedTime g1 = gens gb Ilex;
│ │ │ - -- 1.68769s elapsed
│ │ │ + -- 1.73684s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o18 : Matrix Rlex  <-- Rlex
│ │ │  
│ │ │  i19 : Ilex = ideal(Ilex_*) -- clear out the previous Groebner basis
│ │ │  
│ │ │                  5      4       4       4       3 2      3         3     
│ │ │ @@ -293,15 +293,15 @@
│ │ │        -----------------------------------------------------------------------
│ │ │             5
│ │ │        + 12w )
│ │ │  
│ │ │  o19 : Ideal of Rlex
│ │ │  
│ │ │  i20 : elapsedTime g2 = gens gb(Ilex, Hilbert => hf);
│ │ │ - -- .850458s elapsed
│ │ │ + -- .781425s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o20 : Matrix Rlex  <-- Rlex
│ │ │  
│ │ │  i21 : g1 == g2
│ │ │  
│ │ │  o21 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out
│ │ │ @@ -18,57 +18,57 @@
│ │ │  
│ │ │  o4 = <>
│ │ │  
│ │ │  o4 : Task
│ │ │  
│ │ │  i5 : n
│ │ │  
│ │ │ -o5 = 336394
│ │ │ +o5 = 720619
│ │ │  
│ │ │  i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
│ │ │  
│ │ │  i7 : t
│ │ │  
│ │ │  o7 = <>
│ │ │  
│ │ │  o7 : Task
│ │ │  
│ │ │  i8 : n
│ │ │  
│ │ │ -o8 = 929987
│ │ │ +o8 = 1698282
│ │ │  
│ │ │  i9 : isReady t
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : cancelTask t
│ │ │  
│ │ │  i11 : sleep 2
│ │ │ -stdio:2:25:(3):[1]: error: interrupted
│ │ │ +stdio:2:39:(3):[1]: error: interrupted
│ │ │  
│ │ │  o11 = 0
│ │ │  
│ │ │  i12 : t
│ │ │  
│ │ │  o12 = <>
│ │ │  
│ │ │  o12 : Task
│ │ │  
│ │ │  i13 : n
│ │ │  
│ │ │ -o13 = 930227
│ │ │ +o13 = 1698491
│ │ │  
│ │ │  i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
│ │ │  
│ │ │  i15 : n
│ │ │  
│ │ │ -o15 = 930227
│ │ │ +o15 = 1698491
│ │ │  
│ │ │  i16 : isReady t
│ │ │  
│ │ │  o16 = false
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8535510246140175278
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11296-0/0
│ │ │ +o1 = /tmp/M2-11765-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11296-0/0
│ │ │ +o2 = /tmp/M2-11765-0/0
│ │ │  
│ │ │  i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-11296-0/0/
│ │ │ +o3 = /tmp/M2-11765-0/0/
│ │ │  
│ │ │  i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-11296-0/0/
│ │ │ +o4 = /tmp/M2-11765-0/0/
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out
│ │ │ @@ -4,51 +4,51 @@
│ │ │  
│ │ │  o1 = FirstPackage
│ │ │  
│ │ │  o1 : Package
│ │ │  
│ │ │  i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .313321s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .162154s elapsed
│ │ │  
│ │ │  i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .282764s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283285s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .140014s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .143268s elapsed
│ │ │  
│ │ │  i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .275704s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .142442s elapsed
│ │ │  
│ │ │  i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .246295s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .21342s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .137938s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .139401s elapsed
│ │ │  
│ │ │  i6 : tests(1, "FirstPackage")
│ │ │  
│ │ │  o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]
│ │ │  
│ │ │  o6 : TestInput
│ │ │  
│ │ │  i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .302726s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .147789s elapsed
│ │ │  
│ │ │  i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │       {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o8 : NumberedVerticalList
│ │ │  
│ │ │  i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .28407s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .282385s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .139483s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .136428s elapsed
│ │ │  
│ │ │  i10 : tests "FirstPackage"
│ │ │  
│ │ │  o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │        {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]}
│ │ │  
│ │ │  o10 : NumberedVerticalList
│ │ │  
│ │ │  i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283414s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .137466s elapsed
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out
│ │ │ @@ -1,26 +1,26 @@
│ │ │  -- -*- M2-comint -*- hash: 10365735446967377456
│ │ │  
│ │ │  i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close
│ │ │   ba 
│ │ │   ad 
│ │ │  
│ │ │  o2 = !grep a
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ -     6.12.113-1 (2026-10-15) x86_64 GNU/Linux\n"
│ │ │ +o3 = "Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +     6.12.100-1 (2026-07-30) x86_64 GNU/Linux\n"
│ │ │  
│ │ │  i4 : f = openInOut "!grep -E '^in'"
│ │ │  
│ │ │  o4 = !grep -E '^in'
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │ @@ -74,15 +74,15 @@
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : isReady f
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : while not isReady f do (sleep 1; << "." << flush)
│ │ │ -......
│ │ │ +.....
│ │ │  i11 : read f
│ │ │  
│ │ │  o11 = the answer is 4
│ │ │  
│ │ │  i12 : isReady f
│ │ │  
│ │ │  o12 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │                  ZZ
│ │ │  o23 : Ideal of ----[x..z, w]
│ │ │                 1277
│ │ │  
│ │ │  i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f57805dd000
│ │ │ +   -- registering gb 5 at 0x7f10dc0f6000
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -177,15 +177,15 @@
│ │ │  
│ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │  
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
│ │ │  
│ │ │  i33 : time betti gb f
│ │ │ - -- used 0.266722s (cpu); 0.187826s (thread); 0s (gc)
│ │ │ + -- used 0.172018s (cpu); 0.172731s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -208,15 +208,15 @@
│ │ │  
│ │ │              3    5     8     9    12     14    17
│ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │  
│ │ │  o35 : ZZ[T]
│ │ │  
│ │ │  i36 : time betti gb f
│ │ │ - -- used 0.00399931s (cpu); 0.00644121s (thread); 0s (gc)
│ │ │ + -- used 0.00699234s (cpu); 0.00660643s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__Directory_lp__String_cm__String_rp.out
│ │ │ @@ -1,76 +1,76 @@
│ │ │  -- -*- M2-comint -*- hash: 11422793294564310273
│ │ │  
│ │ │  i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12075-0/0/
│ │ │ +o1 = /tmp/M2-13345-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12075-0/1/
│ │ │ +o2 = /tmp/M2-13345-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12075-0/0/a/
│ │ │ +o3 = /tmp/M2-13345-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12075-0/0/b/
│ │ │ +o4 = /tmp/M2-13345-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12075-0/0/b/c/
│ │ │ +o5 = /tmp/M2-13345-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12075-0/0/a/f
│ │ │ +o6 = /tmp/M2-13345-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12075-0/0/a/g
│ │ │ +o7 = /tmp/M2-13345-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12075-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13345-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-12075-0/0/
│ │ │ -     /tmp/M2-12075-0/0/b/
│ │ │ -     /tmp/M2-12075-0/0/b/c/
│ │ │ -     /tmp/M2-12075-0/0/b/c/g
│ │ │ -     /tmp/M2-12075-0/0/a/
│ │ │ -     /tmp/M2-12075-0/0/a/g
│ │ │ -     /tmp/M2-12075-0/0/a/f
│ │ │ +o9 = /tmp/M2-13345-0/0/
│ │ │ +     /tmp/M2-13345-0/0/a/
│ │ │ +     /tmp/M2-13345-0/0/a/g
│ │ │ +     /tmp/M2-13345-0/0/a/f
│ │ │ +     /tmp/M2-13345-0/0/b/
│ │ │ +     /tmp/M2-13345-0/0/b/c/
│ │ │ +     /tmp/M2-13345-0/0/b/c/g
│ │ │  
│ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-12075-0/0/b/c/g -> /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/g -> /tmp/M2-12075-0/1/a/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/f -> /tmp/M2-12075-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/g -> /tmp/M2-13345-0/1/a/g
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/f -> /tmp/M2-13345-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13345-0/0/b/c/g -> /tmp/M2-13345-0/1/b/c/g
│ │ │  
│ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-12075-0/0/b/c/g not newer than /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/g not newer than /tmp/M2-12075-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/f not newer than /tmp/M2-12075-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/g not newer than /tmp/M2-13345-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/f not newer than /tmp/M2-13345-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13345-0/0/b/c/g not newer than /tmp/M2-13345-0/1/b/c/g
│ │ │  
│ │ │  i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-12075-0/1/
│ │ │ -      /tmp/M2-12075-0/1/a/
│ │ │ -      /tmp/M2-12075-0/1/a/f
│ │ │ -      /tmp/M2-12075-0/1/a/g
│ │ │ -      /tmp/M2-12075-0/1/b/
│ │ │ -      /tmp/M2-12075-0/1/b/c/
│ │ │ -      /tmp/M2-12075-0/1/b/c/g
│ │ │ +o12 = /tmp/M2-13345-0/1/
│ │ │ +      /tmp/M2-13345-0/1/a/
│ │ │ +      /tmp/M2-13345-0/1/a/g
│ │ │ +      /tmp/M2-13345-0/1/a/f
│ │ │ +      /tmp/M2-13345-0/1/b/
│ │ │ +      /tmp/M2-13345-0/1/b/c/
│ │ │ +      /tmp/M2-13345-0/1/b/c/g
│ │ │  
│ │ │  i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
│ │ │  
│ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out
│ │ │ @@ -1,41 +1,41 @@
│ │ │  -- -*- M2-comint -*- hash: 11539475420155775110
│ │ │  
│ │ │  i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11841-0/0
│ │ │ +o1 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11841-0/1
│ │ │ +o2 = /tmp/M2-12871-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11841-0/0
│ │ │ +o3 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11841-0/0 -> /tmp/M2-11841-0/1
│ │ │ + -- copying: /tmp/M2-12871-0/0 -> /tmp/M2-12871-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │  
│ │ │  i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11841-0/0
│ │ │ +o7 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │  
│ │ │  i9 : get dst
│ │ │  
│ │ │  o9 = hi there
│ │ │  
│ │ │  i10 : removeFile src
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out
│ │ │ @@ -1,23 +1,23 @@
│ │ │  -- -*- M2-comint -*- hash: 15508153783232232453
│ │ │  
│ │ │  i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 611.770723009
│ │ │ +o1 = 487.823398062
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │  
│ │ │  i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 613.313432871
│ │ │ +o3 = 488.878609747
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.542709862000038
│ │ │ +o4 = 1.055211684999961
│ │ │  
│ │ │  o4 : RR (of precision 53)
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out
│ │ │ @@ -1,24 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 3660839476107967259
│ │ │  
│ │ │  i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1792811657
│ │ │ +o1 = 1793275644
│ │ │  
│ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 56.81193979063632
│ │ │ +o2 = 56.82664295334988
│ │ │  
│ │ │  o2 : RR (of precision 53)
│ │ │  
│ │ │  i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 9.743277487635794
│ │ │ +o3 = 9.919715440198502
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : run "date"
│ │ │ -Sat Oct 24 03:14:17 UTC 2026
│ │ │ +Thu Oct 29 12:07:24 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
│ │ │ + -- 1.00022s 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
│ │ │ -     -- 1.00011 seconds
│ │ │ +     -- 1.00119 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00011, 0}
│ │ │ +o2 = Time{1.00119, 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 0.453514s (cpu); 0.254782s (thread); 0s (gc)
│ │ │ + -- used 0.347258s (cpu); 0.147574s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │                 3                   3     2               3
│ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.436423s (cpu); 0.247727s (thread); 0s (gc)
│ │ │ + -- used 0.0813805s (cpu); 0.0813131s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -154,15 +154,15 @@
│ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │  
│ │ │  i11 : I1 = substitute(I,R1);
│ │ │  
│ │ │  o11 : Ideal of R1
│ │ │  
│ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.0611301s (cpu); 0.0611395s (thread); 0s (gc)
│ │ │ + -- used 0.0261903s (cpu); 0.0261896s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -228,15 +228,15 @@
│ │ │  
│ │ │                     3             3     2         3
│ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │  
│ │ │  o16 : RingMap A <-- B
│ │ │  
│ │ │  i17 : time G = kernel F
│ │ │ - -- used 0.450122s (cpu); 0.254702s (thread); 0s (gc)
│ │ │ + -- used 0.319701s (cpu); 0.127183s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -297,23 +297,23 @@
│ │ │  i19 : use ring I
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : PolynomialRing
│ │ │  
│ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00269518s (cpu); 0.00269678s (thread); 0s (gc)
│ │ │ + -- used 0.00140979s (cpu); 0.00140852s (thread); 0s (gc)
│ │ │  
│ │ │           9    9      7    3
│ │ │  o20 = x*t  - t  - z*t  - z
│ │ │  
│ │ │  o20 : R
│ │ │  
│ │ │  i21 : time f2 = resultant(I_1,f1,t)
│ │ │ - -- used 0.0712746s (cpu); 0.0712854s (thread); 0s (gc)
│ │ │ + -- used 0.0279962s (cpu); 0.028001s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out
│ │ │ @@ -59,15 +59,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-11022-0/96-rundir/
│ │ │ +             source directory => /tmp/M2-11242-0/96-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
│ │ │  
│ │ │  i7 : dictionaryPath
│ │ │  
│ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out
│ │ │ @@ -1,20 +1,20 @@
│ │ │  -- -*- M2-comint -*- hash: 7475038936570224899
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11391-0/0
│ │ │ +o1 = /tmp/M2-11960-0/0
│ │ │  
│ │ │  i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11391-0/0
│ │ │ +o3 = /tmp/M2-11960-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : fileExists fn
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Length.out
│ │ │ @@ -1,28 +1,28 @@
│ │ │  -- -*- M2-comint -*- hash: 1216695447195237994
│ │ │  
│ │ │  i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-13078-0/0
│ │ │ +o1 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o1 : File
│ │ │  
│ │ │  i2 : fileLength f
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-13078-0/0
│ │ │ +o3 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-13078-0/0
│ │ │ +o4 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
│ │ │  
│ │ │  i6 : get filename
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__File_rp.out
│ │ │ @@ -1,25 +1,25 @@
│ │ │  -- -*- M2-comint -*- hash: 11202140621123993633
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12284-0/0
│ │ │ +o1 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-12284-0/0
│ │ │ +o2 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode f
│ │ │  
│ │ │  o3 = 420
│ │ │  
│ │ │  i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-12284-0/0
│ │ │ +o4 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : removeFile fn
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 4782570202197464532
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11860-0/0
│ │ │ +o1 = /tmp/M2-12910-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11860-0/0
│ │ │ +o2 = /tmp/M2-12910-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode fn
│ │ │  
│ │ │  o3 = 420
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__File_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 17473878267845575442
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11725-0/0
│ │ │ +o1 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11725-0/0
│ │ │ +o2 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : m = 7 + 7*8 + 7*64
│ │ │  
│ │ │  o3 = 511
│ │ │  
│ │ │ @@ -18,15 +18,15 @@
│ │ │  
│ │ │  i5 : fileMode f
│ │ │  
│ │ │  o5 = 511
│ │ │  
│ │ │  i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-11725-0/0
│ │ │ +o6 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : fileMode fn
│ │ │  
│ │ │  o7 = 511
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 16772784390799334723
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12905-0/0
│ │ │ +o1 = /tmp/M2-15045-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12905-0/0
│ │ │ +o2 = /tmp/M2-15045-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : m = fileMode fn
│ │ │  
│ │ │  o3 = 420
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1331310711075
│ │ │  
│ │ │  i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 111
│ │ │ +o1 = 79
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.out
│ │ │ @@ -29,15 +29,15 @@
│ │ │       {4} | 0     x2-3  y3-1  |
│ │ │  
│ │ │               3      3
│ │ │  o6 : Matrix R  <-- R
│ │ │  
│ │ │  i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f466ed708c0
│ │ │ +   -- registering gb 0 at 0x7fdc1bc7a8c0
│ │ │  
│ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3
│ │ │     -- number of monomials                = 9
│ │ │     -- #reduction steps = 6
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_get.out
│ │ │ @@ -10,11 +10,11 @@
│ │ │  
│ │ │  o2 = hi there
│ │ │  
│ │ │  i3 : removeFile "test-file"
│ │ │  
│ │ │  i4 : get "!date"
│ │ │  
│ │ │ -o4 = Sat Oct 24 03:12:48 UTC 2026
│ │ │ +o4 = Thu Oct 29 12:06:26 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__Canceled_lp__Task_rp.out
│ │ │ @@ -21,15 +21,15 @@
│ │ │  i5 : sleep 1
│ │ │  
│ │ │  o5 = 0
│ │ │  
│ │ │  i6 : cancelTask t
│ │ │  
│ │ │  i7 : sleep 2
│ │ │ -stdio:2:25:(3):[1]: error: interrupted
│ │ │ +stdio:2:39:(3):[1]: error: interrupted
│ │ │  
│ │ │  o7 = 0
│ │ │  
│ │ │  i8 : isCanceled t
│ │ │  
│ │ │  o8 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Directory.out
│ │ │ @@ -2,19 +2,19 @@
│ │ │  
│ │ │  i1 : isDirectory "."
│ │ │  
│ │ │  o1 = true
│ │ │  
│ │ │  i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11213-0/0
│ │ │ +o2 = /tmp/M2-11602-0/0
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11213-0/0
│ │ │ +o3 = /tmp/M2-11602-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : isDirectory fn
│ │ │  
│ │ │  o4 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out
│ │ │ @@ -75,15 +75,15 @@
│ │ │  o17 = false
│ │ │  
│ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │  
│ │ │  o18 = false
│ │ │  
│ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 3.67323s elapsed
│ │ │ + -- 4.32444s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
│ │ │  
│ │ │  i20 : facs = facs//toList/toList
│ │ │  
│ │ │ @@ -97,17 +97,17 @@
│ │ │  
│ │ │  i22 : m3 = nextPrime (m^3)
│ │ │  
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │  
│ │ │  i23 : elapsedTime isPrime m3
│ │ │ - -- .05718s elapsed
│ │ │ + -- .0496438s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000110026s elapsed
│ │ │ + -- .000102673s elapsed
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 4782205245758053629
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-13116-0/0
│ │ │ +o1 = /tmp/M2-15476-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-13116-0/0
│ │ │ +o2 = /tmp/M2-15476-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : isRegularFile fn
│ │ │  
│ │ │  o3 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_lock_lp__Mutex_cm__Function_rp.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i2 : f = i -> x += 1;
│ │ │  
│ │ │  i3 : parallelApply(1..1000, f);
│ │ │  
│ │ │  i4 : x
│ │ │  
│ │ │ -o4 = 333
│ │ │ +o4 = 323
│ │ │  
│ │ │  i5 : x = 0
│ │ │  
│ │ │  o5 = 0
│ │ │  
│ │ │  i6 : f = lock f;
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_make__Directory_lp__String_rp.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 5113372159204571746
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11593-0/0
│ │ │ +o1 = /tmp/M2-12363-0/0
│ │ │  
│ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-11593-0/0/a/b/c
│ │ │ +o2 = /tmp/M2-12363-0/0/a/b/c
│ │ │  
│ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │  
│ │ │  i5 : removeDirectory (dir|"/a")
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max__Allowable__Threads.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1331887830690
│ │ │  
│ │ │  i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
│ │ │ +o1 = 17
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out
│ │ │ @@ -3,31 +3,31 @@
│ │ │  i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2)
│ │ │  
│ │ │  o1 = fib
│ │ │  
│ │ │  o1 : FunctionClosure
│ │ │  
│ │ │  i2 : time fib 28
│ │ │ - -- used 1.47303s (cpu); 0.790587s (thread); 0s (gc)
│ │ │ + -- used 0.868084s (cpu); 0.493642s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
│ │ │  
│ │ │  i3 : fib = memoize fib
│ │ │  
│ │ │  o3 = fib
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : time fib 28
│ │ │ - -- used 6.6374e-05s (cpu); 6.5843e-05s (thread); 0s (gc)
│ │ │ + -- used 6.0802e-05s (cpu); 5.886e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
│ │ │  
│ │ │  i5 : time fib 28
│ │ │ - -- used 4.499e-06s (cpu); 4.078e-06s (thread); 0s (gc)
│ │ │ + -- used 3.269e-06s (cpu); 2.745e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 514229
│ │ │  
│ │ │  i6 : fib = memoize( n -> fib(n-1) + fib(n-2) )
│ │ │  
│ │ │  o6 = fib
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out
│ │ │ @@ -17,20 +17,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ -     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {19 => (codim, BettiTally)                                   }
│ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │       {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {24 => (codim, BettiTally)                                   }
│ │ │ +     {24 => (dual, BettiTally)                                    }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │  
│ │ │  i2 : methods resolution
│ │ │  
│ │ │  o2 = {0 => (freeResolution, Ideal)        }
│ │ │ @@ -60,20 +60,20 @@
│ │ │  
│ │ │  o4 = {0 => (++, Module, Module)}
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (-, Matrix, Matrix)                                   }
│ │ │ -     {1 => (diff', Matrix, Matrix)                               }
│ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ +o5 = {0 => (diff', Matrix, Matrix)                               }
│ │ │ +     {1 => (-, Matrix, Matrix)                                   }
│ │ │ +     {2 => (diff, Matrix, Matrix)                                }
│ │ │       {3 => (contract, Matrix, Matrix)                            }
│ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ +     {4 => (contract', Matrix, Matrix)                           }
│ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -84,22 +84,22 @@
│ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ -     {24 => (%, Matrix, Matrix)                                  }
│ │ │ -     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {25 => (%, Matrix, Matrix)                                  }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ -     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {28 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {30 => (intersect, Matrix, Matrix)                          }
│ │ │ +     {31 => (pullback, Matrix, Matrix)                           }
│ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.20053s elapsed
│ │ │ + -- 1.4371s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -26,44 +26,44 @@
│ │ │  o3 : BettiTally
│ │ │  
│ │ │  i4 : I = ideal I_*;
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .517063s elapsed
│ │ │ + -- .612321s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │  o5 : BettiTally
│ │ │  
│ │ │  i6 : I = ideal I_*;
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0258497s elapsed
│ │ │ + -- .0293113s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
│ │ │  
│ │ │  i8 : I = ideal I_*;
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │  
│ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- .826979s elapsed
│ │ │ + -- .992417s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5
│ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │           0: 1  .   .   .   .    .
│ │ │           1: . 35 140 189  84    .
│ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out
│ │ │ @@ -1,22 +1,22 @@
│ │ │  -- -*- M2-comint -*- hash: 15555226809509933135
│ │ │  
│ │ │  i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11612-0/0/
│ │ │ +o1 = /tmp/M2-12402-0/0/
│ │ │  
│ │ │  i2 : mkdir p
│ │ │  
│ │ │  i3 : isDirectory p
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11612-0/0/foo
│ │ │ +o4 = /tmp/M2-12402-0/0/foo
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : get fn
│ │ │  
│ │ │  o5 = hi there
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_move__File_lp__String_cm__String_rp.out
│ │ │ @@ -1,31 +1,31 @@
│ │ │  -- -*- M2-comint -*- hash: 4857944042471093218
│ │ │  
│ │ │  i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11448-0/0
│ │ │ +o1 = /tmp/M2-12077-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11448-0/1
│ │ │ +o2 = /tmp/M2-12077-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11448-0/0
│ │ │ +o3 = /tmp/M2-12077-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-11448-0/0 -> /tmp/M2-11448-0/1
│ │ │ +--moving: /tmp/M2-12077-0/0 -> /tmp/M2-12077-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-11448-0/1.bak
│ │ │ +--backup file created: /tmp/M2-12077-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-11448-0/1.bak
│ │ │ +o6 = /tmp/M2-12077-0/1.bak
│ │ │  
│ │ │  i7 : removeFile bak
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_nanosleep.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1331114612441
│ │ │  
│ │ │  i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500147s elapsed
│ │ │ + -- .500202s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_options_lp__Function_rp.out
│ │ │ @@ -50,26 +50,26 @@
│ │ │  
│ │ │  i6 : methods intersect
│ │ │  
│ │ │  o6 = {0 => (intersect, List)                          }
│ │ │       {1 => (intersect, Module)                        }
│ │ │       {2 => (intersect, Matrix, Matrix, Matrix, Matrix)}
│ │ │       {3 => (intersect, RRi, RRi)                      }
│ │ │ -     {4 => (intersect, Matrix, Matrix)                }
│ │ │ -     {5 => (intersect, Set, Set)                      }
│ │ │ -     {6 => (intersect, Set)                           }
│ │ │ -     {7 => (intersect, Polyhedron, Cone)              }
│ │ │ -     {8 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {4 => (intersect, Set, Set)                      }
│ │ │ +     {5 => (intersect, Matrix, Matrix)                }
│ │ │ +     {6 => (intersect, Polyhedron, Cone)              }
│ │ │ +     {7 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {8 => (intersect, Set)                           }
│ │ │       {9 => (intersect, Ideal)                         }
│ │ │       {10 => (intersect, Ideal, Ideal)                 }
│ │ │       {11 => (intersect, CCi)                          }
│ │ │       {12 => (intersect, RRi)                          }
│ │ │ -     {13 => (intersect, Polyhedron, Polyhedron)       }
│ │ │ -     {14 => (intersect, Cone, Cone)                   }
│ │ │ -     {15 => (intersect, Matrix)                       }
│ │ │ +     {13 => (intersect, Matrix)                       }
│ │ │ +     {14 => (intersect, Polyhedron, Polyhedron)       }
│ │ │ +     {15 => (intersect, Cone, Cone)                   }
│ │ │       {16 => (intersect, Module, Module)               }
│ │ │       {17 => (intersect, CCi, CCi)                     }
│ │ │       {18 => (intersect, CCi, RRi)                     }
│ │ │       {19 => (intersect, RRi, CCi)                     }
│ │ │       {20 => (intersect, Sequence)                     }
│ │ │  
│ │ │  o6 : NumberedVerticalList
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out
│ │ │ @@ -5,26 +5,26 @@
│ │ │  o1 = {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : L = shuffle toList (1..10000);
│ │ │  
│ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .659735s elapsed
│ │ │ + -- .629005s elapsed
│ │ │  
│ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .211134s elapsed
│ │ │ + -- .166776s 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
│ │ │ - -- 1.15969s elapsed
│ │ │ + -- 1.35597s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -84,15 +84,15 @@
│ │ │  o4 : BettiTally
│ │ │  
│ │ │  i5 : I = ideal I_*;
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.14895s elapsed
│ │ │ + -- 1.36984s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o7 : Ideal of S
│ │ │  
│ │ │  i8 : numTBBThreads = 1
│ │ │  
│ │ │  o8 = 1
│ │ │  
│ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.62269s elapsed
│ │ │ + -- 1.47852s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -132,15 +132,15 @@
│ │ │  o11 = 0
│ │ │  
│ │ │  i12 : I = ideal I_*;
│ │ │  
│ │ │  o12 : Ideal of S
│ │ │  
│ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.37571s elapsed
│ │ │ + -- 1.74995s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
│ │ │ @@ -150,15 +150,15 @@
│ │ │  o14 = 1
│ │ │  
│ │ │  i15 : I = ideal I_*;
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │  
│ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 1.51803s elapsed
│ │ │ + -- 1.72232s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
│ │ │ @@ -174,43 +174,43 @@
│ │ │  o18 : PolynomialRing
│ │ │  
│ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │  
│ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.09157s elapsed
│ │ │ + -- 3.1381s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
│ │ │  
│ │ │  i21 : numTBBThreads = 1
│ │ │  
│ │ │  o21 = 1
│ │ │  
│ │ │  i22 : I = ideal I_*;
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │  
│ │ │  i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.21927s elapsed
│ │ │ + -- 6.88737s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
│ │ │  
│ │ │  i24 : numTBBThreads = 10
│ │ │  
│ │ │  o24 = 10
│ │ │  
│ │ │  i25 : I = ideal I_*;
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.08357s elapsed
│ │ │ + -- 2.97431s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
│ │ │  
│ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │  
│ │ │  o27 = AssociativeAlgebras
│ │ │ @@ -233,15 +233,15 @@
│ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │  
│ │ │                  ZZ
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │  
│ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- .823549s elapsed
│ │ │ + -- .765091s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
│ │ │  
│ │ │  i32 : I = ideal I_*
│ │ │  
│ │ │ @@ -253,14 +253,14 @@
│ │ │                 101
│ │ │  
│ │ │  i33 : numTBBThreads = 1
│ │ │  
│ │ │  o33 = 1
│ │ │  
│ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.45319s elapsed
│ │ │ + -- 1.14689s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
│ │ │  
│ │ │  i35 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out
│ │ │ @@ -146,65 +146,65 @@
│ │ │  o26 : ZZ[T]
│ │ │  
│ │ │  i27 : gbTrace = 3
│ │ │  
│ │ │  o27 = 3
│ │ │  
│ │ │  i28 : time poincare I
│ │ │ - -- used 0.000690775s (cpu); 1.8044e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00113363s (cpu); 1.0099e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
│ │ │  
│ │ │  i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 16 at 0x7fad46272c40
│ │ │ +   -- registering gb 16 at 0x7f958a88fc40
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 4186
│ │ │     -- #reduction steps = 38
│ │ │     -- #spairs done = 11
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 29
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0192888s (cpu); 0.0198812s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00686734s (cpu); 0.00988303s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
│ │ │  
│ │ │  i30 : R = QQ[a..d];
│ │ │  
│ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 17 at 0x7fad46272a80
│ │ │ +   -- registering gb 17 at 0x7f958a88fa80
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │     -- nsaved = 0
│ │ │     -- 
│ │ │  o31 : Ideal of R
│ │ │  
│ │ │  i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 18 at 0x7fad462728c0
│ │ │ +   -- registering gb 18 at 0x7f958a88f8c0
│ │ │  
│ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 267
│ │ │     -- #reduction steps = 236
│ │ │     -- #spairs done = 30
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 20
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0079971s (cpu); 0.00761358s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00398917s (cpu); 0.00373273s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
│ │ │  
│ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ @@ -254,27 +254,27 @@
│ │ │  
│ │ │  i36 : gbTrace = 3
│ │ │  
│ │ │  o36 = 3
│ │ │  
│ │ │  i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7fad46272700
│ │ │ +   -- registering gb 19 at 0x7f958a88f700
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
│ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
│ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
│ │ │     -- number of monomials                = 1051
│ │ │     -- #reduction steps = 284
│ │ │     -- #spairs done = 53
│ │ │     -- ncalls = 46
│ │ │     -- nloop = 54
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0839937s (cpu); 0.0833296s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0439806s (cpu); 0.0443454s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
│ │ │  
│ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │  
│ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_printing_spto_spa_spfile.out
│ │ │ @@ -12,19 +12,19 @@
│ │ │  
│ │ │  o2 = stdio
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11803-0/0
│ │ │ +o3 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11803-0/0
│ │ │ +o4 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o4 : File
│ │ │  
│ │ │  i5 : get fn
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │ @@ -49,27 +49,27 @@
│ │ │  x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x + 1
│ │ │  o8 = stdio
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-11803-0/0
│ │ │ +o9 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o9 : File
│ │ │  
│ │ │  i10 : get fn
│ │ │  
│ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │        + 1
│ │ │  
│ │ │  i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-11803-0/0
│ │ │ +o11 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o11 : File
│ │ │  
│ │ │  i12 : get fn
│ │ │  
│ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │        1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_process__I__D.out
│ │ │ @@ -1,7 +1,7 @@
│ │ │  -- -*- M2-comint -*- hash: 1330513630563
│ │ │  
│ │ │  i1 : processID()
│ │ │  
│ │ │ -o1 = 11022
│ │ │ +o1 = 11242
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out
│ │ │ @@ -9,35 +9,35 @@
│ │ │  
│ │ │                4       5
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.95   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ -     1     92.56   ../../m2/matrix1.m2:275:22-275:43
│ │ │ -     1     44.48   ../../m2/matrix1.m2:183:25-183:52
│ │ │ -     1     31.39   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ -     1     30.23   ../../m2/matrix1.m2:130:10-145:16
│ │ │ -     1     22.58   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ -     1     21.8    ../../m2/matrix1.m2:35:10-39:22  
│ │ │ -     1     21.29   ../../m2/set.m2:129:5-129:61     
│ │ │ -     1     3.45    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ -     1     2.36    ../../m2/matrix1.m2:131:13-131:78
│ │ │ -     1     2.15    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ -     1     1.34    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ -     1     1.21    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ -     1     1.11    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ -     1     1.1     ../../m2/matrix1.m2:172:4-174:74 
│ │ │ -     1     1.09    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ -     20    .52     ../../m2/matrix1.m2:181:14-182:67
│ │ │ -     1     .43     ../../m2/modules.m2:282:4-282:52 
│ │ │ -     20    .36     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ -     1     .0041s  elapsed total                    
│ │ │ +     1     93.94   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ +     1     91.36   ../../m2/matrix1.m2:275:22-275:43
│ │ │ +     1     43.45   ../../m2/matrix1.m2:183:25-183:52
│ │ │ +     1     29.88   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ +     1     28.72   ../../m2/matrix1.m2:130:10-145:16
│ │ │ +     1     23.05   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ +     1     21.72   ../../m2/set.m2:129:5-129:61     
│ │ │ +     1     20.21   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ +     1     3.28    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ +     1     2.41    ../../m2/matrix1.m2:131:13-131:78
│ │ │ +     1     2.26    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ +     1     1.53    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ +     1     1.53    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ +     1     1.38    ../../m2/matrix1.m2:137:20-137:64
│ │ │ +     1     1.32    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ +     1     1.13    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ +     1     1.12    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ +     20    .98     ../../m2/matrix1.m2:181:14-182:67
│ │ │ +     20    .82     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ +     20    .69     ../../m2/matrix1.m2:180:17-180:29
│ │ │ +     1     .0030s  elapsed total                    
│ │ │  
│ │ │  i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │       ../../m2/lists.m2:148:24-148:32
│ │ │       ../../m2/lists.m2:148:34-148:58
│ │ │       ../../m2/matrix.m2:30:5-30:35
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │  i5 : codim I, degree I
│ │ │  
│ │ │  o5 = (2, 10)
│ │ │  
│ │ │  o5 : Sequence
│ │ │  
│ │ │  i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.468269s (cpu); 0.20272s (thread); 0s (gc)
│ │ │ + -- used 0.42246s (cpu); 0.116427s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │               2      3   1     3   0     3
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : R=kk[x_0..x_5];
│ │ │ @@ -33,15 +33,15 @@
│ │ │  i9 : codim I, degree I
│ │ │  
│ │ │  o9 = (3, 10)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : time randomKRationalPoint(I)
│ │ │ - -- used 0.372139s (cpu); 0.315367s (thread); 0s (gc)
│ │ │ + -- used 0.306282s (cpu); 0.194319s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : p=10007,kk=ZZ/p,R=kk[x_0..x_2]
│ │ │ @@ -58,12 +58,12 @@
│ │ │  
│ │ │  i14 : I=ideal random(n,R);
│ │ │  
│ │ │  o14 : Ideal of R
│ │ │  
│ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 5.46061s (cpu); 2.7311s (thread); 0s (gc)
│ │ │ + -- used 4.8567s (cpu); 1.88936s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
│ │ │  
│ │ │  i16 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_read__Directory.out
│ │ │ @@ -1,26 +1,26 @@
│ │ │  -- -*- M2-comint -*- hash: 20910736704070514
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12474-0/0
│ │ │ +o1 = /tmp/M2-14164-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-12474-0/0
│ │ │ +o2 = /tmp/M2-14164-0/0
│ │ │  
│ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-12474-0/0/foo
│ │ │ +o3 = /tmp/M2-14164-0/0/foo
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : removeFile fn
│ │ │  
│ │ │  i6 : removeDirectory dir
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reading_spfiles.out
│ │ │ @@ -1,16 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 13513555104200944796
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11978-0/0
│ │ │ +o1 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  i2 : fn << "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3" << endl << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11978-0/0
│ │ │ +o2 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : get fn
│ │ │  
│ │ │  o3 = z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
│ │ │       +8*y^3
│ │ │ @@ -38,15 +38,15 @@
│ │ │  
│ │ │  o6 : Expression of class Product
│ │ │  
│ │ │  i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11978-0/0
│ │ │ +o7 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : get fn
│ │ │  
│ │ │  o8 = sample = 2^100
│ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_readlink.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 4408639611478781130
│ │ │  
│ │ │  i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-12734-0/0
│ │ │ +o1 = /tmp/M2-14684-0/0
│ │ │  
│ │ │  i2 : symlinkFile ("foo", p)
│ │ │  
│ │ │  i3 : readlink p
│ │ │  
│ │ │  o3 = foo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_realpath.out
│ │ │ @@ -1,39 +1,39 @@
│ │ │  -- -*- M2-comint -*- hash: 324072347213224656
│ │ │  
│ │ │  i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-11022-0/91-rundir/
│ │ │ +o1 = /tmp/M2-11242-0/91-rundir/
│ │ │  
│ │ │  i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-12753-0/0
│ │ │ +o2 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-12753-0/1
│ │ │ +o3 = /tmp/M2-14723-0/1
│ │ │  
│ │ │  i4 : symlinkFile(p,q)
│ │ │  
│ │ │  i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-12753-0/0
│ │ │ +o5 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-12753-0/0
│ │ │ +o6 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-12753-0/0
│ │ │ +o7 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  i8 : removeFile p
│ │ │  
│ │ │  i9 : removeFile q
│ │ │  
│ │ │  i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-11022-0/91-rundir/
│ │ │ +o10 = /tmp/M2-11242-0/91-rundir/
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out
│ │ │ @@ -1,17 +1,16 @@
│ │ │  -- -*- M2-comint -*- hash: 1729384374372662693
│ │ │  
│ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
│ │ │  
│ │ │  i2 : collectGarbage() 
│ │ │ ---finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (1)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (3)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (5)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (6)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (7)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (8)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (9)[5]: -- finalizing sequence #6 --
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8532980310097060089
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11650-0/0
│ │ │ +o1 = /tmp/M2-12480-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11650-0/0
│ │ │ +o2 = /tmp/M2-12480-0/0
│ │ │  
│ │ │  i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : removeDirectory dir
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__Path.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731420232148149387
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11116-0/0
│ │ │ +o1 = /tmp/M2-11405-0/0
│ │ │  
│ │ │  i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-11116-0/0
│ │ │ +o2 = /tmp/M2-11405-0/0
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__U__R__I.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731420231525572968
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12417-0/0
│ │ │ +o1 = /tmp/M2-14047-0/0
│ │ │  
│ │ │  i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-12417-0/0
│ │ │ +o2 = file:///tmp/M2-14047-0/0
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_saving_sppolynomials_spand_spmatrices_spin_spfiles.out
│ │ │ @@ -25,19 +25,19 @@
│ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │  
│ │ │                               1
│ │ │  o4 : R-module, submodule of R
│ │ │  
│ │ │  i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-12265-0/0
│ │ │ +o5 = /tmp/M2-13735-0/0
│ │ │  
│ │ │  i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12265-0/0
│ │ │ +o6 = /tmp/M2-13735-0/0
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : get f
│ │ │  
│ │ │  o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2,
│ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_serial__Number.out
│ │ │ @@ -1,15 +1,15 @@
│ │ │  -- -*- M2-comint -*- hash: 5271760183816554957
│ │ │  
│ │ │  i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1367325
│ │ │ +o1 = 1367343
│ │ │  
│ │ │  i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1367327
│ │ │ +o2 = 1367345
│ │ │  
│ │ │  i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out
│ │ │ @@ -189,18 +189,18 @@
│ │ │  o25 = 40
│ │ │  
│ │ │  i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
│ │ │  
│ │ │  i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
│ │ │  
│ │ │  i30 : time X = solve(A,B);
│ │ │ - -- used 0.000183394s (cpu); 0.000176741s (thread); 0s (gc)
│ │ │ + -- used 0.000178311s (cpu); 0.000163156s (thread); 0s (gc)
│ │ │  
│ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000144531s (cpu); 0.000145172s (thread); 0s (gc)
│ │ │ + -- used 7.3243e-05s (cpu); 7.2932e-05s (thread); 0s (gc)
│ │ │  
│ │ │  i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │  
│ │ │  o32 : RR (of precision 53)
│ │ │  
│ │ │ @@ -209,18 +209,18 @@
│ │ │  o33 = 100
│ │ │  
│ │ │  i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
│ │ │  
│ │ │  i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
│ │ │  
│ │ │  i38 : time X = solve(A,B);
│ │ │ - -- used 0.232067s (cpu); 0.232067s (thread); 0s (gc)
│ │ │ + -- used 0.119727s (cpu); 0.119729s (thread); 0s (gc)
│ │ │  
│ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.215747s (cpu); 0.215685s (thread); 0s (gc)
│ │ │ + -- used 0.12075s (cpu); 0.120758s (thread); 0s (gc)
│ │ │  
│ │ │  i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │  
│ │ │  o40 : RR (of precision 100)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__Directory_lp__String_cm__String_rp.out
│ │ │ @@ -1,60 +1,60 @@
│ │ │  -- -*- M2-comint -*- hash: 2989513528213557691
│ │ │  
│ │ │  i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12037-0/0/
│ │ │ +o1 = /tmp/M2-13267-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12037-0/1/
│ │ │ +o2 = /tmp/M2-13267-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12037-0/0/a/
│ │ │ +o3 = /tmp/M2-13267-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12037-0/0/b/
│ │ │ +o4 = /tmp/M2-13267-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12037-0/0/b/c/
│ │ │ +o5 = /tmp/M2-13267-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12037-0/0/a/f
│ │ │ +o6 = /tmp/M2-13267-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12037-0/0/a/g
│ │ │ +o7 = /tmp/M2-13267-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12037-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13267-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │  
│ │ │  i10 : get (dst|"b/c/g")
│ │ │  
│ │ │  o10 = ho there
│ │ │  
│ │ │  i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │  
│ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │  
│ │ │  o12 = rm
│ │ │  
│ │ │  o12 : FunctionClosure
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__File.out
│ │ │ @@ -1,12 +1,12 @@
│ │ │  -- -*- M2-comint -*- hash: 9343844672940306595
│ │ │  
│ │ │  i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12094-0/0
│ │ │ +o1 = /tmp/M2-13384-0/0
│ │ │  
│ │ │  i2 : symlinkFile("qwert", fn)
│ │ │  
│ │ │  i3 : fileExists fn
│ │ │  
│ │ │  o3 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_temporary__File__Name.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1731926531291302106
│ │ │  
│ │ │  i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-13097-0/0.tex
│ │ │ +o1 = /tmp/M2-15437-0/0.tex
│ │ │  
│ │ │  i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-13097-0/1.html
│ │ │ +o2 = /tmp/M2-15437-0/1.html
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1332435500723
│ │ │  
│ │ │  i1 : time 3^30
│ │ │ - -- used 1.1622e-05s (cpu); 6.291e-06s (thread); 0s (gc)
│ │ │ + -- used 1.5377e-05s (cpu); 4.86e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_timing.out
│ │ │ @@ -1,14 +1,14 @@
│ │ │  -- -*- M2-comint -*- hash: 1730988300469098603
│ │ │  
│ │ │  i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000013355 seconds
│ │ │ +     -- .000014771 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000013355, 205891132094649}
│ │ │ +o2 = Time{.000014771, 205891132094649}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.out
│ │ │ @@ -14,26 +14,26 @@
│ │ │  
│ │ │            4     5    6     9     10     11    14     15    16    20
│ │ │  o3 = 1 - T  - 2T  - T  + 2T  + 2T   + 2T   - T   - 2T   - T   + T
│ │ │  
│ │ │  o3 : ZZ[T]
│ │ │  
│ │ │  i4 : elapsedTime Mgb = gb M
│ │ │ - -- 1.4495s elapsed
│ │ │ + -- 1.64788s elapsed
│ │ │  
│ │ │  o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o4 : GroebnerBasis
│ │ │  
│ │ │  i5 : M = ideal M_*;
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .415018s elapsed
│ │ │ + -- .475007s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o6 : GroebnerBasis
│ │ │  
│ │ │  i7 : S = degreesRing R
│ │ │  
│ │ │ @@ -55,14 +55,14 @@
│ │ │  o9 : S
│ │ │  
│ │ │  i10 : M = ideal M_*;
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .517879s elapsed
│ │ │ + -- .551991s elapsed
│ │ │  
│ │ │  o11 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o11 : GroebnerBasis
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out
│ │ │ @@ -36,15 +36,15 @@
│ │ │                 "memtailor version" => 1.4
│ │ │                 "mpfi version" => 1.5.5
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.1
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.113+deb13-amd64
│ │ │ +               "operating system release" => 6.12.100+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism DirectSummands CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC SimplicialModules MatrixFactorizations PathSignatures MacaulayPosets MRDI EliminationTemplates WittVectors Padic WeierstrassSemigroups ResultantComplexes EuclideanDistanceDegree NeuralIdeals TestAudit LanguageServer
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.14.9
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2023.1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │
    		│ │ │ 
    i3 : (c = Command "date";)
    │ │ │
    │ │ │ 
    i4 : c
│ │ │ -Sat Oct 24 03:11:57 UTC 2026
│ │ │ +Thu Oct 29 12:05:54 UTC 2026
│ │ │  
│ │ │  o4 = 0
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ in a file), then it gets executed with empty argument list. │ │ │ │ i1 : (f = Command ( () -> 2^30 );) │ │ │ │ i2 : f │ │ │ │ │ │ │ │ o2 = 1073741824 │ │ │ │ i3 : (c = Command "date";) │ │ │ │ i4 : c │ │ │ │ -Sat Oct 24 03:11:57 UTC 2026 │ │ │ │ +Thu Oct 29 12:05:54 UTC 2026 │ │ │ │ │ │ │ │ o4 = 0 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_u_n -- run an external command │ │ │ │ * _A_f_t_e_r_E_v_a_l -- top level method applied after evaluation │ │ │ │ ********** MMeetthhooddss tthhaatt uussee aa ccoommmmaanndd:: ********** │ │ │ │ * code(Command) -- see _c_o_d_e -- display source code │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Database.html │ │ │ @@ -57,22 +57,22 @@ │ │ │
    │ │ │

    Description

    │ │ │ A database file is just like a hash table, except both the keys and values have to be strings. In this example we create a database file, store a few entries, remove an entry with remove, close the file, and then remove the file. │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ 2 │ │ │ │ o11 = monomialIdeal (a*b*c, a d, b*c*d) │ │ │ │ │ │ │ │ o11 : MonomialIdeal of R │ │ │ │ i12 : standardPairs I │ │ │ │ │ │ │ │ │ │ │ │ -o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {a, c}}, {1, │ │ │ │ +o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 2 │ │ │ │ {b, a}}, {b, {c}}, {b , {c}}} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : independentSets I │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutable__List.html │ │ │ @@ -145,30 +145,30 @@ │ │ │ │ │ │ o10 : MutableList │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-12569-0/0.dbm
    │ │ │ +o1 = /tmp/M2-14359-0/0.dbm │ │ │
    │ │ │ 
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-12569-0/0.dbm
│ │ │ +o2 = /tmp/M2-14359-0/0.dbm
│ │ │  
│ │ │  o2 : Database
    │ │ │
    │ │ │ 
    i3 : x#"first" = "hi there"
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,18 +7,18 @@
│ │ │ │  ************ DDaattaabbaassee ---- tthhee ccllaassss ooff aallll ddaattaabbaassee ffiilleess ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  A database file is just like a hash table, except both the keys and values have
│ │ │ │  to be strings. In this example we create a database file, store a few entries,
│ │ │ │  remove an entry with _r_e_m_o_v_e, close the file, and then remove the file.
│ │ │ │  i1 : filename = temporaryFileName () | ".dbm"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12569-0/0.dbm
│ │ │ │ +o1 = /tmp/M2-14359-0/0.dbm
│ │ │ │  i2 : x = openDatabaseOut filename
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12569-0/0.dbm
│ │ │ │ +o2 = /tmp/M2-14359-0/0.dbm
│ │ │ │  
│ │ │ │  o2 : Database
│ │ │ │  i3 : x#"first" = "hi there"
│ │ │ │  
│ │ │ │  o3 = hi there
│ │ │ │  i4 : x#"first"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html
│ │ │ @@ -58,33 +58,33 @@
│ │ │          
    Description
    
│ │ │          
    Macaulay2 uses the Hans Boehm garbage collector to reclaim unused memory.  The function GCstats provides information about its status, such as the total number of bytes allocated, the current heap size, the number of garbage collections done, the number of threads used in each collection, the total cpu time spent in garbage collection, etc.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 69379379194        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 69261986618        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 219217920
│ │ │ -               "numGCs" => 1481
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 254820352
│ │ │ +               "numGCs" => 1449
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
    
│ │ │              
    
│ │ │          
    The value returned is a hash table, from which individual bits of information can be easily extracted, as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 219217920
    
│ │ │ +o2 = 254820352
│ │ │              
    
│ │ │          
    Any entries whose keys are all upper case give the values of environment variables affecting the operation of the garbage collector that have been specified by the user.
    
│ │ │          
    For further information about the individual items in the table, we refer the user to the source code and documentation of the garbage collector.
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,28 +9,28 @@
│ │ │ │  Macaulay2 uses the Hans Boehm _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r to reclaim unused memory. The
│ │ │ │  function GCstats provides information about its status, such as the total
│ │ │ │  number of bytes allocated, the current heap size, the number of garbage
│ │ │ │  collections done, the number of threads used in each collection, the total cpu
│ │ │ │  time spent in garbage collection, etc.
│ │ │ │  i1 : s = GCstats()
│ │ │ │  
│ │ │ │ -o1 = HashTable{"bytesAlloc" => 69379379194        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 69261986618        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 219217920
│ │ │ │ -               "numGCs" => 1481
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 254820352
│ │ │ │ +               "numGCs" => 1449
│ │ │ │ +               "numGCThreads" => 16
│ │ │ │  
│ │ │ │  o1 : HashTable
│ │ │ │  The value returned is a hash table, from which individual bits of information
│ │ │ │  can be easily extracted, as follows.
│ │ │ │  i2 : s#"heapSize"
│ │ │ │  
│ │ │ │ -o2 = 219217920
│ │ │ │ +o2 = 254820352
│ │ │ │  Any entries whose keys are all upper case give the values of environment
│ │ │ │  variables affecting the operation of the garbage collector that have been
│ │ │ │  specified by the user.
│ │ │ │  For further information about the individual items in the table, we refer the
│ │ │ │  user to the source code and documentation of the garbage collector.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Minimal__Generators.html
│ │ │ @@ -133,23 +133,23 @@
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │
    │ │ │ 
    i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00860229s (cpu); 0.00859565s (thread); 0s (gc)
│ │ │ + -- used 0.00436071s (cpu); 0.00435728s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
    │ │ │
    │ │ │ 
    i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0658401s (cpu); 0.0658504s (thread); 0s (gc)
│ │ │ + -- used 0.038211s (cpu); 0.0382163s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    │ │ │
    │ │ │ 
    i10 : numgens J
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,19 +46,19 @@
│ │ │ │  o6 = R
│ │ │ │  
│ │ │ │  o6 : PolynomialRing
│ │ │ │  i7 : I = monomialCurveIdeal(R, {1,4,5,9});
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ │ - -- used 0.00860229s (cpu); 0.00859565s (thread); 0s (gc)
│ │ │ │ + -- used 0.00436071s (cpu); 0.00435728s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ │ - -- used 0.0658401s (cpu); 0.0658504s (thread); 0s (gc)
│ │ │ │ + -- used 0.038211s (cpu); 0.0382163s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : numgens J
│ │ │ │  
│ │ │ │  o10 = 1067
│ │ │ │  i11 : numgens K
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Monomial__Ideal.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │
    │ │ │ 
    i12 : standardPairs I
│ │ │  
│ │ │                                                                             
│ │ │ -o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {a, c}}, {1,
│ │ │ +o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1,
│ │ │        -----------------------------------------------------------------------
│ │ │                             2
│ │ │        {b, a}}, {b, {c}}, {b , {c}}}
│ │ │  
│ │ │  o12 : List
    │ │ │
    │ │ │ 
    i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at each step
│ │ │ - -- .00605048s elapsed
    │ │ │ + -- .00421461s elapsed │ │ │
    │ │ │ 
    i12 : t = new MutableList from 1000
│ │ │  
│ │ │  o12 = MutableList{...1000...}
│ │ │  
│ │ │  o12 : MutableList
    │ │ │
    │ │ │ 
    i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ - -- .000712171s elapsed
    │ │ │ + -- .000272994s elapsed │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -43,22 +43,22 @@ │ │ │ │ i10 : s = new MutableList │ │ │ │ │ │ │ │ o10 = MutableList{} │ │ │ │ │ │ │ │ o10 : MutableList │ │ │ │ i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at │ │ │ │ each step │ │ │ │ - -- .00605048s elapsed │ │ │ │ + -- .00421461s elapsed │ │ │ │ i12 : t = new MutableList from 1000 │ │ │ │ │ │ │ │ o12 = MutableList{...1000...} │ │ │ │ │ │ │ │ o12 : MutableList │ │ │ │ i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear │ │ │ │ - -- .000712171s elapsed │ │ │ │ + -- .000272994s elapsed │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _B_a_s_i_c_L_i_s_t -- the class of all basic lists │ │ │ │ ******** MMeennuu ******** │ │ │ │ * _B_a_g -- the class of all bags │ │ │ │ ********** TTyyppeess ooff mmuuttaabbllee lliisstt:: ********** │ │ │ │ * _B_a_g -- the class of all bags │ │ │ │ ********** MMeetthhooddss tthhaatt uussee aa mmuuttaabbllee lliisstt:: ********** │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutex.html │ │ │ @@ -78,45 +78,37 @@ │ │ │ o2 : FunctionClosure │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : T = apply(10, i -> schedule(() -> sayhello i))
│ │ │  
│ │ │ -o3 = {<<task, result available, task done>>, <<task, result available, task
│ │ │ +o3 = {<<task, created>>, <<task, created>>, <<task, created>>, <<task,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     done>>, <<task, result available, task done>>, <<task, result available,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     task done>>, <<task, result available, task done>>, <<task, result
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     available, task done>>, <<task, result available, task done>>, <<task,
│ │ │ +     created>>, <<task, created>>, <<task, created>>, <<task, created>>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     result available, task done>>, <<task, result available, task done>>,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     <<task, result available, task done>>}
│ │ │ +     <<task, created>>, <<task, created>>, <<task, created>>}
│ │ │  
│ │ │  o3 : List
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i4 : while not all(T, isReady) do null
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i5 : stack sort lines msgs
│ │ │  
│ │ │  o5 = hello from thread #0
│ │ │ -     hello from thread #1
│ │ │       hello from thread #2
│ │ │       hello from thread #3
│ │ │       hello from thread #4
│ │ │       hello from thread #5
│ │ │ -     hello from thread #6
│ │ │       hello from thread #7
      │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │

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

      │ │ │
      │ │ │ @@ -137,21 +129,23 @@ │ │ │ o7 = │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i8 : T = apply(10, i -> schedule(() -> (lock m; sayhello i; unlock m)))
│ │ │  
│ │ │ -o8 = {<<task, result available, task done>>, <<task, running>>, <<task,
│ │ │ +o8 = {<<task, result available, task done>>, <<task, result available, task
│ │ │       ------------------------------------------------------------------------
│ │ │ -     running>>, <<task, running>>, <<task, running>>, <<task, created>>,
│ │ │ +     done>>, <<task, result available, task done>>, <<task, result available,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     <<task, created>>, <<task, created>>, <<task, created>>, <<task,
│ │ │ +     task done>>, <<task, result available, task done>>, <<task, result
│ │ │ +     ------------------------------------------------------------------------
│ │ │ +     available, task done>>, <<task, result available, task done>>, <<task,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     created>>}
│ │ │ +     result available, task done>>, <<task, running>>, <<task, running>>}
│ │ │  
│ │ │  o8 : List
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i9 : while not all(T, isReady) do null
      │ │ │ ├── html2text {} │ │ │ │ @@ -23,58 +23,52 @@ │ │ │ │ i2 : sayhello = i -> msgs |= "hello from thread #" | toString i | newline │ │ │ │ │ │ │ │ o2 = sayhello │ │ │ │ │ │ │ │ o2 : FunctionClosure │ │ │ │ i3 : T = apply(10, i -> schedule(() -> sayhello i)) │ │ │ │ │ │ │ │ -o3 = {<>, <>, <>, <>, <>, <>, <>, <>, <>, <>, <>, <>, <>, <>, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - result available, task done>>, <>, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - <>} │ │ │ │ + <>, <>, <>} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : while not all(T, isReady) do null │ │ │ │ i5 : stack sort lines msgs │ │ │ │ │ │ │ │ o5 = hello from thread #0 │ │ │ │ - hello from thread #1 │ │ │ │ hello from thread #2 │ │ │ │ hello from thread #3 │ │ │ │ hello from thread #4 │ │ │ │ hello from thread #5 │ │ │ │ - hello from thread #6 │ │ │ │ hello from thread #7 │ │ │ │ We likely ended up with fewer than the expected number of 10 messages. We can │ │ │ │ get around this issue by using a mutex to lock the string so that only one │ │ │ │ thread can modify it at a time. │ │ │ │ i6 : m = new Mutex │ │ │ │ │ │ │ │ o6 = m │ │ │ │ │ │ │ │ o6 : Mutex │ │ │ │ i7 : msgs = "" │ │ │ │ │ │ │ │ o7 = │ │ │ │ i8 : T = apply(10, i -> schedule(() -> (lock m; sayhello i; unlock m))) │ │ │ │ │ │ │ │ -o8 = {<>, <>, <>, <>, <>, <>, <>, │ │ │ │ + done>>, <>, <>, <>, <>, <>, <>, <>, <>, <>} │ │ │ │ + result available, task done>>, <>, <>} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ i9 : while not all(T, isReady) do null │ │ │ │ i10 : stack sort lines msgs │ │ │ │ │ │ │ │ o10 = hello from thread #0 │ │ │ │ hello from thread #1 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.html │ │ │ @@ -73,21 +73,21 @@ │ │ │ o1 : Matrix RR <-- RR │ │ │ 53 53 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : time SVD(M);
│ │ │ - -- used 0.0228821s (cpu); 0.0228783s (thread); 0s (gc)
      │ │ │ + -- used 0.0307446s (cpu); 0.0307456s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0228804s (cpu); 0.0228876s (thread); 0s (gc)
      │ │ │ + -- used 0.0300902s (cpu); 0.0301002s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Functions with optional argument named DivideConquer:

    │ │ │ ├── html2text {} │ │ │ │ @@ -11,17 +11,17 @@ │ │ │ │ For large matrices, this algorithm is often much faster. │ │ │ │ i1 : M = random(RR^200, RR^200); │ │ │ │ │ │ │ │ 200 200 │ │ │ │ o1 : Matrix RR <-- RR │ │ │ │ 53 53 │ │ │ │ i2 : time SVD(M); │ │ │ │ - -- used 0.0228821s (cpu); 0.0228783s (thread); 0s (gc) │ │ │ │ + -- used 0.0307446s (cpu); 0.0307456s (thread); 0s (gc) │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ │ - -- used 0.0228804s (cpu); 0.0228876s (thread); 0s (gc) │ │ │ │ + -- used 0.0300902s (cpu); 0.0301002s (thread); 0s (gc) │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd DDiivviiddeeCCoonnqquueerr:: ********** │ │ │ │ * _S_V_D_(_._._._,_D_i_v_i_d_e_C_o_n_q_u_e_r_=_>_._._._) -- whether to use the LAPACK divide and │ │ │ │ conquer SVD algorithm │ │ │ │ ********** FFuurrtthheerr iinnffoorrmmaattiioonn ********** │ │ │ │ * Default value: _t_r_u_e │ │ │ │ * Function: _S_V_D -- singular value decomposition of a matrix │ │ │ │ * Option key: _D_i_v_i_d_e_C_o_n_q_u_e_r -- an optional argument │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Set.html │ │ │ @@ -67,28 +67,28 @@ │ │ │ 
    i2 : R = QQ[a..d];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : B = set{a^2-b*c,b*d}
│ │ │  
│ │ │ -           2
│ │ │ -o3 = set {a  - b*c, b*d}
│ │ │ +                2
│ │ │ +o3 = set {b*d, a  - b*c}
│ │ │  
│ │ │  o3 : Set
    │ │ │ │ │ │ │ │ │ │ │ │ Set operations, such as membership, union, intersection, difference, Cartesian product, Cartesian power, and subset are available. For example, │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : toList B
│ │ │  
│ │ │ -       2
│ │ │ -o4 = {a  - b*c, b*d}
│ │ │ +            2
│ │ │ +o4 = {b*d, a  - b*c}
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : member(1,A)
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,24 +7,24 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Elements of sets may be any immutable object, such as integers, ring elements
│ │ │ │  and lists of such. Ideals may also be elements of sets.
│ │ │ │  i1 : A = set {1,2};
│ │ │ │  i2 : R = QQ[a..d];
│ │ │ │  i3 : B = set{a^2-b*c,b*d}
│ │ │ │  
│ │ │ │ -           2
│ │ │ │ -o3 = set {a  - b*c, b*d}
│ │ │ │ +                2
│ │ │ │ +o3 = set {b*d, a  - b*c}
│ │ │ │  
│ │ │ │  o3 : Set
│ │ │ │  Set operations, such as _m_e_m_b_e_r_s_h_i_p, _u_n_i_o_n, _i_n_t_e_r_s_e_c_t_i_o_n, _d_i_f_f_e_r_e_n_c_e, _C_a_r_t_e_s_i_a_n
│ │ │ │  _p_r_o_d_u_c_t, _C_a_r_t_e_s_i_a_n_ _p_o_w_e_r, and _s_u_b_s_e_t are available. For example,
│ │ │ │  i4 : toList B
│ │ │ │  
│ │ │ │ -       2
│ │ │ │ -o4 = {a  - b*c, b*d}
│ │ │ │ +            2
│ │ │ │ +o4 = {b*d, a  - b*c}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : member(1,A)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : member(-b*c+a^2,B)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_a_spfirst_sp__Macaulay2_spsession.html
│ │ │ @@ -827,15 +827,15 @@
│ │ │          
    
│ │ │            
    We may use resolution to produce a projective resolution of it, and time to report the time required.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i59 : time C = resolution M
│ │ │ - -- used 0.00458017s (cpu); 0.00457468s (thread); 0s (gc)
│ │ │ + -- used 0.00308399s (cpu); 0.00306297s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o59 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -386,15 +386,15 @@
│ │ │ │                 | c f i l o r |
│ │ │ │  
│ │ │ │                               3
│ │ │ │  o58 : R-module, quotient of R
│ │ │ │  We may use _r_e_s_o_l_u_t_i_o_n to produce a projective resolution of it, and _t_i_m_e to
│ │ │ │  report the time required.
│ │ │ │  i59 : time C = resolution M
│ │ │ │ - -- used 0.00458017s (cpu); 0.00457468s (thread); 0s (gc)
│ │ │ │ + -- used 0.00308399s (cpu); 0.00306297s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3      6      15      18      6
│ │ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │ │  
│ │ │ │        0      1      2       3       4
│ │ │ │  
│ │ │ │  o59 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  o4 = "hi there"
│ │ │              
    
│ │ │                
    i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
    
│ │ │ +o5 = true
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,13 +23,13 @@
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : peek read f
│ │ │ │  
│ │ │ │  o4 = "hi there"
│ │ │ │  i5 : atEndOfFile f
│ │ │ │  
│ │ │ │ -o5 = false
│ │ │ │ +o5 = true
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _a_t_E_n_d_O_f_F_i_l_e_(_F_i_l_e_) -- test for end of file
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_files.m2:374:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_benchmark.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  Produces an accurate timing for the code contained in the string s.  The value returned is the number of seconds.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0003750952360409743
│ │ │ +o1 = .0003064144001933974
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │  The snippet of code provided will be run enough times to register meaningfully on the clock, and the garbage collector will be called beforehand.      
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds it takes to evaluate the code
│ │ │ │              in s
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Produces an accurate timing for the code contained in the string s. The value
│ │ │ │  returned is the number of seconds.
│ │ │ │  i1 : benchmark "sqrt 2p100000"
│ │ │ │  
│ │ │ │ -o1 = .0003750952360409743
│ │ │ │ +o1 = .0003064144001933974
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  The snippet of code provided will be run enough times to register meaningfully
│ │ │ │  on the clock, and the garbage collector will be called beforehand.
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _b_e_n_c_h_m_a_r_k is a _f_u_n_c_t_i_o_n_ _c_l_o_s_u_r_e.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_caching_spcomputation_spresults.html
│ │ │ @@ -74,23 +74,23 @@
│ │ │                
    i3 : M = coker vars R;
    
│ │ │
    │ │ │ 
    i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .0035172s elapsed
│ │ │ + -- .00338879s elapsed
│ │ │  
│ │ │  o4 = 3
    │ │ │
    │ │ │ 
    i5 : elapsedTime pdim' M
│ │ │ - -- 9.22e-7s elapsed
│ │ │ + -- .000002474s elapsed
│ │ │  
│ │ │  o5 = 3
    │ │ │
    │ │ │ 
    i6 : peek M.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,19 +8,19 @@
│ │ │ │  Here is a simple example of caching a computation in a _C_a_c_h_e_T_a_b_l_e, using the
│ │ │ │  augmented null coalescing operator _?_?_=.
│ │ │ │  i1 : pdim' = M -> M.cache.pdim' ??= ( printerr "computing pdim'"; pdim M );
│ │ │ │  i2 : R = QQ[x,y,z];
│ │ │ │  i3 : M = coker vars R;
│ │ │ │  i4 : elapsedTime pdim' M
│ │ │ │   -- computing pdim'
│ │ │ │ - -- .0035172s elapsed
│ │ │ │ + -- .00338879s elapsed
│ │ │ │  
│ │ │ │  o4 = 3
│ │ │ │  i5 : elapsedTime pdim' M
│ │ │ │ - -- 9.22e-7s elapsed
│ │ │ │ + -- .000002474s elapsed
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : peek M.cache
│ │ │ │  
│ │ │ │  o6 = CacheTable{cache => MutableHashTable{}
│ │ │ │  }
│ │ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_can__Use__Hilbert__Hint.html
│ │ │ @@ -347,15 +347,15 @@
│ │ │  
│ │ │  o17 : Ideal of Rlex
    │ │ │
    │ │ │ 
    i18 : elapsedTime g1 = gens gb Ilex;
│ │ │ - -- 1.68769s elapsed
│ │ │ + -- 1.73684s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o18 : Matrix Rlex  <-- Rlex
    │ │ │
    │ │ │ @@ -435,15 +435,15 @@ │ │ │ │ │ │ o19 : Ideal of Rlex │ │ │
    │ │ │ 
    i20 : elapsedTime g2 = gens gb(Ilex, Hilbert => hf);
│ │ │ - -- .850458s elapsed
│ │ │ + -- .781425s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o20 : Matrix Rlex  <-- Rlex
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -221,15 +221,15 @@ │ │ │ │ 18y*z w + 42y*z*w + 23y*w - 28z + 15z w + 18z w - 16z w - 46z*w │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 5 │ │ │ │ + 12w ) │ │ │ │ │ │ │ │ o17 : Ideal of Rlex │ │ │ │ i18 : elapsedTime g1 = gens gb Ilex; │ │ │ │ - -- 1.68769s elapsed │ │ │ │ + -- 1.73684s elapsed │ │ │ │ │ │ │ │ 1 372 │ │ │ │ o18 : Matrix Rlex <-- Rlex │ │ │ │ i19 : Ilex = ideal(Ilex_*) -- clear out the previous Groebner basis │ │ │ │ │ │ │ │ 5 4 4 4 3 2 3 3 │ │ │ │ o19 = ideal (24x - 36x y - 30x z - 29x w + 19x y + 19x y*z - 10x y*w - │ │ │ │ @@ -301,15 +301,15 @@ │ │ │ │ 18y*z w + 42y*z*w + 23y*w - 28z + 15z w + 18z w - 16z w - 46z*w │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 5 │ │ │ │ + 12w ) │ │ │ │ │ │ │ │ o19 : Ideal of Rlex │ │ │ │ i20 : elapsedTime g2 = gens gb(Ilex, Hilbert => hf); │ │ │ │ - -- .850458s elapsed │ │ │ │ + -- .781425s elapsed │ │ │ │ │ │ │ │ 1 372 │ │ │ │ o20 : Matrix Rlex <-- Rlex │ │ │ │ i21 : g1 == g2 │ │ │ │ │ │ │ │ o21 = true │ │ │ │ ********** CCaavveeaatt ********** │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ o4 : Task │ │ │
    │ │ │ 
    i5 : n
│ │ │  
│ │ │ -o5 = 336394
    │ │ │ +o5 = 720619 │ │ │
    │ │ │ 
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    │ │ │ @@ -132,15 +132,15 @@ │ │ │ o7 : Task │ │ │
    │ │ │ 
    i8 : n
│ │ │  
│ │ │ -o8 = 929987
    │ │ │ +o8 = 1698282 │ │ │
    │ │ │ 
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    │ │ │ @@ -150,15 +150,15 @@ │ │ │     		│ │ │ 
    i10 : cancelTask t
    │ │ │
    │ │ │ 
    i11 : sleep 2
│ │ │ -stdio:2:25:(3):[1]: error: interrupted
│ │ │ +stdio:2:39:(3):[1]: error: interrupted
│ │ │  
│ │ │  o11 = 0
    │ │ │
    │ │ │ 
    i12 : t
│ │ │ @@ -168,29 +168,29 @@
│ │ │  o12 : Task
    │ │ │
    │ │ │ 
    i13 : n
│ │ │  
│ │ │ -o13 = 930227
    │ │ │ +o13 = 1698491 │ │ │
    │ │ │ 
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    │ │ │
    │ │ │ 
    i15 : n
│ │ │  
│ │ │ -o15 = 930227
    │ │ │ +o15 = 1698491 │ │ │
    │ │ │ 
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,48 +28,48 @@ │ │ │ │ i4 : t │ │ │ │ │ │ │ │ o4 = <> │ │ │ │ │ │ │ │ o4 : Task │ │ │ │ i5 : n │ │ │ │ │ │ │ │ -o5 = 336394 │ │ │ │ +o5 = 720619 │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ │ │ o6 = 0 │ │ │ │ i7 : t │ │ │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ │ │ o7 : Task │ │ │ │ i8 : n │ │ │ │ │ │ │ │ -o8 = 929987 │ │ │ │ +o8 = 1698282 │ │ │ │ i9 : isReady t │ │ │ │ │ │ │ │ o9 = false │ │ │ │ i10 : cancelTask t │ │ │ │ i11 : sleep 2 │ │ │ │ -stdio:2:25:(3):[1]: error: interrupted │ │ │ │ +stdio:2:39:(3):[1]: error: interrupted │ │ │ │ │ │ │ │ o11 = 0 │ │ │ │ i12 : t │ │ │ │ │ │ │ │ o12 = <> │ │ │ │ │ │ │ │ o12 : Task │ │ │ │ i13 : n │ │ │ │ │ │ │ │ -o13 = 930227 │ │ │ │ +o13 = 1698491 │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ │ │ o14 = 0 │ │ │ │ i15 : n │ │ │ │ │ │ │ │ -o15 = 930227 │ │ │ │ +o15 = 1698491 │ │ │ │ i16 : isReady t │ │ │ │ │ │ │ │ o16 = false │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _c_a_n_c_e_l_T_a_s_k_(_T_a_s_k_) -- stop a task │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Directory.html │ │ │ @@ -76,36 +76,36 @@ │ │ │

    Change the current working directory to dir.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11296-0/0
    │ │ │ +o1 = /tmp/M2-11765-0/0 │ │ │
    │ │ │ 
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11296-0/0
    │ │ │ +o2 = /tmp/M2-11765-0/0 │ │ │
    │ │ │ 
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-11296-0/0/
    │ │ │ +o3 = /tmp/M2-11765-0/0/ │ │ │
    │ │ │ 
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-11296-0/0/
    │ │ │ +o4 = /tmp/M2-11765-0/0/ │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -11,24 +11,24 @@ │ │ │ │ o dir, a _s_t_r_i_n_g, │ │ │ │ * Outputs: │ │ │ │ o a _s_t_r_i_n_g, the new working directory; │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Change the current working directory to dir. │ │ │ │ i1 : dir = temporaryFileName() │ │ │ │ │ │ │ │ -o1 = /tmp/M2-11296-0/0 │ │ │ │ +o1 = /tmp/M2-11765-0/0 │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ │ │ -o2 = /tmp/M2-11296-0/0 │ │ │ │ +o2 = /tmp/M2-11765-0/0 │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ │ │ -o3 = /tmp/M2-11296-0/0/ │ │ │ │ +o3 = /tmp/M2-11765-0/0/ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ │ │ -o4 = /tmp/M2-11296-0/0/ │ │ │ │ +o4 = /tmp/M2-11765-0/0/ │ │ │ │ If dir is omitted, then the current working directory is changed to the user's │ │ │ │ home directory. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_u_r_r_e_n_t_D_i_r_e_c_t_o_r_y -- current working directory │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_a_n_g_e_D_i_r_e_c_t_o_r_y is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html │ │ │ @@ -100,40 +100,40 @@ │ │ │ o1 : Package │ │ │
    │ │ │ 
    i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .313321s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .162154s elapsed │ │ │
    │ │ │ 
    i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .282764s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283285s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .140014s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .143268s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .275704s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .142442s elapsed │ │ │
    │ │ │ 
    i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .246295s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .21342s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .137938s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .139401s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ @@ -145,15 +145,15 @@ │ │ │ │ │ │ o6 : TestInput │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .302726s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .147789s elapsed │ │ │
    │ │ │ 
    i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ @@ -161,16 +161,16 @@
│ │ │  
│ │ │  o8 : NumberedVerticalList
    │ │ │
    │ │ │ 
    i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .28407s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .282385s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .139483s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .136428s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ @@ -183,15 +183,15 @@ │ │ │ │ │ │ o10 : NumberedVerticalList │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283414s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .137466s elapsed │ │ │
    │ │ │
    │ │ │
    │ │ │

    Caveat

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -42,52 +42,52 @@ │ │ │ │ │ │ │ │ o1 = FirstPackage │ │ │ │ │ │ │ │ o1 : Package │ │ │ │ i2 : check_1 FirstPackage │ │ │ │ -- warning: reloading FirstPackage; recreate instances of types from this │ │ │ │ package │ │ │ │ - -- capturing check(1, "FirstPackage") -- .313321s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .162154s elapsed │ │ │ │ i3 : check FirstPackage │ │ │ │ - -- capturing check(0, "FirstPackage") -- .282764s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .283285s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .140014s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .143268s elapsed │ │ │ │ Alternatively, if the package is installed somewhere accessible, one can do the │ │ │ │ following. │ │ │ │ i4 : check_1 "FirstPackage" │ │ │ │ - -- capturing check(1, "FirstPackage") -- .275704s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .142442s elapsed │ │ │ │ i5 : check "FirstPackage" │ │ │ │ - -- capturing check(0, "FirstPackage") -- .246295s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .21342s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .137938s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .139401s elapsed │ │ │ │ A _T_e_s_t_I_n_p_u_t object (or a list of such objects) can also be run directly. │ │ │ │ i6 : tests(1, "FirstPackage") │ │ │ │ │ │ │ │ o6 = TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3] │ │ │ │ │ │ │ │ o6 : TestInput │ │ │ │ i7 : check oo │ │ │ │ - -- capturing check(1, "FirstPackage") -- .302726s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .147789s elapsed │ │ │ │ i8 : tests "FirstPackage" │ │ │ │ │ │ │ │ o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]} │ │ │ │ {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]} │ │ │ │ │ │ │ │ o8 : NumberedVerticalList │ │ │ │ i9 : check oo │ │ │ │ - -- capturing check(0, "FirstPackage") -- .28407s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .282385s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .139483s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .136428s elapsed │ │ │ │ If only an integer is passed as an argument, then the test with that index from │ │ │ │ the last call to _t_e_s_t_s is run. │ │ │ │ i10 : tests "FirstPackage" │ │ │ │ │ │ │ │ o10 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]} │ │ │ │ {1 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:58:5-60:3]} │ │ │ │ │ │ │ │ o10 : NumberedVerticalList │ │ │ │ i11 : check 1 │ │ │ │ - -- capturing check(1, "FirstPackage") -- .283414s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .137466s elapsed │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Currently, if the package was only partially loaded because the documentation │ │ │ │ was obtainable from a database (see _b_e_g_i_n_D_o_c_u_m_e_n_t_a_t_i_o_n), then the package will │ │ │ │ be reloaded, this time completely, to ensure that all tests are considered; │ │ │ │ this may affect user objects of types declared by the package, as they may be │ │ │ │ not usable by the new instance of the package. In a future version, either the │ │ │ │ tests and the documentation will both be cached, or neither will. │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html │ │ │ @@ -55,15 +55,15 @@ │ │ │
    │ │ │

    communicating with programs

    │ │ │
    │ │ │ The most naive way to interact with another program is simply to run it, let it communicate directly with the user, and wait for it to finish. This is done with the run command. │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
    │ │ │
    │ │ │ To run a program and provide it with input, one way is use the operator <<, with a file name whose first character is an exclamation point; the rest of the file name will be taken as the command to run, as in the following example. │ │ │ │ │ │ @@ -79,16 +79,16 @@ │ │ │ │ │ │
    │ │ │ More often, one wants to write Macaulay2 code to obtain and manipulate the output from the other program. If the program requires no input data, then we can use get with a file name whose first character is an exclamation point. In the following example, we also peek at the string to see whether it includes a newline character. │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : peek get "!uname -a"
│ │ │  
│ │ │ -o3 = "Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ -     6.12.113-1 (2026-10-15) x86_64 GNU/Linux\n"
    │ │ │ +o3 = "Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ + 6.12.100-1 (2026-07-30) x86_64 GNU/Linux\n" │ │ │
    │ │ │ Bidirectional communication with a program is also possible. We use openInOut to create a file that serves as a bidirectional connection to a program. That file is called an input output file. In this example we open a connection to the unix utility grep and use it to locate the symbol names in Macaulay2 that begin with in. │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -22,41 +22,41 @@ │ │ │ │ o npaths, an _i_n_t_e_g_e_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ Output is verbose. For other dynamic strategies, see _M_o_n_o_d_r_o_m_y_S_o_l_v_e_r_O_p_t_i_o_n_s. │ │ │ │ i1 : R = CC[a,b,c,d][x,y]; │ │ │ │ i2 : polys = polySystem {a*x+b*y^2,c*x*y+d}; │ │ │ │ i3 : (p0, x0) = createSeedPair polys; │ │ │ │ i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0}) │ │ │ │ ---backup directory created: /tmp/M2-25512-0/2 │ │ │ │ - -- .00616338s elapsed │ │ │ │ +--backup directory created: /tmp/M2-33134-0/2 │ │ │ │ + -- .00311047s elapsed │ │ │ │ H01: 1 │ │ │ │ - -- .00900117s elapsed │ │ │ │ + -- .00298895s elapsed │ │ │ │ H10: 1 │ │ │ │ - -- .000649635s elapsed │ │ │ │ + -- .000580269s elapsed │ │ │ │ number of paths tracked: 2 │ │ │ │ found 1 points in the fiber so far │ │ │ │ - -- .00324889s elapsed │ │ │ │ + -- .00310472s elapsed │ │ │ │ H01: 1 │ │ │ │ - -- .00435076s elapsed │ │ │ │ + -- .00304615s elapsed │ │ │ │ H10: 1 │ │ │ │ - -- .000544598s elapsed │ │ │ │ + -- .000503438s elapsed │ │ │ │ number of paths tracked: 4 │ │ │ │ found 1 points in the fiber so far │ │ │ │ - -- .00428387s elapsed │ │ │ │ + -- .00308371s elapsed │ │ │ │ H01: 1 │ │ │ │ - -- .00298132s elapsed │ │ │ │ + -- .00302496s elapsed │ │ │ │ H10: 1 │ │ │ │ - -- .000466532s elapsed │ │ │ │ + -- .000458343s elapsed │ │ │ │ number of paths tracked: 6 │ │ │ │ found 1 points in the fiber so far │ │ │ │ - -- .0044506s elapsed │ │ │ │ + -- .00276853s elapsed │ │ │ │ H01: 1 │ │ │ │ - -- .00309306s elapsed │ │ │ │ + -- .002966s elapsed │ │ │ │ H10: 1 │ │ │ │ - -- .000457295s elapsed │ │ │ │ + -- .000500639s elapsed │ │ │ │ number of paths tracked: 8 │ │ │ │ found 1 points in the fiber so far │ │ │ │ │ │ │ │ o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8) │ │ │ │ │ │ │ │ o4 : Sequence │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html │ │ │ @@ -96,22 +96,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : f = openInOut "!grep -E '^in'"
│ │ │ @@ -167,15 +167,15 @@
│ │ │  
│ │ │  o9 = false
    │ │ │
    │ │ │ 
    i10 : while not isReady f do (sleep 1; << "." << flush)
│ │ │ -......
    │ │ │ +..... │ │ │
    │ │ │ 
    i11 : read f
│ │ │  
│ │ │  o11 = the answer is 4
    │ │ │ ├── html2text {} │ │ │ │ @@ -5,16 +5,16 @@ │ │ │ │ _n_e_x_t | _p_r_e_v_i_o_u_s | _f_o_r_w_a_r_d | _b_a_c_k_w_a_r_d | _u_p | _i_n_d_e_x | _t_o_c │ │ │ │ =============================================================================== │ │ │ │ ************ ccoommmmuunniiccaattiinngg wwiitthh pprrooggrraammss ************ │ │ │ │ The most naive way to interact with another program is simply to run it, let it │ │ │ │ communicate directly with the user, and wait for it to finish. This is done │ │ │ │ with the _r_u_n command. │ │ │ │ i1 : run "uname -a" │ │ │ │ -Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 │ │ │ │ -(2026-10-15) x86_64 GNU/Linux │ │ │ │ +Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100- │ │ │ │ +1 (2026-07-30) x86_64 GNU/Linux │ │ │ │ │ │ │ │ o1 = 0 │ │ │ │ To run a program and provide it with input, one way is use the operator _<_<, │ │ │ │ with a file name whose first character is an exclamation point; the rest of the │ │ │ │ file name will be taken as the command to run, as in the following example. │ │ │ │ i2 : "!grep a" << " ba \n bc \n ad \n ef \n" << close │ │ │ │ ba │ │ │ │ @@ -26,16 +26,16 @@ │ │ │ │ More often, one wants to write Macaulay2 code to obtain and manipulate the │ │ │ │ output from the other program. If the program requires no input data, then we │ │ │ │ can use _g_e_t with a file name whose first character is an exclamation point. In │ │ │ │ the following example, we also peek at the string to see whether it includes a │ │ │ │ newline character. │ │ │ │ i3 : peek get "!uname -a" │ │ │ │ │ │ │ │ -o3 = "Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ │ - 6.12.113-1 (2026-10-15) x86_64 GNU/Linux\n" │ │ │ │ +o3 = "Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ │ + 6.12.100-1 (2026-07-30) x86_64 GNU/Linux\n" │ │ │ │ Bidirectional communication with a program is also possible. We use _o_p_e_n_I_n_O_u_t │ │ │ │ to create a file that serves as a bidirectional connection to a program. That │ │ │ │ file is called an input output file. In this example we open a connection to │ │ │ │ the unix utility grep and use it to locate the symbol names in Macaulay2 that │ │ │ │ begin with in. │ │ │ │ i4 : f = openInOut "!grep -E '^in'" │ │ │ │ │ │ │ │ @@ -97,15 +97,15 @@ │ │ │ │ o8 = !sleep 5; echo -n the answer is 4 │ │ │ │ │ │ │ │ o8 : File │ │ │ │ i9 : isReady f │ │ │ │ │ │ │ │ o9 = false │ │ │ │ i10 : while not isReady f do (sleep 1; << "." << flush) │ │ │ │ -...... │ │ │ │ +..... │ │ │ │ i11 : read f │ │ │ │ │ │ │ │ o11 = the answer is 4 │ │ │ │ i12 : isReady f │ │ │ │ │ │ │ │ o12 = true │ │ │ │ i13 : atEndOfFile f │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_sp__Groebner_spbases.html │ │ │ @@ -274,15 +274,15 @@ │ │ │ 1277 │ │ │
    │ │ │ 
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f57805dd000
│ │ │ +   -- registering gb 5 at 0x7f10dc0f6000
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -378,15 +378,15 @@
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
    │ │ │
    │ │ │ 
    i33 : time betti gb f
│ │ │ - -- used 0.266722s (cpu); 0.187826s (thread); 0s (gc)
│ │ │ + -- used 0.172018s (cpu); 0.172731s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -422,15 +422,15 @@
│ │ │  
│ │ │  o35 : ZZ[T]
    │ │ │
    │ │ │ 
    i36 : time betti gb f
│ │ │ - -- used 0.00399931s (cpu); 0.00644121s (thread); 0s (gc)
│ │ │ + -- used 0.00699234s (cpu); 0.00660643s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -142,15 +142,15 @@
│ │ │ │  o23 = ideal (x*y - z , y  - w )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o23 : Ideal of ----[x..z, w]
│ │ │ │                 1277
│ │ │ │  i24 : gb I
│ │ │ │  
│ │ │ │ -   -- registering gb 5 at 0x7f57805dd000
│ │ │ │ +   -- registering gb 5 at 0x7f10dc0f6000
│ │ │ │  
│ │ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │ │     -- number of monomials                = 8
│ │ │ │     -- #reduction steps = 2
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │ @@ -215,15 +215,15 @@
│ │ │ │  
│ │ │ │  o31 : ZZ[T]
│ │ │ │  i32 : f = random(R^1,R^{-3,-3,-5,-6});
│ │ │ │  
│ │ │ │                1      4
│ │ │ │  o32 : Matrix R  <-- R
│ │ │ │  i33 : time betti gb f
│ │ │ │ - -- used 0.266722s (cpu); 0.187826s (thread); 0s (gc)
│ │ │ │ + -- used 0.172018s (cpu); 0.172731s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o33 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ │ │ @@ -247,15 +247,15 @@
│ │ │ │  i35 : poincare cokernel f = (1-T^3)*(1-T^3)*(1-T^5)*(1-T^6) -- cache poincare
│ │ │ │  
│ │ │ │              3    5     8     9    12     14    17
│ │ │ │  o35 = 1 - 2T  - T  + 2T  + 2T  - T   - 2T   + T
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : time betti gb f
│ │ │ │ - -- used 0.00399931s (cpu); 0.00644121s (thread); 0s (gc)
│ │ │ │ + -- used 0.00699234s (cpu); 0.00660643s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o36 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,112 +85,112 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12075-0/0/
    
│ │ │ +o1 = /tmp/M2-13345-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12075-0/1/
    
│ │ │ +o2 = /tmp/M2-13345-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12075-0/0/a/
    
│ │ │ +o3 = /tmp/M2-13345-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12075-0/0/b/
    
│ │ │ +o4 = /tmp/M2-13345-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12075-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-13345-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12075-0/0/a/f
│ │ │ +o6 = /tmp/M2-13345-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12075-0/0/a/g
│ │ │ +o7 = /tmp/M2-13345-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12075-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13345-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-12075-0/0/
│ │ │ -     /tmp/M2-12075-0/0/b/
│ │ │ -     /tmp/M2-12075-0/0/b/c/
│ │ │ -     /tmp/M2-12075-0/0/b/c/g
│ │ │ -     /tmp/M2-12075-0/0/a/
│ │ │ -     /tmp/M2-12075-0/0/a/g
│ │ │ -     /tmp/M2-12075-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-13345-0/0/
│ │ │ +     /tmp/M2-13345-0/0/a/
│ │ │ +     /tmp/M2-13345-0/0/a/g
│ │ │ +     /tmp/M2-13345-0/0/a/f
│ │ │ +     /tmp/M2-13345-0/0/b/
│ │ │ +     /tmp/M2-13345-0/0/b/c/
│ │ │ +     /tmp/M2-13345-0/0/b/c/g
│ │ │              
    
│ │ │                
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-12075-0/0/b/c/g -> /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/g -> /tmp/M2-12075-0/1/a/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/f -> /tmp/M2-12075-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/g -> /tmp/M2-13345-0/1/a/g
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/f -> /tmp/M2-13345-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13345-0/0/b/c/g -> /tmp/M2-13345-0/1/b/c/g
│ │ │              
    
│ │ │                
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-12075-0/0/b/c/g not newer than /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/g not newer than /tmp/M2-12075-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/f not newer than /tmp/M2-12075-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/g not newer than /tmp/M2-13345-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/f not newer than /tmp/M2-13345-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13345-0/0/b/c/g not newer than /tmp/M2-13345-0/1/b/c/g
│ │ │              
    
│ │ │                
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-12075-0/1/
│ │ │ -      /tmp/M2-12075-0/1/a/
│ │ │ -      /tmp/M2-12075-0/1/a/f
│ │ │ -      /tmp/M2-12075-0/1/a/g
│ │ │ -      /tmp/M2-12075-0/1/b/
│ │ │ -      /tmp/M2-12075-0/1/b/c/
│ │ │ -      /tmp/M2-12075-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-13345-0/1/
│ │ │ +      /tmp/M2-13345-0/1/a/
│ │ │ +      /tmp/M2-13345-0/1/a/g
│ │ │ +      /tmp/M2-13345-0/1/a/f
│ │ │ +      /tmp/M2-13345-0/1/b/
│ │ │ +      /tmp/M2-13345-0/1/b/c/
│ │ │ +      /tmp/M2-13345-0/1/b/c/g
│ │ │              
    
│ │ │                
    i13 : get (dst|"b/c/g")
│ │ │  
│ │ │  o13 = ho there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,68 +25,68 @@
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o a copy of the directory tree rooted at src is created, rooted at
│ │ │ │              dst
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12075-0/0/
│ │ │ │ +o1 = /tmp/M2-13345-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12075-0/1/
│ │ │ │ +o2 = /tmp/M2-13345-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12075-0/0/a/
│ │ │ │ +o3 = /tmp/M2-13345-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12075-0/0/b/
│ │ │ │ +o4 = /tmp/M2-13345-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12075-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-13345-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12075-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-13345-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12075-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-13345-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-12075-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-13345-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-12075-0/0/
│ │ │ │ -     /tmp/M2-12075-0/0/b/
│ │ │ │ -     /tmp/M2-12075-0/0/b/c/
│ │ │ │ -     /tmp/M2-12075-0/0/b/c/g
│ │ │ │ -     /tmp/M2-12075-0/0/a/
│ │ │ │ -     /tmp/M2-12075-0/0/a/g
│ │ │ │ -     /tmp/M2-12075-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-13345-0/0/
│ │ │ │ +     /tmp/M2-13345-0/0/a/
│ │ │ │ +     /tmp/M2-13345-0/0/a/g
│ │ │ │ +     /tmp/M2-13345-0/0/a/f
│ │ │ │ +     /tmp/M2-13345-0/0/b/
│ │ │ │ +     /tmp/M2-13345-0/0/b/c/
│ │ │ │ +     /tmp/M2-13345-0/0/b/c/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-12075-0/0/b/c/g -> /tmp/M2-12075-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-12075-0/0/a/g -> /tmp/M2-12075-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-12075-0/0/a/f -> /tmp/M2-12075-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-13345-0/0/a/g -> /tmp/M2-13345-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-13345-0/0/a/f -> /tmp/M2-13345-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-13345-0/0/b/c/g -> /tmp/M2-13345-0/1/b/c/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-12075-0/0/b/c/g not newer than /tmp/M2-12075-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-12075-0/0/a/g not newer than /tmp/M2-12075-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-12075-0/0/a/f not newer than /tmp/M2-12075-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-13345-0/0/a/g not newer than /tmp/M2-13345-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-13345-0/0/a/f not newer than /tmp/M2-13345-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-13345-0/0/b/c/g not newer than /tmp/M2-13345-0/1/b/c/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-12075-0/1/
│ │ │ │ -      /tmp/M2-12075-0/1/a/
│ │ │ │ -      /tmp/M2-12075-0/1/a/f
│ │ │ │ -      /tmp/M2-12075-0/1/a/g
│ │ │ │ -      /tmp/M2-12075-0/1/b/
│ │ │ │ -      /tmp/M2-12075-0/1/b/c/
│ │ │ │ -      /tmp/M2-12075-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-13345-0/1/
│ │ │ │ +      /tmp/M2-13345-0/1/a/
│ │ │ │ +      /tmp/M2-13345-0/1/a/g
│ │ │ │ +      /tmp/M2-13345-0/1/a/f
│ │ │ │ +      /tmp/M2-13345-0/1/b/
│ │ │ │ +      /tmp/M2-13345-0/1/b/c/
│ │ │ │ +      /tmp/M2-13345-0/1/b/c/g
│ │ │ │  i13 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o13 = ho there
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o14 = rm
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__File_lp__String_cm__String_rp.html
│ │ │ @@ -83,65 +83,65 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -515,27 +515,27 @@
│ │ │  o36 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11841-0/0
    
│ │ │ +o1 = /tmp/M2-12871-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11841-0/1
    
│ │ │ +o2 = /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11841-0/0
│ │ │ +o3 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11841-0/0 -> /tmp/M2-11841-0/1
    
│ │ │ + -- copying: /tmp/M2-12871-0/0 -> /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
    
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11841-0/0
│ │ │ +o7 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
    
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i9 : get dst
│ │ │  
│ │ │  o9 = hi there
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,37 +18,37 @@
│ │ │ │            o Verbose => a _B_o_o_l_e_a_n_ _v_a_l_u_e, default value false, whether to report
│ │ │ │              individual file operations
│ │ │ │      * Consequences:
│ │ │ │            o the file may be copied
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11841-0/0
│ │ │ │ +o1 = /tmp/M2-12871-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11841-0/1
│ │ │ │ +o2 = /tmp/M2-12871-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11841-0/0
│ │ │ │ +o3 = /tmp/M2-12871-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11841-0/0 -> /tmp/M2-11841-0/1
│ │ │ │ + -- copying: /tmp/M2-12871-0/0 -> /tmp/M2-12871-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11841-0/0
│ │ │ │ +o7 = /tmp/M2-12871-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │ │  i9 : get dst
│ │ │ │  
│ │ │ │  o9 = hi there
│ │ │ │  i10 : removeFile src
│ │ │ │  i11 : removeFile dst
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_D_i_r_e_c_t_o_r_y
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cpu__Time.html
│ │ │ @@ -69,38 +69,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 611.770723009
│ │ │ +o1 = 487.823398062
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │              
    
│ │ │                
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 613.313432871
│ │ │ +o3 = 488.878609747
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.542709862000038
│ │ │ +o4 = 1.055211684999961
│ │ │  
│ │ │  o4 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,26 +9,26 @@
│ │ │ │              cpuTime()
│ │ │ │      * Outputs:
│ │ │ │            o a _r_e_a_l_ _n_u_m_b_e_r, the number of seconds of cpu time used since the
│ │ │ │              program was started
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : t1 = cpuTime()
│ │ │ │  
│ │ │ │ -o1 = 611.770723009
│ │ │ │ +o1 = 487.823398062
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 613.313432871
│ │ │ │ +o3 = 488.878609747
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 1.542709862000038
│ │ │ │ +o4 = 1.055211684999961
│ │ │ │  
│ │ │ │  o4 : RR (of precision 53)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_current__Time.html
│ │ │ @@ -69,48 +69,48 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1792811657
    
│ │ │ +o1 = 1793275644
│ │ │              
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 56.81193979063632
│ │ │ +o2 = 56.82664295334988
│ │ │  
│ │ │  o2 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    We can also compute how many months account for the fractional part of that number.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 9.743277487635794
│ │ │ +o3 = 9.919715440198502
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : run "date"
│ │ │ -Sat Oct 24 03:14:17 UTC 2026
│ │ │ +Thu Oct 29 12:07:24 UTC 2026
│ │ │  
│ │ │  o4 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,31 +9,31 @@
│ │ │ │              currentTime()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the current time, in seconds since 00:00:00 1970-01-01
│ │ │ │              UTC, the beginning of the epoch
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : currentTime()
│ │ │ │  
│ │ │ │ -o1 = 1792811657
│ │ │ │ +o1 = 1793275644
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 56.81193979063632
│ │ │ │ +o2 = 56.82664295334988
│ │ │ │  
│ │ │ │  o2 : RR (of precision 53)
│ │ │ │  We can also compute how many months account for the fractional part of that
│ │ │ │  number.
│ │ │ │  i3 : 12 * (oo - floor oo)
│ │ │ │  
│ │ │ │ -o3 = 9.743277487635794
│ │ │ │ +o3 = 9.919715440198502
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Sat Oct 24 03:14:17 UTC 2026
│ │ │ │ +Thu Oct 29 12:07:24 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_u_r_r_e_n_t_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:1849:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the value of e.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime sleep 1
│ │ │ - -- 1.00015s elapsed
│ │ │ + -- 1.00022s 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
│ │ │ │ + -- 1.00022s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _G_C_s_t_a_t_s -- information about the status of the garbage collector
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Timing.html
│ │ │ @@ -59,24 +59,24 @@
│ │ │  elapsedTiming e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of time elapsed, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTiming sleep 1
│ │ │  
│ │ │  o1 = 0
│ │ │ -     -- 1.00011 seconds
│ │ │ +     -- 1.00119 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{1.00011, 0}
    
│ │ │ +o2 = Time{1.00119, 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
│ │ │ │ -     -- 1.00011 seconds
│ │ │ │ +     -- 1.00119 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{1.00011, 0}
│ │ │ │ +o2 = Time{1.00119, 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
│ │ │ @@ -71,15 +71,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : time leadTerm gens gb I
│ │ │ - -- used 0.453514s (cpu); 0.254782s (thread); 0s (gc)
│ │ │ + -- used 0.347258s (cpu); 0.147574s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,15 +168,15 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.436423s (cpu); 0.247727s (thread); 0s (gc)
│ │ │ + -- used 0.0813805s (cpu); 0.0813131s (thread); 0s (gc)
│ │ │  
│ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │       ------------------------------------------------------------------------
│ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -251,15 +251,15 @@
│ │ │  
│ │ │  o11 : Ideal of R1
    
│ │ │              
    
│ │ │                
    i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.0611301s (cpu); 0.0611395s (thread); 0s (gc)
│ │ │ + -- used 0.0261903s (cpu); 0.0261896s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7  
│ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2  
│ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -343,15 +343,15 @@
│ │ │  
│ │ │  o16 : RingMap A <-- B
    
│ │ │              
    
│ │ │                
    i17 : time G = kernel F
│ │ │ - -- used 0.450122s (cpu); 0.254702s (thread); 0s (gc)
│ │ │ + -- used 0.319701s (cpu); 0.127183s (thread); 0s (gc)
│ │ │  
│ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8   
│ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │        -----------------------------------------------------------------------
│ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7 
│ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -424,26 +424,26 @@
│ │ │  
│ │ │  o19 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00269518s (cpu); 0.00269678s (thread); 0s (gc)
│ │ │ + -- used 0.00140979s (cpu); 0.00140852s (thread); 0s (gc)
│ │ │  
│ │ │           9    9      7    3
│ │ │  o20 = x*t  - t  - z*t  - z
│ │ │  
│ │ │  o20 : R
    
│ │ │              
    
│ │ │                
    i21 : time f2 = resultant(I_1,f1,t)
│ │ │ - -- used 0.0712746s (cpu); 0.0712854s (thread); 0s (gc)
│ │ │ + -- used 0.0279962s (cpu); 0.028001s (thread); 0s (gc)
│ │ │  
│ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2  
│ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │        -----------------------------------------------------------------------
│ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8   
│ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │        -----------------------------------------------------------------------
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  i2 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o2 = ideal (- s  - s*t + x - 1, - t  - 3t  - t + y, - s*t  + z)
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time leadTerm gens gb I
│ │ │ │ - -- used 0.453514s (cpu); 0.254782s (thread); 0s (gc)
│ │ │ │ + -- used 0.347258s (cpu); 0.147574s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  i7 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
│ │ │ │  
│ │ │ │                 3                   3     2               3
│ │ │ │  o7 = ideal (- s  - s*t + x - 1, - t  - 3t  + y - t, - s*t  + z)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : time G = eliminate(I,{s,t})
│ │ │ │ - -- used 0.436423s (cpu); 0.247727s (thread); 0s (gc)
│ │ │ │ + -- used 0.0813805s (cpu); 0.0813131s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o8 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │           7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │       7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -161,15 +161,15 @@
│ │ │ │  Sometimes giving the variables different degrees will speed up the
│ │ │ │  computations. Here, we set the degrees of x, y, and z to be the total degrees.
│ │ │ │  i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
│ │ │ │  i11 : I1 = substitute(I,R1);
│ │ │ │  
│ │ │ │  o11 : Ideal of R1
│ │ │ │  i12 : time G = eliminate(I1,{s,t})
│ │ │ │ - -- used 0.0611301s (cpu); 0.0611395s (thread); 0s (gc)
│ │ │ │ + -- used 0.0261903s (cpu); 0.0261896s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 6 3       3 6    9     2 8         5 4      2 7
│ │ │ │  o12 = ideal(x y  - 3x y z  + 3x*y z  - z  - 6x y z - 15x*y z  + 21y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │          2 9       2 5 3       6 3    7 3         2 6     3 6       7 2
│ │ │ │        3x y  - 324x y z  + 6x*y z  - y z  - 405x*y z  - 3y z  + 7x*y z  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -232,15 +232,15 @@
│ │ │ │  i16 : F = map(A,B,{s^3+s*t+1, t^3+3*t^2+t, s*t^3})
│ │ │ │  
│ │ │ │                     3             3     2         3
│ │ │ │  o16 = map (A, B, {s  + s*t + 1, t  + 3t  + t, s*t })
│ │ │ │  
│ │ │ │  o16 : RingMap A <-- B
│ │ │ │  i17 : time G = kernel F
│ │ │ │ - -- used 0.450122s (cpu); 0.254702s (thread); 0s (gc)
│ │ │ │ + -- used 0.319701s (cpu); 0.127183s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               3 9     2 9     2 8      2 6 3       9    2 7         8
│ │ │ │  o17 = ideal(x y  - 3x y  - 6x y z - 3x y z  + 3x*y  - x y z + 12x*y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            7 2       2 5 3       6 3    7 3        5 4       3 6    9       7
│ │ │ │        7x*y z  - 324x y z  + 6x*y z  - y z  - 15x*y z  + 3x*y z  - y  + 2x*y z
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -301,22 +301,22 @@
│ │ │ │  involve the variables s and t.
│ │ │ │  i19 : use ring I
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : PolynomialRing
│ │ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ │ - -- used 0.00269518s (cpu); 0.00269678s (thread); 0s (gc)
│ │ │ │ + -- used 0.00140979s (cpu); 0.00140852s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9    9      7    3
│ │ │ │  o20 = x*t  - t  - z*t  - z
│ │ │ │  
│ │ │ │  o20 : R
│ │ │ │  i21 : time f2 = resultant(I_1,f1,t)
│ │ │ │ - -- used 0.0712746s (cpu); 0.0712854s (thread); 0s (gc)
│ │ │ │ + -- used 0.0279962s (cpu); 0.028001s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2
│ │ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8
│ │ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end__Package.html
│ │ │ @@ -159,15 +159,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-11022-0/96-rundir/
│ │ │ +             source directory => /tmp/M2-11242-0/96-rundir/
│ │ │               source file => stdio
│ │ │               test inputs => MutableList{}
    
│ │ │              
    
│ │ │                
    i7 : dictionaryPath
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,15 +77,15 @@
│ │ │ │                                      Version => 0.0
│ │ │ │               package prefix => /usr/
│ │ │ │               PackageIsLoaded => true
│ │ │ │               pkgname => Foo
│ │ │ │               private dictionary => Foo#"private dictionary"
│ │ │ │               processed documentation => MutableHashTable{}
│ │ │ │               raw documentation => MutableHashTable{}
│ │ │ │ -             source directory => /tmp/M2-11022-0/96-rundir/
│ │ │ │ +             source directory => /tmp/M2-11242-0/96-rundir/
│ │ │ │               source file => stdio
│ │ │ │               test inputs => MutableList{}
│ │ │ │  i7 : dictionaryPath
│ │ │ │  
│ │ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Truncations.Dictionary, Polyhedra.Dictionary, Saturation.Dictionary,
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Exists.html
│ │ │ @@ -73,29 +73,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11391-0/0
    
│ │ │ +o1 = /tmp/M2-11960-0/0
│ │ │              
    
│ │ │                
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11391-0/0
│ │ │ +o3 = /tmp/M2-11960-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : fileExists fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether a file with the filename or path fn exists
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11391-0/0
│ │ │ │ +o1 = /tmp/M2-11960-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11391-0/0
│ │ │ │ +o3 = /tmp/M2-11960-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : fileExists fn
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : removeFile fn
│ │ │ │  If fn refers to a symbolic link, then whether the file exists is determined by
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Length.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    Description
    
│ │ │          
    The length of an open output file is determined from the internal count of the number of bytes written so far.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -430,15 +430,15 @@
│ │ │  
│ │ │  o33 = 1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-13078-0/0
│ │ │ +o1 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │              
    
│ │ │                
    i2 : fileLength f
│ │ │ @@ -90,24 +90,24 @@
│ │ │  o2 = 8
    
│ │ │              
    
│ │ │                
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-13078-0/0
│ │ │ +o3 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-13078-0/0
    
│ │ │ +o4 = /tmp/M2-15398-0/0
│ │ │              
    
│ │ │                
    i5 : fileLength filename
│ │ │  
│ │ │  o5 = 8
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,28 +12,28 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the length of the file f or the file whose name is f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The length of an open output file is determined from the internal count of the
│ │ │ │  number of bytes written so far.
│ │ │ │  i1 : f = temporaryFileName() << "hi there"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-13078-0/0
│ │ │ │ +o1 = /tmp/M2-15398-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-13078-0/0
│ │ │ │ +o3 = /tmp/M2-15398-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-13078-0/0
│ │ │ │ +o4 = /tmp/M2-15398-0/0
│ │ │ │  i5 : fileLength filename
│ │ │ │  
│ │ │ │  o5 = 8
│ │ │ │  i6 : get filename
│ │ │ │  
│ │ │ │  o6 = hi there
│ │ │ │  i7 : length oo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__File_rp.html
│ │ │ @@ -74,22 +74,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12284-0/0
    
│ │ │ +o1 = /tmp/M2-13774-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-12284-0/0
│ │ │ +o2 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode f
│ │ │ @@ -97,15 +97,15 @@
│ │ │  o3 = 420
    
│ │ │              
    
│ │ │                
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-12284-0/0
│ │ │ +o4 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,26 +11,26 @@
│ │ │ │      * Inputs:
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Outputs:
│ │ │ │            o the mode of the open file f
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12284-0/0
│ │ │ │ +o1 = /tmp/M2-13774-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12284-0/0
│ │ │ │ +o2 = /tmp/M2-13774-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12284-0/0
│ │ │ │ +o4 = /tmp/M2-13774-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _f_i_l_e_M_o_d_e_(_F_i_l_e_) -- get file mode
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__String_rp.html
│ │ │ @@ -74,22 +74,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11860-0/0
    
│ │ │ +o1 = /tmp/M2-12910-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11860-0/0
│ │ │ +o2 = /tmp/M2-12910-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the mode of the file located at the filename or path fn
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11860-0/0
│ │ │ │ +o1 = /tmp/M2-12910-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11860-0/0
│ │ │ │ +o2 = /tmp/M2-12910-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__File_rp.html
│ │ │ @@ -78,22 +78,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -250,23 +250,23 @@
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11725-0/0
    
│ │ │ +o1 = /tmp/M2-12635-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11725-0/0
│ │ │ +o2 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = 7 + 7*8 + 7*64
│ │ │ @@ -113,15 +113,15 @@
│ │ │  o5 = 511
    
│ │ │              
    
│ │ │                
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-11725-0/0
│ │ │ +o6 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,30 +12,30 @@
│ │ │ │            o mo, an _i_n_t_e_g_e_r
│ │ │ │            o f, a _f_i_l_e
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the open file f is set to mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11725-0/0
│ │ │ │ +o1 = /tmp/M2-12635-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11725-0/0
│ │ │ │ +o2 = /tmp/M2-12635-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = 7 + 7*8 + 7*64
│ │ │ │  
│ │ │ │  o3 = 511
│ │ │ │  i4 : fileMode(m,f)
│ │ │ │  i5 : fileMode f
│ │ │ │  
│ │ │ │  o5 = 511
│ │ │ │  i6 : close f
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11725-0/0
│ │ │ │ +o6 = /tmp/M2-12635-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : fileMode fn
│ │ │ │  
│ │ │ │  o7 = 511
│ │ │ │  i8 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__String_rp.html
│ │ │ @@ -78,22 +78,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12905-0/0
    
│ │ │ +o1 = /tmp/M2-15045-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12905-0/0
│ │ │ +o2 = /tmp/M2-15045-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = fileMode fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o fn, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o the mode of the file located at the filename or path fn is set to
│ │ │ │              mo
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12905-0/0
│ │ │ │ +o1 = /tmp/M2-15045-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12905-0/0
│ │ │ │ +o2 = /tmp/M2-15045-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : fileMode(m|7,fn)
│ │ │ │  i5 : fileMode fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Time.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning of the epoch, so the number of seconds ago a file or directory was modified may be found by using the following code.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 111
    
│ │ │ +o1 = 79
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │              returns null if no error occurs
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning
│ │ │ │  of the epoch, so the number of seconds ago a file or directory was modified may
│ │ │ │  be found by using the following code.
│ │ │ │  i1 : currentTime() - fileTime "."
│ │ │ │  
│ │ │ │ -o1 = 111
│ │ │ │ +o1 = 79
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │      * _f_i_l_e_ _m_a_n_i_p_u_l_a_t_i_o_n -- Unix file manipulation functions
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _f_i_l_e_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │  o6 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f466ed708c0
│ │ │ +   -- registering gb 0 at 0x7fdc1bc7a8c0
│ │ │  
│ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb elements = 3
│ │ │     -- number of monomials                = 9
│ │ │     -- #reduction steps = 6
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       {3} | x2-3  0     -z4+2 |
│ │ │ │       {4} | 0     x2-3  y3-1  |
│ │ │ │  
│ │ │ │               3      3
│ │ │ │  o6 : Matrix R  <-- R
│ │ │ │  i7 : syz f
│ │ │ │  
│ │ │ │ -   -- registering gb 0 at 0x7f466ed708c0
│ │ │ │ +   -- registering gb 0 at 0x7fdc1bc7a8c0
│ │ │ │  
│ │ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb
│ │ │ │  elements = 3
│ │ │ │     -- number of monomials                = 9
│ │ │ │     -- #reduction steps = 6
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_get.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │                
    i3 : removeFile "test-file"
    
│ │ │              
    
│ │ │                
    i4 : get "!date"
│ │ │  
│ │ │ -o4 = Sat Oct 24 03:12:48 UTC 2026
    
│ │ │ +o4 = Thu Oct 29 12:06:26 UTC 2026
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  o1 : File
│ │ │ │  i2 : get "test-file"
│ │ │ │  
│ │ │ │  o2 = hi there
│ │ │ │  i3 : removeFile "test-file"
│ │ │ │  i4 : get "!date"
│ │ │ │  
│ │ │ │ -o4 = Sat Oct 24 03:12:48 UTC 2026
│ │ │ │ +o4 = Thu Oct 29 12:06:26 UTC 2026
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d -- read from a file
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ │ │      * _c_l_o_s_e -- close a file
│ │ │ │      * _F_i_l_e_ _<_<_ _T_h_i_n_g -- print to a file
│ │ │ │  ********** WWaayyss ttoo uussee ggeett:: **********
│ │ │ │      * get(File)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │                 defaultPrecision => 53
│ │ │ │                 engineDebugLevel => 0
│ │ │ │                 errorDepth => 0
│ │ │ │                 gbTrace => 0
│ │ │ │                 interpreterDepth => 1
│ │ │ │                 lineNumber => 2
│ │ │ │                 loadDepth => 3
│ │ │ │ -               maxAllowableThreads => 7
│ │ │ │ +               maxAllowableThreads => 17
│ │ │ │                 maxExponent => 1073741823
│ │ │ │                 minExponent => -1073741824
│ │ │ │                 numTBBThreads => 0
│ │ │ │                 o1 => 2432902008176640000
│ │ │ │                 oo => 2432902008176640000
│ │ │ │                 printingAccuracy => -1
│ │ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Canceled_lp__Task_rp.html
│ │ │ @@ -115,15 +115,15 @@
│ │ │              
    		
│ │ │                
    i6 : cancelTask t
    
│ │ │              
    
│ │ │                
    i7 : sleep 2
│ │ │ -stdio:2:25:(3):[1]: error: interrupted
│ │ │ +stdio:2:39:(3):[1]: error: interrupted
│ │ │  
│ │ │  o7 = 0
    
│ │ │              
    
│ │ │                
    i8 : isCanceled t
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : sleep 1
│ │ │ │  
│ │ │ │  o5 = 0
│ │ │ │  i6 : cancelTask t
│ │ │ │  i7 : sleep 2
│ │ │ │ -stdio:2:25:(3):[1]: error: interrupted
│ │ │ │ +stdio:2:39:(3):[1]: error: interrupted
│ │ │ │  
│ │ │ │  o7 = 0
│ │ │ │  i8 : isCanceled t
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  ********** 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
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Directory.html
│ │ │ @@ -80,22 +80,22 @@
│ │ │  o1 = true
    
│ │ │              
    
│ │ │                
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11213-0/0
    
│ │ │ +o2 = /tmp/M2-11602-0/0
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11213-0/0
│ │ │ +o3 = /tmp/M2-11602-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : isDirectory fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : isDirectory "."
│ │ │ │  
│ │ │ │  o1 = true
│ │ │ │  i2 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11213-0/0
│ │ │ │ +o2 = /tmp/M2-11602-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11213-0/0
│ │ │ │ +o3 = /tmp/M2-11602-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : isDirectory fn
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Pseudoprime_lp__Z__Z_rp.html
│ │ │ @@ -216,15 +216,15 @@
│ │ │  
│ │ │  o18 = false
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 3.67323s elapsed
│ │ │ + -- 4.32444s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime m3
│ │ │ - -- .05718s elapsed
│ │ │ + -- .0496438s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000110026s elapsed
│ │ │ + -- .000102673s elapsed
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │  i17 : isPrime (m*m1)
│ │ │ │  
│ │ │ │  o17 = false
│ │ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ │ - -- 3.67323s elapsed
│ │ │ │ + -- 4.32444s elapsed
│ │ │ │  
│ │ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │ │  
│ │ │ │  o19 : Expression of class Product
│ │ │ │  i20 : facs = facs//toList/toList
│ │ │ │  
│ │ │ │  o20 = {{1000000000000000000000000000057, 1},
│ │ │ │ @@ -98,19 +98,19 @@
│ │ │ │  o20 : List
│ │ │ │  i21 : assert(set facs === set {{m,1}, {m1,1}})
│ │ │ │  i22 : m3 = nextPrime (m^3)
│ │ │ │  
│ │ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │ │        00000000000000185613
│ │ │ │  i23 : elapsedTime isPrime m3
│ │ │ │ - -- .05718s elapsed
│ │ │ │ + -- .0496438s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000110026s elapsed
│ │ │ │ + -- .000102673s elapsed
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_P_r_i_m_e_(_Z_Z_) -- whether a integer or polynomial is prime
│ │ │ │      * _f_a_c_t_o_r_(_Z_Z_) -- factor a ring element
│ │ │ │      * _n_e_x_t_P_r_i_m_e_(_N_u_m_b_e_r_) -- compute the smallest prime greater than or equal to
│ │ │ │        a given number
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Regular__File.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  In UNIX, a regular file is one that is not special in some way.  Special files include symbolic links and directories.  A regular file is a sequence of bytes stored permanently in a file system.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-13116-0/0
    
│ │ │ +o1 = /tmp/M2-15476-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-13116-0/0
│ │ │ +o2 = /tmp/M2-15476-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : isRegularFile fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e, whether fn is the path to a regular file
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  In UNIX, a regular file is one that is not special in some way. Special files
│ │ │ │  include symbolic links and directories. A regular file is a sequence of bytes
│ │ │ │  stored permanently in a file system.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-13116-0/0
│ │ │ │ +o1 = /tmp/M2-15476-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-13116-0/0
│ │ │ │ +o2 = /tmp/M2-15476-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : isRegularFile fn
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_lock_lp__Mutex_cm__Function_rp.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │                
    i3 : parallelApply(1..1000, f);
    
│ │ │              
    
│ │ │                
    i4 : x
│ │ │  
│ │ │ -o4 = 333
    
│ │ │ +o4 = 323
│ │ │              
    
│ │ │          
    
│ │ │            
    You likely see that x ended up less than the expected value of 1000.  This is an example of a race condition, as multiple threads were trying to modify the global variable x at the same time.
    
│ │ │            
    Let's try again using lock.
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  i1 : x = 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : f = i -> x += 1;
│ │ │ │  i3 : parallelApply(1..1000, f);
│ │ │ │  i4 : x
│ │ │ │  
│ │ │ │ -o4 = 333
│ │ │ │ +o4 = 323
│ │ │ │  You likely see that x ended up less than the expected value of 1000. This is an
│ │ │ │  example of a rraaccee ccoonnddiittiioonn, as multiple threads were trying to modify the
│ │ │ │  global variable x at the same time.
│ │ │ │  Let's try again using lock.
│ │ │ │  i5 : x = 0
│ │ │ │  
│ │ │ │  o5 = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_make__Directory_lp__String_rp.html
│ │ │ @@ -81,22 +81,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the name of the newly made directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is made, with as many new path components as needed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11593-0/0
│ │ │ │ +o1 = /tmp/M2-12363-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11593-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-12363-0/0/a/b/c
│ │ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │ │  i5 : removeDirectory (dir|"/a")
│ │ │ │  A filename starting with ~/ will have the tilde replaced by the home directory.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_k_d_i_r
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Allowable__Threads.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11593-0/0
    
│ │ │ +o1 = /tmp/M2-12363-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-11593-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-12363-0/0/a/b/c
│ │ │              
    
│ │ │                
    i3 : removeDirectory (dir|"/a/b/c")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
    
│ │ │ +o1 = 17
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,15 +9,15 @@
│ │ │ │      *   Usage:
│ │ │ │              maxAllowableThreads
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the maximum number to which _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s can be set
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : maxAllowableThreads
│ │ │ │  
│ │ │ │ -o1 = 7
│ │ │ │ +o1 = 17
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _m_a_x_A_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s is an _i_n_t_e_g_e_r.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_threads.m2:502:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html
│ │ │ @@ -66,15 +66,15 @@
│ │ │  
│ │ │  o1 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i2 : time fib 28
│ │ │ - -- used 1.47303s (cpu); 0.790587s (thread); 0s (gc)
│ │ │ + -- used 0.868084s (cpu); 0.493642s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │              
    
│ │ │                
    i3 : fib = memoize fib
│ │ │ @@ -83,23 +83,23 @@
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i4 : time fib 28
│ │ │ - -- used 6.6374e-05s (cpu); 6.5843e-05s (thread); 0s (gc)
│ │ │ + -- used 6.0802e-05s (cpu); 5.886e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │              
    
│ │ │                
    i5 : time fib 28
│ │ │ - -- used 4.499e-06s (cpu); 4.078e-06s (thread); 0s (gc)
│ │ │ + -- used 3.269e-06s (cpu); 2.745e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 514229
    
│ │ │              
    
│ │ │          
    An optional second argument to memoize provides a list of initial values, each of the form x => v, where v is the value to be provided for the argument x.
    
│ │ │          
    Alternatively, values can be provided after defining the memoized function using the syntax f x = v. A slightly more efficient implementation of the above would be
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,28 +11,28 @@
│ │ │ │  arguments are presented.
│ │ │ │  i1 : fib = n -> if n <= 1 then 1 else fib(n-1) + fib(n-2)
│ │ │ │  
│ │ │ │  o1 = fib
│ │ │ │  
│ │ │ │  o1 : FunctionClosure
│ │ │ │  i2 : time fib 28
│ │ │ │ - -- used 1.47303s (cpu); 0.790587s (thread); 0s (gc)
│ │ │ │ + -- used 0.868084s (cpu); 0.493642s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 6.6374e-05s (cpu); 6.5843e-05s (thread); 0s (gc)
│ │ │ │ + -- used 6.0802e-05s (cpu); 5.886e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 4.499e-06s (cpu); 4.078e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.269e-06s (cpu); 2.745e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 514229
│ │ │ │  An optional second argument to memoize provides a list of initial values, each
│ │ │ │  of the form x => v, where v is the value to be provided for the argument x.
│ │ │ │  Alternatively, values can be provided after defining the memoized function
│ │ │ │  using the syntax f x = v. A slightly more efficient implementation of the above
│ │ │ │  would be
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_methods.html
│ │ │ @@ -94,20 +94,20 @@
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ -     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {19 => (codim, BettiTally)                                   }
│ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │       {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {24 => (codim, BettiTally)                                   }
│ │ │ +     {24 => (dual, BettiTally)                                    }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │                
    
│ │ │ @@ -193,20 +193,20 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                  
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (-, Matrix, Matrix)                                   }
│ │ │ -     {1 => (diff', Matrix, Matrix)                               }
│ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ +o5 = {0 => (diff', Matrix, Matrix)                               }
│ │ │ +     {1 => (-, Matrix, Matrix)                                   }
│ │ │ +     {2 => (diff, Matrix, Matrix)                                }
│ │ │       {3 => (contract, Matrix, Matrix)                            }
│ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ +     {4 => (contract', Matrix, Matrix)                           }
│ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ @@ -217,22 +217,22 @@
│ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ -     {24 => (%, Matrix, Matrix)                                  }
│ │ │ -     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {25 => (%, Matrix, Matrix)                                  }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ -     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ -     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {28 => (tensor, Matrix, Matrix)                             }
│ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {30 => (intersect, 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 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ │ -     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {19 => (codim, BettiTally)                                   }
│ │ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {21 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ +     {22 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │       {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ -     {24 => (codim, BettiTally)                                   }
│ │ │ │ +     {24 => (dual, BettiTally)                                    }
│ │ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │  o2 = {0 => (freeResolution, Ideal)        }
│ │ │ │       {1 => (freeResolution, MonomialIdeal)}
│ │ │ │ @@ -85,20 +85,20 @@
│ │ │ │      * Inputs:
│ │ │ │            o X, a _t_y_p_e
│ │ │ │            o Y, a _t_y_p_e
│ │ │ │      * Outputs:
│ │ │ │            o a _v_e_r_t_i_c_a_l_ _l_i_s_t of those methods associated with
│ │ │ │  i5 : methods( Matrix, Matrix )
│ │ │ │  
│ │ │ │ -o5 = {0 => (-, Matrix, Matrix)                                   }
│ │ │ │ -     {1 => (diff', Matrix, Matrix)                               }
│ │ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ │ +o5 = {0 => (diff', Matrix, Matrix)                               }
│ │ │ │ +     {1 => (-, Matrix, Matrix)                                   }
│ │ │ │ +     {2 => (diff, Matrix, Matrix)                                }
│ │ │ │       {3 => (contract, Matrix, Matrix)                            }
│ │ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ │ +     {4 => (contract', Matrix, Matrix)                           }
│ │ │ │ +     {5 => (+, Matrix, Matrix)                                   }
│ │ │ │       {6 => (markedGB, Matrix, Matrix)                            }
│ │ │ │       {7 => (Hom, Matrix, Matrix)                                 }
│ │ │ │       {8 => (==, Matrix, Matrix)                                  }
│ │ │ │       {9 => (*, Matrix, Matrix)                                   }
│ │ │ │       {10 => (|, Matrix, Matrix)                                  }
│ │ │ │       {11 => (||, Matrix, Matrix)                                 }
│ │ │ │       {12 => (subquotient, Matrix, Matrix)                        }
│ │ │ │ @@ -109,22 +109,22 @@
│ │ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ │ -     {24 => (%, Matrix, Matrix)                                  }
│ │ │ │ -     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ │ +     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ │ +     {25 => (%, Matrix, Matrix)                                  }
│ │ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ │ -     {29 => (intersect, Matrix, Matrix)                          }
│ │ │ │ -     {30 => (tensor, Matrix, Matrix)                             }
│ │ │ │ -     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ +     {28 => (tensor, Matrix, Matrix)                             }
│ │ │ │ +     {29 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ +     {30 => (intersect, Matrix, Matrix)                          }
│ │ │ │ +     {31 => (pullback, Matrix, Matrix)                           }
│ │ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.20053s elapsed
│ │ │ + -- 1.4371s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -130,15 +130,15 @@
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .517063s elapsed
│ │ │ + -- .612321s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │ @@ -151,15 +151,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0258497s elapsed
│ │ │ + -- .0293113s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
    
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
    
│ │ │                
    i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- .826979s elapsed
│ │ │ + -- .992417s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5
│ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │           0: 1  .   .   .   .    .
│ │ │           1: . 35 140 189  84    .
│ │ │           2: .  .   . 196 735 1080
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = minimalBetti I
│ │ │ │ - -- 1.20053s elapsed
│ │ │ │ + -- 1.4371s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -60,40 +60,40 @@
│ │ │ │  o3 : BettiTally
│ │ │ │  One can compute smaller parts of the Betti table, by using _D_e_g_r_e_e_L_i_m_i_t and/or
│ │ │ │  _L_e_n_g_t_h_L_i_m_i_t.
│ │ │ │  i4 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ │ - -- .517063s elapsed
│ │ │ │ + -- .612321s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7
│ │ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │ │           0: 1  .   .   .   .    .   .   .
│ │ │ │           1: . 35 140 189  84    .   .   .
│ │ │ │           2: .  .   . 196 735 1080 735 196
│ │ │ │  
│ │ │ │  o5 : BettiTally
│ │ │ │  i6 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ │ - -- .0258497s elapsed
│ │ │ │ + -- .0293113s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3  4
│ │ │ │  o7 = total: 1 35 140 189 84
│ │ │ │           0: 1  .   .   .  .
│ │ │ │           1: . 35 140 189 84
│ │ │ │  
│ │ │ │  o7 : BettiTally
│ │ │ │  i8 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o8 : Ideal of S
│ │ │ │  i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ │ - -- .826979s elapsed
│ │ │ │ + -- .992417s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5
│ │ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │ │           0: 1  .   .   .   .    .
│ │ │ │           1: . 35 140 189  84    .
│ │ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_mkdir.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │          
    Description
    
│ │ │          
    Only one directory will be made, so the components of the path p other than the last must already exist.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -96,15 +96,15 @@
│ │ │  o3 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11612-0/0/
    
│ │ │ +o1 = /tmp/M2-12402-0/0/
│ │ │              
    
│ │ │                
    i2 : mkdir p
    
│ │ │              
    
│ │ │                
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11612-0/0/foo
│ │ │ +o4 = /tmp/M2-12402-0/0/foo
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,22 +12,22 @@
│ │ │ │      * Consequences:
│ │ │ │            o a directory will be created at the path p
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Only one directory will be made, so the components of the path p other than the
│ │ │ │  last must already exist.
│ │ │ │  i1 : p = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11612-0/0/
│ │ │ │ +o1 = /tmp/M2-12402-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11612-0/0/foo
│ │ │ │ +o4 = /tmp/M2-12402-0/0/foo
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : removeFile fn
│ │ │ │  i7 : removeDirectory p
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_move__File_lp__String_cm__String_rp.html
│ │ │ @@ -86,52 +86,52 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,32 +20,32 @@
│ │ │ │            o the name of the backup file if one was created, or _n_u_l_l
│ │ │ │      * Consequences:
│ │ │ │            o the file will be moved by creating a new link to the file and
│ │ │ │              removing the old one
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11448-0/0
│ │ │ │ +o1 = /tmp/M2-12077-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11448-0/1
│ │ │ │ +o2 = /tmp/M2-12077-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11448-0/0
│ │ │ │ +o3 = /tmp/M2-12077-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-11448-0/0 -> /tmp/M2-11448-0/1
│ │ │ │ +--moving: /tmp/M2-12077-0/0 -> /tmp/M2-12077-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-11448-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-12077-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11448-0/1.bak
│ │ │ │ +o6 = /tmp/M2-12077-0/1.bak
│ │ │ │  i7 : removeFile bak
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_F_i_l_e
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * moveFile(String)
│ │ │ │      * _m_o_v_e_F_i_l_e_(_S_t_r_i_n_g_,_S_t_r_i_n_g_)
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_nanosleep.html
│ │ │ @@ -56,15 +56,15 @@
│ │ │        
    nanosleep -- sleep for a given number of nanoseconds
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  nanosleep n -- sleeps for n nanoseconds.        
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11448-0/0
    
│ │ │ +o1 = /tmp/M2-12077-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11448-0/1
    
│ │ │ +o2 = /tmp/M2-12077-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11448-0/0
│ │ │ +o3 = /tmp/M2-12077-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-11448-0/0 -> /tmp/M2-11448-0/1
    
│ │ │ +--moving: /tmp/M2-12077-0/0 -> /tmp/M2-12077-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-11448-0/1.bak
│ │ │ +--backup file created: /tmp/M2-12077-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-11448-0/1.bak
    
│ │ │ +o6 = /tmp/M2-12077-0/1.bak
│ │ │              
    
│ │ │                
    i7 : removeFile bak
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500147s elapsed
│ │ │ + -- .500202s elapsed
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -4,15 +4,15 @@
│ │ │ │  [q                   ]
│ │ │ │  _n_e_x_t | _p_r_e_v_i_o_u_s | _f_o_r_w_a_r_d | _b_a_c_k_w_a_r_d | _u_p | _i_n_d_e_x | _t_o_c
│ │ │ │  ===============================================================================
│ │ │ │  ************ nnaannoosslleeeepp ---- sslleeeepp ffoorr aa ggiivveenn nnuummbbeerr ooff nnaannoosseeccoonnddss ************
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  nanosleep n -- sleeps for n nanoseconds.
│ │ │ │  i1 : elapsedTime nanosleep 500000000
│ │ │ │ - -- .500147s elapsed
│ │ │ │ + -- .500202s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_l_e_e_p -- sleep for a while
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _n_a_n_o_s_l_e_e_p is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_options_lp__Function_rp.html
│ │ │ @@ -144,26 +144,26 @@
│ │ │              
    		
│ │ │                
    i6 : methods intersect
│ │ │  
│ │ │  o6 = {0 => (intersect, List)                          }
│ │ │       {1 => (intersect, Module)                        }
│ │ │       {2 => (intersect, Matrix, Matrix, Matrix, Matrix)}
│ │ │       {3 => (intersect, RRi, RRi)                      }
│ │ │ -     {4 => (intersect, Matrix, Matrix)                }
│ │ │ -     {5 => (intersect, Set, Set)                      }
│ │ │ -     {6 => (intersect, Set)                           }
│ │ │ -     {7 => (intersect, Polyhedron, Cone)              }
│ │ │ -     {8 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {4 => (intersect, Set, Set)                      }
│ │ │ +     {5 => (intersect, Matrix, Matrix)                }
│ │ │ +     {6 => (intersect, Polyhedron, Cone)              }
│ │ │ +     {7 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {8 => (intersect, Set)                           }
│ │ │       {9 => (intersect, Ideal)                         }
│ │ │       {10 => (intersect, Ideal, Ideal)                 }
│ │ │       {11 => (intersect, CCi)                          }
│ │ │       {12 => (intersect, RRi)                          }
│ │ │ -     {13 => (intersect, Polyhedron, Polyhedron)       }
│ │ │ -     {14 => (intersect, Cone, Cone)                   }
│ │ │ -     {15 => (intersect, Matrix)                       }
│ │ │ +     {13 => (intersect, Matrix)                       }
│ │ │ +     {14 => (intersect, Polyhedron, Polyhedron)       }
│ │ │ +     {15 => (intersect, Cone, Cone)                   }
│ │ │       {16 => (intersect, Module, Module)               }
│ │ │       {17 => (intersect, CCi, CCi)                     }
│ │ │       {18 => (intersect, CCi, RRi)                     }
│ │ │       {19 => (intersect, RRi, CCi)                     }
│ │ │       {20 => (intersect, Sequence)                     }
│ │ │  
│ │ │  o6 : NumberedVerticalList
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,26 +62,26 @@
│ │ │ │  o5 : NumberedVerticalList
│ │ │ │  i6 : methods intersect
│ │ │ │  
│ │ │ │  o6 = {0 => (intersect, List)                          }
│ │ │ │       {1 => (intersect, Module)                        }
│ │ │ │       {2 => (intersect, Matrix, Matrix, Matrix, Matrix)}
│ │ │ │       {3 => (intersect, RRi, RRi)                      }
│ │ │ │ -     {4 => (intersect, Matrix, Matrix)                }
│ │ │ │ -     {5 => (intersect, Set, Set)                      }
│ │ │ │ -     {6 => (intersect, Set)                           }
│ │ │ │ -     {7 => (intersect, Polyhedron, Cone)              }
│ │ │ │ -     {8 => (intersect, Cone, Polyhedron)              }
│ │ │ │ +     {4 => (intersect, Set, Set)                      }
│ │ │ │ +     {5 => (intersect, Matrix, Matrix)                }
│ │ │ │ +     {6 => (intersect, Polyhedron, Cone)              }
│ │ │ │ +     {7 => (intersect, Cone, Polyhedron)              }
│ │ │ │ +     {8 => (intersect, Set)                           }
│ │ │ │       {9 => (intersect, Ideal)                         }
│ │ │ │       {10 => (intersect, Ideal, Ideal)                 }
│ │ │ │       {11 => (intersect, CCi)                          }
│ │ │ │       {12 => (intersect, RRi)                          }
│ │ │ │ -     {13 => (intersect, Polyhedron, Polyhedron)       }
│ │ │ │ -     {14 => (intersect, Cone, Cone)                   }
│ │ │ │ -     {15 => (intersect, Matrix)                       }
│ │ │ │ +     {13 => (intersect, Matrix)                       }
│ │ │ │ +     {14 => (intersect, Polyhedron, Polyhedron)       }
│ │ │ │ +     {15 => (intersect, Cone, Cone)                   }
│ │ │ │       {16 => (intersect, Module, Module)               }
│ │ │ │       {17 => (intersect, CCi, CCi)                     }
│ │ │ │       {18 => (intersect, CCi, RRi)                     }
│ │ │ │       {19 => (intersect, RRi, CCi)                     }
│ │ │ │       {20 => (intersect, Sequence)                     }
│ │ │ │  
│ │ │ │  o6 : NumberedVerticalList
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html
│ │ │ @@ -77,21 +77,21 @@
│ │ │                  		
│ │ │                
    i2 : L = shuffle toList (1..10000);
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .659735s elapsed
    
│ │ │ + -- .629005s elapsed
│ │ │              
    
│ │ │                
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .211134s elapsed
    
│ │ │ + -- .166776s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    You will have to try it on your examples to see how much they speed up.
    
│ │ │            
    Warning: Threads computing in parallel can give wrong answers if their code is not "thread safe", meaning they make modifications to memory without ensuring the modifications get safely communicated to other threads. (Thread safety can slow computations some.) Currently, modifications to Macaulay2 variables and mutable hash tables are thread safe, but not changes inside mutable lists. Also, access to external libraries such as singular, etc., may not currently be thread safe.
    
│ │ │            
    The rest of this document describes how to control parallel tasks more directly.
    
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o5 = 5
    
│ │ │              
    
│ │ │                
    i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
    
│ │ │ +o6 = 17
│ │ │              
    
│ │ │          
    
│ │ │            
    To run a function in another thread use schedule, as in the following example.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -155,15 +155,15 @@
│ │ │            
    Note that schedule returns a task, not the result of the computation, which will be accessible only after the task has completed the computation.
    
│ │ │          
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : t
│ │ │  
│ │ │ -o11 = <<task, running>>
│ │ │ +o11 = <<task, created>>
│ │ │  
│ │ │  o11 : Task
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Use isReady to check whether the result is available yet.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,17 +17,17 @@
│ │ │ │  big computation. If the list is long, it will be split into chunks for each
│ │ │ │  core, reducing the overhead. But the speedup is still limited by the different
│ │ │ │  threads competing for memory, including cpu caches; it is like running
│ │ │ │  Macaulay2 on a computer that is running other big programs at the same time. We
│ │ │ │  can see this using _e_l_a_p_s_e_d_T_i_m_e.
│ │ │ │  i2 : L = shuffle toList (1..10000);
│ │ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ │ - -- .659735s elapsed
│ │ │ │ + -- .629005s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .211134s elapsed
│ │ │ │ + -- .166776s 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
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 1.15969s elapsed
│ │ │ + -- 1.35597s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -165,15 +165,15 @@
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.14895s elapsed
│ │ │ + -- 1.36984s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -195,15 +195,15 @@
│ │ │  
│ │ │  o8 = 1
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.62269s elapsed
│ │ │ + -- 1.47852s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -236,15 +236,15 @@
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.37571s elapsed
│ │ │ + -- 1.74995s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
    
│ │ │ @@ -263,15 +263,15 @@
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │              
    
│ │ │                
    i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 1.51803s elapsed
│ │ │ + -- 1.72232s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
    
│ │ │ @@ -304,15 +304,15 @@
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │              
    
│ │ │                
    i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.09157s elapsed
│ │ │ + -- 3.1381s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -327,15 +327,15 @@
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │              
    
│ │ │                
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.21927s elapsed
│ │ │ + -- 6.88737s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -350,15 +350,15 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │              
    
│ │ │                
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.08357s elapsed
│ │ │ + -- 2.97431s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -401,15 +401,15 @@
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │              
    
│ │ │                
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- .823549s elapsed
│ │ │ + -- .765091s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │                
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.45319s elapsed
│ │ │ + -- 1.14689s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -93,30 +93,30 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i3 : S = ring I
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : elapsedTime minimalBetti I
│ │ │ │ - -- 1.15969s elapsed
│ │ │ │ + -- 1.35597s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │           4: .  .   .   .   .    .   .   .   .  .  1
│ │ │ │  
│ │ │ │  o4 : BettiTally
│ │ │ │  i5 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ │ - -- 1.14895s elapsed
│ │ │ │ + -- 1.36984s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -126,15 +126,15 @@
│ │ │ │  i7 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o7 : Ideal of S
│ │ │ │  i8 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o8 = 1
│ │ │ │  i9 : elapsedTime minimalBetti(I)
│ │ │ │ - -- 1.62269s elapsed
│ │ │ │ + -- 1.47852s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ │ @@ -149,15 +149,15 @@
│ │ │ │  i11 : numTBBThreads = 0
│ │ │ │  
│ │ │ │  o11 = 0
│ │ │ │  i12 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o12 : Ideal of S
│ │ │ │  i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 2.37571s elapsed
│ │ │ │ + -- 1.74995s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -168,15 +168,15 @@
│ │ │ │  i14 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o14 = 1
│ │ │ │  i15 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o15 : Ideal of S
│ │ │ │  i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ │ - -- 1.51803s elapsed
│ │ │ │ + -- 1.72232s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -195,37 +195,37 @@
│ │ │ │  o18 = S
│ │ │ │  
│ │ │ │  o18 : PolynomialRing
│ │ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │ │  
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.09157s elapsed
│ │ │ │ + -- 3.1381s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o20 : Matrix S  <-- S
│ │ │ │  i21 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o21 = 1
│ │ │ │  i22 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o22 : Ideal of S
│ │ │ │  i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 7.21927s elapsed
│ │ │ │ + -- 6.88737s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o23 : Matrix S  <-- S
│ │ │ │  i24 : numTBBThreads = 10
│ │ │ │  
│ │ │ │  o24 = 10
│ │ │ │  i25 : I = ideal I_*;
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.08357s elapsed
│ │ │ │ + -- 2.97431s elapsed
│ │ │ │  
│ │ │ │                1      108
│ │ │ │  o26 : Matrix S  <-- S
│ │ │ │  For Gröbner basis computation in associative algebras, ParallelizeByDegree is
│ │ │ │  not relevant. In this case, use numTBBThreads to control the amount of
│ │ │ │  parallelism.
│ │ │ │  i27 : needsPackage "AssociativeAlgebras"
│ │ │ │ @@ -246,15 +246,15 @@
│ │ │ │                 2                         2                         2
│ │ │ │  o30 = ideal (5a  + 2b*c + 3c*b, 3a*c + 5b  + 2c*a, 2a*b + 3b*a + 5c )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i31 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- .823549s elapsed
│ │ │ │ + -- .765091s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  i32 : I = ideal I_*
│ │ │ │  
│ │ │ │                 2                         2                         2
│ │ │ │ @@ -263,15 +263,15 @@
│ │ │ │                  ZZ
│ │ │ │  o32 : Ideal of ---<|a, b, c|>
│ │ │ │                 101
│ │ │ │  i33 : numTBBThreads = 1
│ │ │ │  
│ │ │ │  o33 = 1
│ │ │ │  i34 : elapsedTime NCGB(I, 22);
│ │ │ │ - -- 1.45319s elapsed
│ │ │ │ + -- 1.14689s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_i_m_a_l_B_e_t_t_i -- minimal betti numbers of (the minimal free resolution of)
│ │ │ │        a homogeneous ideal or module
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_poincare.html
│ │ │ @@ -375,36 +375,36 @@
│ │ │  
│ │ │  o27 = 3
    
│ │ │              
    
│ │ │                
    i28 : time poincare I
│ │ │ - -- used 0.000690775s (cpu); 1.8044e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00113363s (cpu); 1.0099e-05s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 16 at 0x7fad46272c40
│ │ │ +   -- registering gb 16 at 0x7f958a88fc40
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 4186
│ │ │     -- #reduction steps = 38
│ │ │     -- #spairs done = 11
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 29
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0192888s (cpu); 0.0198812s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00686734s (cpu); 0.00988303s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -416,15 +416,15 @@
│ │ │                
    i30 : R = QQ[a..d];
    
│ │ │              
    
│ │ │                
    i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 17 at 0x7fad46272a80
│ │ │ +   -- registering gb 17 at 0x7f958a88fa80
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -433,24 +433,24 @@
│ │ │  o31 : Ideal of R
    
│ │ │              
    
│ │ │                
    i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 18 at 0x7fad462728c0
│ │ │ +   -- registering gb 18 at 0x7f958a88f8c0
│ │ │  
│ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
│ │ │     -- number of monomials                = 267
│ │ │     -- #reduction steps = 236
│ │ │     -- #spairs done = 30
│ │ │     -- ncalls = 10
│ │ │     -- nloop = 20
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0079971s (cpu); 0.00761358s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00398917s (cpu); 0.00373273s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7fad46272700
│ │ │ +   -- registering gb 19 at 0x7f958a88f700
│ │ │  
│ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
│ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
│ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
│ │ │     -- number of monomials                = 1051
│ │ │     -- #reduction steps = 284
│ │ │     -- #spairs done = 53
│ │ │     -- ncalls = 46
│ │ │     -- nloop = 54
│ │ │     -- nsaved = 0
│ │ │ -   --  -- used 0.0839937s (cpu); 0.0833296s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0439806s (cpu); 0.0443454s (thread); 0s (gc)
│ │ │  
│ │ │                1      39
│ │ │  o37 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -177,66 +177,66 @@
│ │ │ │  o26 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o26 : ZZ[T]
│ │ │ │  i27 : gbTrace = 3
│ │ │ │  
│ │ │ │  o27 = 3
│ │ │ │  i28 : time poincare I
│ │ │ │ - -- used 0.000690775s (cpu); 1.8044e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00113363s (cpu); 1.0099e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 16 at 0x7fad46272c40
│ │ │ │ +   -- registering gb 16 at 0x7f958a88fc40
│ │ │ │  
│ │ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number of
│ │ │ │  (nonminimal) gb elements = 11
│ │ │ │     -- number of monomials                = 4186
│ │ │ │     -- #reduction steps = 38
│ │ │ │     -- #spairs done = 11
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 29
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0192888s (cpu); 0.0198812s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00686734s (cpu); 0.00988303s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      11
│ │ │ │  o29 : Matrix R  <-- R
│ │ │ │  In this case, the savings is minimal, but often it can be dramatic. Another
│ │ │ │  important situation is to compute a Gröbner basis using a different monomial
│ │ │ │  order.
│ │ │ │  i30 : R = QQ[a..d];
│ │ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │ │  
│ │ │ │ -   -- registering gb 17 at 0x7fad46272a80
│ │ │ │ +   -- registering gb 17 at 0x7f958a88fa80
│ │ │ │  
│ │ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │ │     -- number of monomials                = 0
│ │ │ │     -- #reduction steps = 0
│ │ │ │     -- #spairs done = 0
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │     -- nsaved = 0
│ │ │ │     --
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time p = poincare I
│ │ │ │  
│ │ │ │ -   -- registering gb 18 at 0x7fad462728c0
│ │ │ │ +   -- registering gb 18 at 0x7f958a88f8c0
│ │ │ │  
│ │ │ │     -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of
│ │ │ │  (nonminimal) gb elements = 11
│ │ │ │     -- number of monomials                = 267
│ │ │ │     -- #reduction steps = 236
│ │ │ │     -- #spairs done = 30
│ │ │ │     -- ncalls = 10
│ │ │ │     -- nloop = 20
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0079971s (cpu); 0.00761358s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00398917s (cpu); 0.00373273s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o32 : ZZ[T]
│ │ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ │  
│ │ │ │ @@ -281,30 +281,30 @@
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : gbTrace = 3
│ │ │ │  
│ │ │ │  o36 = 3
│ │ │ │  i37 : time gens gb J;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7fad46272700
│ │ │ │ +   -- registering gb 19 at 0x7f958a88f700
│ │ │ │  
│ │ │ │     -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}
│ │ │ │  (3,9)m
│ │ │ │     -- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
│ │ │ │     -- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m
│ │ │ │  {24}(1,3)m
│ │ │ │     -- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements
│ │ │ │  = 39
│ │ │ │     -- number of monomials                = 1051
│ │ │ │     -- #reduction steps = 284
│ │ │ │     -- #spairs done = 53
│ │ │ │     -- ncalls = 46
│ │ │ │     -- nloop = 54
│ │ │ │     -- nsaved = 0
│ │ │ │ -   --  -- used 0.0839937s (cpu); 0.0833296s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0439806s (cpu); 0.0443454s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      39
│ │ │ │  o37 : Matrix S  <-- S
│ │ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │ │  
│ │ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_printing_spto_spa_spfile.html
│ │ │ @@ -102,22 +102,22 @@
│ │ │  o2 : File
│ │ │              
    
│ │ │                
    i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11803-0/0
    
│ │ │ +o3 = /tmp/M2-12793-0/0
│ │ │              
    
│ │ │                
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11803-0/0
│ │ │ +o4 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ @@ -156,15 +156,15 @@
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-11803-0/0
│ │ │ +o9 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o9 : File
    
│ │ │              
    
│ │ │                
    i10 : get fn
│ │ │ @@ -174,15 +174,15 @@
│ │ │        + 1
    
│ │ │              
    
│ │ │                
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-11803-0/0
│ │ │ +o11 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o11 : File
    
│ │ │              
    
│ │ │                
    i12 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,18 +36,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11803-0/0
│ │ │ │ +o3 = /tmp/M2-12793-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11803-0/0
│ │ │ │ +o4 = /tmp/M2-12793-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : R = QQ[x]
│ │ │ │  
│ │ │ │ @@ -66,25 +66,25 @@
│ │ │ │   10      9      8       7       6       5       4       3      2
│ │ │ │  x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x + 1
│ │ │ │  o8 = stdio
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : fn << f << close
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-11803-0/0
│ │ │ │ +o9 = /tmp/M2-12793-0/0
│ │ │ │  
│ │ │ │  o9 : File
│ │ │ │  i10 : get fn
│ │ │ │  
│ │ │ │  o10 =  10      9      8       7       6       5       4       3      2
│ │ │ │        x   + 10x  + 45x  + 120x  + 210x  + 252x  + 210x  + 120x  + 45x  + 10x
│ │ │ │        + 1
│ │ │ │  i11 : fn << toExternalString f << close
│ │ │ │  
│ │ │ │ -o11 = /tmp/M2-11803-0/0
│ │ │ │ +o11 = /tmp/M2-12793-0/0
│ │ │ │  
│ │ │ │  o11 : File
│ │ │ │  i12 : get fn
│ │ │ │  
│ │ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │ │        1
│ │ │ │  i13 : value get fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_process__I__D.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : processID()
│ │ │  
│ │ │ -o1 = 11022
    
│ │ │ +o1 = 11242
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,15 +8,15 @@
│ │ │ │      *   Usage:
│ │ │ │              processID()
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the process identifier of the current Macaulay2 process
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : processID()
│ │ │ │  
│ │ │ │ -o1 = 11022
│ │ │ │ +o1 = 11242
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_r_o_u_p_I_D -- the process group identifier
│ │ │ │      * _s_e_t_G_r_o_u_p_I_D -- set the process group identifier
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _p_r_o_c_e_s_s_I_D is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_profile.html
│ │ │ @@ -96,35 +96,35 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o9 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -178,15 +178,15 @@
│ │ │  o14 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.95   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ -     1     92.56   ../../m2/matrix1.m2:275:22-275:43
│ │ │ -     1     44.48   ../../m2/matrix1.m2:183:25-183:52
│ │ │ -     1     31.39   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ -     1     30.23   ../../m2/matrix1.m2:130:10-145:16
│ │ │ -     1     22.58   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ -     1     21.8    ../../m2/matrix1.m2:35:10-39:22  
│ │ │ -     1     21.29   ../../m2/set.m2:129:5-129:61     
│ │ │ -     1     3.45    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ -     1     2.36    ../../m2/matrix1.m2:131:13-131:78
│ │ │ -     1     2.15    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ -     1     1.34    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ -     1     1.21    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ -     1     1.11    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ -     1     1.1     ../../m2/matrix1.m2:172:4-174:74 
│ │ │ -     1     1.09    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ -     20    .52     ../../m2/matrix1.m2:181:14-182:67
│ │ │ -     1     .43     ../../m2/modules.m2:282:4-282:52 
│ │ │ -     20    .36     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ -     1     .0041s  elapsed total                    
    
│ │ │ +     1     93.94   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ +     1     91.36   ../../m2/matrix1.m2:275:22-275:43
│ │ │ +     1     43.45   ../../m2/matrix1.m2:183:25-183:52
│ │ │ +     1     29.88   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ +     1     28.72   ../../m2/matrix1.m2:130:10-145:16
│ │ │ +     1     23.05   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ +     1     21.72   ../../m2/set.m2:129:5-129:61     
│ │ │ +     1     20.21   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ +     1     3.28    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ +     1     2.41    ../../m2/matrix1.m2:131:13-131:78
│ │ │ +     1     2.26    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ +     1     1.53    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ +     1     1.53    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ +     1     1.38    ../../m2/matrix1.m2:137:20-137:64
│ │ │ +     1     1.32    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ +     1     1.13    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ +     1     1.12    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ +     20    .98     ../../m2/matrix1.m2:181:14-182:67
│ │ │ +     20    .82     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ +     20    .69     ../../m2/matrix1.m2:180:17-180:29
│ │ │ +     1     .0030s  elapsed total                    
│ │ │              
    
│ │ │                
    i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,35 +25,35 @@
│ │ │ │                4       5
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  Afterwards, running profileSummary and coverageSummary produces easy to read
│ │ │ │  tables summarizing the accumulated data so far in different ways.
│ │ │ │  i2 : profileSummary
│ │ │ │  
│ │ │ │  o2 = #run  %time   position
│ │ │ │ -     1     94.95   ../../m2/matrix1.m2:273:4-276:58
│ │ │ │ -     1     92.56   ../../m2/matrix1.m2:275:22-275:43
│ │ │ │ -     1     44.48   ../../m2/matrix1.m2:183:25-183:52
│ │ │ │ -     1     31.39   ../../m2/matrix1.m2:104:5-146:72
│ │ │ │ -     1     30.23   ../../m2/matrix1.m2:130:10-145:16
│ │ │ │ -     1     22.58   ../../m2/matrix1.m2:171:4-171:42
│ │ │ │ -     1     21.8    ../../m2/matrix1.m2:35:10-39:22
│ │ │ │ -     1     21.29   ../../m2/set.m2:129:5-129:61
│ │ │ │ -     1     3.45    ../../m2/matrix1.m2:102:5-102:29
│ │ │ │ -     1     2.36    ../../m2/matrix1.m2:131:13-131:78
│ │ │ │ -     1     2.15    ../../m2/matrix1.m2:86:5-99:11
│ │ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ │ -     1     1.34    ../../m2/matrix1.m2:275:7-275:16
│ │ │ │ -     1     1.21    ../../m2/matrix1.m2:270:4-271:73
│ │ │ │ -     1     1.11    ../../m2/matrix1.m2:101:5-101:91
│ │ │ │ -     1     1.1     ../../m2/matrix1.m2:172:4-174:74
│ │ │ │ -     1     1.09    ../../m2/matrix1.m2:88:10-88:46
│ │ │ │ -     20    .52     ../../m2/matrix1.m2:181:14-182:67
│ │ │ │ -     1     .43     ../../m2/modules.m2:282:4-282:52
│ │ │ │ -     20    .36     ../../m2/matrix1.m2:37:43-37:71
│ │ │ │ -     1     .0041s  elapsed total
│ │ │ │ +     1     93.94   ../../m2/matrix1.m2:273:4-276:58
│ │ │ │ +     1     91.36   ../../m2/matrix1.m2:275:22-275:43
│ │ │ │ +     1     43.45   ../../m2/matrix1.m2:183:25-183:52
│ │ │ │ +     1     29.88   ../../m2/matrix1.m2:104:5-146:72
│ │ │ │ +     1     28.72   ../../m2/matrix1.m2:130:10-145:16
│ │ │ │ +     1     23.05   ../../m2/matrix1.m2:171:4-171:42
│ │ │ │ +     1     21.72   ../../m2/set.m2:129:5-129:61
│ │ │ │ +     1     20.21   ../../m2/matrix1.m2:35:10-39:22
│ │ │ │ +     1     3.28    ../../m2/matrix1.m2:102:5-102:29
│ │ │ │ +     1     2.41    ../../m2/matrix1.m2:131:13-131:78
│ │ │ │ +     1     2.26    ../../m2/matrix1.m2:86:5-99:11
│ │ │ │ +     1     1.53    ../../m2/matrix1.m2:101:5-101:91
│ │ │ │ +     1     1.53    ../../m2/matrix1.m2:275:7-275:16
│ │ │ │ +     1     1.38    ../../m2/matrix1.m2:137:20-137:64
│ │ │ │ +     1     1.32    ../../m2/matrix1.m2:270:4-271:73
│ │ │ │ +     1     1.13    ../../m2/matrix1.m2:172:4-174:74
│ │ │ │ +     1     1.12    ../../m2/matrix1.m2:88:10-88:46
│ │ │ │ +     20    .98     ../../m2/matrix1.m2:181:14-182:67
│ │ │ │ +     20    .82     ../../m2/matrix1.m2:37:43-37:71
│ │ │ │ +     20    .69     ../../m2/matrix1.m2:180:17-180:29
│ │ │ │ +     1     .0030s  elapsed total
│ │ │ │  i3 : coverageSummary
│ │ │ │  
│ │ │ │  o3 = covered lines:
│ │ │ │       ../../m2/lists.m2:148:24-148:32
│ │ │ │       ../../m2/lists.m2:148:34-148:58
│ │ │ │       ../../m2/matrix.m2:30:5-30:35
│ │ │ │       ../../m2/matrix.m2:31:5-31:46
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
    
│ │ │                
    i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.468269s (cpu); 0.20272s (thread); 0s (gc)
│ │ │ + -- used 0.42246s (cpu); 0.116427s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │               2      3   1     3   0     3
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i10 : time randomKRationalPoint(I)
│ │ │ - -- used 0.372139s (cpu); 0.315367s (thread); 0s (gc)
│ │ │ + -- used 0.306282s (cpu); 0.194319s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │  
│ │ │  o10 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c->degree c);
│ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ - -- used 5.46061s (cpu); 2.7311s (thread); 0s (gc)
│ │ │ + -- used 4.8567s (cpu); 1.88936s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 58
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  o4 : Ideal of R
│ │ │ │  i5 : codim I, degree I
│ │ │ │  
│ │ │ │  o5 = (2, 10)
│ │ │ │  
│ │ │ │  o5 : Sequence
│ │ │ │  i6 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.468269s (cpu); 0.20272s (thread); 0s (gc)
│ │ │ │ + -- used 0.42246s (cpu); 0.116427s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal (x  - 53x , x  + 8x , x  - 4x )
│ │ │ │               2      3   1     3   0     3
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : R=kk[x_0..x_5];
│ │ │ │  i8 : I=minors(3,random(R^5,R^{3:-1}));
│ │ │ │ @@ -45,15 +45,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : codim I, degree I
│ │ │ │  
│ │ │ │  o9 = (3, 10)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  i10 : time randomKRationalPoint(I)
│ │ │ │ - -- used 0.372139s (cpu); 0.315367s (thread); 0s (gc)
│ │ │ │ + -- used 0.306282s (cpu); 0.194319s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = ideal (x  - 27x , x  - 16x , x  - 9x , x  + 44x , x  - 52x )
│ │ │ │                4      5   3      5   2     5   1      5   0      5
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  The claim that $63 \%$ of the intersections contain a K-rational point can be
│ │ │ │  experimentally tested:
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o13 : RR (of precision 53)
│ │ │ │  i14 : I=ideal random(n,R);
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time (#select(apply(100,i->(degs=apply(decompose(I+ideal random(1,R)),c-
│ │ │ │  >degree c);
│ │ │ │                       #select(degs,d->d==1))),f->f>0))
│ │ │ │ - -- used 5.46061s (cpu); 2.7311s (thread); 0s (gc)
│ │ │ │ + -- used 4.8567s (cpu); 1.88936s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 58
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommKKRRaattiioonnaallPPooiinntt:: **********
│ │ │ │      * randomKRationalPoint(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_read__Directory.html
│ │ │ @@ -73,38 +73,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12474-0/0
    
│ │ │ +o1 = /tmp/M2-14164-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-12474-0/0
    
│ │ │ +o2 = /tmp/M2-14164-0/0
│ │ │              
    
│ │ │                
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-12474-0/0/foo
│ │ │ +o3 = /tmp/M2-14164-0/0/foo
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : readDirectory dir
│ │ │  
│ │ │ -o4 = {., .., foo}
│ │ │ +o4 = {.., ., foo}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : removeFile fn
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,26 +10,26 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Outputs:
│ │ │ │            o a _l_i_s_t, the list of filenames stored in the directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12474-0/0
│ │ │ │ +o1 = /tmp/M2-14164-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12474-0/0
│ │ │ │ +o2 = /tmp/M2-14164-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12474-0/0/foo
│ │ │ │ +o3 = /tmp/M2-14164-0/0/foo
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : readDirectory dir
│ │ │ │  
│ │ │ │ -o4 = {., .., foo}
│ │ │ │ +o4 = {.., ., foo}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : removeFile fn
│ │ │ │  i6 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_m_o_v_e_D_i_r_e_c_t_o_r_y -- remove a directory
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_reading_spfiles.html
│ │ │ @@ -57,22 +57,22 @@
│ │ │        
    
│ │ │  Sometimes a file will contain a single expression whose value you wish to have access to.  For example, it might be a polynomial produced by another program.  The function get can be used to obtain the entire contents of a file as a single string.  We illustrate this here with a file whose name is expression.        
    
│ │ │  First we create the file by writing the desired text to it.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11978-0/0
    
│ │ │ +o1 = /tmp/M2-13148-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3" << endl << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11978-0/0
│ │ │ +o2 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │  Now we get the contents of the file, as a single string.        
│ │ │            
│ │ │ @@ -121,15 +121,15 @@
│ │ │  Often a file will contain code written in the Macaulay2 language. Let's create such a file.        
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11978-0/0
│ │ │ +o7 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │  Now verify that it contains the desired text with get.        
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -8,20 +8,20 @@
│ │ │ │  Sometimes a file will contain a single expression whose value you wish to have
│ │ │ │  access to. For example, it might be a polynomial produced by another program.
│ │ │ │  The function _g_e_t can be used to obtain the entire contents of a file as a
│ │ │ │  single string. We illustrate this here with a file whose name is expression.
│ │ │ │  First we create the file by writing the desired text to it.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11978-0/0
│ │ │ │ +o1 = /tmp/M2-13148-0/0
│ │ │ │  i2 : fn <<
│ │ │ │  "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3"
│ │ │ │  << endl << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11978-0/0
│ │ │ │ +o2 = /tmp/M2-13148-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  Now we get the contents of the file, as a single string.
│ │ │ │  i3 : get fn
│ │ │ │  
│ │ │ │  o3 = z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
│ │ │ │       +8*y^3
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  o6 : Expression of class Product
│ │ │ │  Often a file will contain code written in the Macaulay2 language. Let's create
│ │ │ │  such a file.
│ │ │ │  i7 : fn << "sample = 2^100
│ │ │ │       print sample
│ │ │ │       " << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11978-0/0
│ │ │ │ +o7 = /tmp/M2-13148-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  Now verify that it contains the desired text with _g_e_t.
│ │ │ │  i8 : get fn
│ │ │ │  
│ │ │ │  o8 = sample = 2^100
│ │ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_readlink.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the resolved path to a symbolic link, or null if the file
│ │ │ │              was not a symbolic link.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : p = temporaryFileName ()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12734-0/0
│ │ │ │ +o1 = /tmp/M2-14684-0/0
│ │ │ │  i2 : symlinkFile ("foo", p)
│ │ │ │  i3 : readlink p
│ │ │ │  
│ │ │ │  o3 = foo
│ │ │ │  i4 : removeFile p
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_l_p_a_t_h -- convert a filename to one passing through no symbolic links
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html
│ │ │ @@ -73,57 +73,57 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-12734-0/0
    
│ │ │ +o1 = /tmp/M2-14684-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile ("foo", p)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -135,15 +135,15 @@
│ │ │          
    
│ │ │                
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-11022-0/91-rundir/
    
│ │ │ +o1 = /tmp/M2-11242-0/91-rundir/
│ │ │              
    
│ │ │                
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-12753-0/0
    
│ │ │ +o2 = /tmp/M2-14723-0/0
│ │ │              
    
│ │ │                
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-12753-0/1
    
│ │ │ +o3 = /tmp/M2-14723-0/1
│ │ │              
    
│ │ │                
    i4 : symlinkFile(p,q)
    
│ │ │              
    
│ │ │                
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-12753-0/0
│ │ │ +o5 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
    
│ │ │                
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-12753-0/0
    
│ │ │ +o6 = /tmp/M2-14723-0/0
│ │ │              
    
│ │ │                
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-12753-0/0
    
│ │ │ +o7 = /tmp/M2-14723-0/0
│ │ │              
    
│ │ │                
    i8 : removeFile p
    
│ │ │              
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-11022-0/91-rundir/
    
│ │ │ +o10 = /tmp/M2-11242-0/91-rundir/
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  Every component of the path must exist in the file system and be accessible to the user. Terminal slashes will be dropped.  Warning: under most operating systems, the value returned is an absolute path (one starting at the root of the file system), but under Solaris, this system call may, in certain circumstances, return a relative path when given a relative path.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,39 +12,39 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename, or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, a pathname to fn passing through no symbolic links, and
│ │ │ │              ending with a slash if fn refers to a directory
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : realpath "."
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11022-0/91-rundir/
│ │ │ │ +o1 = /tmp/M2-11242-0/91-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12753-0/0
│ │ │ │ +o2 = /tmp/M2-14723-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12753-0/1
│ │ │ │ +o3 = /tmp/M2-14723-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12753-0/0
│ │ │ │ +o5 = /tmp/M2-14723-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12753-0/0
│ │ │ │ +o6 = /tmp/M2-14723-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12753-0/0
│ │ │ │ +o7 = /tmp/M2-14723-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-11022-0/91-rundir/
│ │ │ │ +o10 = /tmp/M2-11242-0/91-rundir/
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Every component of the path must exist in the file system and be accessible to
│ │ │ │  the user. Terminal slashes will be dropped. Warning: under most operating
│ │ │ │  systems, the value returned is an absolute path (one starting at the root of
│ │ │ │  the file system), but under Solaris, this system call may, in certain
│ │ │ │  circumstances, return a relative path when given a relative path.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_register__Finalizer.html
│ │ │ @@ -81,24 +81,23 @@
│ │ │                  		
│ │ │                
    i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
    
│ │ │              
    
│ │ │                
    i2 : collectGarbage() 
│ │ │ ---finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
    
│ │ │ +--finalization: (1)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (3)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (5)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (6)[0]: -- finalizing sequence #1 --
│ │ │ +--finalization: (7)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (8)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (9)[5]: -- finalizing sequence #6 --
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    Caveat
    
│ │ │  This function should mainly be used for debugging.  Having a large number of finalizers might degrade the performance of the program.  Moreover, registering two or more objects that are members of a circular chain of pointers for finalization will result in a memory leak, with none of the objects in the chain being freed, even if nothing else points to any of them.      
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,24 +14,23 @@
│ │ │ │      * Consequences:
│ │ │ │            o A finalizer is registered with the garbage collector to print a
│ │ │ │              string when that object is collected as garbage
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-
│ │ │ │  - finalizing sequence #"|i|" --"))
│ │ │ │  i2 : collectGarbage()
│ │ │ │ ---finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ │ ---finalization: (4)[8]: -- finalizing sequence #9 --
│ │ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ │ ---finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (1)[2]: -- finalizing sequence #3 --
│ │ │ │ +--finalization: (2)[8]: -- finalizing sequence #9 --
│ │ │ │ +--finalization: (3)[6]: -- finalizing sequence #7 --
│ │ │ │ +--finalization: (4)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (5)[7]: -- finalizing sequence #8 --
│ │ │ │ +--finalization: (6)[0]: -- finalizing sequence #1 --
│ │ │ │ +--finalization: (7)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (8)[3]: -- finalizing sequence #4 --
│ │ │ │ +--finalization: (9)[5]: -- finalizing sequence #6 --
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This function should mainly be used for debugging. Having a large number of
│ │ │ │  finalizers might degrade the performance of the program. Moreover, registering
│ │ │ │  two or more objects that are members of a circular chain of pointers for
│ │ │ │  finalization will result in a memory leak, with none of the objects in the
│ │ │ │  chain being freed, even if nothing else points to any of them.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_remove__Directory.html
│ │ │ @@ -76,29 +76,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11650-0/0
    
│ │ │ +o1 = /tmp/M2-12480-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11650-0/0
    
│ │ │ +o2 = /tmp/M2-12480-0/0
│ │ │              
    
│ │ │                
    i3 : readDirectory dir
│ │ │  
│ │ │ -o3 = {., ..}
│ │ │ +o3 = {.., .}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : removeDirectory dir
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,21 +10,21 @@
│ │ │ │      * Inputs:
│ │ │ │            o dir, a _s_t_r_i_n_g, a filename or path to a directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is removed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11650-0/0
│ │ │ │ +o1 = /tmp/M2-12480-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11650-0/0
│ │ │ │ +o2 = /tmp/M2-12480-0/0
│ │ │ │  i3 : readDirectory dir
│ │ │ │  
│ │ │ │ -o3 = {., ..}
│ │ │ │ +o3 = {.., .}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d_D_i_r_e_c_t_o_r_y -- read the contents of a directory
│ │ │ │      * _m_a_k_e_D_i_r_e_c_t_o_r_y -- make a directory
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__Path.html
│ │ │ @@ -70,22 +70,22 @@
│ │ │          
    Description
    
│ │ │          
    This string may be concatenated with an absolute path to get one understandable by external programs.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11116-0/0
    
│ │ │ +o1 = /tmp/M2-11405-0/0
│ │ │              
    
│ │ │                
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-11116-0/0
    
│ │ │ +o2 = /tmp/M2-11405-0/0
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by external programs, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by external programs.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11116-0/0
│ │ │ │ +o1 = /tmp/M2-11405-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11116-0/0
│ │ │ │ +o2 = /tmp/M2-11405-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_U_R_I
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_P_a_t_h is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2025:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__U__R__I.html
│ │ │ @@ -70,22 +70,22 @@
│ │ │          
      Description
      
│ │ │          
      This string may be concatenated with an absolute path to get one understandable by an external browser.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12417-0/0
      
│ │ │ +o1 = /tmp/M2-14047-0/0
│ │ │              
      
│ │ │                
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-12417-0/0
      
│ │ │ +o2 = file:///tmp/M2-14047-0/0
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a _s_t_r_i_n_g, the path, as seen by an external browser, to the root of
│ │ │ │              the file system seen by Macaulay2
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This string may be concatenated with an absolute path to get one understandable
│ │ │ │  by an external browser.
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12417-0/0
│ │ │ │ +o1 = /tmp/M2-14047-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-12417-0/0
│ │ │ │ +o2 = file:///tmp/M2-14047-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_P_a_t_h
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_U_R_I is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2041:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_saving_sppolynomials_spand_spmatrices_spin_spfiles.html
│ │ │ @@ -95,22 +95,22 @@
│ │ │  o4 : R-module, submodule of R
│ │ │              
    
│ │ │                
    i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-12265-0/0
    
│ │ │ +o5 = /tmp/M2-13735-0/0
│ │ │              
    
│ │ │                
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12265-0/0
│ │ │ +o6 = /tmp/M2-13735-0/0
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : get f
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,18 +28,18 @@
│ │ │ │  
│ │ │ │  o4 = image | x2 x2-y2 xyz7 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o4 : R-module, submodule of R
│ │ │ │  i5 : f = temporaryFileName()
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12265-0/0
│ │ │ │ +o5 = /tmp/M2-13735-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12265-0/0
│ │ │ │ +o6 = /tmp/M2-13735-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : get f
│ │ │ │  
│ │ │ │  o7 = (x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix {{x^2,
│ │ │ │       x^2-y^2, x*y*z^7}},image matrix {{x^2, x^2-y^2, x*y*z^7}})
│ │ │ │  i8 : (p',m',M') = value get f
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_serial__Number.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1367325
    
│ │ │ +o1 = 1367343
│ │ │              
    
│ │ │                
    i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1367327
    
│ │ │ +o2 = 1367345
│ │ │              
    
│ │ │                
    i3 : serialNumber ZZ
│ │ │  
│ │ │  o3 = 1000050
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,18 +10,18 @@
│ │ │ │      * Inputs:
│ │ │ │            o x
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the serial number of x
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : serialNumber asdf
│ │ │ │  
│ │ │ │ -o1 = 1367325
│ │ │ │ +o1 = 1367343
│ │ │ │  i2 : serialNumber foo
│ │ │ │  
│ │ │ │ -o2 = 1367327
│ │ │ │ +o2 = 1367345
│ │ │ │  i3 : serialNumber ZZ
│ │ │ │  
│ │ │ │  o3 = 1000050
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _s_e_r_i_a_l_N_u_m_b_e_r is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_solve.html
│ │ │ @@ -371,21 +371,21 @@
│ │ │                  		
│ │ │                
    i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i30 : time X = solve(A,B);
│ │ │ - -- used 0.000183394s (cpu); 0.000176741s (thread); 0s (gc)
    
│ │ │ + -- used 0.000178311s (cpu); 0.000163156s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000144531s (cpu); 0.000145172s (thread); 0s (gc)
    
│ │ │ + -- used 7.3243e-05s (cpu); 7.2932e-05s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │ @@ -416,21 +416,21 @@
│ │ │              
    		
│ │ │                
    i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i38 : time X = solve(A,B);
│ │ │ - -- used 0.232067s (cpu); 0.232067s (thread); 0s (gc)
    
│ │ │ + -- used 0.119727s (cpu); 0.119729s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.215747s (cpu); 0.215685s (thread); 0s (gc)
    
│ │ │ + -- used 0.12075s (cpu); 0.120758s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i40 : norm(A*X-B)
│ │ │  
│ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,33 +192,33 @@
│ │ │ │  i24 : printingPrecision = 4;
│ │ │ │  i25 : N = 40
│ │ │ │  
│ │ │ │  o25 = 40
│ │ │ │  i26 : A = mutableMatrix(CC_53, N, N); fillMatrix A;
│ │ │ │  i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
│ │ │ │  i30 : time X = solve(A,B);
│ │ │ │ - -- used 0.000183394s (cpu); 0.000176741s (thread); 0s (gc)
│ │ │ │ + -- used 0.000178311s (cpu); 0.000163156s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000144531s (cpu); 0.000145172s (thread); 0s (gc)
│ │ │ │ + -- used 7.3243e-05s (cpu); 7.2932e-05s (thread); 0s (gc)
│ │ │ │  i32 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o32 = 5.111850690840453e-15
│ │ │ │  
│ │ │ │  o32 : RR (of precision 53)
│ │ │ │  Over higher precision RR or CC, these routines will be much slower than the
│ │ │ │  lower precision LAPACK routines.
│ │ │ │  i33 : N = 100
│ │ │ │  
│ │ │ │  o33 = 100
│ │ │ │  i34 : A = mutableMatrix(CC_100, N, N); fillMatrix A;
│ │ │ │  i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
│ │ │ │  i38 : time X = solve(A,B);
│ │ │ │ - -- used 0.232067s (cpu); 0.232067s (thread); 0s (gc)
│ │ │ │ + -- used 0.119727s (cpu); 0.119729s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.215747s (cpu); 0.215685s (thread); 0s (gc)
│ │ │ │ + -- used 0.12075s (cpu); 0.120758s (thread); 0s (gc)
│ │ │ │  i40 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ │  
│ │ │ │  o40 : RR (of precision 100)
│ │ │ │  Giving the option ClosestFit=>true, in the case when the field is RR or CC,
│ │ │ │  uses a least squares algorithm to find a best fit solution.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,93 +85,93 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12037-0/0/
    
│ │ │ +o1 = /tmp/M2-13267-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12037-0/1/
    
│ │ │ +o2 = /tmp/M2-13267-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12037-0/0/a/
    
│ │ │ +o3 = /tmp/M2-13267-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12037-0/0/b/
    
│ │ │ +o4 = /tmp/M2-13267-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12037-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-13267-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12037-0/0/a/f
│ │ │ +o6 = /tmp/M2-13267-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12037-0/0/a/g
│ │ │ +o7 = /tmp/M2-13267-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12037-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13267-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
    
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │              
    
│ │ │                
    i10 : get (dst|"b/c/g")
│ │ │  
│ │ │  o10 = ho there
    
│ │ │              
    
│ │ │                
    i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
    
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │              
    
│ │ │  Now we remove the files and directories we created.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,53 +30,53 @@
│ │ │ │            o The directory tree rooted at src is duplicated by a directory tree
│ │ │ │              rooted at dst. The files in the source tree are represented by
│ │ │ │              relative symbolic links in the destination tree to the original
│ │ │ │              files in the source tree.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12037-0/0/
│ │ │ │ +o1 = /tmp/M2-13267-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12037-0/1/
│ │ │ │ +o2 = /tmp/M2-13267-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12037-0/0/a/
│ │ │ │ +o3 = /tmp/M2-13267-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12037-0/0/b/
│ │ │ │ +o4 = /tmp/M2-13267-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12037-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-13267-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12037-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-13267-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12037-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-13267-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-12037-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-13267-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │ │  i10 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o10 = ho there
│ │ │ │  i11 : symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
│ │ │ │ ---unsymlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o12 = rm
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : scan(reverse findFiles src, rm)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__File.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            o dst, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o a symbolic link at the location in the directory tree specified by
│ │ │ │              dst is created, pointing to src
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12094-0/0
│ │ │ │ +o1 = /tmp/M2-13384-0/0
│ │ │ │  i2 : symlinkFile("qwert", fn)
│ │ │ │  i3 : fileExists fn
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : readlink fn
│ │ │ │  
│ │ │ │  o4 = qwert
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_temporary__File__Name.html
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The file name is so unique that even with various suffixes appended, no collision with existing files will occur.  The files will be removed when the program terminates, unless it terminates as the result of an error.        
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12094-0/0
    
│ │ │ +o1 = /tmp/M2-13384-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile("qwert", fn)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-13097-0/0.tex
    
│ │ │ +o1 = /tmp/M2-15437-0/0.tex
│ │ │              
    
│ │ │                
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-13097-0/1.html
    
│ │ │ +o2 = /tmp/M2-15437-0/1.html
│ │ │              
    
│ │ │          
    This function will work under Unix, and also under Windows if you have a directory on the same drive called /tmp.
    
│ │ │          
    If the name of the temporary file will be given to an external program, it may be necessary to concatenate it with rootPath or rootURI to enable the external program to find the file.
    
│ │ │          
    The temporary file name is derived from the value of the environment variable TMPDIR, if it has one.
    
│ │ │          
    If fork is used, then the parent and child Macaulay2 processes will each remove their own temporary files upon termination, with the parent removing any files created before fork was called.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,18 +11,18 @@
│ │ │ │            o a unique temporary file name.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  The file name is so unique that even with various suffixes appended, no
│ │ │ │  collision with existing files will occur. The files will be removed when the
│ │ │ │  program terminates, unless it terminates as the result of an error.
│ │ │ │  i1 : temporaryFileName () | ".tex"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-13097-0/0.tex
│ │ │ │ +o1 = /tmp/M2-15437-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-13097-0/1.html
│ │ │ │ +o2 = /tmp/M2-15437-0/1.html
│ │ │ │  This function will work under Unix, and also under Windows if you have a
│ │ │ │  directory on the same drive called /tmp.
│ │ │ │  If the name of the temporary file will be given to an external program, it may
│ │ │ │  be necessary to concatenate it with _r_o_o_t_P_a_t_h or _r_o_o_t_U_R_I to enable the external
│ │ │ │  program to find the file.
│ │ │ │  The temporary file name is derived from the value of the environment variable
│ │ │ │  TMPDIR, if it has one.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_time.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value of e.  The time used by the the current thread and garbage collection during the evaluation of e is also shown.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time 3^30
│ │ │ - -- used 1.1622e-05s (cpu); 6.291e-06s (thread); 0s (gc)
│ │ │ + -- used 1.5377e-05s (cpu); 4.86e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = 205891132094649
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,15 +7,15 @@
│ │ │ │      *   Usage:
│ │ │ │              time e
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value
│ │ │ │  of e. The time used by the the current thread and garbage collection during the
│ │ │ │  evaluation of e is also shown.
│ │ │ │  i1 : time 3^30
│ │ │ │ - -- used 1.1622e-05s (cpu); 6.291e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.5377e-05s (cpu); 4.86e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_timing.html
│ │ │ @@ -59,24 +59,24 @@
│ │ │  timing e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of cpu timing used, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000013355 seconds
│ │ │ +     -- .000014771 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000013355, 205891132094649}
    
│ │ │ +o2 = Time{.000014771, 205891132094649}
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -10,20 +10,20 @@
│ │ │ │  is the number of seconds of cpu timing used, and v is the value of the
│ │ │ │  expression.
│ │ │ │  The default method for printing such timing results is to display the timing
│ │ │ │  separately in a comment below the computed value.
│ │ │ │  i1 : timing 3^30
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │ -     -- .000013355 seconds
│ │ │ │ +     -- .000014771 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000013355, 205891132094649}
│ │ │ │ +o2 = Time{.000014771, 205891132094649}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _T_i_m_e -- the class of all timing results
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o3 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime Mgb = gb M
│ │ │ - -- 1.4495s elapsed
│ │ │ + -- 1.64788s elapsed
│ │ │  
│ │ │  o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o4 : GroebnerBasis
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .415018s elapsed
│ │ │ + -- .475007s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o6 : GroebnerBasis
    
│ │ │              
    
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o10 : Ideal of R
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .517879s elapsed
│ │ │ + -- .551991s elapsed
│ │ │  
│ │ │  o11 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o11 : GroebnerBasis
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,24 +29,24 @@
│ │ │ │  i3 : hf = poincare ideal (a^4,b^5,c^5,d^6)
│ │ │ │  
│ │ │ │            4     5    6     9     10     11    14     15    16    20
│ │ │ │  o3 = 1 - T  - 2T  - T  + 2T  + 2T   + 2T   - T   - 2T   - T   + T
│ │ │ │  
│ │ │ │  o3 : ZZ[T]
│ │ │ │  i4 : elapsedTime Mgb = gb M
│ │ │ │ - -- 1.4495s elapsed
│ │ │ │ + -- 1.64788s elapsed
│ │ │ │  
│ │ │ │  o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │ │  
│ │ │ │  o4 : GroebnerBasis
│ │ │ │  i5 : M = ideal M_*;
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ │ - -- .415018s elapsed
│ │ │ │ + -- .475007s elapsed
│ │ │ │  
│ │ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │ │  
│ │ │ │  o6 : GroebnerBasis
│ │ │ │  However, obtaining the Hilbert function is not always easy to provide in this
│ │ │ │  way. In this case, one must work with the _d_e_g_r_e_e_s_R_i_n_g of the ring in question.
│ │ │ │  i7 : S = degreesRing R
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  o9 = 1 - T  - 2T  - T  + 2T  + 2T   + 2T   - T   - 2T   - T   + T
│ │ │ │  
│ │ │ │  o9 : S
│ │ │ │  i10 : M = ideal M_*;
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ │ - -- .517879s elapsed
│ │ │ │ + -- .551991s elapsed
│ │ │ │  
│ │ │ │  o11 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │ │  
│ │ │ │  o11 : GroebnerBasis
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_a_n_U_s_e_H_i_l_b_e_r_t_H_i_n_t -- whether certain Groebner computations can make use
│ │ │ │        of the Hilbert function
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_version.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │                 "memtailor version" => 1.4
│ │ │                 "mpfi version" => 1.5.5
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.1
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.113+deb13-amd64
│ │ │ +               "operating system release" => 6.12.100+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism DirectSummands CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC SimplicialModules MatrixFactorizations PathSignatures MacaulayPosets MRDI EliminationTemplates WittVectors Padic WeierstrassSemigroups ResultantComplexes EuclideanDistanceDegree NeuralIdeals TestAudit LanguageServer
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.14.9
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2023.1
│ │ │ ├── html2text {}
│ │ │ │ @@ -67,15 +67,15 @@
│ │ │ │                 "memtailor version" => 1.4
│ │ │ │                 "mpfi version" => 1.5.5
│ │ │ │                 "mpfr version" => 4.2.2
│ │ │ │                 "mpsolve version" => 3.2.2
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.11.1
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.12.113+deb13-amd64
│ │ │ │ +               "operating system release" => 6.12.100+deb13-cloud-amd64
│ │ │ │                 "operating system" => Linux
│ │ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic
│ │ │ │  Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition
│ │ │ │  FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato
│ │ │ │  ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure
│ │ │ │  HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra
│ │ │ │  Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes
│ │ ├── ./usr/share/doc/Macaulay2/MacaulayPosets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  UG9zZXRNYXA=
│ │ │  #:len=820
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNsYXNzIG9mIGFsbCBtb3JwaGlz
│ │ │  bXMgYmV0d2VlbiBwb3NldHMiLCAibGluZW51bSIgPT4gMTYyNCwgU2VlQWxzbyA9PiBESVZ7SEVB
│ │ ├── ./usr/share/doc/Macaulay2/MapleInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  c3RvcmU=
│ │ │  #:len=651
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU3RvcmUgcmVzdWx0IG9mIGEgTWFwbGUg
│ │ │  Y29tcHV0YXRpb24gaW4gYSBmaWxlLiIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsi
│ │ ├── ./usr/share/doc/Macaulay2/Markov/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  Z2F1c3NJZGVhbChSaW5nLExpc3Qp
│ │ │  #:len=239
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjE4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnYXVzc0lkZWFsLFJpbmcsTGlzdCksImdhdXNzSWRl
│ │ ├── ./usr/share/doc/Macaulay2/Markov/example-output/___Markov.out
│ │ │ @@ -70,15 +70,15 @@
│ │ │       |   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2|   1,2,2,2 2,2,2,1    1,2,2,1 2,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │       |- p       p        + p       p       |- p       p        + p       p       |
│ │ │       |   1,1,2,1 1,2,1,1    1,1,1,1 1,2,2,1|   1,1,2,2 1,2,1,2    1,1,1,2 1,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │  
│ │ │  i8 : time netList primaryDecomposition J
│ │ │ - -- used 1.78876s (cpu); 1.24178s (thread); 0s (gc)
│ │ │ + -- used 1.66383s (cpu); 1.18152s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ ├── ./usr/share/doc/Macaulay2/Markov/html/index.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │          
    
│ │ │            
    This ideal has 5 primary components.  The first is the one that has statistical significance. The significance of the other components is still poorly understood.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i8 : time netList primaryDecomposition J
│ │ │ - -- used 1.78876s (cpu); 1.24178s (thread); 0s (gc)
│ │ │ + -- used 1.66383s (cpu); 1.18152s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,15 +102,15 @@
│ │ │ │  1,2,2,2|
│ │ │ │       +-------------------------------------+-----------------------------------
│ │ │ │  --+
│ │ │ │  This ideal has 5 primary components. The first is the one that has statistical
│ │ │ │  significance. The significance of the other components is still poorly
│ │ │ │  understood.
│ │ │ │  i8 : time netList primaryDecomposition J
│ │ │ │ - -- used 1.78876s (cpu); 1.24178s (thread); 0s (gc)
│ │ │ │ + -- used 1.66383s (cpu); 1.18152s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MatchingFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  YWxnZWJyYWljTWF0cm9pZENpcmN1aXRz
│ │ │  #:len=1461
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIGJhc2VzIG9mIHRoZSBhbGdlYnJh
│ │ │  aWMgbWF0cm9pZCIsICJsaW5lbnVtIiA9PiAxOTI3LCBJbnB1dHMgPT4ge1NQQU57VFR7IkwifSwi
│ │ ├── ./usr/share/doc/Macaulay2/MatrixFactorizations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  SG9tKFpaZEZhY3Rvcml6YXRpb25NYXAsWlpkRmFjdG9yaXphdGlvbik=
│ │ │  #:len=413
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzMzMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoSG9tLFpaZEZhY3Rvcml6YXRpb25NYXAsWlpkRmFj
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  aXNBU01VbmlvbihMaXN0KQ==
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzMywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNBU01VbmlvbixMaXN0KSwiaXNBU01VbmlvbihM
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_sp__A__S__M_spvarieties.out
│ │ │ @@ -212,17 +212,17 @@
│ │ │        | 1 -1 1 |
│ │ │        | 0 1  0 |
│ │ │  
│ │ │                 3       3
│ │ │  o22 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i23 : time schubertRegularity B
│ │ │ - -- used 0.118965s (cpu); 0.0492731s (thread); 0s (gc)
│ │ │ + -- used 0.0884852s (cpu); 0.0348172s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0269913s (cpu); 0.026995s (thread); 0s (gc)
│ │ │ + -- used 0.0164596s (cpu); 0.0164661s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/___Investigating_spmatrix_sp__Schubert_spvarieties.out
│ │ │ @@ -178,17 +178,17 @@
│ │ │        z   z   z   , z   z   z    - z   z   , z   z   z    - z   z   )
│ │ │         1,2 1,3 2,4   1,2 1,4 2,2    1,2 2,4   1,2 1,3 2,2    1,2 2,3
│ │ │  
│ │ │  o15 : Ideal of QQ[z   ..z   ]
│ │ │                     1,1   5,5
│ │ │  
│ │ │  i16 : time schubertRegularity p
│ │ │ - -- used 0.000305362s (cpu); 0.000300744s (thread); 0s (gc)
│ │ │ + -- used 0.000443582s (cpu); 0.000441936s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 : time regularity comodule I
│ │ │ - -- used 0.0193429s (cpu); 0.0193488s (thread); 0s (gc)
│ │ │ + -- used 0.0191683s (cpu); 0.0191513s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
│ │ │  
│ │ │  i18 :
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/example-output/_grothendieck__Polynomial.out
│ │ │ @@ -3,25 +3,25 @@
│ │ │  i1 : w = {2,1,4,3}
│ │ │  
│ │ │  o1 = {2, 1, 4, 3}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.0048739s (cpu); 0.00487285s (thread); 0s (gc)
│ │ │ + -- used 0.00387792s (cpu); 0.0038782s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
│ │ │  
│ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.00246725s (cpu); 0.00245375s (thread); 0s (gc)
│ │ │ + -- used 0.00211415s (cpu); 0.00211396s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_sp__A__S__M_spvarieties.html
│ │ │ @@ -388,23 +388,23 @@
│ │ │          
    
│ │ │            
    Additionally, this package facilitates investigating homological invariants of ASM ideals such as regularity (schubertRegularity) and codimension (schubertCodim). efficiently by computing the associated invariants for their antidiagonal initial ideals, which are known to be squarefree by [Wei17]. Therefore the extremal Betti numbers (which encode regularity, depth, and projective dimension) of ASM ideals coincide with those of their antidiagonal initial ideals by [CV20].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i23 : time schubertRegularity B
│ │ │ - -- used 0.118965s (cpu); 0.0492731s (thread); 0s (gc)
│ │ │ + -- used 0.0884852s (cpu); 0.0348172s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
    
│ │ │              
    
│ │ │                
    i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0269913s (cpu); 0.026995s (thread); 0s (gc)
│ │ │ + -- used 0.0164596s (cpu); 0.0164661s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = 1
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Functions for investigating ASM varieties
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -244,19 +244,19 @@
│ │ │ │  ASM ideals such as regularity (_s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y) and codimension
│ │ │ │  (_s_c_h_u_b_e_r_t_C_o_d_i_m). efficiently by computing the associated invariants for their
│ │ │ │  antidiagonal initial ideals, which are known to be squarefree by [Wei17].
│ │ │ │  Therefore the extremal Betti numbers (which encode regularity, depth, and
│ │ │ │  projective dimension) of ASM ideals coincide with those of their antidiagonal
│ │ │ │  initial ideals by [CV20].
│ │ │ │  i23 : time schubertRegularity B
│ │ │ │ - -- used 0.118965s (cpu); 0.0492731s (thread); 0s (gc)
│ │ │ │ + -- used 0.0884852s (cpu); 0.0348172s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.0269913s (cpu); 0.026995s (thread); 0s (gc)
│ │ │ │ + -- used 0.0164596s (cpu); 0.0164661s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = 1
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg AASSMM vvaarriieettiieess **********
│ │ │ │      * _i_s_P_a_r_t_i_a_l_A_S_M_(_M_a_t_r_i_x_) -- whether a matrix is a partial alternating sign
│ │ │ │        matrix
│ │ │ │      * _p_a_r_t_i_a_l_A_S_M_T_o_A_S_M_(_M_a_t_r_i_x_) -- extend a partial alternating sign matrix to an
│ │ │ │        alternating sign matrix
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_spmatrix_sp__Schubert_spvarieties.html
│ │ │ @@ -320,23 +320,23 @@
│ │ │          
    
│ │ │            
    Finally, this package contains functions for investigating homological invariants of matrix Schubert varieties efficiently through combinatorial algorithms produced in [PSW24] via schubertRegularityschubertCodim.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i16 : time schubertRegularity p
│ │ │ - -- used 0.000305362s (cpu); 0.000300744s (thread); 0s (gc)
│ │ │ + -- used 0.000443582s (cpu); 0.000441936s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
    
│ │ │              
    
│ │ │                
    i17 : time regularity comodule I
│ │ │ - -- used 0.0193429s (cpu); 0.0193488s (thread); 0s (gc)
│ │ │ + -- used 0.0191683s (cpu); 0.0191513s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 5
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Functions for investigating matrix Schubert varieties
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -545,19 +545,19 @@
│ │ │ │  
│ │ │ │  o15 : Ideal of QQ[z   ..z   ]
│ │ │ │                     1,1   5,5
│ │ │ │  Finally, this package contains functions for investigating homological
│ │ │ │  invariants of matrix Schubert varieties efficiently through combinatorial
│ │ │ │  algorithms produced in [PSW24] via _s_c_h_u_b_e_r_t_R_e_g_u_l_a_r_i_t_y_s_c_h_u_b_e_r_t_C_o_d_i_m.
│ │ │ │  i16 : time schubertRegularity p
│ │ │ │ - -- used 0.000305362s (cpu); 0.000300744s (thread); 0s (gc)
│ │ │ │ + -- used 0.000443582s (cpu); 0.000441936s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  i17 : time regularity comodule I
│ │ │ │ - -- used 0.0193429s (cpu); 0.0193488s (thread); 0s (gc)
│ │ │ │ + -- used 0.0191683s (cpu); 0.0191513s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 5
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg mmaattrriixx SScchhuubbeerrtt vvaarriieettiieess **********
│ │ │ │      * _a_n_t_i_D_i_a_g_I_n_i_t_(_L_i_s_t_) -- compute the (unique) antidiagonal initial ideal of
│ │ │ │        an ASM ideal
│ │ │ │      * _r_a_n_k_T_a_b_l_e_(_L_i_s_t_) -- compute a table of rank conditions that determines the
│ │ │ │        corresponding ASM or matrix Schubert variety
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html
│ │ │ @@ -85,28 +85,28 @@
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │                
    i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.0048739s (cpu); 0.00487285s (thread); 0s (gc)
│ │ │ + -- used 0.00387792s (cpu); 0.0038782s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o2 : QQ[x ..x ]
│ │ │           1   4
    
│ │ │              
    
│ │ │                
    i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ - -- used 0.00246725s (cpu); 0.00245375s (thread); 0s (gc)
│ │ │ + -- used 0.00211415s (cpu); 0.00211396s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,24 +19,24 @@
│ │ │ │  PipeDream.
│ │ │ │  i1 : w = {2,1,4,3}
│ │ │ │  
│ │ │ │  o1 = {2, 1, 4, 3}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time grothendieckPolynomial w
│ │ │ │ - -- used 0.0048739s (cpu); 0.00487285s (thread); 0s (gc)
│ │ │ │ + -- used 0.00387792s (cpu); 0.0038782s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o2 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o2 : QQ[x ..x ]
│ │ │ │           1   4
│ │ │ │  i3 : time grothendieckPolynomial (w,Algorithm=>"PipeDream")
│ │ │ │ - -- used 0.00246725s (cpu); 0.00245375s (thread); 0s (gc)
│ │ │ │ + -- used 0.00211415s (cpu); 0.00211396s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2        2      2               2
│ │ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │ │  
│ │ │ │  o3 : QQ[x ..x ]
│ │ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  ZmxhdHMoTWF0cm9pZCxaWixTdHJpbmcp
│ │ │  #:len=255
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzOSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxhdHMsTWF0cm9pZCxaWixTdHJpbmcpLCJmbGF0
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid.out
│ │ │ @@ -51,20 +51,20 @@
│ │ │  i9 : keys R10.cache
│ │ │  
│ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time isWellDefined R10
│ │ │ - -- used 0.0514784s (cpu); 0.0514772s (thread); 0s (gc)
│ │ │ + -- used 0.0538582s (cpu); 0.0538596s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.0421595s (cpu); 0.0421627s (thread); 0s (gc)
│ │ │ + -- used 0.0445116s (cpu); 0.0445173s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -76,15 +76,15 @@
│ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │        -----------------------------------------------------------------------
│ │ │        groundSet, dual, storedRepresentation}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : time fVector R10
│ │ │ - -- used 0.00021336s (cpu); 0.000213109s (thread); 0s (gc)
│ │ │ + -- used 0.000280983s (cpu); 0.000278407s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_all__Minors.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │  
│ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │  
│ │ │  o2 : Matroid
│ │ │  
│ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0437144s elapsed
│ │ │ + -- .0410016s elapsed
│ │ │  
│ │ │  i4 : #L
│ │ │  
│ │ │  o4 = 64
│ │ │  
│ │ │  i5 : netList L_{0..4}
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_get__Isos.out
│ │ │ @@ -33,14 +33,14 @@
│ │ │  i6 : F7 = specificMatroid "fano"
│ │ │  
│ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │  
│ │ │  o6 : Matroid
│ │ │  
│ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.0390737s (cpu); 0.0390572s (thread); 0s (gc)
│ │ │ + -- used 0.0368011s (cpu); 0.0368016s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : #autF7
│ │ │  
│ │ │  o8 = 168
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out
│ │ │ @@ -9,12 +9,12 @@
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : time hasMinor(M6, M5)
│ │ │ - -- used 2.4626s (cpu); 1.80647s (thread); 0s (gc)
│ │ │ + -- used 1.7926s (cpu); 1.14424s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_isomorphism_lp__Matroid_cm__Matroid_rp.out
│ │ │ @@ -19,15 +19,15 @@
│ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │  
│ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │  
│ │ │  o4 : Matroid
│ │ │  
│ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0170407s (cpu); 0.017038s (thread); 0s (gc)
│ │ │ + -- used 0.0158704s (cpu); 0.0158695s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -56,15 +56,15 @@
│ │ │  i7 : N = relabel M6
│ │ │  
│ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │  
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 3.84691s (cpu); 3.25885s (thread); 0s (gc)
│ │ │ + -- used 2.61257s (cpu); 2.25682s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_quick__Isomorphism__Test.out
│ │ │ @@ -37,15 +37,15 @@
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000721082s (cpu); 0.000717165s (thread); 0s (gc)
│ │ │ + -- used 0.000531086s (cpu); 0.000529562s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
│ │ │  
│ │ │  i11 : value oo === false
│ │ │  
│ │ │  o11 = true
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_set__Representation.out
│ │ │ @@ -35,15 +35,15 @@
│ │ │  i5 : keys M.cache
│ │ │  
│ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime fVector M
│ │ │ - -- .0127952s elapsed
│ │ │ + -- .0120837s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
│ │ │ @@ -153,23 +153,23 @@
│ │ │  
│ │ │  o9 : List
│ │ │              
    
│ │ │                
    i10 : time isWellDefined R10
│ │ │ - -- used 0.0514784s (cpu); 0.0514772s (thread); 0s (gc)
│ │ │ + -- used 0.0538582s (cpu); 0.0538596s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │              
    
│ │ │                
    i11 : time fVector R10
│ │ │ - -- used 0.0421595s (cpu); 0.0421627s (thread); 0s (gc)
│ │ │ + -- used 0.0445116s (cpu); 0.0445173s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -187,15 +187,15 @@
│ │ │  
│ │ │  o12 : List
    
│ │ │              
    
│ │ │                
    i13 : time fVector R10
│ │ │ - -- used 0.00021336s (cpu); 0.000213109s (thread); 0s (gc)
│ │ │ + -- used 0.000280983s (cpu); 0.000278407s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -71,19 +71,19 @@
│ │ │ │  o8 : Matroid
│ │ │ │  i9 : keys R10.cache
│ │ │ │  
│ │ │ │  o9 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o9 : List
│ │ │ │  i10 : time isWellDefined R10
│ │ │ │ - -- used 0.0514784s (cpu); 0.0514772s (thread); 0s (gc)
│ │ │ │ + -- used 0.0538582s (cpu); 0.0538596s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.0421595s (cpu); 0.0421627s (thread); 0s (gc)
│ │ │ │ + -- used 0.0445116s (cpu); 0.0445173s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ │ │ @@ -93,15 +93,15 @@
│ │ │ │  
│ │ │ │  o12 = {hyperplanes, flatsRelationsTable, rankFunction, ideal, ranks, flats,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        groundSet, dual, storedRepresentation}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : time fVector R10
│ │ │ │ - -- used 0.00021336s (cpu); 0.000213109s (thread); 0s (gc)
│ │ │ │ + -- used 0.000280983s (cpu); 0.000278407s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_all__Minors.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : Matroid
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0437144s elapsed
    
│ │ │ + -- .0410016s elapsed
│ │ │              
    
│ │ │                
    i4 : #L
│ │ │  
│ │ │  o4 = 64
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o1 : Matroid
│ │ │ │  i2 : U25 = uniformMatroid(2,5)
│ │ │ │  
│ │ │ │  o2 = a "matroid" of rank 2 on 5 elements
│ │ │ │  
│ │ │ │  o2 : Matroid
│ │ │ │  i3 : elapsedTime L = allMinors(V, U25);
│ │ │ │ - -- .0437144s elapsed
│ │ │ │ + -- .0410016s elapsed
│ │ │ │  i4 : #L
│ │ │ │  
│ │ │ │  o4 = 64
│ │ │ │  i5 : netList L_{0..4}
│ │ │ │  
│ │ │ │       +----------+-------+
│ │ │ │  o5 = |set {5, 3}|set {2}|
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_get__Isos.html
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o6 : Matroid
│ │ │              
    
│ │ │                
    i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.0390737s (cpu); 0.0390572s (thread); 0s (gc)
    
│ │ │ + -- used 0.0368011s (cpu); 0.0368016s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i8 : #autF7
│ │ │  
│ │ │  o8 = 168
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  symmetric group S_7:
│ │ │ │  i6 : F7 = specificMatroid "fano"
│ │ │ │  
│ │ │ │  o6 = a "matroid" of rank 3 on 7 elements
│ │ │ │  
│ │ │ │  o6 : Matroid
│ │ │ │  i7 : time autF7 = getIsos(F7, F7);
│ │ │ │ - -- used 0.0390737s (cpu); 0.0390572s (thread); 0s (gc)
│ │ │ │ + -- used 0.0368011s (cpu); 0.0368016s (thread); 0s (gc)
│ │ │ │  i8 : #autF7
│ │ │ │  
│ │ │ │  o8 = 168
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ │ │        isomorphic matroids
│ │ │ │      * _q_u_i_c_k_I_s_o_m_o_r_p_h_i_s_m_T_e_s_t -- quick checks for isomorphism between matroids
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_has__Minor.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o2 = false
│ │ │              
    
│ │ │                
    i3 : time hasMinor(M6, M5)
│ │ │ - -- used 2.4626s (cpu); 1.80647s (thread); 0s (gc)
│ │ │ + -- used 1.7926s (cpu); 1.14424s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │       elements, a "matroid" of rank 5 on 15 elements)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : time hasMinor(M6, M5)
│ │ │ │ - -- used 2.4626s (cpu); 1.80647s (thread); 0s (gc)
│ │ │ │ + -- used 1.7926s (cpu); 1.14424s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r -- minor of matroid
│ │ │ │      * _i_s_B_i_n_a_r_y -- whether a matroid is representable over F_2
│ │ │ │  ********** WWaayyss ttoo uussee hhaassMMiinnoorr:: **********
│ │ │ │      * hasMinor(Matroid,Matroid)
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_isomorphism_lp__Matroid_cm__Matroid_rp.html
│ │ │ @@ -123,15 +123,15 @@
│ │ │  
│ │ │  o4 : Matroid
    
│ │ │              
    
│ │ │                
    i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0170407s (cpu); 0.017038s (thread); 0s (gc)
│ │ │ + -- used 0.0158704s (cpu); 0.0158695s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -169,15 +169,15 @@
│ │ │  
│ │ │  o7 : Matroid
    
│ │ │              
    
│ │ │                
    i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 3.84691s (cpu); 3.25885s (thread); 0s (gc)
│ │ │ + -- used 2.61257s (cpu); 2.25682s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HashTable{0 => 11 }
│ │ │                 1 => 0
│ │ │                 2 => 1
│ │ │                 3 => 6
│ │ │                 4 => 9
│ │ │                 5 => 8
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : minorM6 = minor(M6, set{8}, set{4,5,6,7})
│ │ │ │  
│ │ │ │  o4 = a "matroid" of rank 4 on 10 elements
│ │ │ │  
│ │ │ │  o4 : Matroid
│ │ │ │  i5 : time isomorphism(M5, minorM6)
│ │ │ │ - -- used 0.0170407s (cpu); 0.017038s (thread); 0s (gc)
│ │ │ │ + -- used 0.0158704s (cpu); 0.0158695s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{0 => 1}
│ │ │ │                 1 => 0
│ │ │ │                 2 => 3
│ │ │ │                 3 => 2
│ │ │ │                 4 => 6
│ │ │ │                 5 => 5
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  o6 : HashTable
│ │ │ │  i7 : N = relabel M6
│ │ │ │  
│ │ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ │ - -- used 3.84691s (cpu); 3.25885s (thread); 0s (gc)
│ │ │ │ + -- used 2.61257s (cpu); 2.25682s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HashTable{0 => 11 }
│ │ │ │                 1 => 0
│ │ │ │                 2 => 1
│ │ │ │                 3 => 6
│ │ │ │                 4 => 9
│ │ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_quick__Isomorphism__Test.html
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000721082s (cpu); 0.000717165s (thread); 0s (gc)
│ │ │ + -- used 0.000531086s (cpu); 0.000529562s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = false
    
│ │ │              
    
│ │ │                
    i11 : value oo === false
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,15 +51,15 @@
│ │ │ │  o7 = a "matroid" of rank 7 on 11 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : R = ZZ[x,y]; tuttePolynomial(M0, R) == tuttePolynomial(M1, R)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time quickIsomorphismTest(M0, M1)
│ │ │ │ - -- used 0.000721082s (cpu); 0.000717165s (thread); 0s (gc)
│ │ │ │ + -- used 0.000531086s (cpu); 0.000529562s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : value oo === false
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_set__Representation.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime fVector M
│ │ │ - -- .0127952s elapsed
│ │ │ + -- .0120837s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -48,15 +48,15 @@
│ │ │ │  o4 : Matrix QQ  <-- QQ
│ │ │ │  i5 : keys M.cache
│ │ │ │  
│ │ │ │  o5 = {groundSet, rankFunction, storedRepresentation}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : elapsedTime fVector M
│ │ │ │ - -- .0127952s elapsed
│ │ │ │ + -- .0120837s elapsed
│ │ │ │  
│ │ │ │  o6 = HashTable{0 => 1 }
│ │ │ │                 1 => 6
│ │ │ │                 2 => 15
│ │ │ │                 3 => 20
│ │ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/MergeTeX/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  bWVyZ2VUZVgoLi4uLFBhdGg9Pi4uLik=
│ │ │  #:len=236
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1ttZXJnZVRlWCxQYXRoXSwibWVyZ2VUZVgoLi4uLFBh
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  ZGVjb21wb3NlKElkZWFsLFN0cmF0ZWd5PT4uLi4p
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzA4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1soZGVjb21wb3NlLElkZWFsKSxTdHJhdGVneV0sImRl
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/___Hybrid.out
│ │ │ @@ -5,16 +5,16 @@
│ │ │  i2 : R = ZZ/101[w..z];
│ │ │  
│ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .00103587)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0131375)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000340419)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024566)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .000967235)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0320218)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000343505)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000016715)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000630212
│ │ │ - -- .0384912s elapsed
│ │ │ + --  Time taken : .000756921
│ │ │ + -- .0252913s elapsed
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical.out
│ │ │ @@ -30,21 +30,21 @@
│ │ │  
│ │ │               2        2   3     2
│ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000442257s elapsed
│ │ │ + -- .00055725s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0121996s elapsed
│ │ │ + -- .0145965s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical__Containment.out
│ │ │ @@ -29,22 +29,22 @@
│ │ │  o5 = 840
│ │ │  
│ │ │  i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .151496s elapsed
│ │ │ + -- .0799346s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00157389s elapsed
│ │ │ + -- .00190731s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00123922s elapsed
│ │ │ + -- .00131179s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -77,21 +77,21 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .00103587)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0131375)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000340419)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024566)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .000967235)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0320218)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000343505)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000016715)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000630212
│ │ │ - -- .0384912s elapsed
    
│ │ │ + --  Time taken : .000756921
│ │ │ + -- .0252913s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,23 +11,24 @@
│ │ │ │  i1 : debug MinimalPrimes
│ │ │ │  i2 : R = ZZ/101[w..z];
│ │ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid
│ │ │ │  {Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ │ -  Strategy: Linear            (time .00103587)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Birational        (time .0131375)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Factorization     (time .000340419)  #primes = 0 #prunedViaCodim =
│ │ │ │ +  Strategy: Linear            (time .000967235)  #primes = 0 #prunedViaCodim =
│ │ │ │  0
│ │ │ │ -  Strategy: DecomposeMonomials(time .000024566)  #primes = 1 #prunedViaCodim =
│ │ │ │ +  Strategy: Birational        (time .0320218)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Factorization     (time .000343505)  #primes = 0 #prunedViaCodim =
│ │ │ │ +0
│ │ │ │ +  Strategy: DecomposeMonomials(time .000016715)  #primes = 1 #prunedViaCodim =
│ │ │ │  0
│ │ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ │ - --  Time taken : .000630212
│ │ │ │ - -- .0384912s elapsed
│ │ │ │ + --  Time taken : .000756921
│ │ │ │ + -- .0252913s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n_(_._._._,_S_t_r_a_t_e_g_y_=_>_._._._)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _H_y_b_r_i_d is a _s_e_l_f_ _i_n_i_t_i_a_l_i_z_i_n_g_ _t_y_p_e, with ancestor classes _L_i_s_t <
│ │ │ │  _V_i_s_i_b_l_e_L_i_s_t < _B_a_s_i_c_L_i_s_t < _T_h_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html
│ │ │ @@ -136,25 +136,25 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000442257s elapsed
│ │ │ + -- .00055725s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0121996s elapsed
│ │ │ + -- .0145965s elapsed
│ │ │  
│ │ │  o7 = ideal (c, b, a)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -62,21 +62,21 @@
│ │ │ │  i5 : I = intersect(ideal(a^2,b^2,c), ideal(a,b^3,c^2))
│ │ │ │  
│ │ │ │               2        2   3     2
│ │ │ │  o5 = ideal (c , a*c, a , b , a*b )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ │ - -- .000442257s elapsed
│ │ │ │ + -- .00055725s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0121996s elapsed
│ │ │ │ + -- .0145965s elapsed
│ │ │ │  
│ │ │ │  o7 = ideal (c, b, a)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  For another example, see _P_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n.
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  Eisenbud, Huneke, Vasconcelos, Invent. Math. 110 207-235 (1992).
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html
│ │ │ @@ -130,31 +130,31 @@
│ │ │  
│ │ │  o6 = true
│ │ │              
    
│ │ │                
    i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .151496s elapsed
│ │ │ + -- .0799346s elapsed
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00157389s elapsed
│ │ │ + -- .00190731s elapsed
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00123922s elapsed
│ │ │ + -- .00131179s elapsed
│ │ │  
│ │ │  o9 = false
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,23 +50,23 @@
│ │ │ │  i5 : D = product(I_*/degree/sum)
│ │ │ │  
│ │ │ │  o5 = 840
│ │ │ │  i6 : x_0^(D-1) % I != 0 and x_0^D % I == 0
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ │ - -- .151496s elapsed
│ │ │ │ + -- .0799346s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ │ - -- .00157389s elapsed
│ │ │ │ + -- .00190731s elapsed
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ │ - -- .00123922s elapsed
│ │ │ │ + -- .00131179s elapsed
│ │ │ │  
│ │ │ │  o9 = false
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_d_i_c_a_l -- the radical of an ideal
│ │ │ │  ********** WWaayyss ttoo uussee rraaddiiccaallCCoonnttaaiinnmmeenntt:: **********
│ │ │ │      * radicalContainment(Ideal,Ideal)
│ │ │ │      * radicalContainment(RingElement,Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/Miura/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  c2NhbGFyTXVsdGlwbGljYXRpb24=
│ │ │  #:len=1318
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQWRkIFJlZHVjZWQgSWRlYWwgTXVsdGlw
│ │ │  bGUgVGltZXMiLCAibGluZW51bSIgPT4gMjAwLCBJbnB1dHMgPT4ge1NQQU57VFR7IkoifSwiLCAi
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bXVsdGlSZWVzSWRlYWwoSWRlYWwp
│ │ │  #:len=280
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjA3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtdWx0aVJlZXNJZGVhbCxJZGVhbCksIm11bHRpUmVl
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/example-output/_multi__Rees__Ideal.out
│ │ │ @@ -57,29 +57,29 @@
│ │ │  i9 : J = ideal vars U
│ │ │  
│ │ │  o9 = ideal (a, b, c)
│ │ │  
│ │ │  o9 : Ideal of U
│ │ │  
│ │ │  i10 : time multiReesIdeal J
│ │ │ - -- used 0.165102s (cpu); 0.107224s (thread); 0s (gc)
│ │ │ + -- used 0.406946s (cpu); 0.106978s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │  
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
│ │ │  
│ │ │  i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0216638s (cpu); 0.0211096s (thread); 0s (gc)
│ │ │ + -- used 0.0821452s (cpu); 0.0216715s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o9 : Ideal of U
    │ │ │
    │ │ │ 
    i10 : time multiReesIdeal J
│ │ │ - -- used 0.165102s (cpu); 0.107224s (thread); 0s (gc)
│ │ │ + -- used 0.406946s (cpu); 0.106978s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ @@ -195,15 +195,15 @@
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
    │ │ │
    │ │ │ 
    i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0216638s (cpu); 0.0211096s (thread); 0s (gc)
│ │ │ + -- used 0.0821452s (cpu); 0.0216715s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,28 +79,28 @@
│ │ │ │  i8 : U = T/minors(2,m);
│ │ │ │  i9 : J = ideal vars U
│ │ │ │  
│ │ │ │  o9 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o9 : Ideal of U
│ │ │ │  i10 : time multiReesIdeal J
│ │ │ │ - -- used 0.165102s (cpu); 0.107224s (thread); 0s (gc)
│ │ │ │ + -- used 0.406946s (cpu); 0.106978s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ │           0      2   1    0 2   0 1    2   0    1 2
│ │ │ │  
│ │ │ │  o10 : Ideal of U[X ..X ]
│ │ │ │                    0   2
│ │ │ │  i11 : time multiReesIdeal (J, a)
│ │ │ │ - -- used 0.0216638s (cpu); 0.0211096s (thread); 0s (gc)
│ │ │ │ + -- used 0.0821452s (cpu); 0.0216715s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │ │                  1      2     1      2     1      2     0      2     0      2
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                      2                 2   2
│ │ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  ZGVmb3JtTUNNTW9kdWxl
│ │ │  #:len=2237
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmVyc2FsIGRlZm9ybWF0aW9uIG9mIE1D
│ │ │  TS1tb2R1bGUgb24gaHlwZXJzdXJmYWNlIiwgRGVzY3JpcHRpb24gPT4gKCJUaGlzIGlzIHRoZSBt
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  
│ │ │  o7 = image | x2 y2 |
│ │ │  
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
│ │ │  
│ │ │  i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.684142s (cpu); 0.379036s (thread); 0s (gc)
│ │ │ + -- used 0.652199s (cpu); 0.343835s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -70,15 +70,15 @@
│ │ │  o10 = cokernel | x2 y2  |
│ │ │                 | -y -x2 |
│ │ │  
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
│ │ │  
│ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.682333s (cpu); 0.475269s (thread); 0s (gc)
│ │ │ + -- used 0.644887s (cpu); 0.461098s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
    │ │ │
    │ │ │ 
    i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.684142s (cpu); 0.379036s (thread); 0s (gc)
│ │ │ + -- used 0.652199s (cpu); 0.343835s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -191,15 +191,15 @@
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
    │ │ │
    │ │ │ 
    i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.682333s (cpu); 0.475269s (thread); 0s (gc)
│ │ │ + -- used 0.644887s (cpu); 0.461098s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ │ ├── html2text {}
│ │ │ │ @@ -70,15 +70,15 @@
│ │ │ │  i7 : N0 = module ideal (x^2,y^2)
│ │ │ │  
│ │ │ │  o7 = image | x2 y2 |
│ │ │ │  
│ │ │ │                               1
│ │ │ │  o7 : R-module, submodule of R
│ │ │ │  i8 : (S,N) = time deformMCMModule N0
│ │ │ │ - -- used 0.684142s (cpu); 0.379036s (thread); 0s (gc)
│ │ │ │ + -- used 0.652199s (cpu); 0.343835s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │ │                    {8} | xxi_4-y+xi_3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │ │       -x2-xxi_2-xi_1                                           |
│ │ │ │  
│ │ │ │ @@ -103,15 +103,15 @@
│ │ │ │  
│ │ │ │  o10 = cokernel | x2 y2  |
│ │ │ │                 | -y -x2 |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o10 : R-module, quotient of R
│ │ │ │  i11 : (S',N') = time deformMCMModule N0'
│ │ │ │ - -- used 0.682333s (cpu); 0.475269s (thread); 0s (gc)
│ │ │ │ + -- used 0.644887s (cpu); 0.461098s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │ │                      | xxi_4-y+xi_3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=46
│ │ │  c3BhcnNlTW9ub2Ryb215U29sdmUoLi4uLE51bWJlck9mUmVwZWF0cz0+Li4uKQ==
│ │ │  #:len=345
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDAwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzcGFyc2VNb25vZHJvbXlTb2x2ZSxOdW1iZXJPZlJl
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out
│ │ │ @@ -3,41 +3,41 @@
│ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │  
│ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │  
│ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │  
│ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ ---backup directory created: /tmp/M2-25512-0/2
│ │ │ - -- .00616338s elapsed
│ │ │ +--backup directory created: /tmp/M2-33134-0/2
│ │ │ + -- .00311047s elapsed
│ │ │    H01: 1
│ │ │ - -- .00900117s elapsed
│ │ │ + -- .00298895s elapsed
│ │ │    H10: 1
│ │ │ - -- .000649635s elapsed
│ │ │ + -- .000580269s elapsed
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00324889s elapsed
│ │ │ + -- .00310472s elapsed
│ │ │    H01: 1
│ │ │ - -- .00435076s elapsed
│ │ │ + -- .00304615s elapsed
│ │ │    H10: 1
│ │ │ - -- .000544598s elapsed
│ │ │ + -- .000503438s elapsed
│ │ │  number of paths tracked: 4
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00428387s elapsed
│ │ │ + -- .00308371s elapsed
│ │ │    H01: 1
│ │ │ - -- .00298132s elapsed
│ │ │ + -- .00302496s elapsed
│ │ │    H10: 1
│ │ │ - -- .000466532s elapsed
│ │ │ + -- .000458343s elapsed
│ │ │  number of paths tracked: 6
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .0044506s elapsed
│ │ │ + -- .00276853s elapsed
│ │ │    H01: 1
│ │ │ - -- .00309306s elapsed
│ │ │ + -- .002966s elapsed
│ │ │    H10: 1
│ │ │ - -- .000457295s elapsed
│ │ │ + -- .000500639s elapsed
│ │ │  number of paths tracked: 8
│ │ │  found 1 points in the fiber so far
│ │ │  
│ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │  
│ │ │  o4 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_get__Track__Time.out
│ │ │ @@ -6,16 +6,16 @@
│ │ │  i2 : declareVariable \ {A,B,C,D,X,Y};
│ │ │  
│ │ │  i3 : F = gateSystem(matrix{{A,B,C,D}},matrix{{X,Y}},matrix{{A*(X-1)^2-B}, {C*(Y+2)^2+D}});
│ │ │  
│ │ │  i4 : (p0,x0) = createSeedPair F;
│ │ │  
│ │ │  i5 : time (V,npaths) = monodromySolve(F,p0,{x0},NumberOfEdges=>3,NumberOfNodes=>5,TargetSolutionCount=>4,Verbose=>false);
│ │ │ - -- used 0.0535401s (cpu); 0.0535372s (thread); 0s (gc)
│ │ │ + -- used 0.062251s (cpu); 0.0622453s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : getTrackTime V.Graph
│ │ │  
│ │ │ -o6 = .006001457999999999
│ │ │ +o6 = .010547476
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
│ │ │ @@ -101,41 +101,41 @@
│ │ │
    		│ │ │ 
    i3 : (p0, x0) = createSeedPair polys;
    │ │ │
    │ │ │ 
    i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ ---backup directory created: /tmp/M2-25512-0/2
│ │ │ - -- .00616338s elapsed
│ │ │ +--backup directory created: /tmp/M2-33134-0/2
│ │ │ + -- .00311047s elapsed
│ │ │    H01: 1
│ │ │ - -- .00900117s elapsed
│ │ │ + -- .00298895s elapsed
│ │ │    H10: 1
│ │ │ - -- .000649635s elapsed
│ │ │ + -- .000580269s elapsed
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00324889s elapsed
│ │ │ + -- .00310472s elapsed
│ │ │    H01: 1
│ │ │ - -- .00435076s elapsed
│ │ │ + -- .00304615s elapsed
│ │ │    H10: 1
│ │ │ - -- .000544598s elapsed
│ │ │ + -- .000503438s elapsed
│ │ │  number of paths tracked: 4
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00428387s elapsed
│ │ │ + -- .00308371s elapsed
│ │ │    H01: 1
│ │ │ - -- .00298132s elapsed
│ │ │ + -- .00302496s elapsed
│ │ │    H10: 1
│ │ │ - -- .000466532s elapsed
│ │ │ + -- .000458343s elapsed
│ │ │  number of paths tracked: 6
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .0044506s elapsed
│ │ │ + -- .00276853s elapsed
│ │ │    H01: 1
│ │ │ - -- .00309306s elapsed
│ │ │ + -- .002966s elapsed
│ │ │    H10: 1
│ │ │ - -- .000457295s elapsed
│ │ │ + -- .000500639s elapsed
│ │ │  number of paths tracked: 8
│ │ │  found 1 points in the fiber so far
│ │ │  
│ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │  
│ │ │  o4 : Sequence
    │ │ │     		│ │ │ 
    i4 : (p0,x0) = createSeedPair F;
    │ │ │
    │ │ │ 
    i5 : time (V,npaths) = monodromySolve(F,p0,{x0},NumberOfEdges=>3,NumberOfNodes=>5,TargetSolutionCount=>4,Verbose=>false);
│ │ │ - -- used 0.0535401s (cpu); 0.0535372s (thread); 0s (gc)
    │ │ │ + -- used 0.062251s (cpu); 0.0622453s (thread); 0s (gc) │ │ │
    │ │ │ 
    i6 : getTrackTime V.Graph
│ │ │  
│ │ │ -o6 = .006001457999999999
│ │ │ +o6 = .010547476
│ │ │  
│ │ │  o6 : RR (of precision 53)
    │ │ │
    │ │ │
    │ │ │

    time measures the elapsed time for the whole call, including graph construction, and other overhead. getTrackTime reports only the cumulative time spent tracking paths along edges of the homotopy graph.

    │ │ │ ├── html2text {} │ │ │ │ @@ -16,18 +16,18 @@ │ │ │ │ -- setting random seed to 0 │ │ │ │ i2 : declareVariable \ {A,B,C,D,X,Y}; │ │ │ │ i3 : F = gateSystem(matrix{{A,B,C,D}},matrix{{X,Y}},matrix{{A*(X-1)^2-B}, {C* │ │ │ │ (Y+2)^2+D}}); │ │ │ │ i4 : (p0,x0) = createSeedPair F; │ │ │ │ i5 : time (V,npaths) = monodromySolve(F,p0, │ │ │ │ {x0},NumberOfEdges=>3,NumberOfNodes=>5,TargetSolutionCount=>4,Verbose=>false); │ │ │ │ - -- used 0.0535401s (cpu); 0.0535372s (thread); 0s (gc) │ │ │ │ + -- used 0.062251s (cpu); 0.0622453s (thread); 0s (gc) │ │ │ │ i6 : getTrackTime V.Graph │ │ │ │ │ │ │ │ -o6 = .006001457999999999 │ │ │ │ +o6 = .010547476 │ │ │ │ │ │ │ │ o6 : RR (of precision 53) │ │ │ │ time measures the elapsed time for the whole call, including graph │ │ │ │ construction, and other overhead. getTrackTime reports only the cumulative time │ │ │ │ spent tracking paths along edges of the homotopy graph. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed │ │ ├── ./usr/share/doc/Macaulay2/MonomialAlgebras/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=28 │ │ │ ZGVjb21wb3NlSG9tb2dlbmVvdXNNQShMaXN0KQ== │ │ │ #:len=308 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTY1Nywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZGVjb21wb3NlSG9tb2dlbmVvdXNNQSxMaXN0KSwi │ │ ├── ./usr/share/doc/Macaulay2/MonomialIntegerPrograms/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ R3JhZGVkQmV0dGlz │ │ │ #:len=431 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiB7fSwgImxpbmVudW0iID0+IDExNDYs │ │ │ IHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IkdyYWRlZEJldHRp │ │ ├── ./usr/share/doc/Macaulay2/MonomialOrbits/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ aGlsYmVydFJlcHJlc2VudGF0aXZlcw== │ │ │ #:len=3397 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZCByZXByZXNlbnRhdGl2ZXMgb2Yg │ │ │ bW9ub21pYWwgaWRlYWxzIHVuZGVyIHBlcm11dGF0aW9ucyBvZiB0aGUgdmFyaWFibGVzIiwgImxp │ │ ├── ./usr/share/doc/Macaulay2/Msolve/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ bXNvbHZlUlVSKElkZWFsKQ== │ │ │ #:len=227 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQ0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtc29sdmVSVVIsSWRlYWwpLCJtc29sdmVSVVIoSWRl │ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/___Msolve.out │ │ │ @@ -9,16 +9,16 @@ │ │ │ i2 : I = ideal(x, y, z) │ │ │ │ │ │ o2 = ideal (x, y, z) │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ i3 : msolveGB(I, Verbosity => 2, Threads => 6) │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-38527-0/0-in.ms -o /tmp/M2-38527-0/0-out.ms │ │ │ -Initial seed for pseudo-random number generator is 1792814436 │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-54065-0/0-in.ms -o /tmp/M2-54065-0/0-out.ms │ │ │ +Initial seed for pseudo-random number generator is 1793277392 │ │ │ │ │ │ --------------- INPUT DATA --------------- │ │ │ #variables 3 │ │ │ #equations 3 │ │ │ #invalid equations 0 │ │ │ field characteristic 0 │ │ │ homogeneous input? 1 │ │ │ @@ -29,15 +29,15 @@ │ │ │ initial hash table size 131072 (2^17) │ │ │ max pair selection ALL │ │ │ reduce gb 1 │ │ │ #threads 6 │ │ │ info level 2 │ │ │ generate pbm files 0 │ │ │ ------------------------------------------ │ │ │ -Initial prime = 1099986659 │ │ │ +Initial prime = 1098871429 │ │ │ │ │ │ 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) │ │ │ @@ -52,19 +52,19 @@ │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ reduce final basis 3 x 3 33.33% 3 new 0 zero 0.00 | 0.00 │ │ │ ------------------------------------------------------------------------------------------------------ │ │ │ │ │ │ ---------------- TIMINGS ---------------- │ │ │ overall(elapsed) 0.00 sec │ │ │ -overall(cpu) 0.00 sec │ │ │ +overall(cpu) 0.01 sec │ │ │ select 0.00 sec 0.0% │ │ │ -symbolic prep. 0.00 sec 0.0% │ │ │ -update 0.00 sec 96.6% │ │ │ -convert 0.00 sec 0.6% │ │ │ +symbolic prep. 0.00 sec 0.1% │ │ │ +update 0.00 sec 90.7% │ │ │ +convert 0.00 sec 0.4% │ │ │ linear algebra 0.00 sec 0.2% │ │ │ reduce gb 0.00 sec 0.0% │ │ │ ----------------------------------------- │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ size of basis 3 │ │ │ #terms in basis 3 │ │ │ @@ -79,18 +79,18 @@ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ [3] │ │ │ #polynomials to lift 3 │ │ │ ----------------------------------------- │ │ │ -New prime = 1120193111 │ │ │ +New prime = 1173221953 │ │ │ │ │ │ ---------------- 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%> │ │ │ @@ -106,15 +106,15 @@ │ │ │ ---------------- TIMINGS ---------------- │ │ │ CRT (elapsed) 0.00 sec │ │ │ ratrecon(elapsed) 0.00 sec │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ │ ------------------------------------------------------------------------------------ │ │ │ -msolve overall time 0.08 sec (elapsed) / 0.27 sec (cpu) │ │ │ +msolve overall time 0.01 sec (elapsed) / 0.05 sec (cpu) │ │ │ ------------------------------------------------------------------------------------ │ │ │ │ │ │ o3 = | z y x | │ │ │ │ │ │ 1 3 │ │ │ o3 : Matrix R <-- R │ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/index.html │ │ │ @@ -83,16 +83,16 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-38527-0/0-in.ms -o /tmp/M2-38527-0/0-out.ms
│ │ │ -Initial seed for pseudo-random number generator is 1792814436
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-54065-0/0-in.ms -o /tmp/M2-54065-0/0-out.ms
│ │ │ +Initial seed for pseudo-random number generator is 1793277392
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -103,15 +103,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1099986659
│ │ │ +Initial prime = 1098871429
│ │ │  
│ │ │  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)
│ │ │ @@ -126,19 +126,19 @@
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │  overall(elapsed)        0.00 sec
│ │ │ -overall(cpu)            0.00 sec
│ │ │ +overall(cpu)            0.01 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.00 sec  96.6%
│ │ │ -convert                 0.00 sec   0.6%
│ │ │ +symbolic prep.          0.00 sec   0.1%
│ │ │ +update                  0.00 sec  90.7%
│ │ │ +convert                 0.00 sec   0.4%
│ │ │  linear algebra          0.00 sec   0.2%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │ @@ -153,18 +153,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1120193111
│ │ │ +New prime = 1173221953
│ │ │  
│ │ │  ---------------- 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%> 
│ │ │ @@ -180,15 +180,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.08 sec (elapsed) /  0.27 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.05 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -31,17 +31,17 @@ │ │ │ │ o1 : PolynomialRing │ │ │ │ i2 : I = ideal(x, y, z) │ │ │ │ │ │ │ │ o2 = ideal (x, y, z) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : msolveGB(I, Verbosity => 2, Threads => 6) │ │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-38527-0/0-in.ms -o /tmp/ │ │ │ │ -M2-38527-0/0-out.ms │ │ │ │ -Initial seed for pseudo-random number generator is 1792814436 │ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-54065-0/0-in.ms -o /tmp/ │ │ │ │ +M2-54065-0/0-out.ms │ │ │ │ +Initial seed for pseudo-random number generator is 1793277392 │ │ │ │ │ │ │ │ --------------- INPUT DATA --------------- │ │ │ │ #variables 3 │ │ │ │ #equations 3 │ │ │ │ #invalid equations 0 │ │ │ │ field characteristic 0 │ │ │ │ homogeneous input? 1 │ │ │ │ @@ -52,15 +52,15 @@ │ │ │ │ initial hash table size 131072 (2^17) │ │ │ │ max pair selection ALL │ │ │ │ reduce gb 1 │ │ │ │ #threads 6 │ │ │ │ info level 2 │ │ │ │ generate pbm files 0 │ │ │ │ ------------------------------------------ │ │ │ │ -Initial prime = 1099986659 │ │ │ │ +Initial prime = 1098871429 │ │ │ │ │ │ │ │ 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) │ │ │ │ @@ -80,19 +80,19 @@ │ │ │ │ reduce final basis 3 x 3 33.33% 3 new 0 zero │ │ │ │ 0.00 | 0.00 │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----------------------- │ │ │ │ │ │ │ │ ---------------- TIMINGS ---------------- │ │ │ │ overall(elapsed) 0.00 sec │ │ │ │ -overall(cpu) 0.00 sec │ │ │ │ +overall(cpu) 0.01 sec │ │ │ │ select 0.00 sec 0.0% │ │ │ │ -symbolic prep. 0.00 sec 0.0% │ │ │ │ -update 0.00 sec 96.6% │ │ │ │ -convert 0.00 sec 0.6% │ │ │ │ +symbolic prep. 0.00 sec 0.1% │ │ │ │ +update 0.00 sec 90.7% │ │ │ │ +convert 0.00 sec 0.4% │ │ │ │ linear algebra 0.00 sec 0.2% │ │ │ │ reduce gb 0.00 sec 0.0% │ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ │ size of basis 3 │ │ │ │ #terms in basis 3 │ │ │ │ @@ -107,18 +107,18 @@ │ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ │ [3] │ │ │ │ #polynomials to lift 3 │ │ │ │ ----------------------------------------- │ │ │ │ -New prime = 1120193111 │ │ │ │ +New prime = 1173221953 │ │ │ │ │ │ │ │ ---------------- 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 │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----------------------- │ │ │ │ @@ -137,15 +137,15 @@ │ │ │ │ CRT (elapsed) 0.00 sec │ │ │ │ ratrecon(elapsed) 0.00 sec │ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----- │ │ │ │ -msolve overall time 0.08 sec (elapsed) / 0.27 sec (cpu) │ │ │ │ +msolve overall time 0.01 sec (elapsed) / 0.05 sec (cpu) │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----- │ │ │ │ │ │ │ │ o3 = | z y x | │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o3 : Matrix R <-- R │ │ ├── ./usr/share/doc/Macaulay2/MultiGradedRationalMap/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ ZGVncmVlT2ZNYXAoSWRlYWwp │ │ │ #:len=283 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjk3LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkZWdyZWVPZk1hcCxJZGVhbCksImRlZ3JlZU9mTWFw │ │ ├── ./usr/share/doc/Macaulay2/MultigradedBGG/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ ZGVncmVlKERpZmZlcmVudGlhbE1vZHVsZSk= │ │ │ #:len=1348 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgZGVncmVlIG9mIHRo │ │ │ ZSBkaWZmZXJlbnRpYWwiLCAibGluZW51bSIgPT4gNDgzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkQi │ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ TXVsdGlncmFkZWRJbXBsaWNpdGl6YXRpb24= │ │ │ #:len=1401 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUGFja2FnZSBmb3IgbGV2YXJhZ2luZyBt │ │ │ dWx0aWdyYWRpbmdzIHRvIHNvbHZlIGltcGxpY2l0aXphdGlvbiBwcm9ibGVtcyIsIERlc2NyaXB0 │ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/example-output/_components__Of__Kernel.out │ │ │ @@ -23,19 +23,19 @@ │ │ │ o4 : RingMap S <-- R │ │ │ │ │ │ i5 : peek componentsOfKernel(2, F) │ │ │ warning: computation begun over finite field. resulting polynomials may not lie in the ideal │ │ │ computing total degree: 1 │ │ │ number of monomials = 6 │ │ │ number of distinct multidegrees = 6 │ │ │ - -- .00225028s elapsed │ │ │ + -- .00201864s elapsed │ │ │ computing total degree: 2 │ │ │ number of monomials = 21 │ │ │ number of distinct multidegrees = 18 │ │ │ - -- .00849938s elapsed │ │ │ + -- .00846963s elapsed │ │ │ │ │ │ o5 = MutableHashTable{{0, 1, 0, 0, 1} => {} } │ │ │ {0, 1, 0, 1, 0} => {} │ │ │ {0, 1, 1, 0, 0} => {} │ │ │ {0, 2, 0, 0, 2} => {} │ │ │ {0, 2, 0, 1, 1} => {} │ │ │ {0, 2, 0, 2, 0} => {} │ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ @@ -122,19 +122,19 @@ │ │ │ │ │ │ │ │ │ 
    i5 : peek componentsOfKernel(2, F)
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 6
│ │ │  number of distinct multidegrees = 6
│ │ │ - -- .00225028s elapsed
│ │ │ + -- .00201864s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00849938s elapsed
│ │ │ + -- .00846963s elapsed
│ │ │  
│ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │                        {0, 2, 0, 2, 0} => {}
│ │ │ ├── html2text {}
│ │ │ │ @@ -51,19 +51,19 @@
│ │ │ │  o4 : RingMap S <-- R
│ │ │ │  i5 : peek componentsOfKernel(2, F)
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 6
│ │ │ │  number of distinct multidegrees = 6
│ │ │ │ - -- .00225028s elapsed
│ │ │ │ + -- .00201864s elapsed
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 21
│ │ │ │  number of distinct multidegrees = 18
│ │ │ │ - -- .00849938s elapsed
│ │ │ │ + -- .00846963s elapsed
│ │ │ │  
│ │ │ │  o5 = MutableHashTable{{0, 1, 0, 0, 1} => {}                   }
│ │ │ │                        {0, 1, 0, 1, 0} => {}
│ │ │ │                        {0, 1, 1, 0, 0} => {}
│ │ │ │                        {0, 2, 0, 0, 2} => {}
│ │ │ │                        {0, 2, 0, 1, 1} => {}
│ │ │ │                        {0, 2, 0, 2, 0} => {}
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  Z3JHcihJZGVhbCk=
│ │ │  #:len=249
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzc4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnckdyLElkZWFsKSwiZ3JHcihJZGVhbCkiLCJNdWx0
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_j__Mult.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : I = ideal"xy,yz,zx"
│ │ │  
│ │ │  o2 = ideal (x*y, y*z, x*z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime jMult I
│ │ │ - -- .0242277s elapsed
│ │ │ + -- .0249312s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .159695s elapsed
│ │ │ + -- .0593157s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .270321s elapsed
│ │ │ + -- .166721s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out
│ │ │ @@ -10,12 +10,12 @@
│ │ │  
│ │ │               2        3
│ │ │  o2 = ideal (x , x*y, y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .0952293s elapsed
│ │ │ + -- .0694347s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out
│ │ │ @@ -13,17 +13,17 @@
│ │ │       ------------------------------------------------------------------------
│ │ │        10 11   8 12   9 11   10 10   11 9   12 8
│ │ │       x  y  , x y  , x y  , x  y  , x  y , x  y )
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime monjMult I
│ │ │ - -- .189553s elapsed
│ │ │ + -- .08533s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.87284s elapsed
│ │ │ + -- 1.28091s elapsed
│ │ │  
│ │ │  o4 = 192
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
│ │ │ @@ -93,31 +93,31 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime jMult I
│ │ │ - -- .0242277s elapsed
│ │ │ + -- .0249312s elapsed
│ │ │  
│ │ │  o3 = 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime monjMult I
│ │ │ - -- .159695s elapsed
│ │ │ + -- .0593157s elapsed
│ │ │  
│ │ │  o4 = 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .270321s elapsed
│ │ │ + -- .166721s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -20,23 +20,23 @@ │ │ │ │ o1 : PolynomialRing │ │ │ │ i2 : I = ideal"xy,yz,zx" │ │ │ │ │ │ │ │ o2 = ideal (x*y, y*z, x*z) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime jMult I │ │ │ │ - -- .0242277s elapsed │ │ │ │ + -- .0249312s elapsed │ │ │ │ │ │ │ │ o3 = 2 │ │ │ │ i4 : elapsedTime monjMult I │ │ │ │ - -- .159695s elapsed │ │ │ │ + -- .0593157s elapsed │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : elapsedTime multiplicitySequence I │ │ │ │ - -- .270321s elapsed │ │ │ │ + -- .166721s elapsed │ │ │ │ │ │ │ │ o5 = HashTable{2 => 3} │ │ │ │ 3 => 2 │ │ │ │ │ │ │ │ o5 : HashTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal │ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .0952293s elapsed
│ │ │ + -- .0694347s elapsed
│ │ │  
│ │ │  o3 = 2
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,15 +22,15 @@ │ │ │ │ i2 : I = ideal"x2,xy,y3" │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o2 = ideal (x , x*y, y ) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime monAnalyticSpread I │ │ │ │ - -- .0952293s elapsed │ │ │ │ + -- .0694347s elapsed │ │ │ │ │ │ │ │ o3 = 2 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _N_P -- the Newton polyhedron of a monomial ideal │ │ │ │ ********** WWaayyss ttoo uussee mmoonnAAnnaallyyttiiccSSpprreeaadd:: ********** │ │ │ │ * monAnalyticSpread(Ideal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ @@ -97,23 +97,23 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime monjMult I
│ │ │ - -- .189553s elapsed
│ │ │ + -- .08533s elapsed
│ │ │  
│ │ │  o3 = 192
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime jMult I
│ │ │ - -- 1.87284s elapsed
│ │ │ + -- 1.28091s elapsed
│ │ │  
│ │ │  o4 = 192
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -24,19 +24,19 @@ │ │ │ │ o2 = ideal (x y , x y , x y , x y , x y , x y , x y , x y , x y , │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 10 11 8 12 9 11 10 10 11 9 12 8 │ │ │ │ x y , x y , x y , x y , x y , x y ) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime monjMult I │ │ │ │ - -- .189553s elapsed │ │ │ │ + -- .08533s elapsed │ │ │ │ │ │ │ │ o3 = 192 │ │ │ │ i4 : elapsedTime jMult I │ │ │ │ - -- 1.87284s elapsed │ │ │ │ + -- 1.28091s elapsed │ │ │ │ │ │ │ │ o4 = 192 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal │ │ │ │ * _j_M_u_l_t -- the j-multiplicity of an ideal │ │ │ │ * _m_o_n_R_e_d_u_c_t_i_o_n -- the minimal monomial reduction of a monomial ideal │ │ │ │ * _N_P -- the Newton polyhedron of a monomial ideal │ │ ├── ./usr/share/doc/Macaulay2/MultiplierIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=41 │ │ │ bG9nQ2Fub25pY2FsVGhyZXNob2xkKENlbnRyYWxBcnJhbmdlbWVudCk= │ │ │ #:len=361 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA3MCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobG9nQ2Fub25pY2FsVGhyZXNob2xkLENlbnRyYWxB │ │ ├── ./usr/share/doc/Macaulay2/MultiplierIdealsDim2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ TXVsdElkZWFs │ │ │ #:len=1363 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIG11bHRpcGxpZXIg │ │ │ aWRlYWwgb2YgYSBnaXZlbiBudW1iZXIuIiwgImxpbmVudW0iID0+IDY0OCwgSW5wdXRzID0+IHtT │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ dGV4TWF0aChSQVQp │ │ │ #:len=209 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDE5OCwgInVuZG9jdW1lbnRlZCIgPT4g │ │ │ dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4TWF0 │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out │ │ │ @@ -13,15 +13,15 @@ │ │ │ o4 : ProjectiveVariety, curve in PP^8 │ │ │ │ │ │ i5 : ? X │ │ │ │ │ │ o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 │ │ │ │ │ │ i6 : time f = X ===> Y; │ │ │ - -- used 3.4873s (cpu); 1.91387s (thread); 0s (gc) │ │ │ + -- used 3.31275s (cpu); 1.74919s (thread); 0s (gc) │ │ │ │ │ │ o6 : MultirationalMap (automorphism of PP^8) │ │ │ │ │ │ i7 : f X │ │ │ │ │ │ o7 = Y │ │ │ │ │ │ @@ -38,15 +38,15 @@ │ │ │ o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8 │ │ │ │ │ │ i10 : W = random({{2},{1}},Y); │ │ │ │ │ │ o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8 │ │ │ │ │ │ i11 : time g = V ===> W; │ │ │ - -- used 3.66337s (cpu); 2.10606s (thread); 0s (gc) │ │ │ + -- used 3.55646s (cpu); 1.93062s (thread); 0s (gc) │ │ │ │ │ │ o11 : MultirationalMap (automorphism of PP^8) │ │ │ │ │ │ i12 : g||W │ │ │ │ │ │ o12 = multi-rational map consisting of one single rational map │ │ │ source variety: 6-dimensional subvariety of PP^8 cut out by 2 hypersurfaces of degrees 1^1 2^1 │ │ │ @@ -129,15 +129,15 @@ │ │ │ o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9 │ │ │ │ │ │ i16 : ? Z │ │ │ │ │ │ o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2 │ │ │ │ │ │ i17 : time h = Z ===> GG_K(1,4) │ │ │ - -- used 8.41326s (cpu); 5.16056s (thread); 0s (gc) │ │ │ + -- used 6.0498s (cpu); 4.2659s (thread); 0s (gc) │ │ │ │ │ │ o17 = h │ │ │ │ │ │ o17 : MultirationalMap (isomorphism from PP^9 to PP^9) │ │ │ │ │ │ i18 : h || GG_K(1,4) │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out │ │ │ @@ -7,15 +7,15 @@ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random({1,1},ring target Phi)); │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4 │ │ │ │ │ │ i4 : time X = Phi^* Y; │ │ │ - -- used 5.22987s (cpu); 3.71826s (thread); 0s (gc) │ │ │ + -- used 3.92653s (cpu); 3.24009s (thread); 0s (gc) │ │ │ │ │ │ o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension 2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 ) │ │ │ │ │ │ i5 : dim X, degree X, degrees X │ │ │ │ │ │ o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1), │ │ │ ------------------------------------------------------------------------ │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out │ │ │ @@ -11,26 +11,26 @@ │ │ │ o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^7 x PP^7) │ │ │ │ │ │ i4 : Z = source Phi; │ │ │ │ │ │ o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7 │ │ │ │ │ │ i5 : time Phi Z; │ │ │ - -- used 0.20103s (cpu); 0.125952s (thread); 0s (gc) │ │ │ + -- used 0.20242s (cpu); 0.114275s (thread); 0s (gc) │ │ │ │ │ │ o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7 │ │ │ │ │ │ i6 : dim oo, degree oo, degrees oo │ │ │ │ │ │ o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)}) │ │ │ │ │ │ o6 : Sequence │ │ │ │ │ │ i7 : time Phi (point Z + point Z + point Z) │ │ │ - -- used 2.02267s (cpu); 1.47252s (thread); 0s (gc) │ │ │ + -- used 1.85571s (cpu); 1.17141s (thread); 0s (gc) │ │ │ │ │ │ o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 │ │ │ │ │ │ o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7 │ │ │ │ │ │ i8 : dim oo, degree oo, degrees oo │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out │ │ │ @@ -11,22 +11,22 @@ │ │ │ o3 = multi-rational map consisting of one single rational map │ │ │ source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of │ │ │ target variety: hypersurface in PP^4 defined by a form of degree 2 │ │ │ ------------------------------------------------------------------------ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ │ │ i4 : time degree(Phi,Strategy=>"random point") │ │ │ - -- used 4.01388s (cpu); 2.8819s (thread); 0s (gc) │ │ │ + -- used 3.78363s (cpu); 2.19192s (thread); 0s (gc) │ │ │ │ │ │ o4 = 2 │ │ │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ - -- used 0.504268s (cpu); 0.357315s (thread); 0s (gc) │ │ │ + -- used 0.410892s (cpu); 0.282598s (thread); 0s (gc) │ │ │ │ │ │ o5 = 2 │ │ │ │ │ │ i6 : time degree Phi │ │ │ - -- used 0.330552s (cpu); 0.2545s (thread); 0s (gc) │ │ │ + -- used 0.208755s (cpu); 0.20931s (thread); 0s (gc) │ │ │ │ │ │ o6 = 2 │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out │ │ │ @@ -3,12 +3,12 @@ │ │ │ i1 : ZZ/300007[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, x_3^2}; │ │ │ │ │ │ i2 : Phi = last graph rationalMap {f,g}; │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ i3 : time degree Phi │ │ │ - -- used 0.43902s (cpu); 0.366735s (thread); 0s (gc) │ │ │ + -- used 0.378111s (cpu); 0.300798s (thread); 0s (gc) │ │ │ │ │ │ o3 = 1 │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out │ │ │ @@ -1,52 +1,52 @@ │ │ │ -- -*- M2-comint -*- hash: 11533721324852072161 │ │ │ │ │ │ i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4); │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5) │ │ │ │ │ │ i2 : time ? Phi │ │ │ - -- used 0.00332575s (cpu); 0.000166912s (thread); 0s (gc) │ │ │ + -- used 0.00178985s (cpu); 0.000175919s (thread); 0s (gc) │ │ │ │ │ │ o2 = multi-rational map consisting of 2 rational maps │ │ │ source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 │ │ │ target variety: PP^4 x PP^5 │ │ │ ------------------------------------------------------------------------ │ │ │ hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ │ │ │ │ i3 : image Phi; │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5 │ │ │ │ │ │ i4 : time ? Phi │ │ │ - -- used 0.00316389s (cpu); 0.000237795s (thread); 0s (gc) │ │ │ + -- used 0.00184043s (cpu); 0.000207385s (thread); 0s (gc) │ │ │ │ │ │ o4 = multi-rational map consisting of 2 rational maps │ │ │ source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ │ target variety: PP^4 x PP^5 │ │ │ dominance: false │ │ │ image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ │ │ │ │ i5 : time describe Phi │ │ │ - -- used 1.3359s (cpu); 1.00639s (thread); 0s (gc) │ │ │ + -- used 1.0703s (cpu); 0.891102s (thread); 0s (gc) │ │ │ │ │ │ o5 = multi-rational map consisting of 2 rational maps │ │ │ source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ │ target variety: PP^4 x PP^5 │ │ │ base locus: empty subscheme of PP^4 x PP^5 │ │ │ dominance: false │ │ │ image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ │ multidegree: {51, 51, 51, 51, 51} │ │ │ degree: 1 │ │ │ degree sequence (map 1/2): [(1,0), (0,2)] │ │ │ degree sequence (map 2/2): [(0,1), (2,0)] │ │ │ coefficient ring: ZZ/65521 │ │ │ │ │ │ i6 : time ? Phi │ │ │ - -- used 0.000157475s (cpu); 0.000414858s (thread); 0s (gc) │ │ │ + -- used 0.000122081s (cpu); 0.000415221s (thread); 0s (gc) │ │ │ │ │ │ o6 = multi-rational map consisting of 2 rational maps │ │ │ source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ │ target variety: PP^4 x PP^5 │ │ │ base locus: empty subscheme of PP^4 x PP^5 │ │ │ dominance: false │ │ │ image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out │ │ │ @@ -3,45 +3,45 @@ │ │ │ i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true) │ │ │ │ │ │ o1 = Phi │ │ │ │ │ │ o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5) │ │ │ │ │ │ i2 : time (Phi1,Phi2) = graph Phi │ │ │ - -- used 0.0281486s (cpu); 0.0267017s (thread); 0s (gc) │ │ │ + -- used 0.035115s (cpu); 0.0226179s (thread); 0s (gc) │ │ │ │ │ │ o2 = (Phi1, Phi2) │ │ │ │ │ │ o2 : Sequence │ │ │ │ │ │ i3 : Phi1; │ │ │ │ │ │ o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4) │ │ │ │ │ │ i4 : Phi2; │ │ │ │ │ │ o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ i5 : time (Phi21,Phi22) = graph Phi2 │ │ │ - -- used 0.134338s (cpu); 0.060805s (thread); 0s (gc) │ │ │ + -- used 0.172939s (cpu); 0.0589556s (thread); 0s (gc) │ │ │ │ │ │ o5 = (Phi21, Phi22) │ │ │ │ │ │ o5 : Sequence │ │ │ │ │ │ i6 : Phi21; │ │ │ │ │ │ o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5) │ │ │ │ │ │ i7 : Phi22; │ │ │ │ │ │ o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ i8 : time (Phi211,Phi212) = graph Phi21 │ │ │ - -- used 0.124474s (cpu); 0.123543s (thread); 0s (gc) │ │ │ + -- used 0.150489s (cpu); 0.118313s (thread); 0s (gc) │ │ │ │ │ │ o8 = (Phi211, Phi212) │ │ │ │ │ │ o8 : Sequence │ │ │ │ │ │ i9 : Phi211; │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out │ │ │ @@ -11,25 +11,25 @@ │ │ │ o3 : RationalMap (quadratic rational map from PP^4 to PP^4) │ │ │ │ │ │ i4 : Phi = rationalMap {f,g}; │ │ │ │ │ │ o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4) │ │ │ │ │ │ i5 : time Z = image Phi; │ │ │ - -- used 0.103109s (cpu); 0.102376s (thread); 0s (gc) │ │ │ + -- used 0.126629s (cpu); 0.114496s (thread); 0s (gc) │ │ │ │ │ │ o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4 │ │ │ │ │ │ i6 : dim Z, degree Z, degrees Z │ │ │ │ │ │ o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)}) │ │ │ │ │ │ o6 : Sequence │ │ │ │ │ │ i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4"); │ │ │ - -- used 5.47152s (cpu); 2.82366s (thread); 0s (gc) │ │ │ + -- used 8.78127s (cpu); 2.43693s (thread); 0s (gc) │ │ │ │ │ │ o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4 │ │ │ │ │ │ i8 : assert(Z == Z') │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out │ │ │ @@ -4,25 +4,25 @@ │ │ │ │ │ │ i2 : -- map defined by the cubics through the secant variety to the rational normal curve of degree 6 │ │ │ Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal PP_K([6],2)); │ │ │ │ │ │ o2 : MultirationalMap (rational map from PP^6 to GG(2,4)) │ │ │ │ │ │ i3 : time Psi = inverse2 Phi; │ │ │ - -- used 0.418413s (cpu); 0.331336s (thread); 0s (gc) │ │ │ + -- used 0.394943s (cpu); 0.308668s (thread); 0s (gc) │ │ │ │ │ │ o3 : MultirationalMap (birational map from GG(2,4) to PP^6) │ │ │ │ │ │ i4 : assert(Phi * Psi == 1) │ │ │ │ │ │ i5 : Phi' = Phi || Phi; │ │ │ │ │ │ o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4)) │ │ │ │ │ │ i6 : time Psi' = inverse2 Phi'; │ │ │ - -- used 1.45295s (cpu); 1.03172s (thread); 0s (gc) │ │ │ + -- used 1.16076s (cpu); 0.910865s (thread); 0s (gc) │ │ │ │ │ │ o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6) │ │ │ │ │ │ i7 : assert(Phi' * Psi' == 1) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out │ │ │ @@ -7,33 +7,33 @@ │ │ │ │ │ │ i2 : -- we see Phi as a dominant map │ │ │ Phi = rationalMap(Phi,image Phi); │ │ │ │ │ │ o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5) │ │ │ │ │ │ i3 : time inverse Phi; │ │ │ - -- used 0.157948s (cpu); 0.0767566s (thread); 0s (gc) │ │ │ + -- used 0.149381s (cpu); 0.0697705s (thread); 0s (gc) │ │ │ │ │ │ o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4) │ │ │ │ │ │ i4 : Psi = last graph Phi; │ │ │ │ │ │ o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ i5 : time inverse Psi; │ │ │ - -- used 0.186263s (cpu); 0.107471s (thread); 0s (gc) │ │ │ + -- used 0.280991s (cpu); 0.110831s (thread); 0s (gc) │ │ │ │ │ │ o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5) │ │ │ │ │ │ i6 : Eta = first graph Psi; │ │ │ │ │ │ o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5) │ │ │ │ │ │ i7 : time inverse Eta; │ │ │ - -- used 0.48562s (cpu); 0.313057s (thread); 0s (gc) │ │ │ + -- used 0.543884s (cpu); 0.300887s (thread); 0s (gc) │ │ │ │ │ │ o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5) │ │ │ │ │ │ i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1) │ │ │ │ │ │ i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1) │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out │ │ │ @@ -6,32 +6,32 @@ │ │ │ o2 : RationalMap (quadratic rational map from PP^3 to PP^2) │ │ │ │ │ │ i3 : Phi = rationalMap {f,f}; │ │ │ │ │ │ o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2) │ │ │ │ │ │ i4 : time isIsomorphism Phi │ │ │ - -- used 0.000119785s (cpu); 1.2293e-05s (thread); 0s (gc) │ │ │ + -- used 0.00399681s (cpu); 2.776e-06s (thread); 0s (gc) │ │ │ │ │ │ o4 = false │ │ │ │ │ │ i5 : Psi = first graph Phi; │ │ │ │ │ │ o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3) │ │ │ │ │ │ i6 : time isIsomorphism Psi │ │ │ - -- used 0.396692s (cpu); 0.213364s (thread); 0s (gc) │ │ │ + -- used 0.455076s (cpu); 0.189026s (thread); 0s (gc) │ │ │ │ │ │ o6 = false │ │ │ │ │ │ i7 : Eta = first graph Psi; │ │ │ │ │ │ o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x PP^3 to threefold in PP^3 x PP^2 x PP^2) │ │ │ │ │ │ i8 : time isIsomorphism Eta │ │ │ - -- used 1.74335s (cpu); 0.941708s (thread); 0s (gc) │ │ │ + -- used 1.63417s (cpu); 0.798647s (thread); 0s (gc) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : assert(o8 and (not o6) and (not o4)) │ │ │ │ │ │ i10 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out │ │ │ @@ -3,24 +3,24 @@ │ │ │ i1 : ZZ/300007[a..e], f = first graph rationalMap ideal(c^2-b*d,b*c-a*d,b^2-a*c,e), g = rationalMap submatrix(matrix f,{0..2}); │ │ │ │ │ │ i2 : Phi = rationalMap {f,g}; │ │ │ │ │ │ o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2) │ │ │ │ │ │ i3 : time isMorphism Phi │ │ │ - -- used 0.386427s (cpu); 0.269828s (thread); 0s (gc) │ │ │ + -- used 0.37364s (cpu); 0.217378s (thread); 0s (gc) │ │ │ │ │ │ o3 = false │ │ │ │ │ │ i4 : time Psi = first graph Phi; │ │ │ - -- used 0.168774s (cpu); 0.0862052s (thread); 0s (gc) │ │ │ + -- used 0.187741s (cpu); 0.0892678s (thread); 0s (gc) │ │ │ │ │ │ o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7) │ │ │ │ │ │ i5 : time isMorphism Psi │ │ │ - -- used 4.42649s (cpu); 3.16319s (thread); 0s (gc) │ │ │ + -- used 3.37906s (cpu); 2.77213s (thread); 0s (gc) │ │ │ │ │ │ o5 = true │ │ │ │ │ │ i6 : assert((not o3) and o5) │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_linearly__Normal__Embedding.out │ │ │ @@ -3,24 +3,24 @@ │ │ │ i1 : K = ZZ/333331; │ │ │ │ │ │ i2 : X = PP_K^(1,7); -- rational normal curve of degree 7 │ │ │ │ │ │ o2 : ProjectiveVariety, curve in PP^7 │ │ │ │ │ │ i3 : time f = linearlyNormalEmbedding X; │ │ │ - -- used 0.00800583s (cpu); 0.00919368s (thread); 0s (gc) │ │ │ + -- used 0.00811821s (cpu); 0.00956031s (thread); 0s (gc) │ │ │ │ │ │ o3 : MultirationalMap (automorphism of X) │ │ │ │ │ │ i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3 │ │ │ │ │ │ o4 : ProjectiveVariety, curve in PP^3 │ │ │ │ │ │ i5 : time g = linearlyNormalEmbedding Y; │ │ │ - -- used 0.709324s (cpu); 0.523301s (thread); 0s (gc) │ │ │ + -- used 0.447966s (cpu); 0.376408s (thread); 0s (gc) │ │ │ │ │ │ o5 : MultirationalMap (birational map from Y to curve in PP^7) │ │ │ │ │ │ i6 : assert(isIsomorphism g) │ │ │ │ │ │ i7 : describe g │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multirational__Map_rp.out │ │ │ @@ -3,15 +3,15 @@ │ │ │ i1 : ZZ/300007[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, x_3^2}; │ │ │ │ │ │ i2 : Phi = last graph rationalMap {f,g}; │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ i3 : time multidegree Phi │ │ │ - -- used 0.453684s (cpu); 0.376266s (thread); 0s (gc) │ │ │ + -- used 0.431685s (cpu); 0.312135s (thread); 0s (gc) │ │ │ │ │ │ o3 = {66, 46, 31, 20} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : (degree source Phi,degree image Phi) │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out │ │ │ @@ -1,21 +1,21 @@ │ │ │ -- -*- M2-comint -*- hash: 16199733219210081214 │ │ │ │ │ │ i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4); │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5) │ │ │ │ │ │ i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi) │ │ │ - -- used 0.00102199s (cpu); 0.00133739s (thread); 0s (gc) │ │ │ - -- used 0.222006s (cpu); 0.14506s (thread); 0s (gc) │ │ │ - -- used 0.340129s (cpu); 0.192965s (thread); 0s (gc) │ │ │ - -- used 0.227149s (cpu); 0.156054s (thread); 0s (gc) │ │ │ - -- used 0.198029s (cpu); 0.122886s (thread); 0s (gc) │ │ │ + -- used 0.00403651s (cpu); 0.00150351s (thread); 0s (gc) │ │ │ + -- used 0.18919s (cpu); 0.127475s (thread); 0s (gc) │ │ │ + -- used 0.303532s (cpu); 0.176257s (thread); 0s (gc) │ │ │ + -- used 0.200725s (cpu); 0.14139s (thread); 0s (gc) │ │ │ + -- used 0.185499s (cpu); 0.118796s (thread); 0s (gc) │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ - -- used 0.164641s (cpu); 0.0837544s (thread); 0s (gc) │ │ │ + -- used 0.183934s (cpu); 0.0820254s (thread); 0s (gc) │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out │ │ │ @@ -3,26 +3,26 @@ │ │ │ i1 : K = ZZ/1000003; │ │ │ │ │ │ i2 : X = PP_K^({1,1,2},{3,2,3}); │ │ │ │ │ │ o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9 │ │ │ │ │ │ i3 : time p := point X │ │ │ - -- used 0.0200872s (cpu); 0.0193746s (thread); 0s (gc) │ │ │ + -- used 0.0667782s (cpu); 0.0233333s (thread); 0s (gc) │ │ │ │ │ │ o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1]) │ │ │ │ │ │ o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ i4 : Y = random({2,1,2},X); │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ i5 : time q = point Y │ │ │ - -- used 2.20856s (cpu); 1.14282s (thread); 0s (gc) │ │ │ + -- used 1.49071s (cpu); 0.91067s (thread); 0s (gc) │ │ │ │ │ │ o5 = q │ │ │ │ │ │ o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ i6 : assert(isSubset(p,X) and isSubset(q,Y)) │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ o4 : RationalMap (quadratic rational map from PP^4 to PP^4) │ │ │ │ │ │ i5 : Phi = rationalMap {f,g,h}; │ │ │ │ │ │ o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4) │ │ │ │ │ │ i6 : time segre Phi; │ │ │ - -- used 0.774927s (cpu); 0.592841s (thread); 0s (gc) │ │ │ + -- used 1.10483s (cpu); 0.545907s (thread); 0s (gc) │ │ │ │ │ │ o6 : RationalMap (rational map from PP^4 to PP^149) │ │ │ │ │ │ i7 : describe segre Phi │ │ │ │ │ │ o7 = rational map defined by forms of degree 6 │ │ │ source variety: PP^4 │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_show_lp__Multirational__Map_rp.out │ │ │ @@ -3,15 +3,15 @@ │ │ │ i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3) │ │ │ │ │ │ o1 = Phi │ │ │ │ │ │ o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2) │ │ │ │ │ │ i2 : time describe Phi │ │ │ - -- used 0.410659s (cpu); 0.270055s (thread); 0s (gc) │ │ │ + -- used 0.300024s (cpu); 0.161044s (thread); 0s (gc) │ │ │ │ │ │ o2 = multi-rational map consisting of 3 rational maps │ │ │ source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1) │ │ │ target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 │ │ │ base locus: empty subscheme of PP^3 x PP^2 │ │ │ dominance: true │ │ │ multidegree: {10, 14, 19, 25} │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html │ │ │ @@ -108,15 +108,15 @@ │ │ │ │ │ │ o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time f = X ===> Y;
│ │ │ - -- used 3.4873s (cpu); 1.91387s (thread); 0s (gc)
│ │ │ + -- used 3.31275s (cpu); 1.74919s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : f X
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time g = V ===> W;
│ │ │ - -- used 3.66337s (cpu); 2.10606s (thread); 0s (gc)
│ │ │ + -- used 3.55646s (cpu); 1.93062s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : g||W
│ │ │ @@ -257,15 +257,15 @@
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 8.41326s (cpu); 5.16056s (thread); 0s (gc)
│ │ │ + -- used 6.0498s (cpu); 4.2659s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -33,15 +33,15 @@ │ │ │ │ take(N,-2)); │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, curve in PP^8 │ │ │ │ i5 : ? X │ │ │ │ │ │ │ │ o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 │ │ │ │ i6 : time f = X ===> Y; │ │ │ │ - -- used 3.4873s (cpu); 1.91387s (thread); 0s (gc) │ │ │ │ + -- used 3.31275s (cpu); 1.74919s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 : MultirationalMap (automorphism of PP^8) │ │ │ │ i7 : f X │ │ │ │ │ │ │ │ o7 = Y │ │ │ │ │ │ │ │ o7 : ProjectiveVariety, curve in PP^8 │ │ │ │ @@ -53,15 +53,15 @@ │ │ │ │ i9 : V = random({{2},{1}},X); │ │ │ │ │ │ │ │ o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8 │ │ │ │ i10 : W = random({{2},{1}},Y); │ │ │ │ │ │ │ │ o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8 │ │ │ │ i11 : time g = V ===> W; │ │ │ │ - -- used 3.66337s (cpu); 2.10606s (thread); 0s (gc) │ │ │ │ + -- used 3.55646s (cpu); 1.93062s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 : MultirationalMap (automorphism of PP^8) │ │ │ │ i12 : g||W │ │ │ │ │ │ │ │ o12 = multi-rational map consisting of one single rational map │ │ │ │ source variety: 6-dimensional subvariety of PP^8 cut out by 2 │ │ │ │ hypersurfaces of degrees 1^1 2^1 │ │ │ │ @@ -144,15 +144,15 @@ │ │ │ │ i15 : Z = projectiveVariety pfaffians(4,A); │ │ │ │ │ │ │ │ o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9 │ │ │ │ i16 : ? Z │ │ │ │ │ │ │ │ o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2 │ │ │ │ i17 : time h = Z ===> GG_K(1,4) │ │ │ │ - -- used 8.41326s (cpu); 5.16056s (thread); 0s (gc) │ │ │ │ + -- used 6.0498s (cpu); 4.2659s (thread); 0s (gc) │ │ │ │ │ │ │ │ o17 = h │ │ │ │ │ │ │ │ o17 : MultirationalMap (isomorphism from PP^9 to PP^9) │ │ │ │ i18 : h || GG_K(1,4) │ │ │ │ │ │ │ │ o18 = multi-rational map consisting of one single rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time X = Phi^* Y;
│ │ │ - -- used 5.22987s (cpu); 3.71826s (thread); 0s (gc)
│ │ │ + -- used 3.92653s (cpu); 3.24009s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension 2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : dim X, degree X, degrees X
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random(
│ │ │ │  {1,1},ring target Phi));
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4
│ │ │ │  i4 : time X = Phi^* Y;
│ │ │ │ - -- used 5.22987s (cpu); 3.71826s (thread); 0s (gc)
│ │ │ │ + -- used 3.92653s (cpu); 3.24009s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension
│ │ │ │  2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-
│ │ │ │  degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
│ │ │ │  i5 : dim X, degree X, degrees X
│ │ │ │  
│ │ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time Phi Z;
│ │ │ - -- used 0.20103s (cpu); 0.125952s (thread); 0s (gc)
│ │ │ + -- used 0.20242s (cpu); 0.114275s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : dim oo, degree oo, degrees oo
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o6 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 2.02267s (cpu); 1.47252s (thread); 0s (gc)
│ │ │ + -- used 1.85571s (cpu); 1.17141s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 
│ │ │  
│ │ │  o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -25,24 +25,24 @@ │ │ │ │ │ │ │ │ o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^7 to PP^7 x PP^7) │ │ │ │ i4 : Z = source Phi; │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7 │ │ │ │ i5 : time Phi Z; │ │ │ │ - -- used 0.20103s (cpu); 0.125952s (thread); 0s (gc) │ │ │ │ + -- used 0.20242s (cpu); 0.114275s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7 │ │ │ │ i6 : dim oo, degree oo, degrees oo │ │ │ │ │ │ │ │ o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)}) │ │ │ │ │ │ │ │ o6 : Sequence │ │ │ │ i7 : time Phi (point Z + point Z + point Z) │ │ │ │ - -- used 2.02267s (cpu); 1.47252s (thread); 0s (gc) │ │ │ │ + -- used 1.85571s (cpu); 1.17141s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of │ │ │ │ multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 │ │ │ │ │ │ │ │ o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7 │ │ │ │ i8 : dim oo, degree oo, degrees oo │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_cm__Option_rp.html │ │ │ @@ -98,31 +98,31 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 4.01388s (cpu); 2.8819s (thread); 0s (gc)
│ │ │ + -- used 3.78363s (cpu); 2.19192s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.504268s (cpu); 0.357315s (thread); 0s (gc)
│ │ │ + -- used 0.410892s (cpu); 0.282598s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time degree Phi
│ │ │ - -- used 0.330552s (cpu); 0.2545s (thread); 0s (gc)
│ │ │ + -- used 0.208755s (cpu); 0.20931s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = 2
    │ │ │ │ │ │ │ │ │ │ │ │

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

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -27,23 +27,23 @@ │ │ │ │ │ │ │ │ o3 = multi-rational map consisting of one single rational map │ │ │ │ source variety: threefold in PP^4 x PP^4 cut out by 13 hypersurfaces of │ │ │ │ target variety: hypersurface in PP^4 defined by a form of degree 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ i4 : time degree(Phi,Strategy=>"random point") │ │ │ │ - -- used 4.01388s (cpu); 2.8819s (thread); 0s (gc) │ │ │ │ + -- used 3.78363s (cpu); 2.19192s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ │ - -- used 0.504268s (cpu); 0.357315s (thread); 0s (gc) │ │ │ │ + -- used 0.410892s (cpu); 0.282598s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 2 │ │ │ │ i6 : time degree Phi │ │ │ │ - -- used 0.330552s (cpu); 0.2545s (thread); 0s (gc) │ │ │ │ + -- used 0.208755s (cpu); 0.20931s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = 2 │ │ │ │ Note, as in the example above, that calculation times may vary depending on the │ │ │ │ strategy used. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- degree of a multi-rational map │ │ │ │ * _d_e_g_r_e_e_M_a_p_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_rp.html │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time degree Phi
│ │ │ - -- used 0.43902s (cpu); 0.366735s (thread); 0s (gc)
│ │ │ + -- used 0.378111s (cpu); 0.300798s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = 1
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -18,15 +18,15 @@ │ │ │ │ x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, │ │ │ │ x_3^2}; │ │ │ │ i2 : Phi = last graph rationalMap {f,g}; │ │ │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to │ │ │ │ PP^2 x PP^4) │ │ │ │ i3 : time degree Phi │ │ │ │ - -- used 0.43902s (cpu); 0.366735s (thread); 0s (gc) │ │ │ │ + -- used 0.378111s (cpu); 0.300798s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = 1 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_,_O_p_t_i_o_n_) -- degree of a multi-rational map using a │ │ │ │ probabilistic approach │ │ │ │ * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multirational__Map_rp.html │ │ │ @@ -82,15 +82,15 @@ │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time ? Phi
│ │ │ - -- used 0.00332575s (cpu); 0.000166912s (thread); 0s (gc)
│ │ │ + -- used 0.00178985s (cpu); 0.000175919s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    │ │ │ │ │ │ @@ -101,27 +101,27 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time ? Phi
│ │ │ - -- used 0.00316389s (cpu); 0.000237795s (thread); 0s (gc)
│ │ │ + -- used 0.00184043s (cpu); 0.000207385s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time describe Phi
│ │ │ - -- used 1.3359s (cpu); 1.00639s (thread); 0s (gc)
│ │ │ + -- used 1.0703s (cpu); 0.891102s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │ @@ -131,15 +131,15 @@
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time ? Phi
│ │ │ - -- used 0.000157475s (cpu); 0.000414858s (thread); 0s (gc)
│ │ │ + -- used 0.000122081s (cpu); 0.000415221s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,36 +16,36 @@
│ │ │ │  ? Phi is a lite version of describe Phi. The latter has a different behavior
│ │ │ │  than _d_e_s_c_r_i_b_e_(_R_a_t_i_o_n_a_l_M_a_p_), since it performs computations.
│ │ │ │  i1 : Phi = multirationalMap graph rationalMap PP_(ZZ/65521)^(1,4);
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to PP^4 x PP^5)
│ │ │ │  i2 : time ? Phi
│ │ │ │ - -- used 0.00332575s (cpu); 0.000166912s (thread); 0s (gc)
│ │ │ │ + -- used 0.00178985s (cpu); 0.000175919s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i3 : image Phi;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5
│ │ │ │  i4 : time ? Phi
│ │ │ │ - -- used 0.00316389s (cpu); 0.000237795s (thread); 0s (gc)
│ │ │ │ + -- used 0.00184043s (cpu); 0.000207385s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │  i5 : time describe Phi
│ │ │ │ - -- used 1.3359s (cpu); 1.00639s (thread); 0s (gc)
│ │ │ │ + -- used 1.0703s (cpu); 0.891102s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │  of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       multidegree: {51, 51, 51, 51, 51}
│ │ │ │       degree: 1
│ │ │ │       degree sequence (map 1/2): [(1,0), (0,2)]
│ │ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │ │       coefficient ring: ZZ/65521
│ │ │ │  i6 : time ? Phi
│ │ │ │ - -- used 0.000157475s (cpu); 0.000414858s (thread); 0s (gc)
│ │ │ │ + -- used 0.000122081s (cpu); 0.000415221s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_graph_lp__Multirational__Map_rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0281486s (cpu); 0.0267017s (thread); 0s (gc)
│ │ │ + -- used 0.035115s (cpu); 0.0226179s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -112,15 +112,15 @@ │ │ │ │ │ │ o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.134338s (cpu); 0.060805s (thread); 0s (gc)
│ │ │ + -- used 0.172939s (cpu); 0.0589556s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -136,15 +136,15 @@ │ │ │ │ │ │ o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.124474s (cpu); 0.123543s (thread); 0s (gc)
│ │ │ + -- used 0.150489s (cpu); 0.118313s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (Phi211, Phi212)
│ │ │  
│ │ │  o8 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -19,43 +19,43 @@ │ │ │ │ Phi)^-1 * (last graph Phi) == Phi are always satisfied. │ │ │ │ i1 : Phi = rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true) │ │ │ │ │ │ │ │ o1 = Phi │ │ │ │ │ │ │ │ o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5) │ │ │ │ i2 : time (Phi1,Phi2) = graph Phi │ │ │ │ - -- used 0.0281486s (cpu); 0.0267017s (thread); 0s (gc) │ │ │ │ + -- used 0.035115s (cpu); 0.0226179s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = (Phi1, Phi2) │ │ │ │ │ │ │ │ o2 : Sequence │ │ │ │ i3 : Phi1; │ │ │ │ │ │ │ │ o3 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^5 to PP^4) │ │ │ │ i4 : Phi2; │ │ │ │ │ │ │ │ o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of │ │ │ │ PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ i5 : time (Phi21,Phi22) = graph Phi2 │ │ │ │ - -- used 0.134338s (cpu); 0.060805s (thread); 0s (gc) │ │ │ │ + -- used 0.172939s (cpu); 0.0589556s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = (Phi21, Phi22) │ │ │ │ │ │ │ │ o5 : Sequence │ │ │ │ i6 : Phi21; │ │ │ │ │ │ │ │ o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5) │ │ │ │ i7 : Phi22; │ │ │ │ │ │ │ │ o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of │ │ │ │ PP^4 x PP^5 x PP^5 to hypersurface in PP^5) │ │ │ │ i8 : time (Phi211,Phi212) = graph Phi21 │ │ │ │ - -- used 0.124474s (cpu); 0.123543s (thread); 0s (gc) │ │ │ │ + -- used 0.150489s (cpu); 0.118313s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = (Phi211, Phi212) │ │ │ │ │ │ │ │ o8 : Sequence │ │ │ │ i9 : Phi211; │ │ │ │ │ │ │ │ o9 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_image_lp__Multirational__Map_rp.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ │ │ │ o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time Z = image Phi;
│ │ │ - -- used 0.103109s (cpu); 0.102376s (thread); 0s (gc)
│ │ │ + -- used 0.126629s (cpu); 0.114496s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : dim Z, degree Z, degrees Z
│ │ │ @@ -120,15 +120,15 @@
│ │ │            
│ │ │          
│ │ │          
    Alternatively, the calculation can be performed using the Segre embedding as follows:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,15 +19,15 @@
│ │ │ │  x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3,
│ │ │ │  x_3^2};
│ │ │ │  i2 : Phi = last graph rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to
│ │ │ │  PP^2 x PP^4)
│ │ │ │  i3 : time multidegree Phi
│ │ │ │ - -- used 0.453684s (cpu); 0.376266s (thread); 0s (gc)
│ │ │ │ + -- used 0.431685s (cpu); 0.312135s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {66, 46, 31, 20}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (degree source Phi,degree image Phi)
│ │ │ │  
│ │ │ │  o4 = (66, 20)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.html
│ │ │ @@ -82,29 +82,29 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 5.47152s (cpu); 2.82366s (thread); 0s (gc)
│ │ │ + -- used 8.78127s (cpu); 2.43693s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    
│ │ │              
    
│ │ │                
    i8 : assert(Z == Z')
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,26 +23,26 @@
│ │ │ │  3*x_2^2+2*x_1*x_3+x_0*x_4, 2*x_1*x_2-2*x_0*x_3, -x_1^2+x_0*x_2};
│ │ │ │  
│ │ │ │  o3 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i4 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4)
│ │ │ │  i5 : time Z = image Phi;
│ │ │ │ - -- used 0.103109s (cpu); 0.102376s (thread); 0s (gc)
│ │ │ │ + -- used 0.126629s (cpu); 0.114496s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i6 : dim Z, degree Z, degrees Z
│ │ │ │  
│ │ │ │  o6 = (4, 151, {({1, 1}, 4), ({1, 2}, 3), ({2, 0}, 5), ({2, 1}, 13)})
│ │ │ │  
│ │ │ │  o6 : Sequence
│ │ │ │  Alternatively, the calculation can be performed using the Segre embedding as
│ │ │ │  follows:
│ │ │ │  i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ │ - -- used 5.47152s (cpu); 2.82366s (thread); 0s (gc)
│ │ │ │ + -- used 8.78127s (cpu); 2.43693s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i8 : assert(Z == Z')
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y -- direct image via a multi-
│ │ │ │        rational map
│ │ │ │      * _i_m_a_g_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- closure of the image of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse2.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │              
    
│ │ │                
    i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.418413s (cpu); 0.331336s (thread); 0s (gc)
│ │ │ + -- used 0.394943s (cpu); 0.308668s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
    
│ │ │              
    
│ │ │                
    i4 : assert(Phi * Psi == 1)
    
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │              
    
│ │ │                
    i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.45295s (cpu); 1.03172s (thread); 0s (gc)
│ │ │ + -- used 1.16076s (cpu); 0.910865s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
    
│ │ │              
    
│ │ │                
    i7 : assert(Phi' * Psi' == 1)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,23 +24,23 @@
│ │ │ │  i2 : -- map defined by the cubics through the secant variety to the rational
│ │ │ │  normal curve of degree 6
│ │ │ │       Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal
│ │ │ │  PP_K([6],2));
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │ │  i3 : time Psi = inverse2 Phi;
│ │ │ │ - -- used 0.418413s (cpu); 0.331336s (thread); 0s (gc)
│ │ │ │ + -- used 0.394943s (cpu); 0.308668s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
│ │ │ │  i4 : assert(Phi * Psi == 1)
│ │ │ │  i5 : Phi' = Phi || Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │ │  i6 : time Psi' = inverse2 Phi';
│ │ │ │ - -- used 1.45295s (cpu); 1.03172s (thread); 0s (gc)
│ │ │ │ + -- used 1.16076s (cpu); 0.910865s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _<_=_=_>_ _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p -- equality of multi-rational maps
│ │ │ │        with checks on internal data
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse_lp__Multirational__Map_rp.html
│ │ │ @@ -93,45 +93,45 @@
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time inverse Phi;
│ │ │ - -- used 0.157948s (cpu); 0.0767566s (thread); 0s (gc)
│ │ │ + -- used 0.149381s (cpu); 0.0697705s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
    
│ │ │              
    
│ │ │                
    i4 : Psi = last graph Phi;
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i5 : time inverse Psi;
│ │ │ - -- used 0.186263s (cpu); 0.107471s (thread); 0s (gc)
│ │ │ + -- used 0.280991s (cpu); 0.110831s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
    
│ │ │              
    
│ │ │                
    i6 : Eta = first graph Psi;
│ │ │  
│ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
    
│ │ │              
    
│ │ │                
    i7 : time inverse Eta;
│ │ │ - -- used 0.48562s (cpu); 0.313057s (thread); 0s (gc)
│ │ │ + -- used 0.543884s (cpu); 0.300887s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
    
│ │ │              
    
│ │ │                
    i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,32 +24,32 @@
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (rational map from PP^4 to PP^5)
│ │ │ │  i2 : -- we see Phi as a dominant map
│ │ │ │       Phi = rationalMap(Phi,image Phi);
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │ │  i3 : time inverse Phi;
│ │ │ │ - -- used 0.157948s (cpu); 0.0767566s (thread); 0s (gc)
│ │ │ │ + -- used 0.149381s (cpu); 0.0697705s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4)
│ │ │ │  i4 : Psi = last graph Phi;
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to hypersurface in PP^5)
│ │ │ │  i5 : time inverse Psi;
│ │ │ │ - -- used 0.186263s (cpu); 0.107471s (thread); 0s (gc)
│ │ │ │ + -- used 0.280991s (cpu); 0.110831s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from hypersurface in PP^5 to 4-
│ │ │ │  dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i6 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5)
│ │ │ │  i7 : time inverse Eta;
│ │ │ │ - -- used 0.48562s (cpu); 0.313057s (thread); 0s (gc)
│ │ │ │ + -- used 0.543884s (cpu); 0.300887s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
│ │ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ │ │  i10 : assert(Eta * Eta^-1 == 1 and Eta^-1 * Eta == 1)
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Isomorphism_lp__Multirational__Map_rp.html
│ │ │ @@ -88,45 +88,45 @@
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │              
    
│ │ │                
    i4 : time isIsomorphism Phi
│ │ │ - -- used 0.000119785s (cpu); 1.2293e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00399681s (cpu); 2.776e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    
│ │ │              
    
│ │ │                
    i5 : Psi = first graph Phi;
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
    
│ │ │              
    
│ │ │                
    i6 : time isIsomorphism Psi
│ │ │ - -- used 0.396692s (cpu); 0.213364s (thread); 0s (gc)
│ │ │ + -- used 0.455076s (cpu); 0.189026s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = false
    
│ │ │              
    
│ │ │                
    i7 : Eta = first graph Psi;
│ │ │  
│ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x PP^3 to threefold in PP^3 x PP^2 x PP^2)
    
│ │ │              
    
│ │ │                
    i8 : time isIsomorphism Eta
│ │ │ - -- used 1.74335s (cpu); 0.941708s (thread); 0s (gc)
│ │ │ + -- used 1.63417s (cpu); 0.798647s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : assert(o8 and (not o6) and (not o4))
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,31 +17,31 @@
│ │ │ │       ZZ/33331[a..d]; f = rationalMap {c^2-b*d,b*c-a*d,b^2-a*c};
│ │ │ │  
│ │ │ │  o2 : RationalMap (quadratic rational map from PP^3 to PP^2)
│ │ │ │  i3 : Phi = rationalMap {f,f};
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │ │  i4 : time isIsomorphism Phi
│ │ │ │ - -- used 0.000119785s (cpu); 1.2293e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00399681s (cpu); 2.776e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : Psi = first graph Phi;
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to
│ │ │ │  PP^3)
│ │ │ │  i6 : time isIsomorphism Psi
│ │ │ │ - -- used 0.396692s (cpu); 0.213364s (thread); 0s (gc)
│ │ │ │ + -- used 0.455076s (cpu); 0.189026s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  i7 : Eta = first graph Psi;
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x
│ │ │ │  PP^3 to threefold in PP^3 x PP^2 x PP^2)
│ │ │ │  i8 : time isIsomorphism Eta
│ │ │ │ - -- used 1.74335s (cpu); 0.941708s (thread); 0s (gc)
│ │ │ │ + -- used 1.63417s (cpu); 0.798647s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a multi-rational map is a
│ │ │ │        morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Morphism_lp__Multirational__Map_rp.html
│ │ │ @@ -85,31 +85,31 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
│ │ │              
    
│ │ │                
    i3 : time isMorphism Phi
│ │ │ - -- used 0.386427s (cpu); 0.269828s (thread); 0s (gc)
│ │ │ + -- used 0.37364s (cpu); 0.217378s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │              
    
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.168774s (cpu); 0.0862052s (thread); 0s (gc)
│ │ │ + -- used 0.187741s (cpu); 0.0892678s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
    
│ │ │              
    
│ │ │                
    i5 : time isMorphism Psi
│ │ │ - -- used 4.42649s (cpu); 3.16319s (thread); 0s (gc)
│ │ │ + -- used 3.37906s (cpu); 2.77213s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
    
│ │ │                
    i6 : assert((not o3) and o5)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,24 +17,24 @@
│ │ │ │  i1 : ZZ/300007[a..e], f = first graph rationalMap ideal(c^2-b*d,b*c-a*d,b^2-
│ │ │ │  a*c,e), g = rationalMap submatrix(matrix f,{0..2});
│ │ │ │  i2 : Phi = rationalMap {f,g};
│ │ │ │  
│ │ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 to PP^4 x PP^2)
│ │ │ │  i3 : time isMorphism Phi
│ │ │ │ - -- used 0.386427s (cpu); 0.269828s (thread); 0s (gc)
│ │ │ │ + -- used 0.37364s (cpu); 0.217378s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.168774s (cpu); 0.0862052s (thread); 0s (gc)
│ │ │ │ + -- used 0.187741s (cpu); 0.0892678s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^7 x PP^4 x PP^2 to 4-dimensional subvariety of PP^4 x PP^7)
│ │ │ │  i5 : time isMorphism Psi
│ │ │ │ - -- used 4.42649s (cpu); 3.16319s (thread); 0s (gc)
│ │ │ │ + -- used 3.37906s (cpu); 2.77213s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : assert((not o3) and o5)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_I_s_o_m_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a birational map is an
│ │ │ │        isomorphism
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_R_a_t_i_o_n_a_l_M_a_p_) -- whether a rational map is a morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linearly__Normal__Embedding.html
│ │ │ @@ -84,30 +84,30 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │              
    
│ │ │                
    i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.00800583s (cpu); 0.00919368s (thread); 0s (gc)
│ │ │ + -- used 0.00811821s (cpu); 0.00956031s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (automorphism of X)
    
│ │ │              
    
│ │ │                
    i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3
│ │ │  
│ │ │  o4 : ProjectiveVariety, curve in PP^3
    
│ │ │              
    
│ │ │                
    i5 : time g = linearlyNormalEmbedding Y;
│ │ │ - -- used 0.709324s (cpu); 0.523301s (thread); 0s (gc)
│ │ │ + -- used 0.447966s (cpu); 0.376408s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
    
│ │ │              
    
│ │ │                
    i6 : assert(isIsomorphism g)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,23 +13,23 @@
│ │ │ │              is a linear projection
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : K = ZZ/333331;
│ │ │ │  i2 : X = PP_K^(1,7); -- rational normal curve of degree 7
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │ │  i3 : time f = linearlyNormalEmbedding X;
│ │ │ │ - -- used 0.00800583s (cpu); 0.00919368s (thread); 0s (gc)
│ │ │ │ + -- used 0.00811821s (cpu); 0.00956031s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (automorphism of X)
│ │ │ │  i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an
│ │ │ │  isomorphic projection of X in PP^3
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3
│ │ │ │  i5 : time g = linearlyNormalEmbedding Y;
│ │ │ │ - -- used 0.709324s (cpu); 0.523301s (thread); 0s (gc)
│ │ │ │ + -- used 0.447966s (cpu); 0.376408s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │ │  i6 : assert(isIsomorphism g)
│ │ │ │  i7 : describe g
│ │ │ │  
│ │ │ │  o7 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: curve in PP^3 cut out by 6 hypersurfaces of degree 4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Multirational__Map_rp.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │              
    
│ │ │                
    i3 : time multidegree Phi
│ │ │ - -- used 0.453684s (cpu); 0.376266s (thread); 0s (gc)
│ │ │ + -- used 0.431685s (cpu); 0.312135s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {66, 46, 31, 20}
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi)
│ │ │ - -- used 0.00102199s (cpu); 0.00133739s (thread); 0s (gc)
│ │ │ - -- used 0.222006s (cpu); 0.14506s (thread); 0s (gc)
│ │ │ - -- used 0.340129s (cpu); 0.192965s (thread); 0s (gc)
│ │ │ - -- used 0.227149s (cpu); 0.156054s (thread); 0s (gc)
│ │ │ - -- used 0.198029s (cpu); 0.122886s (thread); 0s (gc)
│ │ │ + -- used 0.00403651s (cpu); 0.00150351s (thread); 0s (gc)
│ │ │ + -- used 0.18919s (cpu); 0.127475s (thread); 0s (gc)
│ │ │ + -- used 0.303532s (cpu); 0.176257s (thread); 0s (gc)
│ │ │ + -- used 0.200725s (cpu); 0.14139s (thread); 0s (gc)
│ │ │ + -- used 0.185499s (cpu); 0.118796s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.164641s (cpu); 0.0837544s (thread); 0s (gc)
    
│ │ │ + -- used 0.183934s (cpu); 0.0820254s (thread); 0s (gc)
│ │ │              
    
│ │ │
    │ │ │
    │ │ │

    References

    │ │ │ ArXiv preprint: Computations with rational maps between multi-projective varieties.
    │ │ │ ├── html2text {} │ │ │ │ @@ -17,25 +17,25 @@ │ │ │ │ This is calculated by means of the inverse image of an appropriate random │ │ │ │ subvariety of the target. │ │ │ │ i1 : Phi = last graph rationalMap PP_(ZZ/300007)^(1,4); │ │ │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x │ │ │ │ PP^5 to PP^5) │ │ │ │ i2 : for i in {4,3,2,1,0} list time multidegree(i,Phi) │ │ │ │ - -- used 0.00102199s (cpu); 0.00133739s (thread); 0s (gc) │ │ │ │ - -- used 0.222006s (cpu); 0.14506s (thread); 0s (gc) │ │ │ │ - -- used 0.340129s (cpu); 0.192965s (thread); 0s (gc) │ │ │ │ - -- used 0.227149s (cpu); 0.156054s (thread); 0s (gc) │ │ │ │ - -- used 0.198029s (cpu); 0.122886s (thread); 0s (gc) │ │ │ │ + -- used 0.00403651s (cpu); 0.00150351s (thread); 0s (gc) │ │ │ │ + -- used 0.18919s (cpu); 0.127475s (thread); 0s (gc) │ │ │ │ + -- used 0.303532s (cpu); 0.176257s (thread); 0s (gc) │ │ │ │ + -- used 0.200725s (cpu); 0.14139s (thread); 0s (gc) │ │ │ │ + -- used 0.185499s (cpu); 0.118796s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ │ - -- used 0.164641s (cpu); 0.0837544s (thread); 0s (gc) │ │ │ │ + -- used 0.183934s (cpu); 0.0820254s (thread); 0s (gc) │ │ │ │ ********** RReeffeerreenncceess ********** │ │ │ │ ArXiv preprint: _C_o_m_p_u_t_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _m_a_p_s_ _b_e_t_w_e_e_n_ _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e │ │ │ │ _v_a_r_i_e_t_i_e_s. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ │ │ map │ │ │ │ * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s_(_R_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_point_lp__Multiprojective__Variety_rp.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time p := point X
│ │ │ - -- used 0.0200872s (cpu); 0.0193746s (thread); 0s (gc)
│ │ │ + -- used 0.0667782s (cpu); 0.0233333s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -102,15 +102,15 @@ │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time q = point Y
│ │ │ - -- used 2.20856s (cpu); 1.14282s (thread); 0s (gc)
│ │ │ + -- used 1.49071s (cpu); 0.91067s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = q
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -14,25 +14,25 @@ │ │ │ │ o a _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e_ _v_a_r_i_e_t_y, a random rational point on $X$ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : K = ZZ/1000003; │ │ │ │ i2 : X = PP_K^({1,1,2},{3,2,3}); │ │ │ │ │ │ │ │ o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9 │ │ │ │ i3 : time p := point X │ │ │ │ - -- used 0.0200872s (cpu); 0.0193746s (thread); 0s (gc) │ │ │ │ + -- used 0.0667782s (cpu); 0.0233333s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1], │ │ │ │ [3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1]) │ │ │ │ │ │ │ │ o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ i4 : Y = random({2,1,2},X); │ │ │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ i5 : time q = point Y │ │ │ │ - -- used 2.20856s (cpu); 1.14282s (thread); 0s (gc) │ │ │ │ + -- used 1.49071s (cpu); 0.91067s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = q │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ i6 : assert(isSubset(p,X) and isSubset(q,Y)) │ │ │ │ The list of homogeneous coordinates can be obtained with the operator |-. │ │ │ │ i7 : |- p │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html │ │ │ @@ -106,15 +106,15 @@ │ │ │ │ │ │ o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time segre Phi;
│ │ │ - -- used 0.774927s (cpu); 0.592841s (thread); 0s (gc)
│ │ │ + -- used 1.10483s (cpu); 0.545907s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : describe segre Phi
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o4 : RationalMap (quadratic rational map from PP^4 to PP^4)
│ │ │ │  i5 : Phi = rationalMap {f,g,h};
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x
│ │ │ │  PP^4)
│ │ │ │  i6 : time segre Phi;
│ │ │ │ - -- used 0.774927s (cpu); 0.592841s (thread); 0s (gc)
│ │ │ │ + -- used 1.10483s (cpu); 0.545907s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │ │  i7 : describe segre Phi
│ │ │ │  
│ │ │ │  o7 = rational map defined by forms of degree 6
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^149
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time describe Phi
│ │ │ - -- used 0.410659s (cpu); 0.270055s (thread); 0s (gc)
│ │ │ + -- used 0.300024s (cpu); 0.161044s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1)
│ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 
│ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ │       dominance: true
│ │ │       multidegree: {10, 14, 19, 25}
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  i1 : Phi = inverse first graph last graph rationalMap PP_(ZZ/33331)^(1,3)
│ │ │ │  
│ │ │ │  o1 = Phi
│ │ │ │  
│ │ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to
│ │ │ │  threefold in PP^3 x PP^2 x PP^2)
│ │ │ │  i2 : time describe Phi
│ │ │ │ - -- used 0.410659s (cpu); 0.270055s (thread); 0s (gc)
│ │ │ │ + -- used 0.300024s (cpu); 0.161044s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = multi-rational map consisting of 3 rational maps
│ │ │ │       source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of
│ │ │ │  multi-degree (1,1)
│ │ │ │       target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces
│ │ │ │  of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2
│ │ │ │       base locus: empty subscheme of PP^3 x PP^2
│ │ ├── ./usr/share/doc/Macaulay2/NAGtypes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  UG9seVNwYWNl
│ │ │  #:len=827
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwb2x5bm9taWFsIHZlY3RvciBzdWJz
│ │ │  cGFjZSIsICJsaW5lbnVtIiA9PiA4NTcsIFNlZUFsc28gPT4gRElWe0hFQURFUjJ7IlNlZSBhbHNv
│ │ ├── ./usr/share/doc/Macaulay2/NCAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  UVEgJSBOQ0dyb2VibmVyQmFzaXM=
│ │ │  #:len=317
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTY3MCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sICUsUVEsTkNHcm9lYm5lckJhc2lzKSwi
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  Z2VuZXJhdGVSYW5kb21HcmFwaHMoWlosWlosWlop
│ │ │  #:len=275
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTExMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2VuZXJhdGVSYW5kb21HcmFwaHMsWlosWlosWlop
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
│ │ │ @@ -26,22 +26,22 @@
│ │ │  
│ │ │  i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true};
│ │ │  
│ │ │  i8 : prob = n -> log(n)/n;
│ │ │  
│ │ │  i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (63, 89, 84, 84, 91, 94, 100, 96, 95, 95, 95, 98, 98, 96, 100, 99, 96,
│ │ │ +o9 = (68, 81, 91, 90, 94, 92, 96, 97, 96, 99, 99, 96, 97, 97, 99, 99, 97, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     100, 97, 99, 98, 97, 100, 99, 94, 95, 98, 99, 99)
│ │ │ +     99, 100, 99, 98, 100, 100, 100, 98, 100, 100, 99)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (12, 12, 6, 4, 3, 5, 2, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 0,
│ │ │ +o10 = (14, 5, 2, 4, 2, 4, 1, 3, 0, 3, 1, 1, 0, 4, 0, 0, 0, 1, 1, 0, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 0, 0, 0, 0)
│ │ │ +      1, 0, 0, 1, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Graphs.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │  
│ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {D~{, D`{, DFC, D^S, D{G}
│ │ │ +o2 = {Dfc, Dmw, Di[, Dq?, DS{}
│ │ │  
│ │ │  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
│ │ │ @@ -4,18 +4,18 @@
│ │ │  
│ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │  o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out
│ │ │ @@ -13,13 +13,13 @@
│ │ │  i3 : graphComplement "Dhc"
│ │ │  
│ │ │  o3 = DUW
│ │ │  
│ │ │  i4 : G = generateBipartiteGraphs 7;
│ │ │  
│ │ │  i5 : time graphComplement G;
│ │ │ - -- used 0.000443261s (cpu); 0.000438352s (thread); 0s (gc)
│ │ │ + -- used 0.000497855s (cpu); 0.000483802s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0520587s (cpu); 0.0501016s (thread); 0s (gc)
│ │ │ + -- used 0.0595181s (cpu); 0.0571237s (thread); 0s (gc)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -122,28 +122,28 @@
│ │ │                
    i8 : prob = n -> log(n)/n;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (63, 89, 84, 84, 91, 94, 100, 96, 95, 95, 95, 98, 98, 96, 100, 99, 96,
│ │ │ +o9 = (68, 81, 91, 90, 94, 92, 96, 97, 96, 99, 99, 96, 97, 97, 99, 99, 97, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     100, 97, 99, 98, 97, 100, 99, 94, 95, 98, 99, 99)
│ │ │ +     99, 100, 99, 98, 100, 100, 100, 98, 100, 100, 99)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (12, 12, 6, 4, 3, 5, 2, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 0,
│ │ │ +o10 = (14, 5, 2, 4, 2, 4, 1, 3, 0, 3, 1, 1, 0, 4, 0, 0, 0, 1, 1, 0, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 0, 0, 0, 0)
│ │ │ +      1, 0, 0, 1, 0, 0)
│ │ │  
│ │ │  o10 : Sequence
    
│ │ │              
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -38,25 +38,25 @@ │ │ │ │ connected, at least as $n$ tends to infinity. │ │ │ │ i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => │ │ │ │ true}; │ │ │ │ i8 : prob = n -> log(n)/n; │ │ │ │ i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o9 = (63, 89, 84, 84, 91, 94, 100, 96, 95, 95, 95, 98, 98, 96, 100, 99, 96, │ │ │ │ +o9 = (68, 81, 91, 90, 94, 92, 96, 97, 96, 99, 99, 96, 97, 97, 99, 99, 97, 98, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 100, 97, 99, 98, 97, 100, 99, 94, 95, 98, 99, 99) │ │ │ │ + 99, 100, 99, 98, 100, 100, 100, 98, 100, 100, 99) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (12, 12, 6, 4, 3, 5, 2, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 0, │ │ │ │ +o10 = (14, 5, 2, 4, 2, 4, 1, 3, 0, 3, 1, 1, 0, 4, 0, 0, 0, 1, 1, 0, 0, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 1, 0, 0, 0, 0, 0) │ │ │ │ + 1, 0, 0, 1, 0, 0) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with │ │ │ │ filterGraphs and countGraphs │ │ │ │ * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given │ │ │ │ properties │ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Graphs.html │ │ │ @@ -105,15 +105,15 @@ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {D~{, D`{, DFC, D^S, D{G}
│ │ │ +o2 = {Dfc, Dmw, Di[, Dq?, DS{}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {D~{, D`{, DFC, D^S, D{G}
│ │ │ │ +o2 = {Dfc, Dmw, Di[, Dq?, DS{}
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -96,19 +96,19 @@
│ │ │              
│ │ │                
    i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │  o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}}}
│ │ │             "ring" => R
│ │ │             "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,19 +28,19 @@
│ │ │ │  i1 : R = QQ[a..e];
│ │ │ │  i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │ │  
│ │ │ │  o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │ │              "ring" => R
│ │ │ │              "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ │ +     Graph{"edges" => {{b, c}, {a, d}, {c, d}, {a, e}, {b, e}}},
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}}}
│ │ │ │ +     Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, e}, {d, e}}}}
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The number of vertices $n$ must be positive as nauty cannot handle graphs with
│ │ │ │  zero vertices.
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_graph__Complement.html
│ │ │ @@ -121,21 +121,21 @@
│ │ │              
│ │ │                
    i4 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time graphComplement G;
│ │ │ - -- used 0.000443261s (cpu); 0.000438352s (thread); 0s (gc)
    
│ │ │ + -- used 0.000497855s (cpu); 0.000483802s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0520587s (cpu); 0.0501016s (thread); 0s (gc)
    │ │ │ + -- used 0.0595181s (cpu); 0.0571237s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -41,17 +41,17 @@ │ │ │ │ │ │ │ │ o3 = DUW │ │ │ │ Batch calls can be performed considerably faster when using the List input │ │ │ │ format. However, care should be taken as the returned list is entirely in │ │ │ │ Graph6 or Sparse6 format. │ │ │ │ i4 : G = generateBipartiteGraphs 7; │ │ │ │ i5 : time graphComplement G; │ │ │ │ - -- used 0.000443261s (cpu); 0.000438352s (thread); 0s (gc) │ │ │ │ + -- used 0.000497855s (cpu); 0.000483802s (thread); 0s (gc) │ │ │ │ i6 : time (graphComplement \ G); │ │ │ │ - -- used 0.0520587s (cpu); 0.0501016s (thread); 0s (gc) │ │ │ │ + -- used 0.0595181s (cpu); 0.0571237s (thread); 0s (gc) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_o_m_p_l_e_m_e_n_t_G_r_a_p_h -- returns the complement of a graph or hypergraph │ │ │ │ ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: ********** │ │ │ │ * graphComplement(Graph) │ │ │ │ * graphComplement(List) │ │ │ │ * graphComplement(String) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ YWRkRWRnZXMoLi4uLE5vTmV3NUN5Y2xlcz0+Li4uKQ== │ │ │ #:len=261 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1thZGRFZGdlcyxOb05ldzVDeWNsZXNdLCJhZGRFZGdl │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out │ │ │ @@ -26,22 +26,22 @@ │ │ │ │ │ │ i7 : connected = buildGraphFilter {"Connectivity" => 0, "NegateConnectivity" => true}; │ │ │ │ │ │ i8 : prob = n -> log(n)/n; │ │ │ │ │ │ i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected)) │ │ │ │ │ │ -o9 = (71, 75, 84, 96, 95, 97, 96, 98, 96, 94, 93, 98, 98, 99, 96, 95, 96, 96, │ │ │ +o9 = (67, 85, 91, 91, 93, 96, 95, 95, 94, 98, 98, 98, 95, 98, 97, 95, 100, │ │ │ ------------------------------------------------------------------------ │ │ │ - 97, 98, 99, 98, 97, 97, 98, 94, 100, 98, 98) │ │ │ + 98, 97, 99, 97, 98, 99, 99, 98, 99, 99, 98, 97) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected)) │ │ │ │ │ │ -o10 = (11, 14, 6, 2, 2, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, │ │ │ +o10 = (17, 10, 10, 1, 8, 4, 1, 1, 2, 1, 3, 4, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, │ │ │ ----------------------------------------------------------------------- │ │ │ - 0, 1, 0, 0, 0, 1) │ │ │ + 0, 1, 0, 0, 0, 0, 0) │ │ │ │ │ │ o10 : Sequence │ │ │ │ │ │ i11 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ │ │ │ o1 = {DDO, Dx_, Dlw, Dx{, D_K} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : generateRandomGraphs(5, 5) │ │ │ │ │ │ -o2 = {DJc, DRK, DhS, D_s, DkS} │ │ │ +o2 = {D^k, Dqo, Da[, DvS, Dt[} │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : generateRandomGraphs(5, 5, RandomSeed => 314159) │ │ │ │ │ │ o3 = {DDO, Dx_, Dlw, Dx{, D_K} │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out │ │ │ @@ -1,9 +1,9 @@ │ │ │ -- -*- M2-comint -*- hash: 1331287392268 │ │ │ │ │ │ i1 : generateRandomRegularGraphs(5, 3, 2) │ │ │ │ │ │ -o1 = {DkK, DUW, DLo} │ │ │ +o1 = {Dhc, Dbg, DpS} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out │ │ │ @@ -13,13 +13,13 @@ │ │ │ 4 => {2, 1} │ │ │ │ │ │ o2 : Graph │ │ │ │ │ │ i3 : G = generateBipartiteGraphs 7; │ │ │ │ │ │ i4 : time graphComplement G; │ │ │ - -- used 0.000506259s (cpu); 0.000493143s (thread); 0s (gc) │ │ │ + -- used 0.000506182s (cpu); 0.000490205s (thread); 0s (gc) │ │ │ │ │ │ i5 : time (graphComplement \ G); │ │ │ - -- used 0.0534439s (cpu); 0.0512922s (thread); 0s (gc) │ │ │ + -- used 0.0585763s (cpu); 0.056209s (thread); 0s (gc) │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ @@ -122,28 +122,28 @@ │ │ │ 
      i8 : prob = n -> log(n)/n;
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (71, 75, 84, 96, 95, 97, 96, 98, 96, 94, 93, 98, 98, 99, 96, 95, 96, 96,
│ │ │ +o9 = (67, 85, 91, 91, 93, 96, 95, 95, 94, 98, 98, 98, 95, 98, 97, 95, 100,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 98, 99, 98, 97, 97, 98, 94, 100, 98, 98)
│ │ │ +     98, 97, 99, 97, 98, 99, 99, 98, 99, 99, 98, 97)
│ │ │  
│ │ │  o9 : Sequence
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (11, 14, 6, 2, 2, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,
│ │ │ +o10 = (17, 10, 10, 1, 8, 4, 1, 1, 2, 1, 3, 4, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 1, 0, 0, 0, 1)
│ │ │ +      0, 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 = (71, 75, 84, 96, 95, 97, 96, 98, 96, 94, 93, 98, 98, 99, 96, 95, 96, 96, │ │ │ │ +o9 = (67, 85, 91, 91, 93, 96, 95, 95, 94, 98, 98, 98, 95, 98, 97, 95, 100, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 97, 98, 99, 98, 97, 97, 98, 94, 100, 98, 98) │ │ │ │ + 98, 97, 99, 97, 98, 99, 99, 98, 99, 99, 98, 97) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (11, 14, 6, 2, 2, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, │ │ │ │ +o10 = (17, 10, 10, 1, 8, 4, 1, 1, 2, 1, 3, 4, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 0, 1, 0, 0, 0, 1) │ │ │ │ + 0, 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/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ @@ -98,15 +98,15 @@ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DJc, DRK, DhS, D_s, DkS}
│ │ │ +o2 = {D^k, Dqo, Da[, DvS, Dt[}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,15 +30,15 @@
│ │ │ │  i1 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o1 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : generateRandomGraphs(5, 5)
│ │ │ │  
│ │ │ │ -o2 = {DJc, DRK, DhS, D_s, DkS}
│ │ │ │ +o2 = {D^k, Dqo, Da[, DvS, Dt[}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │            
    This method generates a specified number of random graphs on a given number of vertices with a given regularity. Note that some graphs may be isomorphic.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │  
│ │ │ -o1 = {DkK, DUW, DLo}
│ │ │ +o1 = {Dhc, Dbg, DpS}
│ │ │  
│ │ │  o1 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -18,15 +18,15 @@ │ │ │ │ * Outputs: │ │ │ │ o G, a _l_i_s_t, the randomly generated regular graphs │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ This method generates a specified number of random graphs on a given number of │ │ │ │ vertices with a given regularity. Note that some graphs may be isomorphic. │ │ │ │ i1 : generateRandomRegularGraphs(5, 3, 2) │ │ │ │ │ │ │ │ -o1 = {DkK, DUW, DLo} │ │ │ │ +o1 = {Dhc, Dbg, DpS} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The number of vertices $n$ must be positive as nauty cannot handle graphs with │ │ │ │ zero vertices. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_a_t_e_R_a_n_d_o_m_G_r_a_p_h_s -- generates random graphs on a given number of │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ @@ -115,21 +115,21 @@ │ │ │ │ │ │ 
    i3 : G = generateBipartiteGraphs 7;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time graphComplement G;
│ │ │ - -- used 0.000506259s (cpu); 0.000493143s (thread); 0s (gc)
    │ │ │ + -- used 0.000506182s (cpu); 0.000490205s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.0534439s (cpu); 0.0512922s (thread); 0s (gc)
    │ │ │ + -- used 0.0585763s (cpu); 0.056209s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use graphComplement:

    │ │ │ ├── html2text {} │ │ │ │ @@ -38,17 +38,17 @@ │ │ │ │ │ │ │ │ o2 : Graph │ │ │ │ Batch calls can be performed considerably faster when using the List input │ │ │ │ format. However, care should be taken as the returned list is entirely in │ │ │ │ Graph6 or Sparse6 format. │ │ │ │ i3 : G = generateBipartiteGraphs 7; │ │ │ │ i4 : time graphComplement G; │ │ │ │ - -- used 0.000506259s (cpu); 0.000493143s (thread); 0s (gc) │ │ │ │ + -- used 0.000506182s (cpu); 0.000490205s (thread); 0s (gc) │ │ │ │ i5 : time (graphComplement \ G); │ │ │ │ - -- used 0.0534439s (cpu); 0.0512922s (thread); 0s (gc) │ │ │ │ + -- used 0.0585763s (cpu); 0.056209s (thread); 0s (gc) │ │ │ │ ********** WWaayyss ttoo uussee ggrraapphhCCoommpplleemmeenntt:: ********** │ │ │ │ * graphComplement(Graph) │ │ │ │ * graphComplement(List) │ │ │ │ * graphComplement(String) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _g_r_a_p_h_C_o_m_p_l_e_m_e_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/NeuralIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ aXNDYW5vbmljYWwoLi4uLFBvbGFyaXplZD0+Li4uKQ== │ │ │ #:len=270 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTI5LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tpc0Nhbm9uaWNhbCxQb2xhcml6ZWRdLCJpc0Nhbm9u │ │ ├── ./usr/share/doc/Macaulay2/NoetherNormalization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=38 │ │ │ bm9ldGhlck5vcm1hbGl6YXRpb24oLi4uLFZlcmJvc2U9Pi4uLik= │ │ │ #:len=326 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzIwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tub2V0aGVyTm9ybWFsaXphdGlvbixWZXJib3NlXSwi │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ bm9ldGhlcmlhbk9wZXJhdG9ycyhJZGVhbCxJZGVhbCk= │ │ │ #:len=2583 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTm9ldGhlcmlhbiBvcGVyYXRvcnMgb2Yg │ │ │ YSBwcmltYXJ5IGNvbXBvbmVudCIsICJsaW5lbnVtIiA9PiAyNzQ5LCBJbnB1dHMgPT4ge1NQQU57 │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ @@ -47,15 +47,15 @@ │ │ │ o4 : Ideal of R │ │ │ │ │ │ i5 : isPrimary Q │ │ │ │ │ │ o5 = true │ │ │ │ │ │ i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot") │ │ │ - -- .167714s elapsed │ │ │ + -- .0856696s elapsed │ │ │ │ │ │ o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, | │ │ │ ------------------------------------------------------------------------ │ │ │ 2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |} │ │ │ │ │ │ o6 : List │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ @@ -125,15 +125,15 @@ │ │ │ │ │ │ o5 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .167714s elapsed
│ │ │ + -- .0856696s elapsed
│ │ │  
│ │ │  o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, |
│ │ │       ------------------------------------------------------------------------
│ │ │       2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |}
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -51,15 +51,15 @@ │ │ │ │ 1 2 3 2 3 │ │ │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : isPrimary Q │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot") │ │ │ │ - -- .167714s elapsed │ │ │ │ + -- .0856696s elapsed │ │ │ │ │ │ │ │ o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/NonPrincipalTestIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=24 │ │ │ aXNJbnZlcnRpYmxlSWRlYWwoSWRlYWwp │ │ │ #:len=308 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIwNiwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNJbnZlcnRpYmxlSWRlYWwsSWRlYWwpLCJpc0lu │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ VG9yaWNEaXZpc29yID09IFpa │ │ │ #:len=349 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTQ0MSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sID09LFRvcmljRGl2aXNvcixaWiksIlRv │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ @@ -78,15 +78,15 @@ │ │ │ i13 : for i to dim X list hilbertFunction (i, A1) │ │ │ │ │ │ o13 = {1, 2, 3, 3, 2, 1} │ │ │ │ │ │ o13 : List │ │ │ │ │ │ i14 : Y = time smoothFanoToricVariety(5,100); │ │ │ - -- used 0.212369s (cpu); 0.212369s (thread); 0s (gc) │ │ │ + -- used 0.226858s (cpu); 0.226859s (thread); 0s (gc) │ │ │ │ │ │ i15 : A2 = intersectionRing Y; │ │ │ │ │ │ i16 : assert (# rays Y === numgens A2) │ │ │ │ │ │ i17 : ideal A2 │ │ │ │ │ │ @@ -110,19 +110,19 @@ │ │ │ 2 2 2 2 2 2 2 2 3 2 │ │ │ (t + t t , t t + t , t + t t , t t , t t + t , t - t t - 3t t + t t + 2t , - t t + t + 2t t , t t , - t t + t , t t ) │ │ │ 3 3 5 3 5 5 5 5 6 3 6 5 6 6 8 8 9 8 10 9 10 10 8 9 9 9 10 8 9 8 10 10 8 10 │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ │ │ i19 : for i to dim Y list time hilbertFunction (i, A2) │ │ │ - -- used 0.00146338s (cpu); 0.00146077s (thread); 0s (gc) │ │ │ - -- used 8.8586e-05s (cpu); 8.8917e-05s (thread); 0s (gc) │ │ │ - -- used 8.0521e-05s (cpu); 8.0932e-05s (thread); 0s (gc) │ │ │ - -- used 8.6783e-05s (cpu); 8.7574e-05s (thread); 0s (gc) │ │ │ - -- used 8.3096e-05s (cpu); 8.3456e-05s (thread); 0s (gc) │ │ │ - -- used 8.5891e-05s (cpu); 8.6282e-05s (thread); 0s (gc) │ │ │ + -- used 0.00134762s (cpu); 0.00134452s (thread); 0s (gc) │ │ │ + -- used 0.000112183s (cpu); 0.000112429s (thread); 0s (gc) │ │ │ + -- used 9.8641e-05s (cpu); 9.8918e-05s (thread); 0s (gc) │ │ │ + -- used 8.1925e-05s (cpu); 8.2077e-05s (thread); 0s (gc) │ │ │ + -- used 9.0115e-05s (cpu); 9.0323e-05s (thread); 0s (gc) │ │ │ + -- used 8.5594e-05s (cpu); 8.5917e-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 1792812560 │ │ │ + -- setting random seed to 1793276221 │ │ │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ -o3 = {2, 4, 6} │ │ │ +o3 = {1, 2, 5} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i7 : q │ │ │ │ │ │ -o7 = {1, 5, 6, 9, 9} │ │ │ +o7 = {2, 3, 3, 6, 7} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ @@ -6,61 +6,61 @@ │ │ │ │ │ │ o2 = 5*PP2 │ │ │ 0 │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │ │ │ │ i3 : M1 = elapsedTime monomials D1 │ │ │ - -- .033906s elapsed │ │ │ + -- .027659s elapsed │ │ │ │ │ │ 5 4 4 2 3 3 2 3 3 2 2 2 2 2 3 2 4 │ │ │ o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x , │ │ │ 2 1 2 0 2 1 2 0 1 2 0 2 1 2 0 1 2 0 1 2 0 2 1 2 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 2 2 3 4 5 4 2 3 3 2 4 5 │ │ │ x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x } │ │ │ 0 1 2 0 1 2 0 1 2 0 2 1 0 1 0 1 0 1 0 1 0 │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1)) │ │ │ - -- .0131253s elapsed │ │ │ + -- .00117399s elapsed │ │ │ │ │ │ i5 : FF2 = hirzebruchSurface 2; │ │ │ │ │ │ i6 : D2 = 2*FF2_0 + 3 * FF2_1 │ │ │ │ │ │ o6 = 2*FF2 + 3*FF2 │ │ │ 0 1 │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ │ │ i7 : M2 = elapsedTime monomials D2 │ │ │ - -- .134092s elapsed │ │ │ + -- .0684969s elapsed │ │ │ │ │ │ 2 3 2 3 2 3 │ │ │ o7 = {x x , x x , x x x , x x } │ │ │ 1 3 1 2 0 1 2 0 1 │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2)) │ │ │ - -- .00103874s elapsed │ │ │ + -- .00111672s elapsed │ │ │ │ │ │ i9 : X = kleinschmidt (5, {1,2,3}); │ │ │ │ │ │ i10 : D3 = 3*X_0 + 5*X_1 │ │ │ │ │ │ o10 = 3*X + 5*X │ │ │ 0 1 │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ │ │ i11 : m3 = elapsedTime # monomials D3 │ │ │ - -- 46.8661s elapsed │ │ │ + -- 25.6719s elapsed │ │ │ │ │ │ o11 = 7909 │ │ │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3)) │ │ │ - -- .183473s elapsed │ │ │ + -- .0310593s elapsed │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ @@ -24,19 +24,19 @@ │ │ │ o3 : List │ │ │ │ │ │ i4 : X = normalToricVariety F; │ │ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ - -- used 5.1586e-05s (cpu); 4.5906e-05s (thread); 0s (gc) │ │ │ + -- used 3.2976e-05s (cpu); 2.584e-05s (thread); 0s (gc) │ │ │ │ │ │ o6 = X1 │ │ │ │ │ │ o6 : NormalToricVariety │ │ │ │ │ │ i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}}; │ │ │ - -- used 0.0871239s (cpu); 0.08713s (thread); 0s (gc) │ │ │ + -- used 0.0484872s (cpu); 0.0484955s (thread); 0s (gc) │ │ │ │ │ │ i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2) │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ @@ -88,15 +88,15 @@ │ │ │ o18 = | 0 1 0 | │ │ │ | 0 0 1 | │ │ │ │ │ │ 2 3 │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ │ │ i19 : X1 = time normalToricVariety convexHull (vertMatrix); │ │ │ - -- used 0.023116s (cpu); 0.0231153s (thread); 0s (gc) │ │ │ + -- used 0.024974s (cpu); 0.0249746s (thread); 0s (gc) │ │ │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ - -- used 0.00248959s (cpu); 0.00249056s (thread); 0s (gc) │ │ │ + -- used 0.00263933s (cpu); 0.00264461s (thread); 0s (gc) │ │ │ │ │ │ i21 : assert (set rays X2 === set rays X1 and max X1 === max X2) │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ @@ -212,15 +212,15 @@ │ │ │
    │ │ │

    We end with a slightly larger example.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -259,20 +259,20 @@ │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.212369s (cpu); 0.212369s (thread); 0s (gc)
    │ │ │ + -- used 0.226858s (cpu); 0.226859s (thread); 0s (gc) │ │ │
    │ │ │ 
    i15 : A2 = intersectionRing Y;
    │ │ │
    │ │ │ 
    i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00146338s (cpu); 0.00146077s (thread); 0s (gc)
│ │ │ - -- used 8.8586e-05s (cpu); 8.8917e-05s (thread); 0s (gc)
│ │ │ - -- used 8.0521e-05s (cpu); 8.0932e-05s (thread); 0s (gc)
│ │ │ - -- used 8.6783e-05s (cpu); 8.7574e-05s (thread); 0s (gc)
│ │ │ - -- used 8.3096e-05s (cpu); 8.3456e-05s (thread); 0s (gc)
│ │ │ - -- used 8.5891e-05s (cpu); 8.6282e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00134762s (cpu); 0.00134452s (thread); 0s (gc)
│ │ │ + -- used 0.000112183s (cpu); 0.000112429s (thread); 0s (gc)
│ │ │ + -- used 9.8641e-05s (cpu); 9.8918e-05s (thread); 0s (gc)
│ │ │ + -- used 8.1925e-05s (cpu); 8.2077e-05s (thread); 0s (gc)
│ │ │ + -- used 9.0115e-05s (cpu); 9.0323e-05s (thread); 0s (gc)
│ │ │ + -- used 8.5594e-05s (cpu); 8.5917e-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 0.212369s (cpu); 0.212369s (thread); 0s (gc) │ │ │ │ + -- used 0.226858s (cpu); 0.226859s (thread); 0s (gc) │ │ │ │ i15 : A2 = intersectionRing Y; │ │ │ │ i16 : assert (# rays Y === numgens A2) │ │ │ │ i17 : ideal A2 │ │ │ │ │ │ │ │ o17 = ideal (t t , t t , t t , t t , t t , t t , t t , t t , t t t , │ │ │ │ 2 3 2 5 4 5 3 6 4 6 1 7 7 9 8 9 0 1 10 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ @@ -129,20 +129,20 @@ │ │ │ │ (t + t t , t t + t , t + t t , t t , t t + t , t - t t - 3t t + t │ │ │ │ t + 2t , - t t + t + 2t t , t t , - t t + t , t t ) │ │ │ │ 3 3 5 3 5 5 5 5 6 3 6 5 6 6 8 8 9 8 10 │ │ │ │ 9 10 10 8 9 9 9 10 8 9 8 10 10 8 10 │ │ │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ i19 : for i to dim Y list time hilbertFunction (i, A2) │ │ │ │ - -- used 0.00146338s (cpu); 0.00146077s (thread); 0s (gc) │ │ │ │ - -- used 8.8586e-05s (cpu); 8.8917e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.0521e-05s (cpu); 8.0932e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.6783e-05s (cpu); 8.7574e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.3096e-05s (cpu); 8.3456e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.5891e-05s (cpu); 8.6282e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00134762s (cpu); 0.00134452s (thread); 0s (gc) │ │ │ │ + -- used 0.000112183s (cpu); 0.000112429s (thread); 0s (gc) │ │ │ │ + -- used 9.8641e-05s (cpu); 9.8918e-05s (thread); 0s (gc) │ │ │ │ + -- used 8.1925e-05s (cpu); 8.2077e-05s (thread); 0s (gc) │ │ │ │ + -- used 9.0115e-05s (cpu); 9.0323e-05s (thread); 0s (gc) │ │ │ │ + -- used 8.5594e-05s (cpu); 8.5917e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ │ │ o19 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _s_h_e_a_v_e_s -- information about coherent sheaves and total │ │ │ │ coordinate rings (a.k.a. Cox rings) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_is__Well__Defined_lp__Normal__Toric__Variety_rp.html │ │ │ @@ -98,22 +98,22 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1792812560
    │ │ │ + -- setting random seed to 1793276221 │ │ │
    │ │ │ 
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {2, 4, 6}
│ │ │ +o3 = {1, 2, 5}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i4 : assert isWellDefined kleinschmidt (4,a)
    │ │ │ @@ -131,15 +131,15 @@ │ │ │ 
    i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
    │ │ │
    │ │ │ 
    i7 : q
│ │ │  
│ │ │ -o7 = {1, 5, 6, 9, 9}
│ │ │ +o7 = {2, 3, 3, 6, 7}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ 
    i8 : assert isWellDefined weightedProjectiveSpace q
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,27 +28,27 @@ │ │ │ │ * the intersection of the cones associated to two elements of coneList is a │ │ │ │ face of each cone. │ │ │ │ The first examples illustrate that small projective spaces are well-defined. │ │ │ │ i1 : assert all (5, d -> isWellDefined toricProjectiveSpace (d+1)) │ │ │ │ The second examples show that a randomly selected Kleinschmidt toric variety │ │ │ │ and a weighted projective space are also well-defined. │ │ │ │ i2 : setRandomSeed (currentTime ()); │ │ │ │ - -- setting random seed to 1792812560 │ │ │ │ + -- setting random seed to 1793276221 │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ │ │ -o3 = {2, 4, 6} │ │ │ │ +o3 = {1, 2, 5} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j │ │ │ │ -> random (1,9)); │ │ │ │ i7 : q │ │ │ │ │ │ │ │ -o7 = {1, 5, 6, 9, 9} │ │ │ │ +o7 = {2, 3, 3, 6, 7} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : assert isWellDefined weightedProjectiveSpace q │ │ │ │ The next ten examples illustrate various ways that two lists can fail to define │ │ │ │ a normal toric variety. By making the current debugging level greater than one, │ │ │ │ one gets some addition information about the nature of the failure. │ │ │ │ i9 : X = new MutableHashTable; │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_monomials_lp__Toric__Divisor_rp.html │ │ │ @@ -101,15 +101,15 @@ │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │
    │ │ │ 
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .033906s elapsed
│ │ │ + -- .027659s elapsed
│ │ │  
│ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4   
│ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2 
│ │ │       ------------------------------------------------------------------------
│ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .0131253s elapsed
    │ │ │ + -- .00117399s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ @@ -143,27 +143,27 @@ │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -176,23 +176,23 @@ │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .134092s elapsed
│ │ │ + -- .0684969s elapsed
│ │ │  
│ │ │         2     3 2     3     2 3
│ │ │  o7 = {x x , x x , x x x , x x }
│ │ │         1 3   1 2   0 1 2   0 1
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ 
    i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring variety D2))
│ │ │ - -- .00103874s elapsed
    │ │ │ + -- .00111672s elapsed │ │ │
    │ │ │ 
    i9 : X = kleinschmidt (5, {1,2,3});
    │ │ │
    │ │ │ 
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 46.8661s elapsed
│ │ │ + -- 25.6719s elapsed
│ │ │  
│ │ │  o11 = 7909
    │ │ │
    │ │ │ 
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .183473s elapsed
    │ │ │ + -- .0310593s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -27,61 +27,61 @@ │ │ │ │ i2 : D1 = 5*PP2_0 │ │ │ │ │ │ │ │ o2 = 5*PP2 │ │ │ │ 0 │ │ │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │ │ i3 : M1 = elapsedTime monomials D1 │ │ │ │ - -- .033906s elapsed │ │ │ │ + -- .027659s elapsed │ │ │ │ │ │ │ │ 5 4 4 2 3 3 2 3 3 2 2 2 2 2 3 2 4 │ │ │ │ o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x , │ │ │ │ 2 1 2 0 2 1 2 0 1 2 0 2 1 2 0 1 2 0 1 2 0 2 1 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 2 2 3 4 5 4 2 3 3 2 4 5 │ │ │ │ x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x } │ │ │ │ 0 1 2 0 1 2 0 1 2 0 2 1 0 1 0 1 0 1 0 1 0 │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring │ │ │ │ variety D1)) │ │ │ │ - -- .0131253s elapsed │ │ │ │ + -- .00117399s elapsed │ │ │ │ Toric varieties of Picard-rank 2 are slightly more interesting. │ │ │ │ i5 : FF2 = hirzebruchSurface 2; │ │ │ │ i6 : D2 = 2*FF2_0 + 3 * FF2_1 │ │ │ │ │ │ │ │ o6 = 2*FF2 + 3*FF2 │ │ │ │ 0 1 │ │ │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ i7 : M2 = elapsedTime monomials D2 │ │ │ │ - -- .134092s elapsed │ │ │ │ + -- .0684969s elapsed │ │ │ │ │ │ │ │ 2 3 2 3 2 3 │ │ │ │ o7 = {x x , x x , x x x , x x } │ │ │ │ 1 3 1 2 0 1 2 0 1 │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : elapsedTime assert (set M2 === set first entries basis (degree D2, ring │ │ │ │ variety D2)) │ │ │ │ - -- .00103874s elapsed │ │ │ │ + -- .00111672s elapsed │ │ │ │ i9 : X = kleinschmidt (5, {1,2,3}); │ │ │ │ i10 : D3 = 3*X_0 + 5*X_1 │ │ │ │ │ │ │ │ o10 = 3*X + 5*X │ │ │ │ 0 1 │ │ │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ i11 : m3 = elapsedTime # monomials D3 │ │ │ │ - -- 46.8661s elapsed │ │ │ │ + -- 25.6719s elapsed │ │ │ │ │ │ │ │ o11 = 7909 │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety │ │ │ │ D3)) │ │ │ │ - -- .183473s elapsed │ │ │ │ + -- .0310593s elapsed │ │ │ │ By exploiting _l_a_t_t_i_c_e_P_o_i_n_t_s, this method function avoids using the _b_a_s_i_s │ │ │ │ function. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _d_i_v_i_s_o_r_s -- information about toric divisors and their │ │ │ │ related groups │ │ │ │ * _r_i_n_g_(_N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_) -- make the total coordinate ring (a.k.a. Cox │ │ │ │ ring) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ @@ -130,25 +130,25 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -48,22 +48,22 @@ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ The recommended method for creating a _N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y from a fan is │ │ │ │ _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_L_i_s_t_,_L_i_s_t_). In fact, this package avoids using objects from │ │ │ │ the _P_o_l_y_h_e_d_r_a package whenever possible. Here is a trivial example, namely │ │ │ │ projective 2-space, illustrating the substantial increase in time resulting │ │ │ │ from the use of a _P_o_l_y_h_e_d_r_a fan. │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ │ - -- used 5.1586e-05s (cpu); 4.5906e-05s (thread); 0s (gc) │ │ │ │ + -- used 3.2976e-05s (cpu); 2.584e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = X1 │ │ │ │ │ │ │ │ o6 : NormalToricVariety │ │ │ │ i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull │ │ │ │ matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}}; │ │ │ │ - -- used 0.0871239s (cpu); 0.08713s (thread); 0s (gc) │ │ │ │ + -- used 0.0484872s (cpu); 0.0484955s (thread); 0s (gc) │ │ │ │ i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y -- make a normal toric variety │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_F_a_n_) -- make a normal toric variety from a 'Polyhedra' │ │ │ │ fan │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ @@ -238,21 +238,21 @@ │ │ │ 2 3 │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -102,17 +102,17 @@ │ │ │ │ │ │ │ │ o18 = | 0 1 0 | │ │ │ │ | 0 0 1 | │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ i19 : X1 = time normalToricVariety convexHull (vertMatrix); │ │ │ │ - -- used 0.023116s (cpu); 0.0231153s (thread); 0s (gc) │ │ │ │ + -- used 0.024974s (cpu); 0.0249746s (thread); 0s (gc) │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ │ - -- used 0.00248959s (cpu); 0.00249056s (thread); 0s (gc) │ │ │ │ + -- used 0.00263933s (cpu); 0.00264461s (thread); 0s (gc) │ │ │ │ i21 : assert (set rays X2 === set rays X1 and max X1 === max X2) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_M_a_t_r_i_x_) -- make a normal toric variety from a polytope │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_P_o_l_y_h_e_d_r_o_n_) -- make a normal toric variety from a │ │ │ │ 'Polyhedra' polyhedron │ │ ├── ./usr/share/doc/Macaulay2/Normaliz/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ aW50Y2xNb25JZGVhbChJZGVhbCxSaW5nRWxlbWVudCk= │ │ │ #:len=2753 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibm9ybWFsaXphdGlvbiBvZiBSZWVzIGFs │ │ │ Z2VicmEiLCAibGluZW51bSIgPT4gMTg3NiwgSW5wdXRzID0+IHtTUEFOe1NQQU57ImFuICIsVE8y │ │ ├── ./usr/share/doc/Macaulay2/NumericSolutions/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ am9yZGFuRm9ybShNYXRyaXgp │ │ │ #:len=263 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODE0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhqb3JkYW5Gb3JtLE1hdHJpeCksImpvcmRhbkZvcm0o │ │ ├── ./usr/share/doc/Macaulay2/NumericalAlgebraicGeometry/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=43 │ │ │ aXNTdWJzZXQoTnVtZXJpY2FsVmFyaWV0eSxOdW1lcmljYWxWYXJpZXR5KQ== │ │ │ #:len=1399 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2sgY29udGFpbm1lbnQiLCAibGlu │ │ │ ZW51bSIgPT4gNzg5LCBJbnB1dHMgPT4ge1NQQU57VFR7IlYifSwiLCAiLFNQQU57ImEgIixUTzJ7 │ │ ├── ./usr/share/doc/Macaulay2/NumericalCertification/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=23 │ │ │ aW50ZXJ2YWxDQ2koUlJpLE51bWJlcik= │ │ │ #:len=261 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDQsICJ1bmRvY3VtZW50ZWQiID0+IHRy │ │ │ dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KGludGVydmFs │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:32 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=42 │ │ │ bnVtZXJpY2FsSGlsYmVydEZ1bmN0aW9uKFJpbmdNYXAsSWRlYWwsWlop │ │ │ #:len=367 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDUwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhudW1lcmljYWxIaWxiZXJ0RnVuY3Rpb24sUmluZ01h │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/___Convert__To__Cone.out │ │ │ @@ -21,21 +21,21 @@ │ │ │ │ │ │ i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue == 0 │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ Sampling image points ... │ │ │ - -- used .00691559 seconds │ │ │ + -- used .00696398 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .005626 seconds │ │ │ + -- used .00541614 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00273274 seconds │ │ │ + -- used .00256543 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000418905 seconds │ │ │ + -- used .000394582 seconds │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ │ │ │ │ i6 : extractImageEquations(T, AttemptZZ => true) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out │ │ │ @@ -11,21 +11,21 @@ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true) │ │ │ Sampling image points ... │ │ │ - -- used .00350208 seconds │ │ │ + -- used .00415073 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .00258173 seconds │ │ │ + -- used .00302792 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00100088 seconds │ │ │ + -- used .00115586 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000278743 seconds │ │ │ + -- used .000291325 seconds │ │ │ │ │ │ o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 | │ │ │ │ │ │ 1 3 │ │ │ o3 : Matrix (CC [y ..y ]) <-- (CC [y ..y ]) │ │ │ 53 0 3 53 0 3 │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out │ │ │ @@ -11,40 +11,40 @@ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ Sampling image points ... │ │ │ - -- used .0119364 seconds │ │ │ + -- used .0121076 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0107729 seconds │ │ │ + -- used .011188 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00683732 seconds │ │ │ + -- used .0063946 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000713648 seconds │ │ │ + -- used .000669884 seconds │ │ │ │ │ │ o3 = a "numerical interpolation table", indicating │ │ │ the space of degree 4 forms in the ideal of the image has dimension 22 │ │ │ │ │ │ o3 : NumericalInterpolationTable │ │ │ │ │ │ i4 : R = CC[x_(1,1)..x_(2,4)]; │ │ │ │ │ │ i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*; │ │ │ │ │ │ i6 : S = numericalImageSample(F, ideal 0_R, 60); │ │ │ │ │ │ i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true) │ │ │ Creating interpolation matrix ... │ │ │ - -- used .00320623 seconds │ │ │ + -- used .00278009 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00838546 seconds │ │ │ + -- used .00785936 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000876183 seconds │ │ │ + -- used .000802133 seconds │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Image__Dim.out │ │ │ @@ -20,12 +20,12 @@ │ │ │ │ │ │ i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5)))); │ │ │ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ i9 : time numericalImageDim(F, ideal 0_R) │ │ │ - -- used 0.117425s (cpu); 0.11742s (thread); 0s (gc) │ │ │ + -- used 0.0720523s (cpu); 0.0720474s (thread); 0s (gc) │ │ │ │ │ │ o9 = 69 │ │ │ │ │ │ i10 : │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_real__Point.out │ │ │ @@ -31,15 +31,15 @@ │ │ │ o5 : Ideal of R │ │ │ │ │ │ i6 : I = I1 + I2; │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ i7 : elapsedTime p = realPoint(I, Iterations => 100) │ │ │ - -- .642071s elapsed │ │ │ + -- .421241s elapsed │ │ │ │ │ │ o7 = p │ │ │ │ │ │ o7 : Point │ │ │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ @@ -105,21 +105,21 @@ │ │ │ o4 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -33,21 +33,21 @@ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue │ │ │ │ == 0 │ │ │ │ │ │ │ │ o4 = true │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .00691559 seconds │ │ │ │ + -- used .00696398 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .005626 seconds │ │ │ │ + -- used .00541614 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00273274 seconds │ │ │ │ + -- used .00256543 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000418905 seconds │ │ │ │ + -- used .000394582 seconds │ │ │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ │ │ i6 : extractImageEquations(T, AttemptZZ => true) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html │ │ │ @@ -107,21 +107,21 @@ │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,21 +38,21 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .00350208 seconds │ │ │ │ + -- used .00415073 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .00258173 seconds │ │ │ │ + -- used .00302792 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00100088 seconds │ │ │ │ + -- used .00115586 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000278743 seconds │ │ │ │ + -- used .000291325 seconds │ │ │ │ │ │ │ │ o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 | │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o3 : Matrix (CC [y ..y ]) <-- (CC [y ..y ]) │ │ │ │ 53 0 3 53 0 3 │ │ │ │ Here is how to do the same computation symbolically. │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html │ │ │ @@ -112,21 +112,21 @@ │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -152,19 +152,19 @@ │ │ │ 
    i6 : S = numericalImageSample(F, ideal 0_R, 60);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -57,39 +57,39 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .0119364 seconds │ │ │ │ + -- used .0121076 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0107729 seconds │ │ │ │ + -- used .011188 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00683732 seconds │ │ │ │ + -- used .0063946 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000713648 seconds │ │ │ │ + -- used .000669884 seconds │ │ │ │ │ │ │ │ o3 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 4 forms in the ideal of the image has dimension 22 │ │ │ │ │ │ │ │ o3 : NumericalInterpolationTable │ │ │ │ The following example computes the dimension of Plücker quadrics in the │ │ │ │ defining ideal of the Grassmannian $Gr(2,4)$ of $P^1$'s in $P^3$, in the │ │ │ │ ambient space $P^5$. │ │ │ │ i4 : R = CC[x_(1,1)..x_(2,4)]; │ │ │ │ i5 : F = (minors(2, genericMatrix(R, 2, 4)))_*; │ │ │ │ i6 : S = numericalImageSample(F, ideal 0_R, 60); │ │ │ │ i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true) │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .00320623 seconds │ │ │ │ + -- used .00278009 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00838546 seconds │ │ │ │ + -- used .00785936 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000876183 seconds │ │ │ │ + -- used .000802133 seconds │ │ │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _N_u_m_e_r_i_c_a_l_I_n_t_e_r_p_o_l_a_t_i_o_n_T_a_b_l_e -- the class of all │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ @@ -145,15 +145,15 @@ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 5.1586e-05s (cpu); 4.5906e-05s (thread); 0s (gc)
│ │ │ + -- used 3.2976e-05s (cpu); 2.584e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = X1
│ │ │  
│ │ │  o6 : NormalToricVariety
    │ │ │
    │ │ │ 
    i7 : X2 = time normalToricVariety fan {posHull matrix {{-1,1},{-1,0}}, posHull matrix {{1,0},{0,1}}, posHull matrix{{-1,0},{-1,1}}};
│ │ │ - -- used 0.0871239s (cpu); 0.08713s (thread); 0s (gc)
    │ │ │ + -- used 0.0484872s (cpu); 0.0484955s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    │ │ │
    │ │ │ 
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.023116s (cpu); 0.0231153s (thread); 0s (gc)
    │ │ │ + -- used 0.024974s (cpu); 0.0249746s (thread); 0s (gc) │ │ │
    │ │ │ 
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00248959s (cpu); 0.00249056s (thread); 0s (gc)
    │ │ │ + -- used 0.00263933s (cpu); 0.00264461s (thread); 0s (gc) │ │ │
    │ │ │ 
    i21 : assert (set rays X2 === set rays X1 and max X1 === max X2)
    │ │ │
    │ │ │ 
    i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00691559 seconds
│ │ │ +     -- used .00696398 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .005626 seconds
│ │ │ +     -- used .00541614 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00273274 seconds
│ │ │ +     -- used .00256543 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000418905 seconds
│ │ │ +     -- used .000394582 seconds
│ │ │  
│ │ │  o5 = a "numerical interpolation table", indicating
│ │ │       the space of degree 3 forms in the ideal of the image has dimension 3
│ │ │  
│ │ │  o5 : NumericalInterpolationTable
    │ │ │
    │ │ │ 
    i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true)
│ │ │  Sampling image points ...
│ │ │ -     -- used .00350208 seconds
│ │ │ +     -- used .00415073 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00258173 seconds
│ │ │ +     -- used .00302792 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00100088 seconds
│ │ │ +     -- used .00115586 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000278743 seconds
│ │ │ +     -- used .000291325 seconds
│ │ │  
│ │ │  o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 |
│ │ │  
│ │ │                            1                   3
│ │ │  o3 : Matrix (CC  [y ..y ])  <-- (CC  [y ..y ])
│ │ │                 53  0   3           53  0   3
    │ │ │
    │ │ │ 
    i3 : numericalHilbertFunction(F, ideal 0_R, 4)
│ │ │  Sampling image points ...
│ │ │ -     -- used .0119364 seconds
│ │ │ +     -- used .0121076 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0107729 seconds
│ │ │ +     -- used .011188 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00683732 seconds
│ │ │ +     -- used .0063946 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000713648 seconds
│ │ │ +     -- used .000669884 seconds
│ │ │  
│ │ │  o3 = a "numerical interpolation table", indicating
│ │ │       the space of degree 4 forms in the ideal of the image has dimension 22
│ │ │  
│ │ │  o3 : NumericalInterpolationTable
    │ │ │
    │ │ │ 
    i7 : numericalHilbertFunction(F, ideal 0_R, S, 2, UseSLP => true)
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00320623 seconds
│ │ │ +     -- used .00278009 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00838546 seconds
│ │ │ +     -- used .00785936 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000876183 seconds
│ │ │ +     -- used .000802133 seconds
│ │ │  
│ │ │  o7 = a "numerical interpolation table", indicating
│ │ │       the space of degree 2 forms in the ideal of the image has dimension 1
│ │ │  
│ │ │  o7 : NumericalInterpolationTable
    │ │ │
    │ │ │ 
    i9 : time numericalImageDim(F, ideal 0_R)
│ │ │ - -- used 0.117425s (cpu); 0.11742s (thread); 0s (gc)
│ │ │ + -- used 0.0720523s (cpu); 0.0720474s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = 69
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -43,15 +43,15 @@ │ │ │ │ 201-222. We numerically verify this below. │ │ │ │ i7 : R = CC[a_(1,1)..a_(14,5)]; │ │ │ │ i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5)))); │ │ │ │ │ │ │ │ 1 70 │ │ │ │ o8 : Matrix R <-- R │ │ │ │ i9 : time numericalImageDim(F, ideal 0_R) │ │ │ │ - -- used 0.117425s (cpu); 0.11742s (thread); 0s (gc) │ │ │ │ + -- used 0.0720523s (cpu); 0.0720474s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 = 69 │ │ │ │ ********** WWaayyss ttoo uussee nnuummeerriiccaallIImmaaggeeDDiimm:: ********** │ │ │ │ * numericalImageDim(List,Ideal) │ │ │ │ * numericalImageDim(List,Ideal,Point) │ │ │ │ * numericalImageDim(Matrix,Ideal) │ │ │ │ * numericalImageDim(Matrix,Ideal,Point) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ @@ -137,15 +137,15 @@ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .642071s elapsed
│ │ │ + -- .421241s elapsed
│ │ │  
│ │ │  o7 = p
│ │ │  
│ │ │  o7 : Point
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -49,15 +49,15 @@ │ │ │ │ i5 : I2 = ideal apply(entries transpose A, row -> sum(row, v -> v^2) - 1); │ │ │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ i6 : I = I1 + I2; │ │ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ i7 : elapsedTime p = realPoint(I, Iterations => 100) │ │ │ │ - -- .642071s elapsed │ │ │ │ + -- .421241s elapsed │ │ │ │ │ │ │ │ o7 = p │ │ │ │ │ │ │ │ o7 : Point │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ │ │ │ │ │ │ o8 = | .722359 .289465 -.295808 .591752 -.454678 | │ │ ├── ./usr/share/doc/Macaulay2/NumericalLinearAlgebra/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ bnVtZXJpY2FsS2VybmVsKC4uLixUb2xlcmFuY2U9Pi4uLik= │ │ │ #:len=352 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjI0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tudW1lcmljYWxLZXJuZWwsVG9sZXJhbmNlXSwibnVt │ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ UGllcmlSb290Q291bnQoLi4uLFZlcmJvc2U9Pi4uLik= │ │ │ #:len=334 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmVxdWVzdCB2ZXJib3NlIGZlZWRiYWNr │ │ │ IiwgRGVzY3JpcHRpb24gPT4ge30sICJsaW5lbnVtIiA9PiAzMzMsIHN5bWJvbCBEb2N1bWVudFRh │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ c2VtaWdyb3VwRnJvbU11KExpc3Qp │ │ │ #:len=289 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzAwNCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2VtaWdyb3VwRnJvbU11LExpc3QpLCJzZW1pZ3Jv │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=29 │ │ │ dG9TdHJpbmcoUG9seW5vbWlhbE9JQWxnZWJyYSk= │ │ │ #:len=1044 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheSBhIHBvbHlub21pYWwgT0kt │ │ │ YWxnZWJyYSBpbiBjb25kZW5zZWQgZm9ybSIsICJsaW5lbnVtIiA9PiAxNTE2LCBJbnB1dHMgPT4g │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1) │ │ │ - -- used 0.0877009s (cpu); 0.0877049s (thread); 0s (gc) │ │ │ + -- used 0.0890212s (cpu); 0.0890212s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ │ │ o5 : OIResolution │ │ │ │ │ │ i6 : C.dd_0 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution_sp_us_sp__Z__Z.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.224314s (cpu); 0.152688s (thread); 0s (gc) │ │ │ + -- used 0.194055s (cpu); 0.112459s (thread); 0s (gc) │ │ │ │ │ │ i6 : C_0 │ │ │ │ │ │ o6 = Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Top__Nonminimal.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time oiRes({b}, 2, TopNonminimal => true) │ │ │ - -- used 0.459073s (cpu); 0.290457s (thread); 0s (gc) │ │ │ + -- used 0.436135s (cpu); 0.274183s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe__Full.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.0985988s (cpu); 0.0985983s (thread); 0s (gc) │ │ │ + -- used 0.0952108s (cpu); 0.0951496s (thread); 0s (gc) │ │ │ │ │ │ i6 : describeFull C │ │ │ │ │ │ o6 = 0: Module: Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__O__I__Resolution_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.0855533s (cpu); 0.0855538s (thread); 0s (gc) │ │ │ + -- used 0.0877281s (cpu); 0.0877285s (thread); 0s (gc) │ │ │ │ │ │ i6 : describe C │ │ │ │ │ │ o6 = 0: Module: Basis symbol: e0 │ │ │ Basis element widths: {2} │ │ │ Degree shifts: {-2} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Complex.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 2, TopNonminimal => true) │ │ │ - -- used 0.357846s (cpu); 0.284765s (thread); 0s (gc) │ │ │ + -- used 0.343991s (cpu); 0.274009s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__O__I__G__B.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3); │ │ │ │ │ │ i10 : isOIGB {b1, b2} │ │ │ │ │ │ o10 = false │ │ │ │ │ │ i11 : time B = oiGB {b1, b2} │ │ │ - -- used 0.167028s (cpu); 0.0579109s (thread); 0s (gc) │ │ │ + -- used 0.132359s (cpu); 0.0437382s (thread); 0s (gc) │ │ │ │ │ │ o11 = {x e + x e , x x e + x x e , │ │ │ 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ ----------------------------------------------------------------------- │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_minimize__O__I__G__B.out │ │ │ @@ -11,15 +11,15 @@ │ │ │ i5 : installGeneratorsInWidth(F, 3); │ │ │ │ │ │ i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2); │ │ │ │ │ │ i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3); │ │ │ │ │ │ i10 : time B = oiGB {b1, b2} │ │ │ - -- used 0.0295377s (cpu); 0.0295415s (thread); 0s (gc) │ │ │ + -- used 0.0266728s (cpu); 0.0266733s (thread); 0s (gc) │ │ │ │ │ │ o10 = {x e + x e , x x e + x x e , │ │ │ 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ ----------------------------------------------------------------------- │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.368596s (cpu); 0.179956s (thread); 0s (gc) │ │ │ + -- used 0.306623s (cpu); 0.14031s (thread); 0s (gc) │ │ │ │ │ │ i6 : net C │ │ │ │ │ │ o6 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__G__B.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2); │ │ │ │ │ │ i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3); │ │ │ │ │ │ i9 : time oiGB {b1, b2} │ │ │ - -- used 0.0272125s (cpu); 0.0272119s (thread); 0s (gc) │ │ │ + -- used 0.030305s (cpu); 0.0303045s (thread); 0s (gc) │ │ │ │ │ │ o9 = {x e + x e , x x e + x x e , │ │ │ 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ ------------------------------------------------------------------------ │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Res.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time oiRes({b}, 2, TopNonminimal => true) │ │ │ - -- used 0.652096s (cpu); 0.338501s (thread); 0s (gc) │ │ │ + -- used 0.549138s (cpu); 0.295833s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2); │ │ │ │ │ │ i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2); │ │ │ │ │ │ i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal) │ │ │ - -- used 0.133994s (cpu); 0.133994s (thread); 0s (gc) │ │ │ + -- used 0.122582s (cpu); 0.122583s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ o9 = {x e + x e , x x e + x x e , │ │ │ 2,1 1,{1},2 1,1 1,{1},2 1,2 1,1 2,{2},1 2,2 1,2 2,{2},2 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 │ │ │ x x e - x x e } │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution.html │ │ │ @@ -79,15 +79,15 @@ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1)
│ │ │ - -- used 0.0877009s (cpu); 0.0877049s (thread); 0s (gc)
│ │ │ + -- used 0.0890212s (cpu); 0.0890212s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │  
│ │ │  o5 : OIResolution
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -11,15 +11,15 @@ │ │ │ │ complex, use _i_s_C_o_m_p_l_e_x. To get the $n$th differential in an OI-resolution C, │ │ │ │ use C.dd_n. │ │ │ │ i1 : P = makePolynomialOIAlgebra(2, x, QQ); │ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ i5 : time C = oiRes({b}, 1) │ │ │ │ - -- used 0.0877009s (cpu); 0.0877049s (thread); 0s (gc) │ │ │ │ + -- used 0.0890212s (cpu); 0.0890212s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ │ 1: (e1, {4, 4}, {-4, -4}) │ │ │ │ │ │ │ │ o5 : OIResolution │ │ │ │ i6 : C.dd_0 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution_sp_us_sp__Z__Z.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.224314s (cpu); 0.152688s (thread); 0s (gc)
    │ │ │ + -- used 0.194055s (cpu); 0.112459s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : C_0
│ │ │  
│ │ │  o6 = Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Returns the free OI-module of $C$ in homological degree $n$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.224314s (cpu); 0.152688s (thread); 0s (gc)
│ │ │ │ + -- used 0.194055s (cpu); 0.112459s (thread); 0s (gc)
│ │ │ │  i6 : C_0
│ │ │ │  
│ │ │ │  o6 = Basis symbol: e0
│ │ │ │       Basis element widths: {2}
│ │ │ │       Degree shifts: {-2}
│ │ │ │       Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │       Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Top__Nonminimal.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.459073s (cpu); 0.290457s (thread); 0s (gc)
│ │ │ + -- used 0.436135s (cpu); 0.274183s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │  homological degree $n-1$ to be minimized. Therefore, use TopNonminimal => true
│ │ │ │  for no minimization of the basis in degree $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.459073s (cpu); 0.290457s (thread); 0s (gc)
│ │ │ │ + -- used 0.436135s (cpu); 0.274183s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd TTooppNNoonnmmiinniimmaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe__Full.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0985988s (cpu); 0.0985983s (thread); 0s (gc)
    
│ │ │ + -- used 0.0952108s (cpu); 0.0951496s (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 0.0985988s (cpu); 0.0985983s (thread); 0s (gc)
│ │ │ │ + -- used 0.0952108s (cpu); 0.0951496s (thread); 0s (gc)
│ │ │ │  i6 : describeFull C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0855533s (cpu); 0.0855538s (thread); 0s (gc)
    
│ │ │ + -- used 0.0877281s (cpu); 0.0877285s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : describe C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,15 +14,15 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Displays the free OI-modules and differentials of an OI-resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.0855533s (cpu); 0.0855538s (thread); 0s (gc)
│ │ │ │ + -- used 0.0877281s (cpu); 0.0877285s (thread); 0s (gc)
│ │ │ │  i6 : describe C
│ │ │ │  
│ │ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ │                  Basis element widths: {2}
│ │ │ │                  Degree shifts: {-2}
│ │ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │                  Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__Complex.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.357846s (cpu); 0.284765s (thread); 0s (gc)
│ │ │ + -- used 0.343991s (cpu); 0.274009s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,15 +17,15 @@
│ │ │ │  option must be either true or false, depending on whether one wants debug
│ │ │ │  information printed.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.357846s (cpu); 0.284765s (thread); 0s (gc)
│ │ │ │ + -- used 0.343991s (cpu); 0.274009s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : isComplex C
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__O__I__G__B.html
│ │ │ @@ -121,15 +121,15 @@
│ │ │  
│ │ │  o10 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.167028s (cpu); 0.0579109s (thread); 0s (gc)
│ │ │ + -- used 0.132359s (cpu); 0.0437382s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i10 : isOIGB {b1, b2}
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.167028s (cpu); 0.0579109s (thread); 0s (gc)
│ │ │ │ + -- used 0.132359s (cpu); 0.0437382s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │              
│ │ │                
    i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0295377s (cpu); 0.0295415s (thread); 0s (gc)
│ │ │ + -- used 0.0266728s (cpu); 0.0266733s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i10 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.0295377s (cpu); 0.0295415s (thread); 0s (gc)
│ │ │ │ + -- used 0.0266728s (cpu); 0.0266733s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.368596s (cpu); 0.179956s (thread); 0s (gc)
    
│ │ │ + -- used 0.306623s (cpu); 0.14031s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : net C
│ │ │  
│ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ ├── html2text {}
│ │ │ │ @@ -15,15 +15,15 @@
│ │ │ │  Displays the basis element widths and degree shifts of the free OI-modules in
│ │ │ │  an OI-resolution.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time C = oiRes({b}, 1);
│ │ │ │ - -- used 0.368596s (cpu); 0.179956s (thread); 0s (gc)
│ │ │ │ + -- used 0.306623s (cpu); 0.14031s (thread); 0s (gc)
│ │ │ │  i6 : net C
│ │ │ │  
│ │ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_e_t_(_O_I_R_e_s_o_l_u_t_i_o_n_) -- display an OI-resolution
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time oiGB {b1, b2}
│ │ │ - -- used 0.0272125s (cpu); 0.0272119s (thread); 0s (gc)
│ │ │ + -- used 0.030305s (cpu); 0.0303045s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   x   e           - x   x   x   e          }
│ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i9 : time oiGB {b1, b2}
│ │ │ │ - -- used 0.0272125s (cpu); 0.0272119s (thread); 0s (gc)
│ │ │ │ + -- used 0.030305s (cpu); 0.0303045s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   x   x   e           - x   x   x   e          }
│ │ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.652096s (cpu); 0.338501s (thread); 0s (gc)
│ │ │ + -- used 0.549138s (cpu); 0.295833s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │       1: (e1, {4}, {-4})
│ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │  
│ │ │  o5 : OIResolution
    
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  Therefore, use TopNonminimal => true for no minimization of the basis in degree
│ │ │ │  $n-1$.
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 2);
│ │ │ │  i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
│ │ │ │  i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ │ - -- used 0.652096s (cpu); 0.338501s (thread); 0s (gc)
│ │ │ │ + -- used 0.549138s (cpu); 0.295833s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** WWaayyss ttoo uussee ooiiRReess:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ - -- used 0.133994s (cpu); 0.133994s (thread); 0s (gc)
│ │ │ + -- used 0.122582s (cpu); 0.122583s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2 
│ │ │       ------------------------------------------------------------------------
│ │ │        2                  2
│ │ │       x   x   e        - x   x   e       }
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
│ │ │ │  i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ │ - -- used 0.133994s (cpu); 0.133994s (thread); 0s (gc)
│ │ │ │ + -- used 0.122582s (cpu); 0.122583s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e       ,
│ │ │ │         2,1 1,{1},2    1,1 1,{1},2   1,2 1,1 2,{2},1    2,2 1,2 2,{2},2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2                  2
│ │ │ │       x   x   e        - x   x   e       }
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  Z3JhZGVkTW9kdWxlKE1vZHVsZSk=
│ │ │  #:len=294
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODEsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KGdyYWRlZE1vZHVsZSxNb2R1bGUpLCJncmFkZWRNb2R1
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/___Fast__Nonminimal.out
│ │ │ @@ -9,25 +9,25 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.66263s elapsed
│ │ │ + -- 1.68651s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
│ │ │  
│ │ │  i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ - -- 1.80662s elapsed
│ │ │ + -- 1.14711s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out
│ │ │ @@ -9,15 +9,15 @@
│ │ │  i2 : S = ring I
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.7181s elapsed
│ │ │ + -- 1.66646s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/example-output/_computing_spresolutions.out
│ │ │ @@ -36,22 +36,19 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.906586s (cpu); 0.748056s (thread); 0s (gc)
│ │ │ - -- used 1.12461s (cpu); 0.850193s (thread); 0s (gc)
│ │ │ - -- used 0.0418703s (cpu); 0.0433256s (thread); 0s (gc)
│ │ │ + -- used 1.09321s (cpu); 0.954232s (thread); 0s (gc)
│ │ │ + -- used 0.588315s (cpu); 0.511504s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │ --- computation interrupted
│ │ │ --- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ │                                           
│ │ │  0      1       2       3        4       5
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/___Fast__Nonminimal.html
│ │ │ @@ -94,28 +94,28 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.66263s elapsed
│ │ │ + -- 1.68651s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ - -- 1.80662s elapsed
│ │ │ + -- 1.14711s elapsed
│ │ │  
│ │ │        1      35      140      385      819      1080      819      385      140      35      1
│ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S    <-- S   <-- S  <-- 0
│ │ │                                                                                                      
│ │ │       0      1       2        3        4        5         6        7        8        9       10     11
│ │ │  
│ │ │  o4 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 1.66263s elapsed
│ │ │ │ + -- 1.68651s elapsed
│ │ │ │  
│ │ │ │        1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S
│ │ │ │  <-- S    <-- S   <-- 0
│ │ │ │  
│ │ │ │  
│ │ │ │       0      1       2        3        4         5         6         7         8
│ │ │ │  9        10      11
│ │ │ │  
│ │ │ │  o3 : ChainComplex
│ │ │ │  i4 : elapsedTime C1 = res ideal(I_*)
│ │ │ │ - -- 1.80662s elapsed
│ │ │ │ + -- 1.14711s elapsed
│ │ │ │  
│ │ │ │        1      35      140      385      819      1080      819      385      140
│ │ │ │  35      1
│ │ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S
│ │ │ │  <-- S   <-- S  <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.7181s elapsed
│ │ │ + -- 1.66646s elapsed
│ │ │  
│ │ │        1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S   <-- 0
│ │ │                                                                                                           
│ │ │       0      1       2        3        4         5         6         7         8        9        10      11
│ │ │  
│ │ │  o3 : ChainComplex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i2 : S = ring I
│ │ │ │  
│ │ │ │  o2 = S
│ │ │ │  
│ │ │ │  o2 : PolynomialRing
│ │ │ │  i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ │ - -- 1.7181s elapsed
│ │ │ │ + -- 1.66646s elapsed
│ │ │ │  
│ │ │ │        1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S
│ │ │ │  <-- S    <-- S   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html
│ │ │ @@ -117,22 +117,19 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.906586s (cpu); 0.748056s (thread); 0s (gc)
│ │ │ - -- used 1.12461s (cpu); 0.850193s (thread); 0s (gc)
│ │ │ - -- used 0.0418703s (cpu); 0.0433256s (thread); 0s (gc)
│ │ │ + -- used 1.09321s (cpu); 0.954232s (thread); 0s (gc)
│ │ │ + -- used 0.588315s (cpu); 0.511504s (thread); 0s (gc)
│ │ │  -- computation started: 
│ │ │  -- computation interrupted
│ │ │  -- continuing the computation
│ │ │ --- computation interrupted
│ │ │ --- continuing the computation
│ │ │  -- computation complete
│ │ │   4      11      89      122      40
│ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ │                                           
│ │ │  0      1       2       3        4       5
    
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -50,22 +50,19 @@
│ │ │ │            << res M << endl << endl;
│ │ │ │            break;
│ │ │ │            ) else (
│ │ │ │            << "-- computation interrupted" << endl;
│ │ │ │            status M.cache.resolution;
│ │ │ │            << "-- continuing the computation" << endl;
│ │ │ │            ))
│ │ │ │ - -- used 0.906586s (cpu); 0.748056s (thread); 0s (gc)
│ │ │ │ - -- used 1.12461s (cpu); 0.850193s (thread); 0s (gc)
│ │ │ │ - -- used 0.0418703s (cpu); 0.0433256s (thread); 0s (gc)
│ │ │ │ + -- used 1.09321s (cpu); 0.954232s (thread); 0s (gc)
│ │ │ │ + -- used 0.588315s (cpu); 0.511504s (thread); 0s (gc)
│ │ │ │  -- computation started:
│ │ │ │  -- computation interrupted
│ │ │ │  -- continuing the computation
│ │ │ │ --- computation interrupted
│ │ │ │ --- continuing the computation
│ │ │ │  -- computation complete
│ │ │ │   4      11      89      122      40
│ │ │ │  R  <-- R   <-- R   <-- R    <-- R   <-- 0
│ │ │ │  
│ │ │ │  0      1       2       3        4       5
│ │ │ │  If the user has a chain complex in hand that is known to be a projective
│ │ │ │  resolution of M, then it can be installed with M.cache.resolution = C.
│ │ ├── ./usr/share/doc/Macaulay2/OnlineLookup/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  b2Vpcw==
│ │ │  #:len=753
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT0VJUyBsb29rdXAiLCBEZXNjcmlwdGlv
│ │ │  biA9PiAoRElWe1BBUkF7VEVYeyJUaGlzIGZ1bmN0aW9uIGxvb2tzIHVwIHRoZSBhcmd1bWVudCAo
│ │ ├── ./usr/share/doc/Macaulay2/OpenMath/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  T3Blbk1hdGg=
│ │ │  #:len=478
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3Blbk1hdGggc3VwcG9ydCIsICJsaW5l
│ │ │  bnVtIiA9PiA4NSwgU2VlQWxzbyA9PiBESVZ7SEVBREVSMnsiU2VlIGFsc28ifSxVTHtMSXtUT0h7
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  b3NjUmluZyguLi4sQ29lZmZpY2llbnRSaW5nPT4uLi4p
│ │ │  #:len=277
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tvc2NSaW5nLENvZWZmaWNpZW50UmluZ10sIm9zY1Jp
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.out
│ │ │ @@ -182,25 +182,25 @@
│ │ │  o15 = 4
│ │ │  
│ │ │  i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .284219s elapsed
│ │ │ - -- .401541s elapsed
│ │ │ - -- .867569s elapsed
│ │ │ - -- .245534s elapsed
│ │ │ - -- .354798s elapsed
│ │ │ - -- .429779s elapsed
│ │ │ - -- .666427s elapsed
│ │ │ - -- .418819s elapsed
│ │ │ - -- .478799s elapsed
│ │ │ - -- .409628s elapsed
│ │ │ - -- .282237s elapsed
│ │ │ + -- .211184s elapsed
│ │ │ + -- .243263s elapsed
│ │ │ + -- .385711s elapsed
│ │ │ + -- .205387s elapsed
│ │ │ + -- .231241s elapsed
│ │ │ + -- .258153s elapsed
│ │ │ + -- .491836s elapsed
│ │ │ + -- .37767s elapsed
│ │ │ + -- .416397s elapsed
│ │ │ + -- .293586s elapsed
│ │ │ + -- .213893s elapsed
│ │ │  
│ │ │  i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │        +---------------+---------------+
│ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ @@ -242,75 +242,75 @@
│ │ │  o22 = 15
│ │ │  
│ │ │  i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .539129s elapsed
│ │ │ - -- .645006s elapsed
│ │ │ - -- 1.05918s elapsed
│ │ │ - -- 1.23488s elapsed
│ │ │ - -- .813801s elapsed
│ │ │ + -- .310823s elapsed
│ │ │ + -- .376704s elapsed
│ │ │ + -- .768631s elapsed
│ │ │ + -- .979926s elapsed
│ │ │ + -- .609071s elapsed
│ │ │ + -- .763568s elapsed
│ │ │ + -- .836899s elapsed
│ │ │ + -- .878904s elapsed
│ │ │ + -- .616915s elapsed
│ │ │ + -- .61028s elapsed
│ │ │ + -- .28669s elapsed
│ │ │ + -- .33432s elapsed
│ │ │ + -- .411242s elapsed
│ │ │ + -- .53354s elapsed
│ │ │ + -- .704534s elapsed
│ │ │ + -- 1.02456s elapsed
│ │ │ + -- .783083s elapsed
│ │ │ + -- .706565s elapsed
│ │ │ + -- .991695s elapsed
│ │ │ + -- .837744s elapsed
│ │ │ + -- .66319s elapsed
│ │ │ + -- .746018s elapsed
│ │ │ + -- 1.13548s elapsed
│ │ │ + -- .943452s elapsed
│ │ │ + -- .381898s elapsed
│ │ │ + -- .52052s elapsed
│ │ │   -- 1.01888s elapsed
│ │ │ - -- 1.18556s elapsed
│ │ │ - -- 1.19237s elapsed
│ │ │ - -- .809367s elapsed
│ │ │ - -- .688287s elapsed
│ │ │ - -- .34871s elapsed
│ │ │ - -- .470565s elapsed
│ │ │ - -- .796532s elapsed
│ │ │ - -- .592092s elapsed
│ │ │ - -- .870412s elapsed
│ │ │ - -- 1.45151s elapsed
│ │ │ - -- 1.08533s elapsed
│ │ │ - -- .929525s elapsed
│ │ │ - -- 1.53034s elapsed
│ │ │ - -- 1.33678s elapsed
│ │ │ - -- .760184s elapsed
│ │ │ - -- 1.28518s elapsed
│ │ │ - -- 1.71666s elapsed
│ │ │ - -- 1.22854s elapsed
│ │ │ - -- .674943s elapsed
│ │ │ - -- .894164s elapsed
│ │ │ - -- 1.57127s elapsed
│ │ │ - -- .743101s elapsed
│ │ │ - -- .756756s elapsed
│ │ │ - -- 1.13138s elapsed
│ │ │ - -- 1.25775s elapsed
│ │ │ - -- 1.00643s elapsed
│ │ │ - -- .580285s elapsed
│ │ │ - -- 1.47228s elapsed
│ │ │ - -- 1.02334s elapsed
│ │ │ - -- 1.53993s elapsed
│ │ │ - -- .942582s elapsed
│ │ │ - -- 1.21686s elapsed
│ │ │ - -- 1.59056s elapsed
│ │ │ - -- 1.01279s elapsed
│ │ │ - -- .781942s elapsed
│ │ │ - -- 1.63071s elapsed
│ │ │ - -- 1.80895s elapsed
│ │ │ - -- 2.00814s elapsed
│ │ │ - -- 1.15234s elapsed
│ │ │ - -- 1.32068s elapsed
│ │ │ - -- 1.58339s elapsed
│ │ │ - -- 1.20196s elapsed
│ │ │ - -- 1.10322s elapsed
│ │ │ - -- 1.215s elapsed
│ │ │ - -- 1.22183s elapsed
│ │ │ - -- .920406s elapsed
│ │ │ - -- .900219s elapsed
│ │ │ - -- 1.32942s elapsed
│ │ │ - -- .609367s elapsed
│ │ │ - -- 1.44447s elapsed
│ │ │ - -- 1.35121s elapsed
│ │ │ - -- 1.47404s elapsed
│ │ │ - -- .758204s elapsed
│ │ │ - -- .533039s elapsed
│ │ │ - -- .40806s elapsed
│ │ │ + -- .555376s elapsed
│ │ │ + -- .456873s elapsed
│ │ │ + -- .668581s elapsed
│ │ │ + -- .847484s elapsed
│ │ │ + -- .681135s elapsed
│ │ │ + -- .464221s elapsed
│ │ │ + -- .866599s elapsed
│ │ │ + -- .651717s elapsed
│ │ │ + -- .921585s elapsed
│ │ │ + -- .786177s elapsed
│ │ │ + -- .92925s elapsed
│ │ │ + -- 1.00142s elapsed
│ │ │ + -- .609246s elapsed
│ │ │ + -- .595969s elapsed
│ │ │ + -- .917554s elapsed
│ │ │ + -- 1.14664s elapsed
│ │ │ + -- 1.40924s elapsed
│ │ │ + -- 1.0083s elapsed
│ │ │ + -- .97673s elapsed
│ │ │ + -- 1.1517s elapsed
│ │ │ + -- .989014s elapsed
│ │ │ + -- .804033s elapsed
│ │ │ + -- .87493s elapsed
│ │ │ + -- .887405s elapsed
│ │ │ + -- .650366s elapsed
│ │ │ + -- .669484s elapsed
│ │ │ + -- .802823s elapsed
│ │ │ + -- .500618s elapsed
│ │ │ + -- .911672s elapsed
│ │ │ + -- 1.00836s elapsed
│ │ │ + -- 1.12274s elapsed
│ │ │ + -- .621319s elapsed
│ │ │ + -- .397177s elapsed
│ │ │ + -- .29607s elapsed
│ │ │  
│ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │  o24 = |codimensions            |degrees                 |
│ │ │        +------------------------+------------------------+
│ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.out
│ │ │ @@ -39,15 +39,15 @@
│ │ │       .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201,
│ │ │       ------------------------------------------------------------------------
│ │ │       .954}}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .859s elapsed
│ │ │ + -- .827s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -60,15 +60,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │ - -- .901s elapsed
│ │ │ + -- .869s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │  
│ │ │  o6 : List
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  
│ │ │  i5 : printingPrecision = 3
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 82, 85, 86, 87, 88, 89, 90}
│ │ │ - -- .646s elapsed
│ │ │ + -- .57s elapsed
│ │ │  warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97}
│ │ │ - -- .578s elapsed
│ │ │ - -- .694s elapsed
│ │ │ - -- .859s elapsed
│ │ │ + -- .562s elapsed
│ │ │ + -- .716s elapsed
│ │ │ + -- .921s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |1    |1   |1   |1    |0    |0    |0    |0    |
│ │ │ @@ -69,20 +69,20 @@
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │ - -- .966s elapsed
│ │ │ - -- 1.19s elapsed
│ │ │ + -- 1.1s elapsed
│ │ │ + -- 1.12s elapsed
│ │ │  warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, 47, 48, 53, 59, 60, 61}
│ │ │ - -- 1.27s elapsed
│ │ │ + -- 1.37s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34}
│ │ │ - -- 1.07s elapsed
│ │ │ - -- 1.05s elapsed
│ │ │ + -- 1.14s elapsed
│ │ │ + -- 1.14s elapsed
│ │ │  warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │ - -- 1.21s elapsed
│ │ │ + -- 1.22s elapsed
│ │ │  warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, 75, 76, 77}
│ │ │ - -- 1.03s elapsed
│ │ │ + -- 1.15s elapsed
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 1729328129346969841
│ │ │  
│ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │  
│ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │  warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21}
│ │ │ - -- .215945s elapsed
│ │ │ + -- .189996s elapsed
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │             2 => {1, 3}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 3}
│ │ │  
│ │ │  o1 : Graph
│ │ │  
│ │ │  i2 : showExoticSolutions G
│ │ │ - -- .722313s elapsed
│ │ │ + -- .808396s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -48,14 +48,14 @@
│ │ │             2 => {1, 3, 4}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 2, 3}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : showExoticSolutions G
│ │ │ - -- .986724s elapsed
│ │ │ + -- 1.06047s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html
│ │ │ @@ -300,25 +300,25 @@
│ │ │            
│ │ │              
│ │ │                
    i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .284219s elapsed
│ │ │ - -- .401541s elapsed
│ │ │ - -- .867569s elapsed
│ │ │ - -- .245534s elapsed
│ │ │ - -- .354798s elapsed
│ │ │ - -- .429779s elapsed
│ │ │ - -- .666427s elapsed
│ │ │ - -- .418819s elapsed
│ │ │ - -- .478799s elapsed
│ │ │ - -- .409628s elapsed
│ │ │ - -- .282237s elapsed
    
│ │ │ + -- .211184s elapsed
│ │ │ + -- .243263s elapsed
│ │ │ + -- .385711s elapsed
│ │ │ + -- .205387s elapsed
│ │ │ + -- .231241s elapsed
│ │ │ + -- .258153s elapsed
│ │ │ + -- .491836s elapsed
│ │ │ + -- .37767s elapsed
│ │ │ + -- .416397s elapsed
│ │ │ + -- .293586s elapsed
│ │ │ + -- .213893s elapsed
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │ @@ -385,75 +385,75 @@
│ │ │            
│ │ │              
│ │ │                
    i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .539129s elapsed
│ │ │ - -- .645006s elapsed
│ │ │ - -- 1.05918s elapsed
│ │ │ - -- 1.23488s elapsed
│ │ │ - -- .813801s elapsed
│ │ │ + -- .310823s elapsed
│ │ │ + -- .376704s elapsed
│ │ │ + -- .768631s elapsed
│ │ │ + -- .979926s elapsed
│ │ │ + -- .609071s elapsed
│ │ │ + -- .763568s elapsed
│ │ │ + -- .836899s elapsed
│ │ │ + -- .878904s elapsed
│ │ │ + -- .616915s elapsed
│ │ │ + -- .61028s elapsed
│ │ │ + -- .28669s elapsed
│ │ │ + -- .33432s elapsed
│ │ │ + -- .411242s elapsed
│ │ │ + -- .53354s elapsed
│ │ │ + -- .704534s elapsed
│ │ │ + -- 1.02456s elapsed
│ │ │ + -- .783083s elapsed
│ │ │ + -- .706565s elapsed
│ │ │ + -- .991695s elapsed
│ │ │ + -- .837744s elapsed
│ │ │ + -- .66319s elapsed
│ │ │ + -- .746018s elapsed
│ │ │ + -- 1.13548s elapsed
│ │ │ + -- .943452s elapsed
│ │ │ + -- .381898s elapsed
│ │ │ + -- .52052s elapsed
│ │ │   -- 1.01888s elapsed
│ │ │ - -- 1.18556s elapsed
│ │ │ - -- 1.19237s elapsed
│ │ │ - -- .809367s elapsed
│ │ │ - -- .688287s elapsed
│ │ │ - -- .34871s elapsed
│ │ │ - -- .470565s elapsed
│ │ │ - -- .796532s elapsed
│ │ │ - -- .592092s elapsed
│ │ │ - -- .870412s elapsed
│ │ │ - -- 1.45151s elapsed
│ │ │ - -- 1.08533s elapsed
│ │ │ - -- .929525s elapsed
│ │ │ - -- 1.53034s elapsed
│ │ │ - -- 1.33678s elapsed
│ │ │ - -- .760184s elapsed
│ │ │ - -- 1.28518s elapsed
│ │ │ - -- 1.71666s elapsed
│ │ │ - -- 1.22854s elapsed
│ │ │ - -- .674943s elapsed
│ │ │ - -- .894164s elapsed
│ │ │ - -- 1.57127s elapsed
│ │ │ - -- .743101s elapsed
│ │ │ - -- .756756s elapsed
│ │ │ - -- 1.13138s elapsed
│ │ │ - -- 1.25775s elapsed
│ │ │ - -- 1.00643s elapsed
│ │ │ - -- .580285s elapsed
│ │ │ - -- 1.47228s elapsed
│ │ │ - -- 1.02334s elapsed
│ │ │ - -- 1.53993s elapsed
│ │ │ - -- .942582s elapsed
│ │ │ - -- 1.21686s elapsed
│ │ │ - -- 1.59056s elapsed
│ │ │ - -- 1.01279s elapsed
│ │ │ - -- .781942s elapsed
│ │ │ - -- 1.63071s elapsed
│ │ │ - -- 1.80895s elapsed
│ │ │ - -- 2.00814s elapsed
│ │ │ - -- 1.15234s elapsed
│ │ │ - -- 1.32068s elapsed
│ │ │ - -- 1.58339s elapsed
│ │ │ - -- 1.20196s elapsed
│ │ │ - -- 1.10322s elapsed
│ │ │ - -- 1.215s elapsed
│ │ │ - -- 1.22183s elapsed
│ │ │ - -- .920406s elapsed
│ │ │ - -- .900219s elapsed
│ │ │ - -- 1.32942s elapsed
│ │ │ - -- .609367s elapsed
│ │ │ - -- 1.44447s elapsed
│ │ │ - -- 1.35121s elapsed
│ │ │ - -- 1.47404s elapsed
│ │ │ - -- .758204s elapsed
│ │ │ - -- .533039s elapsed
│ │ │ - -- .40806s elapsed
    
│ │ │ + -- .555376s elapsed
│ │ │ + -- .456873s elapsed
│ │ │ + -- .668581s elapsed
│ │ │ + -- .847484s elapsed
│ │ │ + -- .681135s elapsed
│ │ │ + -- .464221s elapsed
│ │ │ + -- .866599s elapsed
│ │ │ + -- .651717s elapsed
│ │ │ + -- .921585s elapsed
│ │ │ + -- .786177s elapsed
│ │ │ + -- .92925s elapsed
│ │ │ + -- 1.00142s elapsed
│ │ │ + -- .609246s elapsed
│ │ │ + -- .595969s elapsed
│ │ │ + -- .917554s elapsed
│ │ │ + -- 1.14664s elapsed
│ │ │ + -- 1.40924s elapsed
│ │ │ + -- 1.0083s elapsed
│ │ │ + -- .97673s elapsed
│ │ │ + -- 1.1517s elapsed
│ │ │ + -- .989014s elapsed
│ │ │ + -- .804033s elapsed
│ │ │ + -- .87493s elapsed
│ │ │ + -- .887405s elapsed
│ │ │ + -- .650366s elapsed
│ │ │ + -- .669484s elapsed
│ │ │ + -- .802823s elapsed
│ │ │ + -- .500618s elapsed
│ │ │ + -- .911672s elapsed
│ │ │ + -- 1.00836s elapsed
│ │ │ + -- 1.12274s elapsed
│ │ │ + -- .621319s elapsed
│ │ │ + -- .397177s elapsed
│ │ │ + -- .29607s elapsed
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │  
│ │ │        +------------------------+------------------------+
│ │ │ ├── html2text {}
│ │ │ │ @@ -180,25 +180,25 @@
│ │ │ │  
│ │ │ │  o15 = 4
│ │ │ │  i16 : for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .284219s elapsed
│ │ │ │ - -- .401541s elapsed
│ │ │ │ - -- .867569s elapsed
│ │ │ │ - -- .245534s elapsed
│ │ │ │ - -- .354798s elapsed
│ │ │ │ - -- .429779s elapsed
│ │ │ │ - -- .666427s elapsed
│ │ │ │ - -- .418819s elapsed
│ │ │ │ - -- .478799s elapsed
│ │ │ │ - -- .409628s elapsed
│ │ │ │ - -- .282237s elapsed
│ │ │ │ + -- .211184s elapsed
│ │ │ │ + -- .243263s elapsed
│ │ │ │ + -- .385711s elapsed
│ │ │ │ + -- .205387s elapsed
│ │ │ │ + -- .231241s elapsed
│ │ │ │ + -- .258153s elapsed
│ │ │ │ + -- .491836s elapsed
│ │ │ │ + -- .37767s elapsed
│ │ │ │ + -- .416397s elapsed
│ │ │ │ + -- .293586s elapsed
│ │ │ │ + -- .213893s elapsed
│ │ │ │  i17 : netList oo
│ │ │ │  
│ │ │ │        +---------------+---------------+
│ │ │ │  o17 = |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │        |{3, 4, 4}      |{2, 3, 5}      |
│ │ │ │        +---------------+---------------+
│ │ │ │ @@ -233,75 +233,75 @@
│ │ │ │  
│ │ │ │  o22 = 15
│ │ │ │  i23 : allcomps = for G in Gs list (
│ │ │ │            IG = oscQuadrics(G, R);
│ │ │ │            elapsedTime comps := decompose IG;
│ │ │ │            {comps/codim, comps/degree}
│ │ │ │            );
│ │ │ │ - -- .539129s elapsed
│ │ │ │ - -- .645006s elapsed
│ │ │ │ - -- 1.05918s elapsed
│ │ │ │ - -- 1.23488s elapsed
│ │ │ │ - -- .813801s elapsed
│ │ │ │ + -- .310823s elapsed
│ │ │ │ + -- .376704s elapsed
│ │ │ │ + -- .768631s elapsed
│ │ │ │ + -- .979926s elapsed
│ │ │ │ + -- .609071s elapsed
│ │ │ │ + -- .763568s elapsed
│ │ │ │ + -- .836899s elapsed
│ │ │ │ + -- .878904s elapsed
│ │ │ │ + -- .616915s elapsed
│ │ │ │ + -- .61028s elapsed
│ │ │ │ + -- .28669s elapsed
│ │ │ │ + -- .33432s elapsed
│ │ │ │ + -- .411242s elapsed
│ │ │ │ + -- .53354s elapsed
│ │ │ │ + -- .704534s elapsed
│ │ │ │ + -- 1.02456s elapsed
│ │ │ │ + -- .783083s elapsed
│ │ │ │ + -- .706565s elapsed
│ │ │ │ + -- .991695s elapsed
│ │ │ │ + -- .837744s elapsed
│ │ │ │ + -- .66319s elapsed
│ │ │ │ + -- .746018s elapsed
│ │ │ │ + -- 1.13548s elapsed
│ │ │ │ + -- .943452s elapsed
│ │ │ │ + -- .381898s elapsed
│ │ │ │ + -- .52052s elapsed
│ │ │ │   -- 1.01888s elapsed
│ │ │ │ - -- 1.18556s elapsed
│ │ │ │ - -- 1.19237s elapsed
│ │ │ │ - -- .809367s elapsed
│ │ │ │ - -- .688287s elapsed
│ │ │ │ - -- .34871s elapsed
│ │ │ │ - -- .470565s elapsed
│ │ │ │ - -- .796532s elapsed
│ │ │ │ - -- .592092s elapsed
│ │ │ │ - -- .870412s elapsed
│ │ │ │ - -- 1.45151s elapsed
│ │ │ │ - -- 1.08533s elapsed
│ │ │ │ - -- .929525s elapsed
│ │ │ │ - -- 1.53034s elapsed
│ │ │ │ - -- 1.33678s elapsed
│ │ │ │ - -- .760184s elapsed
│ │ │ │ - -- 1.28518s elapsed
│ │ │ │ - -- 1.71666s elapsed
│ │ │ │ - -- 1.22854s elapsed
│ │ │ │ - -- .674943s elapsed
│ │ │ │ - -- .894164s elapsed
│ │ │ │ - -- 1.57127s elapsed
│ │ │ │ - -- .743101s elapsed
│ │ │ │ - -- .756756s elapsed
│ │ │ │ - -- 1.13138s elapsed
│ │ │ │ - -- 1.25775s elapsed
│ │ │ │ - -- 1.00643s elapsed
│ │ │ │ - -- .580285s elapsed
│ │ │ │ - -- 1.47228s elapsed
│ │ │ │ - -- 1.02334s elapsed
│ │ │ │ - -- 1.53993s elapsed
│ │ │ │ - -- .942582s elapsed
│ │ │ │ - -- 1.21686s elapsed
│ │ │ │ - -- 1.59056s elapsed
│ │ │ │ - -- 1.01279s elapsed
│ │ │ │ - -- .781942s elapsed
│ │ │ │ - -- 1.63071s elapsed
│ │ │ │ - -- 1.80895s elapsed
│ │ │ │ - -- 2.00814s elapsed
│ │ │ │ - -- 1.15234s elapsed
│ │ │ │ - -- 1.32068s elapsed
│ │ │ │ - -- 1.58339s elapsed
│ │ │ │ - -- 1.20196s elapsed
│ │ │ │ - -- 1.10322s elapsed
│ │ │ │ - -- 1.215s elapsed
│ │ │ │ - -- 1.22183s elapsed
│ │ │ │ - -- .920406s elapsed
│ │ │ │ - -- .900219s elapsed
│ │ │ │ - -- 1.32942s elapsed
│ │ │ │ - -- .609367s elapsed
│ │ │ │ - -- 1.44447s elapsed
│ │ │ │ - -- 1.35121s elapsed
│ │ │ │ - -- 1.47404s elapsed
│ │ │ │ - -- .758204s elapsed
│ │ │ │ - -- .533039s elapsed
│ │ │ │ - -- .40806s elapsed
│ │ │ │ + -- .555376s elapsed
│ │ │ │ + -- .456873s elapsed
│ │ │ │ + -- .668581s elapsed
│ │ │ │ + -- .847484s elapsed
│ │ │ │ + -- .681135s elapsed
│ │ │ │ + -- .464221s elapsed
│ │ │ │ + -- .866599s elapsed
│ │ │ │ + -- .651717s elapsed
│ │ │ │ + -- .921585s elapsed
│ │ │ │ + -- .786177s elapsed
│ │ │ │ + -- .92925s elapsed
│ │ │ │ + -- 1.00142s elapsed
│ │ │ │ + -- .609246s elapsed
│ │ │ │ + -- .595969s elapsed
│ │ │ │ + -- .917554s elapsed
│ │ │ │ + -- 1.14664s elapsed
│ │ │ │ + -- 1.40924s elapsed
│ │ │ │ + -- 1.0083s elapsed
│ │ │ │ + -- .97673s elapsed
│ │ │ │ + -- 1.1517s elapsed
│ │ │ │ + -- .989014s elapsed
│ │ │ │ + -- .804033s elapsed
│ │ │ │ + -- .87493s elapsed
│ │ │ │ + -- .887405s elapsed
│ │ │ │ + -- .650366s elapsed
│ │ │ │ + -- .669484s elapsed
│ │ │ │ + -- .802823s elapsed
│ │ │ │ + -- .500618s elapsed
│ │ │ │ + -- .911672s elapsed
│ │ │ │ + -- 1.00836s elapsed
│ │ │ │ + -- 1.12274s elapsed
│ │ │ │ + -- .621319s elapsed
│ │ │ │ + -- .397177s elapsed
│ │ │ │ + -- .29607s elapsed
│ │ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │ │  
│ │ │ │        +------------------------+------------------------+
│ │ │ │  o24 = |codimensions            |degrees                 |
│ │ │ │        +------------------------+------------------------+
│ │ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ │ │        +------------------------+------------------------+
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html
│ │ │ @@ -115,15 +115,15 @@
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .859s elapsed
│ │ │ + -- .827s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -136,15 +136,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │ - -- .901s elapsed
│ │ │ + -- .869s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -43,15 +43,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ .954}} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : elapsedTime stablesolsPent = showExoticSolutions Pent │ │ │ │ - -- .859s elapsed │ │ │ │ + -- .827s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {1, 4}} -- │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ +----+-----+-----+----+-----+-----+-----+-----+ │ │ │ │ |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951| │ │ │ │ @@ -64,15 +64,15 @@ │ │ │ │ +---+---+---+---+ │ │ │ │ |72 |144|216|288| │ │ │ │ +---+---+---+---+ │ │ │ │ |0 |0 |0 |0 | │ │ │ │ +---+---+---+---+ │ │ │ │ |288|216|144|72 | │ │ │ │ +---+---+---+---+ │ │ │ │ - -- .901s elapsed │ │ │ │ + -- .869s elapsed │ │ │ │ │ │ │ │ o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ Computing the (linearly) stable solutions for K5C5 takes a minute or two: │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ @@ -120,19 +120,19 @@ │ │ │ o5 = 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 82, 85, 86, 87, 88, 89, 90}
│ │ │ - -- .646s elapsed
│ │ │ + -- .57s elapsed
│ │ │  warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97}
│ │ │ - -- .578s elapsed
│ │ │ - -- .694s elapsed
│ │ │ - -- .859s elapsed
│ │ │ + -- .562s elapsed
│ │ │ + -- .716s elapsed
│ │ │ + -- .921s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |1    |1   |1   |1    |0    |0    |0    |0    |
│ │ │ @@ -145,25 +145,25 @@
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │ - -- .966s elapsed
│ │ │ - -- 1.19s elapsed
│ │ │ + -- 1.1s elapsed
│ │ │ + -- 1.12s elapsed
│ │ │  warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, 47, 48, 53, 59, 60, 61}
│ │ │ - -- 1.27s elapsed
│ │ │ + -- 1.37s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34}
│ │ │ - -- 1.07s elapsed
│ │ │ - -- 1.05s elapsed
│ │ │ + -- 1.14s elapsed
│ │ │ + -- 1.14s elapsed
│ │ │  warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │ - -- 1.21s elapsed
│ │ │ + -- 1.22s elapsed
│ │ │  warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, 75, 76, 77}
│ │ │ - -- 1.03s elapsed
    │ │ │ + -- 1.15s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,21 +48,21 @@ │ │ │ │ i5 : printingPrecision = 3 │ │ │ │ │ │ │ │ o5 = 3 │ │ │ │ i6 : for G in Gs list showExoticSolutions G; │ │ │ │ warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, │ │ │ │ 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, │ │ │ │ 80, 82, 85, 86, 87, 88, 89, 90} │ │ │ │ - -- .646s elapsed │ │ │ │ + -- .57s elapsed │ │ │ │ warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, │ │ │ │ 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, │ │ │ │ 91, 94, 95, 97} │ │ │ │ - -- .578s elapsed │ │ │ │ - -- .694s elapsed │ │ │ │ - -- .859s elapsed │ │ │ │ + -- .562s elapsed │ │ │ │ + -- .716s elapsed │ │ │ │ + -- .921s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {2, 3}} -- │ │ │ │ 1 => {3, 4} │ │ │ │ 2 => {0, 4} │ │ │ │ 3 => {0, 1} │ │ │ │ 4 => {2, 1} │ │ │ │ +-----+----+----+-----+-----+-----+-----+-----+ │ │ │ │ |1 |1 |1 |1 |0 |0 |0 |0 | │ │ │ │ @@ -75,24 +75,24 @@ │ │ │ │ +---+---+---+---+ │ │ │ │ |0 |0 |0 |0 | │ │ │ │ +---+---+---+---+ │ │ │ │ |216|72 |288|144| │ │ │ │ +---+---+---+---+ │ │ │ │ |144|288|72 |216| │ │ │ │ +---+---+---+---+ │ │ │ │ - -- .966s elapsed │ │ │ │ - -- 1.19s elapsed │ │ │ │ + -- 1.1s elapsed │ │ │ │ + -- 1.12s elapsed │ │ │ │ warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, │ │ │ │ 47, 48, 53, 59, 60, 61} │ │ │ │ - -- 1.27s elapsed │ │ │ │ + -- 1.37s elapsed │ │ │ │ warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, │ │ │ │ 28, 29, 30, 31, 32, 33, 34} │ │ │ │ - -- 1.07s elapsed │ │ │ │ - -- 1.05s elapsed │ │ │ │ + -- 1.14s elapsed │ │ │ │ + -- 1.14s elapsed │ │ │ │ warning: some solutions are not regular: {26, 27, 30, 31, 33} │ │ │ │ - -- 1.21s elapsed │ │ │ │ + -- 1.22s elapsed │ │ │ │ warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, │ │ │ │ 75, 76, 77} │ │ │ │ - -- 1.03s elapsed │ │ │ │ + -- 1.15s elapsed │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:812:0. │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ 
    i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : getLinearlyStableSolutions(G)
│ │ │  warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21}
│ │ │ - -- .215945s elapsed
│ │ │ + -- .189996s elapsed
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ of each oscillator is given by the Kuramoto model. The linear stability of a │ │ │ │ solution is determined by the eigenvalues of the Jacobian matrix of the system │ │ │ │ evaluated at the solution. │ │ │ │ i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}}); │ │ │ │ i2 : getLinearlyStableSolutions(G) │ │ │ │ warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, │ │ │ │ 16, 17, 18, 19, 21} │ │ │ │ - -- .215945s elapsed │ │ │ │ + -- .189996s elapsed │ │ │ │ │ │ │ │ o2 = {{1, 1, 1, 0, 0, 0}} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _f_i_n_d_R_e_a_l_S_o_l_u_t_i_o_n_s -- find real solutions, at least one per component for │ │ │ │ well-conditioned systems │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html │ │ │ @@ -98,15 +98,15 @@ │ │ │ │ │ │ o1 : Graph │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : showExoticSolutions G
│ │ │ - -- .722313s elapsed
│ │ │ + -- .808396s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o3 : Graph
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : showExoticSolutions G
│ │ │ - -- .986724s elapsed
│ │ │ + -- 1.06047s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -34,15 +34,15 @@ │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ │ │ │ │ o1 : Graph │ │ │ │ i2 : showExoticSolutions G │ │ │ │ - -- .722313s elapsed │ │ │ │ + -- .808396s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {1, 4}} -- │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ +-------+--------+--------+-------+--------+--------+--------+--------+ │ │ │ │ |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057| │ │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3, 4} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 2, 3} │ │ │ │ │ │ │ │ o3 : Graph │ │ │ │ i4 : showExoticSolutions G │ │ │ │ - -- .986724s elapsed │ │ │ │ + -- 1.06047s elapsed │ │ │ │ │ │ │ │ o4 = {{1, 1, 1, 1, 0, 0, 0, 0}} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_t_L_i_n_e_a_r_l_y_S_t_a_b_l_e_S_o_l_u_t_i_o_n_s -- Compute linearly stable solutions for the │ │ │ │ Kuramoto oscillator system associated to a graph │ │ ├── ./usr/share/doc/Macaulay2/PHCpack/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=34 │ │ │ ZmFjdG9yV2l0bmVzc1NldCguLi4sVmVyYm9zZT0+Li4uKQ== │ │ │ #:len=595 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uIHRvIHNwZWNpZnkgd2hldGhl │ │ │ ciBhZGRpdGlvbmFsIG91dHB1dCBpcyB3YW50ZWQiLCAibGluZW51bSIgPT4gNDEyLCAiZmlsZW5h │ │ ├── ./usr/share/doc/Macaulay2/PackageCitations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ UGFja2FnZUNpdGF0aW9ucw== │ │ │ #:len=1928 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZhY2lsaXRhdGluZyBj │ │ │ aXRhdGlvbiBvZiBNYWNhdWxheTIgcGFja2FnZXMiLCAibGluZW51bSIgPT4gMjY1LCBTZWVBbHNv │ │ ├── ./usr/share/doc/Macaulay2/PackageTemplate/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ c2Vjb25kRnVuY3Rpb24= │ │ │ #:len=379 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBzaWxseSBzZWNvbmQgZnVuY3Rpb24i │ │ │ LCBEZXNjcmlwdGlvbiA9PiAoIlRoaXMgZnVuY3Rpb24gaXMgcHJvdmlkZWQgYnkgdGhlIHBhY2th │ │ ├── ./usr/share/doc/Macaulay2/Padic/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ aW52ZXJzZShQYWRpY051bWJlcik= │ │ │ #:len=1089 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibXVsdGlwbGljYXRpdmUgaW52ZXJzZSBv │ │ │ ZiBhIHAtYWRpYyBudW1iZXIiLCAibGluZW51bSIgPT4gNzU4LCBJbnB1dHMgPT4ge1NQQU57VFR7 │ │ ├── ./usr/share/doc/Macaulay2/Parametrization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ cGFyYW1ldHJpemVDb25pYw== │ │ │ #:len=1904 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3B0aW9uIHdoZXRoZXIgdG8gcmF0aW9u │ │ │ YWxseSBwYXJhbWV0cml6ZSBjb25pY3MuIiwgImxpbmVudW0iID0+IDE2NzYsICJmaWxlbmFtZSIg │ │ ├── ./usr/share/doc/Macaulay2/Parsing/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ Y2hhckFuYWx5emVy │ │ │ #:len=752 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBsZXhpY2FsIGFuYWx5emVyIHRoYXQg │ │ │ cHJvdmlkZXMgY2hhcmFjdGVycyBmcm9tIGEgc3RyaW5nIG9uZSBhdCBhIHRpbWUiLCAibGluZW51 │ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=7 │ │ │ YW1iaWVudA== │ │ │ #:len=256 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZXhwb3J0ZWQgZnJvbSB0aGUgTkNBTGdl │ │ │ YnJhIHBhY2thZ2UuIiwgRGVzY3JpcHRpb24gPT4ge30sICJsaW5lbnVtIiA9PiAxNTksIHN5bWJv │ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ @@ -80,20 +80,20 @@ │ │ │ i19 : needsPackage "MultigradedImplicitization"; │ │ │ │ │ │ i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S); │ │ │ warning: computation begun over finite field. resulting polynomials may not lie in the ideal │ │ │ computing total degree: 1 │ │ │ number of monomials = 9 │ │ │ number of distinct multidegrees = 1 │ │ │ - -- .0157139s elapsed │ │ │ + -- .00776879s elapsed │ │ │ WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation. │ │ │ computing total degree: 2 │ │ │ number of monomials = 45 │ │ │ number of distinct multidegrees = 1 │ │ │ - -- .717745s elapsed │ │ │ + -- .465207s elapsed │ │ │ │ │ │ o20 : Ideal of S │ │ │ │ │ │ i21 : dim I │ │ │ │ │ │ o21 = 5 │ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ @@ -213,20 +213,20 @@ │ │ │ │ │ │ │ │ │ 
    i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .0157139s elapsed
│ │ │ + -- .00776879s elapsed
│ │ │  WARNING: There are linear relations. You may want to reduce the number of variables to speed up the computation.
│ │ │  computing total degree: 2
│ │ │  number of monomials = 45
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .717745s elapsed
│ │ │ + -- .465207s elapsed
│ │ │  
│ │ │  o20 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : dim I
│ │ │ ├── html2text {}
│ │ │ │ @@ -77,21 +77,21 @@
│ │ │ │  i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix
│ │ │ │  {toList(9:1)}), S);
│ │ │ │  warning: computation begun over finite field. resulting polynomials may not lie
│ │ │ │  in the ideal
│ │ │ │  computing total degree: 1
│ │ │ │  number of monomials = 9
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .0157139s elapsed
│ │ │ │ + -- .00776879s elapsed
│ │ │ │  WARNING: There are linear relations. You may want to reduce the number of
│ │ │ │  variables to speed up the computation.
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 45
│ │ │ │  number of distinct multidegrees = 1
│ │ │ │ - -- .717745s elapsed
│ │ │ │ + -- .465207s elapsed
│ │ │ │  
│ │ │ │  o20 : Ideal of S
│ │ │ │  i21 : dim I
│ │ │ │  
│ │ │ │  o21 = 5
│ │ │ │  i22 : isPrime I
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  cmFuZG9tRXh0ZW5zaW9uKE1hdHJpeCxNYXRyaXgp
│ │ │  #:len=301
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzIxMywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmFuZG9tRXh0ZW5zaW9uLE1hdHJpeCxNYXRyaXgp
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │  i3 : g=3
│ │ │  
│ │ │  o3 = 3
│ │ │  
│ │ │  i4 : kk= ZZ/101;
│ │ │  
│ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- .260086s elapsed
│ │ │ + -- .221428s elapsed
│ │ │  
│ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │  
│ │ │  o6 : CliffordModule
│ │ │  
│ │ │ @@ -67,30 +67,30 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .898827s elapsed
│ │ │ + -- .467505s elapsed
│ │ │  
│ │ │  i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │            -4: 16  .     -2: 32  .
│ │ │            -3: 16  .     -1:  .  .
│ │ │            -2:  .  .      0:  .  .
│ │ │            -1:  . 16      1:  . 32
│ │ │             0:  . 16
│ │ │  
│ │ │  o12 : Sequence
│ │ │  
│ │ │  i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 30.2671s elapsed
│ │ │ + -- 15.2917s elapsed
│ │ │  
│ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │  o14 : Matrix S   <-- S
│ │ │  
│ │ │  i15 : r=2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_search__Ulrich.out
│ │ │ @@ -46,30 +46,30 @@
│ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o11 = CliffordModule{...6...}
│ │ │  
│ │ │  o11 : CliffordModule
│ │ │  
│ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- 1.14431s elapsed
│ │ │ + -- .631573s elapsed
│ │ │  
│ │ │  i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │  o13 = total: 8 16 8
│ │ │            0: 8 16 8
│ │ │  
│ │ │  o13 : BettiTally
│ │ │  
│ │ │  i14 : ann Ulr == ideal qs
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.91837s elapsed
│ │ │ + -- 2.13417s elapsed
│ │ │  
│ │ │  i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │  o16 = total: 12 24 12
│ │ │            0: 12 24 12
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html
│ │ │ @@ -133,15 +133,15 @@
│ │ │              
│ │ │                
    i4 : kk= ZZ/101;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- .260086s elapsed
    
│ │ │ + -- .221428s elapsed
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │ @@ -177,15 +177,15 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .898827s elapsed
    │ │ │ + -- .467505s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o12 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 30.2671s elapsed
    │ │ │ + -- 15.2917s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │  
│ │ │                32      32
│ │ │ ├── html2text {}
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │              -- will give an Ulrich bundle, with betti table
│ │ │ │              -- 16 32 16
│ │ │ │  i3 : g=3
│ │ │ │  
│ │ │ │  o3 = 3
│ │ │ │  i4 : kk= ZZ/101;
│ │ │ │  i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ │ - -- .260086s elapsed
│ │ │ │ + -- .221428s elapsed
│ │ │ │  i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o6 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o6 : CliffordModule
│ │ │ │  i7 : Mor = vectorBundleOnE M.evenCenter;
│ │ │ │  i8 : Mor1= vectorBundleOnE M.oddCenter;
│ │ │ │ @@ -75,29 +75,29 @@
│ │ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │ │            V = vectorBundleOnE m12;
│ │ │ │            Ul=tensorProduct(Mor,V);
│ │ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │ │            d0=unique degrees target Ul.yAction;
│ │ │ │            d1=unique degrees target Ul1.yAction;
│ │ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ │ - -- .898827s elapsed
│ │ │ │ + -- .467505s elapsed
│ │ │ │  i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │ │  
│ │ │ │                 0  1          0  1
│ │ │ │  o12 = (total: 32 32, total: 32 32)
│ │ │ │            -4: 16  .     -2: 32  .
│ │ │ │            -3: 16  .     -1:  .  .
│ │ │ │            -2:  .  .      0:  .  .
│ │ │ │            -1:  . 16      1:  . 32
│ │ │ │             0:  . 16
│ │ │ │  
│ │ │ │  o12 : Sequence
│ │ │ │  i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the
│ │ │ │  actions of generators
│ │ │ │ - -- 30.2671s elapsed
│ │ │ │ + -- 15.2917s elapsed
│ │ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │ │  
│ │ │ │                32      32
│ │ │ │  o14 : Matrix S   <-- S
│ │ │ │  i15 : r=2
│ │ │ │  
│ │ │ │  o15 = 2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_search__Ulrich.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o11 : CliffordModule
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- 1.14431s elapsed
    │ │ │ + -- .631573s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │ @@ -190,15 +190,15 @@
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.91837s elapsed
    │ │ │ + -- 2.13417s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │ ├── html2text {}
│ │ │ │ @@ -64,27 +64,27 @@
│ │ │ │  o10 : Matrix S  <-- S
│ │ │ │  i11 : M=cliffordModule(Mu1,Mu2,R)
│ │ │ │  
│ │ │ │  o11 = CliffordModule{...6...}
│ │ │ │  
│ │ │ │  o11 : CliffordModule
│ │ │ │  i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ │ - -- 1.14431s elapsed
│ │ │ │ + -- .631573s elapsed
│ │ │ │  i13 : betti freeResolution Ulr
│ │ │ │  
│ │ │ │               0  1 2
│ │ │ │  o13 = total: 8 16 8
│ │ │ │            0: 8 16 8
│ │ │ │  
│ │ │ │  o13 : BettiTally
│ │ │ │  i14 : ann Ulr == ideal qs
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ │ - -- 2.91837s elapsed
│ │ │ │ + -- 2.13417s elapsed
│ │ │ │  i16 : betti freeResolution Ulr3
│ │ │ │  
│ │ │ │                0  1  2
│ │ │ │  o16 = total: 12 24 12
│ │ │ │            0: 12 24 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/Permanents/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  cnlzZXI=
│ │ │  #:len=2107
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSBwZXJtYW5lbnQgdXNpbmcg
│ │ │  UnlzZXIncyBmb3JtdWxhIiwgImxpbmVudW0iID0+IDUyMywgSW5wdXRzID0+IHtTUEFOe1RUeyJN
│ │ ├── ./usr/share/doc/Macaulay2/Permutations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZGVzY2VudHMoUGVybXV0YXRpb24p
│ │ │  #:len=276
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkZXNjZW50cyxQZXJtdXRhdGlvbiksImRlc2NlbnRz
│ │ ├── ./usr/share/doc/Macaulay2/PhylogeneticTrees/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  cGh5bG9Ub3JpY1F1YWRz
│ │ │  #:len=2848
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSB0aGUgcXVhZHJhdGljIGlu
│ │ │  dmFyaWFudHMgb2YgYSBncm91cC1iYXNlZCBwaHlsb2dlbmV0aWMgbW9kZWwiLCAibGluZW51bSIg
│ │ ├── ./usr/share/doc/Macaulay2/PieriMaps/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  ZGlzcGxheUxSSW1hZ2UoTGlzdCk=
│ │ │  #:len=258
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkaXNwbGF5TFJJbWFnZSxMaXN0KSwiZGlzcGxheUxS
│ │ ├── ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  YWRkaXRpb24oSWRlYWwsSWRlYWwsSWRlYWwp
│ │ │  #:len=295
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGRpdGlvbixJZGVhbCxJZGVhbCxJZGVhbCksImFk
│ │ ├── ./usr/share/doc/Macaulay2/Points/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  cmFuZG9tUG9pbnRzTWF0KFJpbmcsWlop
│ │ │  #:len=255
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODgzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyYW5kb21Qb2ludHNNYXQsUmluZyxaWiksInJhbmRv
│ │ ├── ./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out
│ │ │ @@ -66,17 +66,17 @@
│ │ │  i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3}
│ │ │  
│ │ │  o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.34235s elapsed
│ │ │ + -- 1.26611s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.80041s elapsed
│ │ │ + -- 3.53481s elapsed
│ │ │  
│ │ │  i12 : G==H
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
│ │ │ @@ -182,21 +182,21 @@
│ │ │  
│ │ │  o9 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.34235s elapsed
    │ │ │ + -- 1.26611s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.80041s elapsed
    │ │ │ + -- 3.53481s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : G==H
│ │ │  
│ │ │  o12 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -81,17 +81,17 @@ │ │ │ │ o8 : Matrix K <-- K │ │ │ │ i9 : mults = {1,2,3,1,2,3,1,2,3,1,2,3} │ │ │ │ │ │ │ │ o9 = {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3} │ │ │ │ │ │ │ │ o9 : List │ │ │ │ i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R); │ │ │ │ - -- 2.34235s elapsed │ │ │ │ + -- 1.26611s elapsed │ │ │ │ i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R); │ │ │ │ - -- 5.80041s elapsed │ │ │ │ + -- 3.53481s elapsed │ │ │ │ i12 : G==H │ │ │ │ │ │ │ │ o12 = true │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For reduced points, this function may be a bit slower than _a_f_f_i_n_e_P_o_i_n_t_s. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _a_f_f_i_n_e_F_a_t_P_o_i_n_t_s_B_y_I_n_t_e_r_s_e_c_t_i_o_n_(_M_a_t_r_i_x_,_L_i_s_t_,_R_i_n_g_) -- computes ideal of fat │ │ ├── ./usr/share/doc/Macaulay2/Polyhedra/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ bWF4Q29uZXM= │ │ │ #:len=1185 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheXMgdGhlIGdlbmVyYXRpbmcg │ │ │ Q29uZXMgb2YgYSBGYW4iLCAibGluZW51bSIgPT4gODQzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkYi │ │ ├── ./usr/share/doc/Macaulay2/Polymake/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ UG9seW1ha2U= │ │ │ #:len=610 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciBpbnRlcmZhY2lu │ │ │ ZyB3aXRoIHBvbHltYWtlIiwgRGVzY3JpcHRpb24gPT4gKEVNeyJQb2x5bWFrZSJ9LCIgaXMgYSBw │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ c3RhbmRhcmRSb29rTnVtYmVy │ │ │ #:len=1159 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU3RhbmRhcmQgcm9vayBudW1iZXIgb2Yg │ │ │ YSBjb2xsZWN0aW9uIG9mIGNlbGxzIiwgImxpbmVudW0iID0+IDE1MzQsIElucHV0cyA9PiB7U1BB │ │ ├── ./usr/share/doc/Macaulay2/Posets/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=17 │ │ │ bWF4aW1hbEFudGljaGFpbnM= │ │ │ #:len=1127 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYWxsIG1heGltYWwgYW50 │ │ │ aWNoYWlucyBvZiBhIHBvc2V0IiwgImxpbmVudW0iID0+IDQ5ODgsIElucHV0cyA9PiB7U1BBTntU │ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out │ │ │ @@ -31,27 +31,27 @@ │ │ │ o5 = CacheTable{name => P} │ │ │ │ │ │ i6 : C == P │ │ │ │ │ │ o6 = true │ │ │ │ │ │ i7 : time isDistributive C │ │ │ - -- used 8.826e-06s (cpu); 6.002e-06s (thread); 0s (gc) │ │ │ + -- used 1.2925e-05s (cpu); 6.759e-06s (thread); 0s (gc) │ │ │ │ │ │ o7 = true │ │ │ │ │ │ i8 : time isDistributive P │ │ │ - -- used 6.82863s (cpu); 4.33499s (thread); 0s (gc) │ │ │ + -- used 6.07607s (cpu); 3.65574s (thread); 0s (gc) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : C' = dual C; │ │ │ │ │ │ i10 : time isDistributive C' │ │ │ - -- used 7.003e-06s (cpu); 6.472e-06s (thread); 0s (gc) │ │ │ + -- used 6.717e-06s (cpu); 1.7056e-05s (thread); 0s (gc) │ │ │ │ │ │ o10 = true │ │ │ │ │ │ i11 : peek C'.cache │ │ │ │ │ │ o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}} } │ │ │ coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4}, {6, 5}, {7, 6}, {8, 7}, {9, 8}} │ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out │ │ │ @@ -7,22 +7,22 @@ │ │ │ o2 = Partition{4, 2} │ │ │ │ │ │ o2 : Partition │ │ │ │ │ │ i3 : D = dominanceLattice 6; │ │ │ │ │ │ i4 : time greeneKleitmanPartition(D, Strategy => "antichains") │ │ │ - -- used 0.435352s (cpu); 0.261753s (thread); 0s (gc) │ │ │ + -- used 0.361411s (cpu); 0.216392s (thread); 0s (gc) │ │ │ │ │ │ o4 = Partition{9, 2} │ │ │ │ │ │ o4 : Partition │ │ │ │ │ │ i5 : time greeneKleitmanPartition(D, Strategy => "chains") │ │ │ - -- used 1.2754e-05s (cpu); 1.2283e-05s (thread); 0s (gc) │ │ │ + -- used 1.2029e-05s (cpu); 1.088e-05s (thread); 0s (gc) │ │ │ │ │ │ o5 = Partition{9, 2} │ │ │ │ │ │ o5 : Partition │ │ │ │ │ │ i6 : greeneKleitmanPartition chain 10 │ │ ├── ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ @@ -112,23 +112,23 @@ │ │ │ │ │ │ o6 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isDistributive C
│ │ │ - -- used 8.826e-06s (cpu); 6.002e-06s (thread); 0s (gc)
│ │ │ + -- used 1.2925e-05s (cpu); 6.759e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time isDistributive P
│ │ │ - -- used 6.82863s (cpu); 4.33499s (thread); 0s (gc)
│ │ │ + -- used 6.07607s (cpu); 3.65574s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │ @@ -138,15 +138,15 @@ │ │ │ │ │ │ 
    i9 : C' = dual C;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time isDistributive C'
│ │ │ - -- used 7.003e-06s (cpu); 6.472e-06s (thread); 0s (gc)
│ │ │ + -- used 6.717e-06s (cpu); 1.7056e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : peek C'.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,26 +41,26 @@
│ │ │ │  i5 : peek P.cache
│ │ │ │  
│ │ │ │  o5 = CacheTable{name => P}
│ │ │ │  i6 : C == P
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : time isDistributive C
│ │ │ │ - -- used 8.826e-06s (cpu); 6.002e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.2925e-05s (cpu); 6.759e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.82863s (cpu); 4.33499s (thread); 0s (gc)
│ │ │ │ + -- used 6.07607s (cpu); 3.65574s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  We also know that the dual of a distributive lattice is again a distributive
│ │ │ │  lattice. Other information is copied when possible.
│ │ │ │  i9 : C' = dual C;
│ │ │ │  i10 : time isDistributive C'
│ │ │ │ - -- used 7.003e-06s (cpu); 6.472e-06s (thread); 0s (gc)
│ │ │ │ + -- used 6.717e-06s (cpu); 1.7056e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : peek C'.cache
│ │ │ │  
│ │ │ │  o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}
│ │ │ │  }
│ │ │ │                   coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4},
│ │ ├── ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html
│ │ │ @@ -107,25 +107,25 @@
│ │ │              
│ │ │                
    i3 : D = dominanceLattice 6;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ - -- used 0.435352s (cpu); 0.261753s (thread); 0s (gc)
│ │ │ + -- used 0.361411s (cpu); 0.216392s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.2754e-05s (cpu); 1.2283e-05s (thread); 0s (gc)
│ │ │ + -- used 1.2029e-05s (cpu); 1.088e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,21 +30,21 @@
│ │ │ │  
│ │ │ │  o2 : Partition
│ │ │ │  The conjugate of $l$ has the same property, but with chains replaced by
│ │ │ │  _a_n_t_i_c_h_a_i_n_s. Because of this, it is often better to count via antichains instead
│ │ │ │  of chains. This can be done by passing "antichains" as the Strategy.
│ │ │ │  i3 : D = dominanceLattice 6;
│ │ │ │  i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ │ - -- used 0.435352s (cpu); 0.261753s (thread); 0s (gc)
│ │ │ │ + -- used 0.361411s (cpu); 0.216392s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 1.2754e-05s (cpu); 1.2283e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.2029e-05s (cpu); 1.088e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o5 : Partition
│ │ │ │  The Greene-Kleitman partition of the $n$ _c_h_a_i_n is the partition of $n$ with $1$
│ │ │ │  part.
│ │ │ │  i6 : greeneKleitmanPartition chain 10
│ │ ├── ./usr/share/doc/Macaulay2/PositivityToricBundles/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  ZHJhd1BhcmxpYW1lbnQyRHRpa3o=
│ │ │  #:len=2626
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmlzdWFsaXNlcyB0aGUgcGFybGlhbWVu
│ │ │  dCBvZiBwb2x5dG9wZXMgZm9yIGEgdmVjdG9yIGJ1bmRsZSBvbiBhIHRvcmljIHN1cmZhY2UgdXNp
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  cHJpbWFyeUNvbXBvbmVudCguLi4sU3RyYXRlZ3k9Pi4uLik=
│ │ │  #:len=316
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzcxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1twcmltYXJ5Q29tcG9uZW50LFN0cmF0ZWd5XSwicHJp
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_associated__Primes.out
│ │ │ @@ -125,24 +125,24 @@
│ │ │        -----------------------------------------------------------------------
│ │ │        ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o20 : Set
│ │ │  
│ │ │  i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o21 : Set
│ │ │  
│ │ │  i22 : assert(M1 === M2)
│ │ │  
│ │ │  i23 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out
│ │ │ @@ -24,35 +24,35 @@
│ │ │                | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                              3
│ │ │  o3 : R-module, quotient of R
│ │ │  
│ │ │  i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .217729s elapsed
│ │ │ + -- .111292s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
│ │ │  
│ │ │  i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0174897s elapsed
│ │ │ + -- .0205101s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
│ │ │  
│ │ │  i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .0178098s elapsed
│ │ │ + -- .0208454s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │       ------------------------------------------------------------------------
│ │ │       x x )
│ │ │        0 4
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .0318276s elapsed
│ │ │ + -- .0311418s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : R = QQ[h,l,s,x,y,z]
│ │ │ @@ -41,15 +41,15 @@
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00758549s elapsed
│ │ │ + -- .00684863s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_associated__Primes.html
│ │ │ @@ -291,28 +291,28 @@
│ │ │  o19 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o20 : Set
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : M2 = set apply(L2, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o21 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a},
│ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, a}, {e, c, b, a}}
│ │ │ +      b, a}, {e, d, c, b, a}}
│ │ │  
│ │ │  o21 : Set
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i22 : assert(M1 === M2)
    │ │ │ ├── html2text {} │ │ │ │ @@ -155,24 +155,24 @@ │ │ │ │ o19 = {ideal (a, e), ideal (a, b, c), ideal (a, b, d), ideal (a, b, c, d), │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ ideal (a, b, c, e), ideal (a, b, d, e), ideal (a, b, c, d, e)} │ │ │ │ │ │ │ │ o19 : List │ │ │ │ i20 : M1 = set apply(L1, I -> sort flatten entries gens I) │ │ │ │ │ │ │ │ -o20 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a}, │ │ │ │ +o20 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {e, a}, {e, c, b, a}} │ │ │ │ + b, a}, {e, d, c, b, a}} │ │ │ │ │ │ │ │ o20 : Set │ │ │ │ i21 : M2 = set apply(L2, I -> sort flatten entries gens I) │ │ │ │ │ │ │ │ -o21 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, b, a}, {e, d, b, a}, │ │ │ │ +o21 = set {{e, c, b, a}, {d, b, a}, {d, c, b, a}, {c, b, a}, {e, a}, {e, d, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {e, a}, {e, c, b, a}} │ │ │ │ + b, a}, {e, d, c, b, a}} │ │ │ │ │ │ │ │ o21 : Set │ │ │ │ i22 : assert(M1 === M2) │ │ │ │ The method using Ext modules comes from Eisenbud-Huneke-Vasconcelos, Invent. │ │ │ │ Math 110 (1992) 207-235. │ │ │ │ Original author (for ideals): _C_._ _Y_a_c_k_e_l. Updated for modules by J. Chen. │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html │ │ │ @@ -112,41 +112,41 @@ │ │ │ 3 │ │ │ o3 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .217729s elapsed
│ │ │ + -- .111292s elapsed
│ │ │  
│ │ │  o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o4 : R-module, subquotient of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime kernelOfLocalization(M, I2)
│ │ │ - -- .0174897s elapsed
│ │ │ + -- .0205101s elapsed
│ │ │  
│ │ │  o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o5 : R-module, subquotient of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime kernelOfLocalization(M, I3)
│ │ │ - -- .0178098s elapsed
│ │ │ + -- .0208454s elapsed
│ │ │  
│ │ │  o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
│ │ │                    | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
│ │ │                    | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |
│ │ │  
│ │ │                                 3
│ │ │  o6 : R-module, subquotient of R
    │ │ │ ├── html2text {} │ │ │ │ @@ -41,39 +41,39 @@ │ │ │ │ | │ │ │ │ | 0 0 0 0 x_1^5- │ │ │ │ x_0x_2^4 | │ │ │ │ │ │ │ │ 3 │ │ │ │ o3 : R-module, quotient of R │ │ │ │ i4 : elapsedTime kernelOfLocalization(M, I1) │ │ │ │ - -- .217729s elapsed │ │ │ │ + -- .111292s elapsed │ │ │ │ │ │ │ │ o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 1 0 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 1 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ │ │ │ │ │ │ 3 │ │ │ │ o4 : R-module, subquotient of R │ │ │ │ i5 : elapsedTime kernelOfLocalization(M, I2) │ │ │ │ - -- .0174897s elapsed │ │ │ │ + -- .0205101s elapsed │ │ │ │ │ │ │ │ o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 0 0 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 1 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ │ │ │ │ │ │ 3 │ │ │ │ o5 : R-module, subquotient of R │ │ │ │ i6 : elapsedTime kernelOfLocalization(M, I3) │ │ │ │ - -- .0178098s elapsed │ │ │ │ + -- .0208454s elapsed │ │ │ │ │ │ │ │ o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 0 1 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 0 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ @@ -107,15 +107,15 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .0318276s elapsed
│ │ │ + -- .0311418s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │ │ │ │ │ │ │ @@ -153,15 +153,15 @@ │ │ │ │ │ │ o6 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00758549s elapsed
│ │ │ + -- .00684863s elapsed
│ │ │  
│ │ │                     2                3    2     2    8    3    2     2
│ │ │  o7 = ideal (h*l - l  - 4l*s + h*y, h  + l s - h x, s  + h  + l s - h x)
│ │ │  
│ │ │  o7 : Ideal of R
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -40,15 +40,15 @@ │ │ │ │ 2 7 0 7 3 6 2 6 1 6 0 6 2 5 0 5 3 4 2 4 1 4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ 0 4 │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime regSeqInIdeal I │ │ │ │ - -- .0318276s elapsed │ │ │ │ + -- .0311418s elapsed │ │ │ │ │ │ │ │ o3 = ideal (x x , x x + x x , x x + x x , x x + x x ) │ │ │ │ 2 7 3 6 0 7 2 5 0 7 1 4 0 7 │ │ │ │ │ │ │ │ o3 : Ideal of R │ │ │ │ If I is the unit ideal, then an ideal of variables of the ring is returned. │ │ │ │ If the codimension of I is already known, then one can specify this, along with │ │ │ │ @@ -70,15 +70,15 @@ │ │ │ │ l , s ) │ │ │ │ │ │ │ │ o5 : Ideal of R │ │ │ │ i6 : isSubset(I, ideal(s,l,h)) -- implies codim I == 3 │ │ │ │ │ │ │ │ o6 = true │ │ │ │ i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1) │ │ │ │ - -- .00758549s elapsed │ │ │ │ + -- .00684863s elapsed │ │ │ │ │ │ │ │ 2 3 2 2 8 3 2 2 │ │ │ │ o7 = ideal (h*l - l - 4l*s + h*y, h + l s - h x, s + h + l s - h x) │ │ │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_d_i_c_a_l -- the radical of an ideal │ │ ├── ./usr/share/doc/Macaulay2/Probability/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ dERpc3RyaWJ1dGlvbihOdW1iZXIp │ │ │ #:len=261 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIyOSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodERpc3RyaWJ1dGlvbixOdW1iZXIpLCJ0RGlzdHJp │ │ ├── ./usr/share/doc/Macaulay2/PseudomonomialPrimaryDecomposition/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=34 │ │ │ UHNldWRvbW9ub21pYWxQcmltYXJ5RGVjb21wb3NpdGlvbg== │ │ │ #:len=1413 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJpbWFyeSBkZWNvbXBvc2l0aW9uIG9m │ │ │ IGEgc3F1YXJlIGZyZWUgcHNldWRvbW9ub21pYWwgaWRlYWwiLCBEZXNjcmlwdGlvbiA9PiAoRElW │ │ ├── ./usr/share/doc/Macaulay2/Pullback/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ aW50ZXJuYWxVc2VEaXJlY3RTdW0oUmluZyxSaW5nKQ== │ │ │ #:len=285 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY2LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpbnRlcm5hbFVzZURpcmVjdFN1bSxSaW5nLFJpbmcp │ │ ├── ./usr/share/doc/Macaulay2/PushForward/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ cHVzaEZ3ZChNb2R1bGUp │ │ │ #:len=267 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwdXNoRndkLE1vZHVsZSksInB1c2hGd2QoTW9kdWxl │ │ ├── ./usr/share/doc/Macaulay2/Python/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=39 │ │ │ YWRkUHlUb00yRnVuY3Rpb24oTGlzdCxGdW5jdGlvbixTdHJpbmcp │ │ │ #:len=293 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDcxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGRQeVRvTTJGdW5jdGlvbixMaXN0LEZ1bmN0aW9u │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_iterator_lp__Python__Object_rp.out │ │ │ @@ -10,12 +10,12 @@ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ -o3 = │ │ │ +o3 = │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_next_lp__Python__Object_rp.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ -o3 = │ │ │ +o3 = │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ │ │ i4 : next i │ │ │ │ │ │ o4 = 0 │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_python__Run__Script.out │ │ │ @@ -1,22 +1,22 @@ │ │ │ -- -*- M2-comint -*- hash: 447449196062331972 │ │ │ │ │ │ i1 : pyfile = temporaryFileName() | ".py" │ │ │ │ │ │ -o1 = /tmp/M2-32830-0/0.py │ │ │ +o1 = /tmp/M2-44289-0/0.py │ │ │ │ │ │ i2 : pyfile << "import math" << endl │ │ │ │ │ │ -o2 = /tmp/M2-32830-0/0.py │ │ │ +o2 = /tmp/M2-44289-0/0.py │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : pyfile << "x = math.sin(3.4)" << endl << close │ │ │ │ │ │ -o3 = /tmp/M2-32830-0/0.py │ │ │ +o3 = /tmp/M2-44289-0/0.py │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : get pyfile │ │ │ │ │ │ o4 = import math │ │ │ x = math.sin(3.4) │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_to__Python.out │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ │ │ o12 = m2sqrt │ │ │ │ │ │ o12 : FunctionClosure │ │ │ │ │ │ i13 : pysqrt = toPython m2sqrt │ │ │ │ │ │ -o13 = │ │ │ +o13 = │ │ │ │ │ │ o13 : PythonObject of class builtin_function_or_method │ │ │ │ │ │ i14 : pysqrt 2 │ │ │ calling Macaulay2 code from Python! │ │ │ │ │ │ o14 = 1.4142135623730951 │ │ ├── ./usr/share/doc/Macaulay2/Python/example-output/_use_lp__Python__Context_rp.out │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ │ │ o7 : Symbol │ │ │ │ │ │ i8 : use ctx │ │ │ │ │ │ i9 : f │ │ │ │ │ │ -o9 = at 0x7f5ec19a1f30> │ │ │ +o9 = at 0x7f6a08621f30> │ │ │ │ │ │ o9 : PythonObject of class function │ │ │ │ │ │ i10 : x │ │ │ │ │ │ o10 = 5 │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f5ec1813cc0>
│ │ │ +o3 = <range_iterator object at 0x7f6a08493a80>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,15 +22,15 @@ │ │ │ │ i2 : x = builtins@@range 3 │ │ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ │ │ -o3 = │ │ │ │ +o3 = │ │ │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _n_e_x_t_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- retrieve the next item from a python iterator │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _i_t_e_r_a_t_o_r_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- get iterator of iterable python object │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_next_lp__Python__Object_rp.html │ │ │ @@ -91,15 +91,15 @@ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f5ec190e9d0>
│ │ │ +o3 = <range_iterator object at 0x7f6a0858e9d0>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : next i
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : PythonObject of class range_iterator
│ │ │ │  i4 : next i
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  
│ │ │ │  o4 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_python__Run__Script.html
│ │ │ @@ -81,31 +81,31 @@
│ │ │            
    The return value is a Python dictionary containing all the variables defined in the global scope.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-32830-0/0.py
    │ │ │ +o1 = /tmp/M2-44289-0/0.py │ │ │
    │ │ │ 
    i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-32830-0/0.py
│ │ │ +o2 = /tmp/M2-44289-0/0.py
│ │ │  
│ │ │  o2 : File
    │ │ │
    │ │ │ 
    i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-32830-0/0.py
│ │ │ +o3 = /tmp/M2-44289-0/0.py
│ │ │  
│ │ │  o3 : File
    │ │ │
    │ │ │ 
    i4 : get pyfile
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,23 +16,23 @@
│ │ │ │  Execute a sequence of statements as if they were read from a Python file. This
│ │ │ │  is for multi-line code that might contain definitions, control structures,
│ │ │ │  imports, etc. It is great for running Python code from a file.
│ │ │ │  The return value is a Python dictionary containing all the variables defined in
│ │ │ │  the global scope.
│ │ │ │  i1 : pyfile = temporaryFileName() | ".py"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-32830-0/0.py
│ │ │ │ +o1 = /tmp/M2-44289-0/0.py
│ │ │ │  i2 : pyfile << "import math" << endl
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-32830-0/0.py
│ │ │ │ +o2 = /tmp/M2-44289-0/0.py
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-32830-0/0.py
│ │ │ │ +o3 = /tmp/M2-44289-0/0.py
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : get pyfile
│ │ │ │  
│ │ │ │  o4 = import math
│ │ │ │       x = math.sin(3.4)
│ │ │ │  i5 : pythonRunScript oo
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_to__Python.html
│ │ │ @@ -186,15 +186,15 @@
│ │ │  o12 : FunctionClosure
    │ │ │
    │ │ │ 
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7f5ec17e36f0>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7f6a084636a0>
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
    │ │ │
    │ │ │ 
    i14 : pysqrt 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │            sqrt x)
│ │ │ │  
│ │ │ │  o12 = m2sqrt
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : pysqrt = toPython m2sqrt
│ │ │ │  
│ │ │ │ -o13 = 
│ │ │ │ +o13 = 
│ │ │ │  
│ │ │ │  o13 : PythonObject of class builtin_function_or_method
│ │ │ │  i14 : pysqrt 2
│ │ │ │  calling Macaulay2 code from Python!
│ │ │ │  
│ │ │ │  o14 = 1.4142135623730951
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_use_lp__Python__Context_rp.html
│ │ │ @@ -129,15 +129,15 @@
│ │ │                
    i8 : use ctx
    
│ │ │
    │ │ │ 
    i9 : f
│ │ │  
│ │ │ -o9 = <function <lambda> at 0x7f5ec19a1f30>
│ │ │ +o9 = <function <lambda> at 0x7f6a08621f30>
│ │ │  
│ │ │  o9 : PythonObject of class function
    │ │ │
    │ │ │ 
    i10 : x
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  
│ │ │ │  o7 = y
│ │ │ │  
│ │ │ │  o7 : Symbol
│ │ │ │  i8 : use ctx
│ │ │ │  i9 : f
│ │ │ │  
│ │ │ │ -o9 =  at 0x7f5ec19a1f30>
│ │ │ │ +o9 =  at 0x7f6a08621f30>
│ │ │ │  
│ │ │ │  o9 : PythonObject of class function
│ │ │ │  i10 : x
│ │ │ │  
│ │ │ │  o10 = 5
│ │ │ │  
│ │ │ │  o10 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/QthPower/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  bWluaW1pemF0aW9u
│ │ │  #:len=2755
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hhbmdlIHRvIGEgYmV0dGVyIE5vZXRo
│ │ │  ZXIgbm9ybWFsaXphdGlvbiBzdWdnZXN0ZWQgYnkgdGhlIGluZHVjZWQgd2VpZ2h0cyIsICJsaW5l
│ │ ├── ./usr/share/doc/Macaulay2/QuadraticIdealExamplesByRoos/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  aGlnaGVyRGVwdGhUYWJsZQ==
│ │ │  #:len=793
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ3JlYXRlcyBoYXNodGFibGUgb2YgSmFu
│ │ │  LUVyaWsgUm9vcycgZXhhbXBsZXMgb2YgcXVhZHJhdGljIGlkZWFscyB3aXRoIHBvc2l0aXZlIGRl
│ │ ├── ./usr/share/doc/Macaulay2/Quasidegrees/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  cXVhc2lkZWdyZWVzTG9jYWxDb2hvbW9sb2d5
│ │ │  #:len=3881
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgcXVhc2lkZWdyZWUg
│ │ │  c2V0cyBvZiBsb2NhbCBjb2hvbW9sb2d5IG1vZHVsZXMiLCAibGluZW51bSIgPT4gNzg1LCBJbnB1
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  W1FRXQ==
│ │ │  #:len=544
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUXVhdGVybmFyeSBRdWFydGljIEZvcm1z
│ │ │  IGFuZCBHb3JlbnN0ZWluIHJpbmdzIChLYXB1c3RrYSwgS2FwdXN0a2EsIFJhbmVzdGFkLCBTY2hl
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out
│ │ │ @@ -180,15 +180,15 @@
│ │ │  i21 : L = trim groebnerStratum F;
│ │ │  
│ │ │  o21 : Ideal of T
│ │ │  
│ │ │  i22 : assert(dim L == 18)
│ │ │  
│ │ │  i23 : elapsedTime isPrime L
│ │ │ - -- 3.65241s elapsed
│ │ │ + -- 2.05354s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : I = pointsIdeal randomPoints(S, 6)
│ │ │  
│ │ │                               2                              2   2          
│ │ │  o24 = ideal (a*c - 7b*c - 49c  + 40a*d - 42b*d + 12c*d + 28d , b  - 36b*c -
│ │ │ @@ -302,15 +302,15 @@
│ │ │  o38 = true
│ │ │  
│ │ │  i39 : L441 = trim(L + ideal M1);
│ │ │  
│ │ │  o39 : Ideal of T
│ │ │  
│ │ │  i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.58155s elapsed
│ │ │ + -- 1.69767s elapsed
│ │ │  
│ │ │  i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
│ │ │  
│ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html
│ │ │ @@ -349,15 +349,15 @@
│ │ │
    		│ │ │ 
    i22 : assert(dim L == 18)
    │ │ │
    │ │ │ 
    i23 : elapsedTime isPrime L
│ │ │ - -- 3.65241s elapsed
│ │ │ + -- 2.05354s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │
    │ │ │
    │ │ │

    The Schreyer resolution and minimal Betti numbers

    │ │ │ @@ -561,15 +561,15 @@ │ │ │ │ │ │ o39 : Ideal of T     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.58155s elapsed
    │ │ │ + -- 1.69767s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
    │ │ │ ├── html2text {} │ │ │ │ @@ -251,15 +251,15 @@ │ │ │ │ | 31 33 32 34 35 36 | │ │ │ │ +--------------------------------------------------------------+ │ │ │ │ i21 : L = trim groebnerStratum F; │ │ │ │ │ │ │ │ o21 : Ideal of T │ │ │ │ i22 : assert(dim L == 18) │ │ │ │ i23 : elapsedTime isPrime L │ │ │ │ - -- 3.65241s elapsed │ │ │ │ + -- 2.05354s elapsed │ │ │ │ │ │ │ │ o23 = true │ │ │ │ ********** TThhee SScchhrreeyyeerr rreessoolluuttiioonn aanndd mmiinniimmaall BBeettttii nnuummbbeerrss ********** │ │ │ │ Schreyer's construction of a nonminimal free resolution starts with a Groebner │ │ │ │ basis. First, one constructs the SScchhrreeyyeerr ffrraammee (see La Scala, Stillman). This │ │ │ │ is determined solely from the initial ideal $J$ and its minimal generators (but │ │ │ │ depends on some choices of ordering, but otherwise is combinatorial). This │ │ │ │ @@ -415,15 +415,15 @@ │ │ │ │ We now compute the locus in $V(L)$ where the Betti diagram has no cancellation. │ │ │ │ This is a closed subscheme of $V(L)$, which is a closed subscheme of the │ │ │ │ Hilbert scheme. Notice that there are two components. │ │ │ │ i39 : L441 = trim(L + ideal M1); │ │ │ │ │ │ │ │ o39 : Ideal of T │ │ │ │ i40 : elapsedTime compsL441 = decompose L441; │ │ │ │ - -- 2.58155s elapsed │ │ │ │ + -- 1.69767s elapsed │ │ │ │ i41 : #compsL441 │ │ │ │ │ │ │ │ o41 = 2 │ │ │ │ i42 : compsL441/dim -- two components, of dimensions 14 and 16. │ │ │ │ │ │ │ │ o42 = {16, 14} │ │ ├── ./usr/share/doc/Macaulay2/QuillenSuslin/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ aG9ycm9ja3M= │ │ │ #:len=4664 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBsb2NhbCBzb2x1dGlv │ │ │ biB0byB0aGUgdW5pbW9kdWxhciByb3cgcHJvYmxlbSBvdmVyIGEgbG9jYWxpemF0aW9uIGF0IGEg │ │ ├── ./usr/share/doc/Macaulay2/RInterface/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ bmV3IFJPYmplY3QgZnJvbSBDQw== │ │ │ #:len=1343 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY3JlYXRlIGFuIFIgY29tcGxleCB2ZWN0 │ │ │ b3IgZnJvbSBhIGNvbXBsZXggbnVtYmVyIiwgImxpbmVudW0iID0+IDI1NSwgSW5wdXRzID0+IHtT │ │ ├── ./usr/share/doc/Macaulay2/RInterface/example-output/___R__Value.out │ │ │ @@ -14,15 +14,15 @@ │ │ │ │ │ │ o3 = [1] 120 │ │ │ │ │ │ o3 : RObject of type double │ │ │ │ │ │ i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ} │ │ │ │ │ │ -o4 = │ │ │ +o4 = │ │ │ │ │ │ o4 : RObject of type environment │ │ │ │ │ │ i5 : RValue("choose(n, k)", Environment => env) │ │ │ │ │ │ o5 = [1] 120 │ │ ├── ./usr/share/doc/Macaulay2/RInterface/example-output/_new_sp__R__Object_spfrom_sp__Hash__Table.out │ │ │ @@ -1,12 +1,12 @@ │ │ │ -- -*- M2-comint -*- hash: 5466188237138105394 │ │ │ │ │ │ i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ} │ │ │ │ │ │ -o1 = │ │ │ +o1 = │ │ │ │ │ │ o1 : RObject of type environment │ │ │ │ │ │ i2 : RValue("x^y", Environment => env) │ │ │ │ │ │ o2 = [1] 25 │ │ ├── ./usr/share/doc/Macaulay2/RInterface/html/___R__Value.html │ │ │ @@ -115,15 +115,15 @@ │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │  
│ │ │ -o4 = <environment: 0x7ff03f169410>
│ │ │ +o4 = <environment: 0x7f6e9316acd0>
│ │ │  
│ │ │  o4 : RObject of type environment
    │ │ │
    │ │ │ 
    i5 : RValue("choose(n, k)", Environment => env)
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │  o3 = [1] 120
│ │ │ │  
│ │ │ │  o3 : RObject of type double
│ │ │ │  The Environment option specifies the R environment in which to evaluate the
│ │ │ │  code.
│ │ │ │  i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │ │  
│ │ │ │ -o4 = 
│ │ │ │ +o4 = 
│ │ │ │  
│ │ │ │  o4 : RObject of type environment
│ │ │ │  i5 : RValue("choose(n, k)", Environment => env)
│ │ │ │  
│ │ │ │  o5 = [1] 120
│ │ │ │  
│ │ │ │  o5 : RObject of type double
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/html/_new_sp__R__Object_spfrom_sp__Hash__Table.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
    Converts a Macaulay2 HashTable with String keys to an R environment, with each key-value pair becoming a variable binding.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │  
│ │ │ -o1 = <environment: 0x7ff03ea9ce20>
│ │ │ +o1 = <environment: 0x7f6e92a9e6e0>
│ │ │  
│ │ │  o1 : RObject of type environment
    
│ │ │              
    
│ │ │                
    i2 : RValue("x^y", Environment => env)
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _R_ _o_b_j_e_c_t, an R environment;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Converts a Macaulay2 _H_a_s_h_T_a_b_l_e with _S_t_r_i_n_g keys to an R eennvviirroonnmmeenntt, with each
│ │ │ │  key-value pair becoming a variable binding.
│ │ │ │  i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │ │  
│ │ │ │ -o1 = 
│ │ │ │ +o1 = 
│ │ │ │  
│ │ │ │  o1 : RObject of type environment
│ │ │ │  i2 : RValue("x^y", Environment => env)
│ │ │ │  
│ │ │ │  o2 = [1] 25
│ │ │ │  
│ │ │ │  o2 : RObject of type double
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bWF4aW1hbEVudHJ5KENvbXBsZXgp
│ │ │  #:len=270
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDAwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYXhpbWFsRW50cnksQ29tcGxleCksIm1heGltYWxF
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out
│ │ │ @@ -7,42 +7,42 @@
│ │ │  
│ │ │  o2 = (10, 20)
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00359595, .00152225)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00180197, .000740737)
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00949277, .0656054)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00438771, .0352486)
│ │ │  
│ │ │  o4 : Sequence
│ │ │  
│ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {(.0124466, .0228281), (.0113995, .00784108), (.0115798, .0132774),
│ │ │ +o5 = {(.00578277, .0115256), (.00538082, .00389377), (.00610737, .00727412),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.0119615, .0184515), (.00929783, .0257456), (.00955183, .0231329),
│ │ │ +     (.00613753, .0110362), (.00575849, .0134107), (.00607651, .012123),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00861791, .0149704), (.00912547, .0138485), (.00780699, .00994154),
│ │ │ +     (.00399843, .00773129), (.00520689, .00775578), (.00379029, .00511795),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00953055, .0147996)}
│ │ │ +     (.00548542, .00764178)}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .0101318
│ │ │ +o6 = .005372452699999997
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .01648367429999995
│ │ │ +o7 = .008751012800000258
│ │ │  
│ │ │  o7 : RR (of precision 53)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html
│ │ │ @@ -98,57 +98,57 @@
│ │ │  o2 : Sequence
    
│ │ │              
    
│ │ │                
    i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00359595, .00152225)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00180197, .000740737)
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
    
│ │ │                
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00949277, .0656054)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00438771, .0352486)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │              
    
│ │ │                
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {(.0124466, .0228281), (.0113995, .00784108), (.0115798, .0132774),
│ │ │ +o5 = {(.00578277, .0115256), (.00538082, .00389377), (.00610737, .00727412),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.0119615, .0184515), (.00929783, .0257456), (.00955183, .0231329),
│ │ │ +     (.00613753, .0110362), (.00575849, .0134107), (.00607651, .012123),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00861791, .0149704), (.00912547, .0138485), (.00780699, .00994154),
│ │ │ +     (.00399843, .00773129), (.00520689, .00775578), (.00379029, .00511795),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00953055, .0147996)}
│ │ │ +     (.00548542, .00764178)}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .0101318
│ │ │ +o6 = .005372452699999997
│ │ │  
│ │ │  o6 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .01648367429999995
│ │ │ +o7 = .008751012800000258
│ │ │  
│ │ │  o7 : RR (of precision 53)
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,41 +25,41 @@
│ │ │ │  i2 : r=10,n=20
│ │ │ │  
│ │ │ │  o2 = (10, 20)
│ │ │ │  
│ │ │ │  o2 : Sequence
│ │ │ │  i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │ │  
│ │ │ │ -o3 = ({5, 2.91596e52, 9}, .00359595, .00152225)
│ │ │ │ +o3 = ({5, 2.91596e52, 9}, .00180197, .000740737)
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │ │  
│ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00949277, .0656054)
│ │ │ │ +o4 = ({50, 2.30853e454, 98}, .00438771, .0352486)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │ │  
│ │ │ │ -o5 = {(.0124466, .0228281), (.0113995, .00784108), (.0115798, .0132774),
│ │ │ │ +o5 = {(.00578277, .0115256), (.00538082, .00389377), (.00610737, .00727412),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.0119615, .0184515), (.00929783, .0257456), (.00955183, .0231329),
│ │ │ │ +     (.00613753, .0110362), (.00575849, .0134107), (.00607651, .012123),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.00861791, .0149704), (.00912547, .0138485), (.00780699, .00994154),
│ │ │ │ +     (.00399843, .00773129), (.00520689, .00775578), (.00379029, .00511795),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.00953055, .0147996)}
│ │ │ │ +     (.00548542, .00764178)}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │ │  
│ │ │ │ -o6 = .0101318
│ │ │ │ +o6 = .005372452699999997
│ │ │ │  
│ │ │ │  o6 : RR (of precision 53)
│ │ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │ │  
│ │ │ │ -o7 = .01648367429999995
│ │ │ │ +o7 = .008751012800000258
│ │ │ │  
│ │ │ │  o7 : RR (of precision 53)
│ │ │ │  ********** WWaayyss ttoo uussee tteessttTTiimmeeFFoorrLLLLLLoonnSSyyzzyyggiieess:: **********
│ │ │ │      * testTimeForLLLonSyzygies(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _t_e_s_t_T_i_m_e_F_o_r_L_L_L_o_n_S_y_z_y_g_i_e_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  a25vd25VbmlyYXRpb25hbENvbXBvbmVudE9mU3BhY2VDdXJ2ZXM=
│ │ │  #:len=2280
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2sgd2hldGhlciB0aGVyZSBpcyBh
│ │ │  IHVuaXJhdGlvbmFsIGNvbnN0cnVjdGlvbiBmb3IgYSBjb21wb25lbnQgb2YgdGhlIEhpbGJlcnQg
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/example-output/_canonical__Curve.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  i2 : g=14;
│ │ │  
│ │ │  i3 : FF=ZZ/10007;
│ │ │  
│ │ │  i4 : R=FF[x_0..x_(g-1)];
│ │ │  
│ │ │  i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 9.30535s (cpu); 6.17515s (thread); 0s (gc)
│ │ │ + -- used 6.78926s (cpu); 5.23017s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/example-output/_random__Curve__Genus14__Degree18in__P6.out
│ │ │ @@ -4,15 +4,15 @@
│ │ │   -- setting random seed to 10206284518
│ │ │  
│ │ │  i2 : FF=ZZ/10007;
│ │ │  
│ │ │  i3 : S=FF[x_0..x_6];
│ │ │  
│ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.96317s (cpu); 1.48421s (thread); 0s (gc)
│ │ │ + -- used 1.51784s (cpu); 1.27132s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
│ │ │  
│ │ │  i5 : betti res I
│ │ │  
│ │ │              0  1  2  3  4 5
│ │ │  o5 = total: 1 13 45 56 25 2
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/html/_canonical__Curve.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │
    		│ │ │ 
    i4 : R=FF[x_0..x_(g-1)];
    │ │ │
    │ │ │ 
    i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 9.30535s (cpu); 6.17515s (thread); 0s (gc)
│ │ │ + -- used 6.78926s (cpu); 5.23017s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
    │ │ │ ├── html2text {} │ │ │ │ @@ -17,15 +17,15 @@ │ │ │ │ unirationality of $M_g$ by Severi, Sernesi, Chang-Ran and Verra. │ │ │ │ i1 : setRandomSeed "alpha"; │ │ │ │ -- setting random seed to 10206284518 │ │ │ │ i2 : g=14; │ │ │ │ i3 : FF=ZZ/10007; │ │ │ │ i4 : R=FF[x_0..x_(g-1)]; │ │ │ │ i5 : time betti(I=(random canonicalCurve)(g,R)) │ │ │ │ - -- used 9.30535s (cpu); 6.17515s (thread); 0s (gc) │ │ │ │ + -- used 6.78926s (cpu); 5.23017s (thread); 0s (gc) │ │ │ │ │ │ │ │ 0 1 │ │ │ │ o5 = total: 1 66 │ │ │ │ 0: 1 . │ │ │ │ 1: . 66 │ │ │ │ │ │ │ │ o5 : BettiTally │ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ @@ -98,15 +98,15 @@ │ │ │     		│ │ │ 
    i3 : S=FF[x_0..x_6];
    │ │ │
    │ │ │ 
    i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.96317s (cpu); 1.48421s (thread); 0s (gc)
│ │ │ + -- used 1.51784s (cpu); 1.27132s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S
    │ │ │
    │ │ │ 
    i5 : betti res I
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  fields of the chosen finite characteristic 10007, for fields of characteristic
│ │ │ │  0 by semi-continuity, and, hence, for all but finitely many primes $p$.
│ │ │ │  i1 : setRandomSeed("alpha");
│ │ │ │   -- setting random seed to 10206284518
│ │ │ │  i2 : FF=ZZ/10007;
│ │ │ │  i3 : S=FF[x_0..x_6];
│ │ │ │  i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ │ - -- used 1.96317s (cpu); 1.48421s (thread); 0s (gc)
│ │ │ │ + -- used 1.51784s (cpu); 1.27132s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : betti res I
│ │ │ │  
│ │ │ │              0  1  2  3  4 5
│ │ │ │  o5 = total: 1 13 45 56 25 2
│ │ │ │           0: 1  .  .  .  . .
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=46
│ │ │  c21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1KC4uLixQcmludGluZz0+Li4uKQ==
│ │ │  #:len=370
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTI4Mywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1LFBy
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 11549527689790345152
│ │ │  
│ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.34101s (cpu); 1.04498s (thread); 0s (gc)
│ │ │ + -- used 1.19724s (cpu); 1.02065s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
│ │ │  
│ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │            
    If the option Printing is set to true then printings about the current step in the construction are displayed.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  For g<=10 the curves are constructed via plane models.
│ │ │ │  For g<=13 the curves are constructed via space models.
│ │ │ │  For g=14 the curves are constructed by Verra's method.
│ │ │ │  For g=15 the curves are constructed via matrix factorizations.
│ │ │ │  If the option Printing is set to true then printings about the current step in
│ │ │ │  the construction are displayed.
│ │ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ │ - -- used 1.34101s (cpu); 1.04498s (thread); 0s (gc)
│ │ │ │ + -- used 1.19724s (cpu); 1.02065s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                ZZ
│ │ │ │  o1 : Ideal of --[t ..t  ]
│ │ │ │                 5  0   10
│ │ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ │ │  
│ │ │ │  o2 = (2, 11, 20)
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  cmFuZG9tU2hlbGxhYmxlSWRlYWxDaGFpbg==
│ │ │  #:len=1799
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUHJvZHVjZXMgYSBjaGFpbiBvZiBpZGVh
│ │ │  bHMgZnJvbSBhIHJhbmRvbSBzaGVsbGluZyIsICJsaW5lbnVtIiA9PiA3NjAsIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out
│ │ │ @@ -1,24 +1,24 @@
│ │ │  -- -*- M2-comint -*- hash: 9542801742429495161
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812652
│ │ │ + -- setting random seed to 1793276281
│ │ │  
│ │ │ -o1 = 1792812652
│ │ │ +o1 = 1793276281
│ │ │  
│ │ │  i2 : kk=ZZ/101;
│ │ │  
│ │ │  i3 : S=kk[vars(0..5)];
│ │ │  
│ │ │  i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 4.13607s (cpu); 2.14846s (thread); 0s (gc)
│ │ │ + -- used 3.4081s (cpu); 1.6422s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 37 }
│ │ │ -           5 => 221
│ │ │ -           6 => 158
│ │ │ -           7 => 75
│ │ │ -           8 => 8
│ │ │ +o4 = Tally{4 => 48 }
│ │ │ +           5 => 200
│ │ │ +           6 => 189
│ │ │ +           7 => 55
│ │ │ +           8 => 7
│ │ │             9 => 1
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 5959465567197821046
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812658
│ │ │ + -- setting random seed to 1793276285
│ │ │  
│ │ │ -o1 = 1792812658
│ │ │ +o1 = 1793276285
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -15,13 +15,12 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = b c
│ │ │ +o4 = a*b*c
│ │ │  
│ │ │  o4 : S
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 8876340562021865447
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812665
│ │ │ + -- setting random seed to 1793276288
│ │ │  
│ │ │ -o1 = 1792812665
│ │ │ +o1 = 1793276288
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -22,18 +22,18 @@
│ │ │  o4 = {3, 5, 7}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*c*d)
│ │ │ +o5 = ideal(b*c*e)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*e, c*d, a*c, b*c, c*e)
│ │ │ +o6 = ideal (b*e, c*d, d*e, 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 1792812666
│ │ │ + -- setting random seed to 1793276290
│ │ │  
│ │ │ -o1 = 1792812666
│ │ │ +o1 = 1793276290
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -15,15 +15,15 @@
│ │ │  
│ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │  
│ │ │  o3 : MonomialIdeal of S
│ │ │  
│ │ │  i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, b*c, a*d), {a*b, b*c, a*d}, {c*d, b*d, a*c}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d, c*d), {a*b, a*d, c*d}, {b*d, b*c, a*c}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : setRandomSeed(1)
│ │ │   -- setting random seed to 1
│ │ │  
│ │ │  o5 = 1
│ │ │ @@ -37,15 +37,15 @@
│ │ │  i7 : J = monomialIdeal 0_S
│ │ │  
│ │ │  o7 = monomialIdeal ()
│ │ │  
│ │ │  o7 : MonomialIdeal of S
│ │ │  
│ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 3.64766s (cpu); 2.71005s (thread); 0s (gc)
│ │ │ + -- used 3.36132s (cpu); 2.54575s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : #T
│ │ │  
│ │ │  o9 = 168
│ │ │  
│ │ │  i10 : T
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html
│ │ │ @@ -76,17 +76,17 @@
│ │ │          
    
│ │ │            
    Chooses a random monomial.
    
│ │ │          
    
│ │ │          
    
│ │ │                
    i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.34101s (cpu); 1.04498s (thread); 0s (gc)
│ │ │ + -- used 1.19724s (cpu); 1.02065s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812658
│ │ │ + -- setting random seed to 1793276285
│ │ │  
│ │ │ -o1 = 1792812658
    
│ │ │ +o1 = 1793276285
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -103,16 +103,15 @@
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = b c
│ │ │ +o4 = a*b*c
│ │ │  
│ │ │  o4 : S
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,31 +11,30 @@
│ │ │ │            o d, an _i_n_t_e_g_e_r, non-negative
│ │ │ │            o S, a _r_i_n_g, polynomial ring
│ │ │ │      * Outputs:
│ │ │ │            o m, a _r_i_n_g_ _e_l_e_m_e_n_t, monomial of S
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Chooses a random monomial.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1792812658
│ │ │ │ + -- setting random seed to 1793276285
│ │ │ │  
│ │ │ │ -o1 = 1792812658
│ │ │ │ +o1 = 1793276285
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : randomMonomial(3,S)
│ │ │ │  
│ │ │ │ -      2
│ │ │ │ -o4 = b c
│ │ │ │ +o4 = a*b*c
│ │ │ │  
│ │ │ │  o4 : S
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_n_d_o_m_M_o_n_o_m_i_a_l_I_d_e_a_l -- random monomial ideal with given degree generators
│ │ │ │      * _r_a_n_d_o_m_S_q_u_a_r_e_F_r_e_e_M_o_n_o_m_i_a_l_I_d_e_a_l -- random square-free monomial ideal with
│ │ │ │        given degree generators
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommMMoonnoommiiaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html
│ │ │ @@ -76,17 +76,17 @@
│ │ │          
    
│ │ │            
    Choose a random square-free monomial ideal whose generators have the degrees specified by the list or sequence L.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812665
│ │ │ + -- setting random seed to 1793276288
│ │ │  
│ │ │ -o1 = 1792812665
    
│ │ │ +o1 = 1793276288
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -113,24 +113,24 @@
│ │ │              
    
│ │ │                
    i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*c*d)
│ │ │ +o5 = ideal(b*c*e)
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*e, c*d, a*c, b*c, c*e)
│ │ │ +o6 = ideal (b*e, c*d, d*e, 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 1792812665
│ │ │ │ + -- setting random seed to 1793276288
│ │ │ │  
│ │ │ │ -o1 = 1792812665
│ │ │ │ +o1 = 1793276288
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a..e]
│ │ │ │  
│ │ │ │ @@ -34,20 +34,20 @@
│ │ │ │  
│ │ │ │  o4 = {3, 5, 7}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │ │  low degree gens generated everything
│ │ │ │  
│ │ │ │ -o5 = ideal(a*c*d)
│ │ │ │ +o5 = ideal(b*c*e)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (a*e, c*d, a*c, b*c, c*e)
│ │ │ │ +o6 = ideal (b*e, c*d, d*e, 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
│ │ │ @@ -84,17 +84,17 @@
│ │ │            
    With probability p the routine takes the Alexander dual of I; the default value of p is .05, and it can be set with the option AlexanderProbility.
    
│ │ │            
    Otherwise uses the Metropolis algorithm to produce a random walk on the space of square-free ideals. Note that there are a LOT of square-free ideals; these are the Dedekind numbers, and the sequence (with 1,2,3,4,5,6,7,8 variables) begins 3,6,20,168,7581, 7828354, 2414682040998, 56130437228687557907788. (see the Online Encyclopedia of Integer Sequences for more information). Given I in a polynomial ring S, we make a list ISocgens of the square-free minimal monomial generators of the socle of S/(squares+I) and a list of minimal generators Igens of I. A candidate "next" ideal J is formed as follows: We choose randomly from the union of these lists; if a socle element is chosen, it's added to I; if a minimal generator is chosen, it's replaced by the square-free part of the maximal ideal times it. the chance of making the given move is then 1/(#ISocgens+#Igens), and the chance of making the move back would be the similar quantity for J, so we make the move or not depending on whether random RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812666
│ │ │ + -- setting random seed to 1793276290
│ │ │  
│ │ │ -o1 = 1792812666
    
│ │ │ +o1 = 1793276290
│ │ │              
    
│ │ │                
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │ @@ -111,15 +111,15 @@
│ │ │  o3 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, b*c, a*d), {a*b, b*c, a*d}, {c*d, b*d, a*c}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d, c*d), {a*b, a*d, c*d}, {b*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.
    
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    │ │ │
    │ │ │ 
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 3.64766s (cpu); 2.71005s (thread); 0s (gc)
    │ │ │ + -- used 3.36132s (cpu); 2.54575s (thread); 0s (gc) │ │ │
    │ │ │ 
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    │ │ │ ├── html2text {} │ │ │ │ @@ -35,30 +35,30 @@ │ │ │ │ choose randomly from the union of these lists; if a socle element is chosen, │ │ │ │ it's added to I; if a minimal generator is chosen, it's replaced by the square- │ │ │ │ free part of the maximal ideal times it. the chance of making the given move is │ │ │ │ then 1/(#ISocgens+#Igens), and the chance of making the move back would be the │ │ │ │ similar quantity for J, so we make the move or not depending on whether random │ │ │ │ RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1]. │ │ │ │ i1 : setRandomSeed(currentTime()) │ │ │ │ - -- setting random seed to 1792812666 │ │ │ │ + -- setting random seed to 1793276290 │ │ │ │ │ │ │ │ -o1 = 1792812666 │ │ │ │ +o1 = 1793276290 │ │ │ │ i2 : S=ZZ/2[vars(0..3)] │ │ │ │ │ │ │ │ o2 = S │ │ │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ i3 : J = monomialIdeal"ab,ad, bcd" │ │ │ │ │ │ │ │ o3 = monomialIdeal (a*b, a*d, b*c*d) │ │ │ │ │ │ │ │ o3 : MonomialIdeal of S │ │ │ │ i4 : randomSquareFreeStep J │ │ │ │ │ │ │ │ -o4 = {monomialIdeal (a*b, b*c, a*d), {a*b, b*c, a*d}, {c*d, b*d, a*c}} │ │ │ │ +o4 = {monomialIdeal (a*b, a*d, c*d), {a*b, a*d, c*d}, {b*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 │ │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ i7 : J = monomialIdeal 0_S │ │ │ │ │ │ │ │ o7 = monomialIdeal () │ │ │ │ │ │ │ │ o7 : MonomialIdeal of S │ │ │ │ i8 : time T=tally for t from 1 to 5000 list first (J=rsfs │ │ │ │ (J,AlexanderProbability => .01)); │ │ │ │ - -- used 3.64766s (cpu); 2.71005s (thread); 0s (gc) │ │ │ │ + -- used 3.36132s (cpu); 2.54575s (thread); 0s (gc) │ │ │ │ i9 : #T │ │ │ │ │ │ │ │ o9 = 168 │ │ │ │ i10 : T │ │ │ │ │ │ │ │ o10 = Tally{monomialIdeal () => 45 } │ │ │ │ monomialIdeal (a*b*c, a*b*d) => 33 │ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ @@ -59,17 +59,17 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -77,21 +77,21 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812652
│ │ │ + -- setting random seed to 1793276281
│ │ │  
│ │ │ -o1 = 1792812652
    │ │ │ +o1 = 1793276281 │ │ │
    │ │ │ 
    i2 : kk=ZZ/101;
    │ │ │
    │ │ │ 
    i3 : S=kk[vars(0..5)];
    │ │ │
    │ │ │ 
    i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S)
│ │ │ - -- used 4.13607s (cpu); 2.14846s (thread); 0s (gc)
│ │ │ + -- used 3.4081s (cpu); 1.6422s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 37 }
│ │ │ -           5 => 221
│ │ │ -           6 => 158
│ │ │ -           7 => 75
│ │ │ -           8 => 8
│ │ │ +o4 = Tally{4 => 48 }
│ │ │ +           5 => 200
│ │ │ +           6 => 189
│ │ │ +           7 => 55
│ │ │ +           8 => 7
│ │ │             9 => 1
│ │ │  
│ │ │  o4 : Tally
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -9,27 +9,27 @@ │ │ │ │ This package can be used to make experiments, trying many ideals, perhaps over │ │ │ │ small fields. For example...what would you expect the regularities of "typical" │ │ │ │ monomial ideals with 10 generators of degree 3 in 6 variables to be? Try a │ │ │ │ bunch of examples -- it's fast. Here we do only 500 -- this takes about a │ │ │ │ second on a fast machine -- but with a little patience, thousands can be done │ │ │ │ conveniently. │ │ │ │ i1 : setRandomSeed(currentTime()) │ │ │ │ - -- setting random seed to 1792812652 │ │ │ │ + -- setting random seed to 1793276281 │ │ │ │ │ │ │ │ -o1 = 1792812652 │ │ │ │ +o1 = 1793276281 │ │ │ │ i2 : kk=ZZ/101; │ │ │ │ i3 : S=kk[vars(0..5)]; │ │ │ │ i4 : time tally for n from 1 to 500 list regularity randomMonomialIdeal(10:3,S) │ │ │ │ - -- used 4.13607s (cpu); 2.14846s (thread); 0s (gc) │ │ │ │ + -- used 3.4081s (cpu); 1.6422s (thread); 0s (gc) │ │ │ │ │ │ │ │ -o4 = Tally{4 => 37 } │ │ │ │ - 5 => 221 │ │ │ │ - 6 => 158 │ │ │ │ - 7 => 75 │ │ │ │ - 8 => 8 │ │ │ │ +o4 = Tally{4 => 48 } │ │ │ │ + 5 => 200 │ │ │ │ + 6 => 189 │ │ │ │ + 7 => 55 │ │ │ │ + 8 => 7 │ │ │ │ 9 => 1 │ │ │ │ │ │ │ │ o4 : Tally │ │ │ │ How does this compare with the case of binomial ideals? or pure binomial │ │ │ │ ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" │ │ │ │ above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger │ │ │ │ numbers of examples. Click the link "Finding Extreme Examples" below to see │ │ ├── ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ VmFyaWFibGVOYW1l │ │ │ #:len=1241 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl │ │ │ IHRoZSBpbmRleGVkIHZhcmlhYmxlIG5hbWUgZm9yIHRoZSBwb2x5bm9taWFsIHJpbmciLCAibGlu │ │ ├── ./usr/share/doc/Macaulay2/RandomObjects/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ QXR0ZW1wdHM= │ │ │ #:len=502 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibnVtYmVyIG9mIGF0dGVtcHRzIGluIHRo │ │ │ ZSBjb25zdHJ1Y3Rpb24gb2YgYSByYW5kb20gb2JqZWN0IiwgImxpbmVudW0iID0+IDI2MiwgImZp │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=46 │ │ │ ZmluZEFOb25aZXJvTWlub3IoLi4uLFBvaW50Q2hlY2tBdHRlbXB0cz0+Li4uKQ== │ │ │ #:len=315 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg4OCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbZmluZEFOb25aZXJvTWlub3IsUG9pbnRDaGVja0F0 │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out │ │ │ @@ -5,17 +5,17 @@ │ │ │ i2 : S=kk[y_0..y_8]; │ │ │ │ │ │ i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2; │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ i4 : elapsedTime dimViaBezout(I) │ │ │ - -- 2.00843s elapsed │ │ │ + -- 1.18708s elapsed │ │ │ │ │ │ o4 = 4 │ │ │ │ │ │ i5 : elapsedTime dim I │ │ │ - -- 3.91258s elapsed │ │ │ + -- 2.61849s elapsed │ │ │ │ │ │ o5 = 4 │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out │ │ │ @@ -35,15 +35,15 @@ │ │ │ i8 : i = 0; │ │ │ │ │ │ i9 : J = I; │ │ │ │ │ │ o9 : Ideal of T │ │ │ │ │ │ i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) ); │ │ │ - -- 2.88102s elapsed │ │ │ + -- 1.6586s elapsed │ │ │ │ │ │ i11 : dim J │ │ │ │ │ │ o11 = 1 │ │ │ │ │ │ i12 : i │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out │ │ │ @@ -27,24 +27,24 @@ │ │ │ i6 : S=ZZ/103[y_0..y_30]; │ │ │ │ │ │ i7 : I=minors(2,random(S^3,S^{3:-1})); │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable) │ │ │ - -- 4.05108s elapsed │ │ │ + -- 2.55696s elapsed │ │ │ │ │ │ o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0, │ │ │ ------------------------------------------------------------------------ │ │ │ -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}} │ │ │ │ │ │ o8 : List │ │ │ │ │ │ i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose) │ │ │ - -- 3.02229s elapsed │ │ │ + -- 1.94318s elapsed │ │ │ │ │ │ o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16, │ │ │ ------------------------------------------------------------------------ │ │ │ 11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}} │ │ │ │ │ │ o9 : List │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html │ │ │ @@ -100,23 +100,23 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │
    │ │ │ 
    i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 2.00843s elapsed
│ │ │ + -- 1.18708s elapsed
│ │ │  
│ │ │  o4 = 4
    │ │ │
    │ │ │ 
    i5 : elapsedTime dim I
│ │ │ - -- 3.91258s elapsed
│ │ │ + -- 2.61849s elapsed
│ │ │  
│ │ │  o5 = 4
    │ │ │
    │ │ │
    │ │ │

    The user may set the MinimumFieldSize to ensure that the field being worked over is big enough. For instance, there are relatively few linear spaces over a field of characteristic 2, and this can cause incorrect results to be provided. If no size is provided, the function tries to guess a good size based on ambient ring.

    │ │ │ ├── html2text {} │ │ │ │ @@ -32,19 +32,19 @@ │ │ │ │ examples, the built in dim function is much faster. │ │ │ │ i1 : kk=ZZ/101; │ │ │ │ i2 : S=kk[y_0..y_8]; │ │ │ │ i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2; │ │ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ i4 : elapsedTime dimViaBezout(I) │ │ │ │ - -- 2.00843s elapsed │ │ │ │ + -- 1.18708s elapsed │ │ │ │ │ │ │ │ o4 = 4 │ │ │ │ i5 : elapsedTime dim I │ │ │ │ - -- 3.91258s elapsed │ │ │ │ + -- 2.61849s elapsed │ │ │ │ │ │ │ │ o5 = 4 │ │ │ │ The user may set the MinimumFieldSize to ensure that the field being worked │ │ │ │ over is big enough. For instance, there are relatively few linear spaces over a │ │ │ │ field of characteristic 2, and this can cause incorrect results to be provided. │ │ │ │ If no size is provided, the function tries to guess a good size based on │ │ │ │ ambient ring. │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html │ │ │ @@ -160,15 +160,15 @@ │ │ │ │ │ │ o9 : Ideal of T │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.88102s elapsed
    │ │ │ + -- 1.6586s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : dim J
│ │ │  
│ │ │  o11 = 1
    │ │ │ ├── html2text {} │ │ │ │ @@ -79,15 +79,15 @@ │ │ │ │ o7 : Matrix T <-- T │ │ │ │ i8 : i = 0; │ │ │ │ i9 : J = I; │ │ │ │ │ │ │ │ o9 : Ideal of T │ │ │ │ i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = │ │ │ │ extendIdealByNonZeroMinor(4, M, J)) ); │ │ │ │ - -- 2.88102s elapsed │ │ │ │ + -- 1.6586s elapsed │ │ │ │ i11 : dim J │ │ │ │ │ │ │ │ o11 = 1 │ │ │ │ i12 : i │ │ │ │ │ │ │ │ o12 = 4 │ │ │ │ In this particular example, there tend to be about 5 associated primes when │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html │ │ │ @@ -149,27 +149,27 @@ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 4.05108s elapsed
│ │ │ + -- 2.55696s elapsed
│ │ │  
│ │ │  o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}
│ │ │  
│ │ │  o8 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
│ │ │ - -- 3.02229s elapsed
│ │ │ + -- 1.94318s elapsed
│ │ │  
│ │ │  o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
│ │ │       ------------------------------------------------------------------------
│ │ │       11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}
│ │ │  
│ │ │  o9 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -66,24 +66,24 @@ │ │ │ │ first in rings with more variables. │ │ │ │ i6 : S=ZZ/103[y_0..y_30]; │ │ │ │ i7 : I=minors(2,random(S^3,S^{3:-1})); │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, │ │ │ │ DecompositionStrategy=>MultiplicationTable) │ │ │ │ - -- 4.05108s elapsed │ │ │ │ + -- 2.55696s elapsed │ │ │ │ │ │ │ │ o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, │ │ │ │ DecompositionStrategy=>Decompose) │ │ │ │ - -- 3.02229s elapsed │ │ │ │ + -- 1.94318s elapsed │ │ │ │ │ │ │ │ o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}} │ │ │ │ │ │ │ │ o9 : List │ │ │ │ ********** WWaayyss ttoo uussee rraannddoommPPooiinnttss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=46 │ │ │ aXNCaXJhdGlvbmFsT250b0ltYWdlKC4uLixBc3N1bWVEb21pbmFudD0+Li4uKQ== │ │ │ #:len=321 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgwNiwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaXNCaXJhdGlvbmFsT250b0ltYWdlLEFzc3VtZURv │ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out │ │ │ @@ -49,15 +49,15 @@ │ │ │ i12 : Q=QQ[x,y,z,t,u]; │ │ │ │ │ │ i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}}); │ │ │ │ │ │ o13 : RingMap Q <-- Q │ │ │ │ │ │ i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0) │ │ │ - -- used 0.341115s (cpu); 0.337449s (thread); 0s (gc) │ │ │ + -- used 0.44598s (cpu); 0.333498s (thread); 0s (gc) │ │ │ │ │ │ 125 124 120 5 124 100 25 104 20 108 15 2 112 10 3 116 5 4 120 5 124 125 4 120 8 115 2 12 110 3 16 105 4 20 100 5 24 95 6 28 90 7 32 85 8 36 80 9 40 75 10 44 70 11 48 65 12 52 60 13 56 55 14 60 50 15 64 45 16 68 40 17 72 35 18 76 30 19 80 25 20 84 20 21 88 15 22 92 10 23 96 5 24 100 25 24 100 28 95 32 90 2 36 85 3 40 80 4 44 75 5 48 70 6 52 65 7 56 60 8 60 55 9 64 50 10 68 45 11 72 40 12 76 35 13 80 30 14 84 25 15 88 20 16 92 15 17 96 10 18 100 5 19 104 20 48 75 2 52 70 2 56 65 2 2 60 60 3 2 64 55 4 2 68 50 5 2 72 45 6 2 76 40 7 2 80 35 8 2 84 30 9 2 88 25 10 2 92 20 11 2 96 15 12 2 100 10 13 2 104 5 14 2 108 15 2 72 50 3 76 45 3 80 40 2 3 84 35 3 3 88 30 4 3 92 25 5 3 96 20 6 3 100 15 7 3 104 10 8 3 108 5 9 3 112 10 3 96 25 4 100 20 4 104 15 2 4 108 10 3 4 112 5 4 4 116 5 4 120 5 124 │ │ │ o14 = Proj Q - - - > Proj Q {x , x y, - x y + x z, x y - 5x y z + 10x y z - 10x y z + 5x y z - x z + x t, - y + 25x y z - 300x y z + 2300x y z - 12650x y z + 53130x y z - 177100x y z + 480700x y z - 1081575x y z + 2042975x y z - 3268760x y z + 4457400x y z - 5200300x y z + 5200300x y z - 4457400x y z + 3268760x y z - 2042975x y z + 1081575x y z - 480700x y z + 177100x y z - 53130x y z + 12650x y z - 2300x y z + 300x y z - 25x y z + x z - 5x y t + 100x y z*t - 950x y z t + 5700x y z t - 24225x y z t + 77520x y z t - 193800x y z t + 387600x y z t - 629850x y z t + 839800x y z t - 923780x y z t + 839800x y z t - 629850x y z t + 387600x y z t - 193800x y z t + 77520x y z t - 24225x y z t + 5700x y z t - 950x y z t + 100x y z t - 5x z t - 10x y t + 150x y z*t - 1050x y z t + 4550x y z t - 13650x y z t + 30030x y z t - 50050x y z t + 64350x y z t - 64350x y z t + 50050x y z t - 30030x y z t + 13650x y z t - 4550x y z t + 1050x y z t - 150x y z t + 10x z t - 10x y t + 100x y z*t - 450x y z t + 1200x y z t - 2100x y z t + 2520x y z t - 2100x y z t + 1200x y z t - 450x y z t + 100x y z t - 10x z t - 5x y t + 25x y z*t - 50x y z t + 50x y z t - 25x y z t + 5x z t - x t + x u} │ │ │ │ │ │ o14 : RationalMapping │ │ │ │ │ │ i15 : R=QQ[x,y,z,t]/(z-2*t); │ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html │ │ │ @@ -194,15 +194,15 @@ │ │ │ │ │ │ o13 : RingMap Q <-- Q │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.341115s (cpu); 0.337449s (thread); 0s (gc)
│ │ │ + -- used 0.44598s (cpu); 0.333498s (thread); 0s (gc)
│ │ │  
│ │ │                                  125   124      120 5    124    100 25     104 20       108 15 2      112 10 3     116 5 4    120 5    124      125      4 120        8 115 2        12 110 3         16 105 4         20 100 5          24 95 6          28 90 7           32 85 8           36 80 9           40 75 10           44 70 11           48 65 12           52 60 13           56 55 14           60 50 15           64 45 16           68 40 17          72 35 18          76 30 19         80 25 20         84 20 21        88 15 22       92 10 23      96 5 24    100 25     24 100        28 95          32 90 2         36 85 3          40 80 4          44 75 5           48 70 6           52 65 7           56 60 8           60 55 9           64 50 10           68 45 11           72 40 12           76 35 13           80 30 14          84 25 15          88 20 16         92 15 17        96 10 18        100 5 19      104 20       48 75 2       52 70   2        56 65 2 2        60 60 3 2         64 55 4 2         68 50 5 2         72 45 6 2         76 40 7 2         80 35 8 2         84 30 9 2         88 25 10 2         92 20 11 2        96 15 12 2        100 10 13 2       104 5 14 2      108 15 2      72 50 3       76 45   3       80 40 2 3        84 35 3 3        88 30 4 3        92 25 5 3        96 20 6 3        100 15 7 3       104 10 8 3       108 5 9 3      112 10 3     96 25 4      100 20   4      104 15 2 4      108 10 3 4      112 5 4 4     116 5 4    120 5    124
│ │ │  o14 = Proj Q - - - > Proj Q   {x   , x   y, - x   y  + x   z, x   y   - 5x   y  z + 10x   y  z  - 10x   y  z  + 5x   y z  - x   z  + x   t, - y    + 25x y   z - 300x y   z  + 2300x  y   z  - 12650x  y   z  + 53130x  y   z  - 177100x  y  z  + 480700x  y  z  - 1081575x  y  z  + 2042975x  y  z  - 3268760x  y  z   + 4457400x  y  z   - 5200300x  y  z   + 5200300x  y  z   - 4457400x  y  z   + 3268760x  y  z   - 2042975x  y  z   + 1081575x  y  z   - 480700x  y  z   + 177100x  y  z   - 53130x  y  z   + 12650x  y  z   - 2300x  y  z   + 300x  y  z   - 25x  y z   + x   z   - 5x  y   t + 100x  y  z*t - 950x  y  z t + 5700x  y  z t - 24225x  y  z t + 77520x  y  z t - 193800x  y  z t + 387600x  y  z t - 629850x  y  z t + 839800x  y  z t - 923780x  y  z  t + 839800x  y  z  t - 629850x  y  z  t + 387600x  y  z  t - 193800x  y  z  t + 77520x  y  z  t - 24225x  y  z  t + 5700x  y  z  t - 950x  y  z  t + 100x   y z  t - 5x   z  t - 10x  y  t  + 150x  y  z*t  - 1050x  y  z t  + 4550x  y  z t  - 13650x  y  z t  + 30030x  y  z t  - 50050x  y  z t  + 64350x  y  z t  - 64350x  y  z t  + 50050x  y  z t  - 30030x  y  z  t  + 13650x  y  z  t  - 4550x  y  z  t  + 1050x   y  z  t  - 150x   y z  t  + 10x   z  t  - 10x  y  t  + 100x  y  z*t  - 450x  y  z t  + 1200x  y  z t  - 2100x  y  z t  + 2520x  y  z t  - 2100x  y  z t  + 1200x   y  z t  - 450x   y  z t  + 100x   y z t  - 10x   z  t  - 5x  y  t  + 25x   y  z*t  - 50x   y  z t  + 50x   y  z t  - 25x   y z t  + 5x   z t  - x   t  + x   u}
│ │ │  
│ │ │  o14 : RationalMapping
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ o11 : Ideal of blowUpSubvar │ │ │ │ The next example is a birational map on $\mathbb{P}^4$. │ │ │ │ i12 : Q=QQ[x,y,z,t,u]; │ │ │ │ i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}}); │ │ │ │ │ │ │ │ o13 : RingMap Q <-- Q │ │ │ │ i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0) │ │ │ │ - -- used 0.341115s (cpu); 0.337449s (thread); 0s (gc) │ │ │ │ + -- used 0.44598s (cpu); 0.333498s (thread); 0s (gc) │ │ │ │ │ │ │ │ 125 124 120 5 124 100 25 104 │ │ │ │ 20 108 15 2 112 10 3 116 5 4 120 5 124 125 4 120 │ │ │ │ 8 115 2 12 110 3 16 105 4 20 100 5 24 95 6 │ │ │ │ 28 90 7 32 85 8 36 80 9 40 75 10 44 70 │ │ │ │ 11 48 65 12 52 60 13 56 55 14 60 50 15 │ │ │ │ 64 45 16 68 40 17 72 35 18 76 30 19 80 25 │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ U29ydEdlbnM= │ │ │ #:len=246 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg3LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTb3J0R2VucyIsIlNvcnRHZW5zIiwiUmF0aW9uYWxQ │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ bmV0KFByb2plY3RpdmVQb2ludCk= │ │ │ #:len=205 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTcsICJ1bmRvY3VtZW50ZWQiID0+IHRy │ │ │ dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KG5ldCxQcm9q │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out │ │ │ @@ -48,15 +48,15 @@ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ ZZ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ 101 0 10 │ │ │ │ │ │ i14 : time rationalPoints(I, Amount => true) │ │ │ - -- used 0.00513051s (cpu); 0.00512835s (thread); 0s (gc) │ │ │ + -- used 0.0031578s (cpu); 0.00315555s (thread); 0s (gc) │ │ │ │ │ │ o14 = 110462212541120451001 │ │ │ │ │ │ i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z); │ │ │ │ │ │ o16 : Ideal of QQ[x..z] │ │ │ │ │ │ @@ -142,23 +142,23 @@ │ │ │ │ │ │ i31 : nodes = I + ideal jacobian I; │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ │ │ i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true); │ │ │ -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25) │ │ │ - -- used 1.28715s (cpu); 1.09718s (thread); 0s (gc) │ │ │ + -- used 0.965208s (cpu); 0.755189s (thread); 0s (gc) │ │ │ │ │ │ i33 : #oo │ │ │ │ │ │ o33 = 31 │ │ │ │ │ │ i34 : nodes' = baseChange_32003 nodes; │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ │ │ i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true) │ │ │ - -- used 0.282073s (cpu); 0.221929s (thread); 0s (gc) │ │ │ + -- used 0.272778s (cpu); 0.21031s (thread); 0s (gc) │ │ │ │ │ │ o35 = 31 │ │ │ │ │ │ i36 : │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html │ │ │ @@ -183,15 +183,15 @@ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ 101 0 10 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00513051s (cpu); 0.00512835s (thread); 0s (gc)
│ │ │ + -- used 0.0031578s (cpu); 0.00315555s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Over number fields

    │ │ │ @@ -353,15 +353,15 @@ │ │ │ o31 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 1.28715s (cpu); 1.09718s (thread); 0s (gc)
    │ │ │ + -- used 0.965208s (cpu); 0.755189s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i33 : #oo
│ │ │  
│ │ │  o33 = 31
    │ │ │ @@ -378,15 +378,15 @@ │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.282073s (cpu); 0.221929s (thread); 0s (gc)
│ │ │ + -- used 0.272778s (cpu); 0.21031s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 31
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -89,15 +89,15 @@ │ │ │ │ o13 = ideal(u + u + u + u + u + u + u + u + u + u + u ) │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ │ 101 0 10 │ │ │ │ i14 : time rationalPoints(I, Amount => true) │ │ │ │ - -- used 0.00513051s (cpu); 0.00512835s (thread); 0s (gc) │ │ │ │ + -- used 0.0031578s (cpu); 0.00315555s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = 110462212541120451001 │ │ │ │ ****** OOvveerr nnuummbbeerr ffiieellddss ****** │ │ │ │ Over a number field one can use the option Bound to specify a maximal │ │ │ │ multiplicative height given by $(x_0:\dots:x_n)\mapsto \prod_{v}\max_i|x_i|_v ^ │ │ │ │ {d_v/d}$ (this is also available as a method _g_l_o_b_a_l_H_e_i_g_h_t). │ │ │ │ i15 : QQ[x,y,z]; I = homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)), z); │ │ │ │ @@ -197,24 +197,24 @@ │ │ │ │ │ │ │ │ o30 : Ideal of R │ │ │ │ i31 : nodes = I + ideal jacobian I; │ │ │ │ │ │ │ │ o31 : Ideal of R │ │ │ │ i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true); │ │ │ │ -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25) │ │ │ │ - -- used 1.28715s (cpu); 1.09718s (thread); 0s (gc) │ │ │ │ + -- used 0.965208s (cpu); 0.755189s (thread); 0s (gc) │ │ │ │ i33 : #oo │ │ │ │ │ │ │ │ o33 = 31 │ │ │ │ Still it runs a lot faster when reduced to a positive characteristic. │ │ │ │ i34 : nodes' = baseChange_32003 nodes; │ │ │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true) │ │ │ │ - -- used 0.282073s (cpu); 0.221929s (thread); 0s (gc) │ │ │ │ + -- used 0.272778s (cpu); 0.21031s (thread); 0s (gc) │ │ │ │ │ │ │ │ o35 = 31 │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ For a number field other than QQ, the enumeration of elements with bounded │ │ │ │ height depends on an algorithm by Doyle–Krumm, which is currently only │ │ │ │ implemented in Sage. │ │ │ │ ******** MMeennuu ******** │ │ ├── ./usr/share/doc/Macaulay2/ReactionNetworks/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ bW9kaWZpY2F0aW9uT2ZUd29TdWJzdHJhdGVzSCgp │ │ │ #:len=344 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg1LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20gezE6KG1vZGlmaWNhdGlvbk9mVHdvU3Vic3RyYXRlc0gp │ │ ├── ./usr/share/doc/Macaulay2/RealRoots/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=17 │ │ │ dHJhY2VDb3VudChJZGVhbCk= │ │ │ #:len=241 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTAyMSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodHJhY2VDb3VudCxJZGVhbCksInRyYWNlQ291bnQo │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ aXNMaW5lYXJUeXBlKE1vZHVsZSk= │ │ │ #:len=257 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE5OSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNMaW5lYXJUeXBlLE1vZHVsZSksImlzTGluZWFy │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/___Plane__Curve__Singularities.out │ │ │ @@ -331,15 +331,15 @@ │ │ │ 2 2 2 2 2 2 2 │ │ │ - p w , p y - p , p w y - p p , p w - p ) │ │ │ 2 1 0 1 0 0 1 2 0 0 2 │ │ │ │ │ │ o47 : Ideal of B2 │ │ │ │ │ │ i48 : time sing2 = ideal singularLocus strictTransform2; │ │ │ - -- used 1.07953s (cpu); 0.841221s (thread); 0s (gc) │ │ │ + -- used 0.811307s (cpu); 0.658795s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o48 : Ideal of -----[p ..p , w ..w , x..y] │ │ │ 32003 0 2 0 1 │ │ │ │ │ │ i49 : saturate(sing2, sub(irrelTot, ring sing2)) │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_expected__Rees__Ideal.out │ │ │ @@ -58,15 +58,15 @@ │ │ │ o5 : Matrix S <-- S │ │ │ │ │ │ i6 : I = minors(n-1, M); │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec. │ │ │ - -- used 1.37298s (cpu); 0.841764s (thread); 0s (gc) │ │ │ + -- used 1.14349s (cpu); 0.752969s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of S[w ..w ] │ │ │ 0 4 │ │ │ │ │ │ i8 : kk = ZZ/101; │ │ │ │ │ │ i9 : S = kk[x,y,z]; │ │ │ @@ -77,19 +77,19 @@ │ │ │ o10 : Matrix S <-- S │ │ │ │ │ │ i11 : I = minors(3,m); │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ │ │ i12 : time reesIdeal (I, I_0); │ │ │ - -- used 1.79026s (cpu); 1.2643s (thread); 0s (gc) │ │ │ + -- used 1.58086s (cpu); 1.24315s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ 0 3 │ │ │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ - -- used 2.01763s (cpu); 1.4125s (thread); 0s (gc) │ │ │ + -- used 1.54773s (cpu); 1.26084s (thread); 0s (gc) │ │ │ │ │ │ o13 : Ideal of S[w ..w ] │ │ │ 0 3 │ │ │ │ │ │ i14 : │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out │ │ │ @@ -13,21 +13,21 @@ │ │ │ 3 2 │ │ │ - x x x , x - x x ) │ │ │ 0 2 4 1 0 4 │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ i4 : time V1 = reesIdeal i; │ │ │ - -- used 0.0295188s (cpu); 0.0275558s (thread); 0s (gc) │ │ │ + -- used 0.0805851s (cpu); 0.0295094s (thread); 0s (gc) │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ 0 6 │ │ │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ - -- used 0.138904s (cpu); 0.138178s (thread); 0s (gc) │ │ │ + -- used 0.150284s (cpu); 0.14039s (thread); 0s (gc) │ │ │ │ │ │ o5 : Ideal of S[w ..w ] │ │ │ 0 6 │ │ │ │ │ │ i6 : S=kk[a,b,c] │ │ │ │ │ │ o6 = S │ │ │ @@ -47,21 +47,21 @@ │ │ │ │ │ │ 2 2 │ │ │ o8 = ideal (a , a*b, b ) │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ i9 : time I1 = reesIdeal i; │ │ │ - -- used 0.023238s (cpu); 0.0222073s (thread); 0s (gc) │ │ │ + -- used 0.0964715s (cpu); 0.0257072s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ - -- used 0.00959305s (cpu); 0.00933152s (thread); 0s (gc) │ │ │ + -- used 0.0429585s (cpu); 0.0118258s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i11 : transpose gens I1 │ │ │ │ │ │ o11 = {-1, -3} | aw_1-bw_2 | │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/___Plane__Curve__Singularities.html │ │ │ @@ -592,15 +592,15 @@ │ │ │
    │ │ │

    We compute the singular locus once again:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -325,15 +325,15 @@ │ │ │ │ 2 2 2 2 2 2 2 │ │ │ │ - p w , p y - p , p w y - p p , p w - p ) │ │ │ │ 2 1 0 1 0 0 1 2 0 0 2 │ │ │ │ │ │ │ │ o47 : Ideal of B2 │ │ │ │ We compute the singular locus once again: │ │ │ │ i48 : time sing2 = ideal singularLocus strictTransform2; │ │ │ │ - -- used 1.07953s (cpu); 0.841221s (thread); 0s (gc) │ │ │ │ + -- used 0.811307s (cpu); 0.658795s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o48 : Ideal of -----[p ..p , w ..w , x..y] │ │ │ │ 32003 0 2 0 1 │ │ │ │ i49 : saturate(sing2, sub(irrelTot, ring sing2)) │ │ │ │ │ │ │ │ o49 = ideal 1 │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_expected__Rees__Ideal.html │ │ │ @@ -156,15 +156,15 @@ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 1.07953s (cpu); 0.841221s (thread); 0s (gc)
│ │ │ + -- used 0.811307s (cpu); 0.658795s (thread); 0s (gc)
│ │ │  
│ │ │                   ZZ
│ │ │  o48 : Ideal of -----[p ..p , w ..w , x..y]
│ │ │                 32003  0   2   0   1
    │ │ │
    │ │ │ 
    i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec.
│ │ │ - -- used 1.37298s (cpu); 0.841764s (thread); 0s (gc)
│ │ │ + -- used 1.14349s (cpu); 0.752969s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
    │ │ │
    │ │ │ @@ -190,24 +190,24 @@ │ │ │ │ │ │ o11 : Ideal of S │ │ │
    │ │ │ 
    i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.79026s (cpu); 1.2643s (thread); 0s (gc)
│ │ │ + -- used 1.58086s (cpu); 1.24315s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    │ │ │
    │ │ │ 
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 2.01763s (cpu); 1.4125s (thread); 0s (gc)
│ │ │ + -- used 1.54773s (cpu); 1.26084s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of S[w ..w ]
│ │ │                    0   3
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -86,34 +86,34 @@ │ │ │ │ │ │ │ │ 5 4 │ │ │ │ o5 : Matrix S <-- S │ │ │ │ i6 : I = minors(n-1, M); │ │ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ i7 : time rI = expectedReesIdeal I; -- n= 5 case takes < 1 sec. │ │ │ │ - -- used 1.37298s (cpu); 0.841764s (thread); 0s (gc) │ │ │ │ + -- used 1.14349s (cpu); 0.752969s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of S[w ..w ] │ │ │ │ 0 4 │ │ │ │ i8 : kk = ZZ/101; │ │ │ │ i9 : S = kk[x,y,z]; │ │ │ │ i10 : m = random(S^3, S^{4:-2}); │ │ │ │ │ │ │ │ 3 4 │ │ │ │ o10 : Matrix S <-- S │ │ │ │ i11 : I = minors(3,m); │ │ │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ i12 : time reesIdeal (I, I_0); │ │ │ │ - -- used 1.79026s (cpu); 1.2643s (thread); 0s (gc) │ │ │ │ + -- used 1.58086s (cpu); 1.24315s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ │ - -- used 2.01763s (cpu); 1.4125s (thread); 0s (gc) │ │ │ │ + -- used 1.54773s (cpu); 1.26084s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_y_m_m_e_t_r_i_c_A_l_g_e_b_r_a_I_d_e_a_l -- Ideal of the symmetric algebra of an ideal or │ │ │ │ module │ │ │ │ * _j_a_c_o_b_i_a_n_D_u_a_l -- Computes the 'jacobian dual', part of a method of finding │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal.html │ │ │ @@ -115,24 +115,24 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0295188s (cpu); 0.0275558s (thread); 0s (gc)
│ │ │ + -- used 0.0805851s (cpu); 0.0295094s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.138904s (cpu); 0.138178s (thread); 0s (gc)
│ │ │ + -- used 0.150284s (cpu); 0.14039s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ @@ -169,24 +169,24 @@ │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.023238s (cpu); 0.0222073s (thread); 0s (gc)
│ │ │ + -- used 0.0964715s (cpu); 0.0257072s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00959305s (cpu); 0.00933152s (thread); 0s (gc)
│ │ │ + -- used 0.0429585s (cpu); 0.0118258s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of S[w ..w ]
│ │ │                    0   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -51,20 +51,20 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 2 │ │ │ │ - x x x , x - x x ) │ │ │ │ 0 2 4 1 0 4 │ │ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ i4 : time V1 = reesIdeal i; │ │ │ │ - -- used 0.0295188s (cpu); 0.0275558s (thread); 0s (gc) │ │ │ │ + -- used 0.0805851s (cpu); 0.0295094s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ │ 0 6 │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.138904s (cpu); 0.138178s (thread); 0s (gc) │ │ │ │ + -- used 0.150284s (cpu); 0.14039s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 : Ideal of S[w ..w ] │ │ │ │ 0 6 │ │ │ │ The following example shows how we handle degrees │ │ │ │ i6 : S=kk[a,b,c] │ │ │ │ │ │ │ │ o6 = S │ │ │ │ @@ -81,20 +81,20 @@ │ │ │ │ i8 : i=minors(2,m) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o8 = ideal (a , a*b, b ) │ │ │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ i9 : time I1 = reesIdeal i; │ │ │ │ - -- used 0.023238s (cpu); 0.0222073s (thread); 0s (gc) │ │ │ │ + -- used 0.0964715s (cpu); 0.0257072s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ │ 0 2 │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.00959305s (cpu); 0.00933152s (thread); 0s (gc) │ │ │ │ + -- used 0.0429585s (cpu); 0.0118258s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 : Ideal of S[w ..w ] │ │ │ │ 0 2 │ │ │ │ i11 : transpose gens I1 │ │ │ │ │ │ │ │ o11 = {-1, -3} | aw_1-bw_2 | │ │ │ │ {-1, -3} | aw_0-bw_1 | │ │ ├── ./usr/share/doc/Macaulay2/ReflexivePolytopesDB/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=7 │ │ │ S1NFbnRyeQ== │ │ │ #:len=2438 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZW50cnkgZnJvbSB0aGUgS3JldXpl │ │ │ ci1Ta2Fya2UgZGF0YWJhc2Ugb2YgZGltZW5zaW9uIDMgYW5kIDQgcmVmbGV4aXZlIHBvbHl0b3Bl │ │ ├── ./usr/share/doc/Macaulay2/Regularity/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ UmVndWxhcml0eQ== │ │ │ #:len=797 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSBDYXN0ZWxudW92by1NdW1m │ │ │ b3JkIHJlZ3VsYXJpdHkgb2YgYSBob21vZ2VuZW91cyBpZGVhbCIsIERlc2NyaXB0aW9uID0+IChQ │ │ ├── ./usr/share/doc/Macaulay2/Regularity/example-output/_m__Regularity.out │ │ │ @@ -71,15 +71,15 @@ │ │ │ x x x , x + x x - x x - x x x , x + x - x x ) │ │ │ 0 1 3 0 0 1 1 2 0 2 5 0 2 0 5 │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ │ │ i8 : benchmark "mRegularity I1" │ │ │ │ │ │ -o8 = .3009249333333335 │ │ │ +o8 = .205549047076923 │ │ │ │ │ │ o8 : RR (of precision 53) │ │ │ │ │ │ i9 : R = QQ[x_0..x_5] │ │ │ │ │ │ o9 = R │ │ │ │ │ │ @@ -87,17 +87,17 @@ │ │ │ │ │ │ i10 : I2 = ideal ( x_0^2+x_5^2, x_0^2+x_0*x_3+x_4^2, x_0^2+x_0*x_5+x_2*x_5, x_0^2-x_0*x_3-x_3*x_5, x_0^2-x_3*x_4, x_0*x_3); │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ i11 : benchmark " mRegularity I2" │ │ │ │ │ │ -o11 = .04256394509756095 │ │ │ +o11 = .04303760846666668 │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ │ │ i12 : time regularity I2 │ │ │ - -- used 0.00348648s (cpu); 0.00348699s (thread); 0s (gc) │ │ │ + -- used 0.00351191s (cpu); 0.00351598s (thread); 0s (gc) │ │ │ │ │ │ o12 = 4 │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html │ │ │ @@ -181,15 +181,15 @@ │ │ │ o7 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .3009249333333335
│ │ │ +o8 = .205549047076923
│ │ │  
│ │ │  o8 : RR (of precision 53)
    │ │ │ │ │ │ │ │ │ │ │ │

    This is an example where regularity is faster than mRegularity.

    │ │ │ │ │ │ @@ -209,23 +209,23 @@ │ │ │ o10 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .04256394509756095
│ │ │ +o11 = .04303760846666668
│ │ │  
│ │ │  o11 : RR (of precision 53)
    │ │ │
    │ │ │ 
    i12 : time regularity I2  
│ │ │ - -- used 0.00348648s (cpu); 0.00348699s (thread); 0s (gc)
│ │ │ + -- used 0.00351191s (cpu); 0.00351598s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = 4
    │ │ │
    │ │ │

    This symbol is provided by the package Regularity.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -94,34 +94,34 @@ │ │ │ │ 3 2 2 3 3 2 │ │ │ │ x x x , x + x x - x x - x x x , x + x - x x ) │ │ │ │ 0 1 3 0 0 1 1 2 0 2 5 0 2 0 5 │ │ │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ i8 : benchmark "mRegularity I1" │ │ │ │ │ │ │ │ -o8 = .3009249333333335 │ │ │ │ +o8 = .205549047076923 │ │ │ │ │ │ │ │ o8 : RR (of precision 53) │ │ │ │ This is an example where regularity is faster than mRegularity. │ │ │ │ i9 : R = QQ[x_0..x_5] │ │ │ │ │ │ │ │ o9 = R │ │ │ │ │ │ │ │ o9 : PolynomialRing │ │ │ │ i10 : I2 = ideal ( x_0^2+x_5^2, x_0^2+x_0*x_3+x_4^2, x_0^2+x_0*x_5+x_2*x_5, │ │ │ │ x_0^2-x_0*x_3-x_3*x_5, x_0^2-x_3*x_4, x_0*x_3); │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : benchmark " mRegularity I2" │ │ │ │ │ │ │ │ -o11 = .04256394509756095 │ │ │ │ +o11 = .04303760846666668 │ │ │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ i12 : time regularity I2 │ │ │ │ - -- used 0.00348648s (cpu); 0.00348699s (thread); 0s (gc) │ │ │ │ + -- used 0.00351191s (cpu); 0.00351598s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 = 4 │ │ │ │ This symbol is provided by the package Regularity. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_g_u_l_a_r_i_t_y -- compute the Castelnuovo-Mumford regularity │ │ │ │ ********** WWaayyss ttoo uussee mmRReegguullaarriittyy:: ********** │ │ │ │ * mRegularity(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ Y2Fub25pY2FsTXVsdGlwbGllcnM= │ │ │ #:len=1705 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNhbm9uaWNhbCBt │ │ │ dWx0aXBsaWVycyBvZiBhIHJhdGlvbmFsIGN1cnZlcyB3aXRoIG5vZGVzIiwgImxpbmVudW0iID0+ │ │ ├── ./usr/share/doc/Macaulay2/ResLengthThree/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ bXVsdFRhYmxlT25lVHdvKFJpbmcp │ │ │ #:len=273 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjYyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtdWx0VGFibGVPbmVUd28sUmluZyksIm11bHRUYWJs │ │ ├── ./usr/share/doc/Macaulay2/ResidualIntersections/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ Z2VuZXJpY1Jlc2lkdWFs │ │ │ #:len=1739 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgZ2VuZXJpYyByZXNpZHVh │ │ │ bCBpbnRlcnNlY3Rpb25zIG9mIGFuIGlkZWFsIiwgImxpbmVudW0iID0+IDg4OSwgSW5wdXRzID0+ │ │ ├── ./usr/share/doc/Macaulay2/ResolutionsOfStanleyReisnerRings/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=41 │ │ │ YmFyeWNlbnRyaWNTdWJkaXZpc2lvbihTaW1wbGljaWFsQ29tcGxleCk= │ │ │ #:len=381 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI5LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhiYXJ5Y2VudHJpY1N1YmRpdmlzaW9uLFNpbXBsaWNp │ │ ├── ./usr/share/doc/Macaulay2/ResultantComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ bWl4ZWRTdWJkaXZpc2lvbg== │ │ │ #:len=1366 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBjb2hlcmVudCBtaXhl │ │ │ ZCBzdWItZGl2aXNpb24iLCAibGluZW51bSIgPT4gNDc5LCBJbnB1dHMgPT4ge1NQQU57VFR7Imdy │ │ ├── ./usr/share/doc/Macaulay2/Resultants/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ aHVyd2l0ekZvcm0oLi4uLFNpbmd1bGFyTG9jdXM9Pi4uLik= │ │ │ #:len=277 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMyOSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaHVyd2l0ekZvcm0sU2luZ3VsYXJMb2N1c10sImh1 │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out │ │ │ @@ -5,18 +5,18 @@ │ │ │ o2 = ideal(x x - x x ) │ │ │ 0 1 2 3 │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2); │ │ │ - -- used 0.187761s (cpu); 0.10659s (thread); 0s (gc) │ │ │ + -- used 0.11934s (cpu); 0.0621734s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ - -- used 0.297065s (cpu); 0.151907s (thread); 0s (gc) │ │ │ + -- used 0.201418s (cpu); 0.08334s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ o4 = (ideal (x x - x x , x x - x x , x x - x x , │ │ │ 0,3 1,2 0,2 1,3 1,0 1,1 1,2 1,3 0,3 1,1 0,1 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ @@ -37,17 +37,17 @@ │ │ │ 2 2 │ │ │ 4x x x x - 2x x x x + x x )) │ │ │ 0,0 0,1 1,2 1,3 0,2 0,3 1,2 1,3 0,2 1,3 │ │ │ │ │ │ o4 : Sequence │ │ │ │ │ │ i5 : time cayleyTrick(P1xP1,1,Duality=>true); │ │ │ - -- used 0.288499s (cpu); 0.151777s (thread); 0s (gc) │ │ │ + -- used 0.224121s (cpu); 0.109781s (thread); 0s (gc) │ │ │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ - -- used 0.340521s (cpu); 0.183633s (thread); 0s (gc) │ │ │ + -- used 0.263383s (cpu); 0.11857s (thread); 0s (gc) │ │ │ │ │ │ i8 : assert(oo == (P1xP1xP2,P1xP1xP2')) │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ o2 = ideal (x + x + x + x , x x + x x + x x ) │ │ │ 0 1 2 3 0 1 1 2 2 3 │ │ │ │ │ │ o2 : Ideal of P3 │ │ │ │ │ │ i3 : -- Chow equations of C │ │ │ time eqsC = chowEquations chowForm C │ │ │ - -- used 0.0407572s (cpu); 0.040758s (thread); 0s (gc) │ │ │ + -- used 0.0423502s (cpu); 0.0423495s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 4 2 2 2 2 │ │ │ o3 = ideal (x x + x x + x x + x , x x x x + x x x + x x , x x x + │ │ │ 0 3 1 3 2 3 3 0 1 2 3 1 2 3 2 3 0 2 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 3 2 2 3 3 2 2 2 2 │ │ │ x x x + x x - 2x x x - 2x x x - x x , x x + 2x x x - x x x + x x │ │ │ @@ -72,15 +72,15 @@ │ │ │ o5 = ideal (x - x x , x - x x x , x x - x x ) │ │ │ 1 0 2 2 0 1 3 1 2 0 3 │ │ │ │ │ │ o5 : Ideal of P3 │ │ │ │ │ │ i6 : -- Chow equations of D │ │ │ time eqsD = chowEquations chowForm D │ │ │ - -- used 0.0401433s (cpu); 0.0401425s (thread); 0s (gc) │ │ │ + -- used 0.0389318s (cpu); 0.0389348s (thread); 0s (gc) │ │ │ │ │ │ 4 3 2 3 2 2 3 2 2 2 2 2 2 │ │ │ o6 = ideal (x x - x x , x x x - x x x , x x x - x x x , x x x - x x x , │ │ │ 2 3 1 3 1 2 3 0 1 3 0 2 3 0 1 3 1 2 3 0 1 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 3 2 3 3 2 4 2 2 2 3 │ │ │ x x x x - x x , x x x - x x , x x - 4x x x x + 3x x x , x x x - │ │ │ @@ -117,24 +117,24 @@ │ │ │ o9 = ideal(x x + x x ) │ │ │ 0 1 2 3 │ │ │ │ │ │ o9 : Ideal of P3 │ │ │ │ │ │ i10 : -- tangential Chow forms of Q │ │ │ time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2)) │ │ │ - -- used 0.160178s (cpu); 0.106293s (thread); 0s (gc) │ │ │ + -- used 0.164697s (cpu); 0.104009s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o10 = (x x + x x , x - 4x x + 2x x + x , x x + │ │ │ 0 1 2 3 0,1 0,2 1,3 0,1 2,3 2,3 0,1,2 0,1,3 │ │ │ ----------------------------------------------------------------------- │ │ │ x x ) │ │ │ 0,2,3 1,2,3 │ │ │ │ │ │ o10 : Sequence │ │ │ │ │ │ i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2)) │ │ │ - -- used 0.0594214s (cpu); 0.0594176s (thread); 0s (gc) │ │ │ + -- used 0.0602164s (cpu); 0.0602231s (thread); 0s (gc) │ │ │ │ │ │ o11 = true │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out │ │ │ @@ -16,15 +16,15 @@ │ │ │ │ │ │ ZZ │ │ │ o2 : Ideal of ----[x ..x ] │ │ │ 3331 0 5 │ │ │ │ │ │ i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian) │ │ │ time ChowV = chowForm(V,AffineChartGrass=>{1,2,3}) │ │ │ - -- used 5.50368s (cpu); 4.92753s (thread); 0s (gc) │ │ │ + -- used 4.50303s (cpu); 4.15467s (thread); 0s (gc) │ │ │ │ │ │ 4 2 2 2 2 │ │ │ o3 = x + 2x x x + x x - 2x x x + │ │ │ 1,2,4 0,2,4 1,2,4 2,3,4 0,2,4 2,3,4 1,2,3 1,2,4 1,2,5 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 │ │ │ x x - x x x + x x x x + │ │ │ @@ -143,19 +143,19 @@ │ │ │ 3331 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 0,1,5 0,2,5 1,2,5 0,3,5 1,3,5 2,3,5 0,4,5 1,4,5 2,4,5 3,4,5 │ │ │ o3 : ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- │ │ │ (x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x - x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x + x x - x x + x x , x x - x x + x x , x x - x x + x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x - x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x ) │ │ │ 2,3,5 1,4,5 1,3,5 2,4,5 1,2,5 3,4,5 2,3,4 1,4,5 1,3,4 2,4,5 1,2,4 3,4,5 2,3,5 0,4,5 0,3,5 2,4,5 0,2,5 3,4,5 1,3,5 0,4,5 0,3,5 1,4,5 0,1,5 3,4,5 1,2,5 0,4,5 0,2,5 1,4,5 0,1,5 2,4,5 2,3,4 0,4,5 0,3,4 2,4,5 0,2,4 3,4,5 1,3,4 0,4,5 0,3,4 1,4,5 0,1,4 3,4,5 1,2,4 0,4,5 0,2,4 1,4,5 0,1,4 2,4,5 1,2,3 0,4,5 0,2,3 1,4,5 0,1,3 2,4,5 0,1,2 3,4,5 2,3,4 1,3,5 1,3,4 2,3,5 1,2,3 3,4,5 1,2,5 0,3,5 0,2,5 1,3,5 0,1,5 2,3,5 2,3,4 0,3,5 0,3,4 2,3,5 0,2,3 3,4,5 1,3,4 0,3,5 0,3,4 1,3,5 0,1,3 3,4,5 1,2,4 0,3,5 0,2,4 1,3,5 0,1,4 2,3,5 0,1,2 3,4,5 1,2,3 0,3,5 0,2,3 1,3,5 0,1,3 2,3,5 2,3,4 1,2,5 1,2,4 2,3,5 1,2,3 2,4,5 1,3,4 1,2,5 1,2,4 1,3,5 1,2,3 1,4,5 0,3,4 1,2,5 0,2,4 1,3,5 0,1,4 2,3,5 0,2,3 1,4,5 0,1,3 2,4,5 0,1,2 3,4,5 2,3,4 0,2,5 0,2,4 2,3,5 0,2,3 2,4,5 1,3,4 0,2,5 0,2,4 1,3,5 0,2,3 1,4,5 0,1,2 3,4,5 0,3,4 0,2,5 0,2,4 0,3,5 0,2,3 0,4,5 1,2,4 0,2,5 0,2,4 1,2,5 0,1,2 2,4,5 1,2,3 0,2,5 0,2,3 1,2,5 0,1,2 2,3,5 2,3,4 0,1,5 0,1,4 2,3,5 0,1,3 2,4,5 0,1,2 3,4,5 1,3,4 0,1,5 0,1,4 1,3,5 0,1,3 1,4,5 0,3,4 0,1,5 0,1,4 0,3,5 0,1,3 0,4,5 1,2,4 0,1,5 0,1,4 1,2,5 0,1,2 1,4,5 0,2,4 0,1,5 0,1,4 0,2,5 0,1,2 0,4,5 1,2,3 0,1,5 0,1,3 1,2,5 0,1,2 1,3,5 0,2,3 0,1,5 0,1,3 0,2,5 0,1,2 0,3,5 1,2,4 0,3,4 0,2,4 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4 0,2,3 1,2,4 0,1,2 2,3,4 1,2,3 0,1,4 0,1,3 1,2,4 0,1,2 1,3,4 0,2,3 0,1,4 0,1,3 0,2,4 0,1,2 0,3,4 │ │ │ │ │ │ i4 : -- equivalently (but faster)... │ │ │ time assert(ChowV === chowForm f) │ │ │ - -- used 1.1406s (cpu); 0.999558s (thread); 0s (gc) │ │ │ + -- used 0.95788s (cpu); 0.885679s (thread); 0s (gc) │ │ │ │ │ │ i5 : -- X-resultant of V │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ - -- used 0.325978s (cpu); 0.196045s (thread); 0s (gc) │ │ │ + -- used 0.266935s (cpu); 0.193022s (thread); 0s (gc) │ │ │ │ │ │ i6 : -- three generic ternary quadrics │ │ │ F = genericPolynomials({2,2,2},ZZ/3331) │ │ │ │ │ │ 2 2 2 2 2 │ │ │ o6 = {a x + a x x + a x + a x x + a x x + a x , b x + b x x + b x + │ │ │ 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 │ │ │ @@ -164,12 +164,12 @@ │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : -- resultant of the three forms │ │ │ time resF = resultant F; │ │ │ - -- used 0.341473s (cpu); 0.215038s (thread); 0s (gc) │ │ │ + -- used 0.294847s (cpu); 0.218844s (thread); 0s (gc) │ │ │ │ │ │ i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres)) │ │ │ │ │ │ i9 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out │ │ │ @@ -4,30 +4,30 @@ │ │ │ │ │ │ 2 2 │ │ │ o2 = a*x + b*x*y + c*y │ │ │ │ │ │ o2 : ZZ[a..c][x..y] │ │ │ │ │ │ i3 : time discriminant F │ │ │ - -- used 0.00924086s (cpu); 0.00924015s (thread); 0s (gc) │ │ │ + -- used 0.0104301s (cpu); 0.0104054s (thread); 0s (gc) │ │ │ │ │ │ 2 │ │ │ o3 = - b + 4a*c │ │ │ │ │ │ o3 : ZZ[a..c] │ │ │ │ │ │ i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3 │ │ │ │ │ │ 3 2 2 3 │ │ │ o5 = a*x + b*x y + c*x*y + d*y │ │ │ │ │ │ o5 : ZZ[a..d][x..y] │ │ │ │ │ │ i6 : time discriminant F │ │ │ - -- used 0.0098778s (cpu); 0.0098795s (thread); 0s (gc) │ │ │ + -- used 0.0108438s (cpu); 0.0108453s (thread); 0s (gc) │ │ │ │ │ │ 2 2 3 3 2 2 │ │ │ o6 = - b c + 4a*c + 4b d - 18a*b*c*d + 27a d │ │ │ │ │ │ o6 : ZZ[a..d] │ │ │ │ │ │ i7 : x=symbol x; R=ZZ/331[x_0..x_3] │ │ │ @@ -59,15 +59,15 @@ │ │ │ 4 3 4 4 3 4 │ │ │ o12 = (t + t )x - t x x + t x + (t - t )x + t x x + t x │ │ │ 0 1 0 1 0 1 0 1 0 1 2 1 2 3 0 3 │ │ │ │ │ │ o12 : R' │ │ │ │ │ │ i13 : time D=discriminant pencil │ │ │ - -- used 0.44332s (cpu); 0.443247s (thread); 0s (gc) │ │ │ + -- used 0.371582s (cpu); 0.371418s (thread); 0s (gc) │ │ │ │ │ │ 108 106 2 102 6 100 8 98 10 96 12 │ │ │ o13 = - 62t + 19t t + 160t t + 91t t + 129t t + 117t t + │ │ │ 0 0 1 0 1 0 1 0 1 0 1 │ │ │ ----------------------------------------------------------------------- │ │ │ 94 14 92 16 90 18 88 20 86 22 84 24 │ │ │ 161t t + 124t t - 82t t - 21t t - 49t t - 123t t + │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out │ │ │ @@ -9,25 +9,25 @@ │ │ │ x x ) │ │ │ 0 3 │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 5 │ │ │ │ │ │ i2 : time V' = dualVariety V │ │ │ - -- used 0.202374s (cpu); 0.134305s (thread); 0s (gc) │ │ │ + -- used 0.178788s (cpu); 0.111587s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 │ │ │ o2 = ideal(x x - x x x + x x + x x - 4x x x ) │ │ │ 2 3 1 2 4 0 4 1 5 0 3 5 │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 5 │ │ │ │ │ │ i3 : time V == dualVariety V' │ │ │ - -- used 0.310587s (cpu); 0.159166s (thread); 0s (gc) │ │ │ + -- used 0.271897s (cpu); 0.148364s (thread); 0s (gc) │ │ │ │ │ │ o3 = true │ │ │ │ │ │ i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331) │ │ │ │ │ │ 3 2 2 3 2 2 2 │ │ │ o4 = a x + a x x + a x x + a x + a x x + a x x x + a x x + a x x + │ │ │ @@ -38,22 +38,22 @@ │ │ │ 8 1 2 9 2 │ │ │ │ │ │ ZZ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ 3331 0 9 0 2 │ │ │ │ │ │ i5 : time discF = ideal discriminant F; │ │ │ - -- used 0.162855s (cpu); 0.0858796s (thread); 0s (gc) │ │ │ + -- used 0.137864s (cpu); 0.0738357s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ 3331 0 9 │ │ │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ - -- used 0.745903s (cpu); 0.673493s (thread); 0s (gc) │ │ │ + -- used 0.60157s (cpu); 0.53061s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o6 : Ideal of ----[x ..x ] │ │ │ 3331 0 9 │ │ │ │ │ │ i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out │ │ │ @@ -6,15 +6,15 @@ │ │ │ o1 = ideal (x - x x , x x - x x , x - x x ) │ │ │ 2 1 3 1 2 0 3 1 0 2 │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ i2 : time fromPluckerToStiefel dualize chowForm C │ │ │ - -- used 0.141834s (cpu); 0.0752672s (thread); 0s (gc) │ │ │ + -- used 0.119929s (cpu); 0.0565095s (thread); 0s (gc) │ │ │ │ │ │ 3 3 2 2 2 2 2 3 │ │ │ o2 = - x x + x x x x - x x x x + x x x - │ │ │ 0,3 1,0 0,2 0,3 1,0 1,1 0,1 0,3 1,0 1,1 0,0 0,3 1,1 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 2 │ │ │ x x x x + 2x x x x + x x x x x x - │ │ │ @@ -56,15 +56,15 @@ │ │ │ x x x x - 2x x x x - x x x x + x x │ │ │ 0,0 0,1 1,1 1,3 0,0 0,2 1,1 1,3 0,0 0,1 1,2 1,3 0,0 1,3 │ │ │ │ │ │ o2 : QQ[x ..x ] │ │ │ 0,0 1,3 │ │ │ │ │ │ i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1}) │ │ │ - -- used 0.177092s (cpu); 0.069066s (thread); 0s (gc) │ │ │ + -- used 0.219055s (cpu); 0.0776059s (thread); 0s (gc) │ │ │ │ │ │ 3 2 2 │ │ │ o3 = - x x + x x x - x x x + x x + 3x x x - │ │ │ 0,3 1,2 0,2 1,2 1,3 0,2 0,3 1,2 0,2 1,3 0,3 1,2 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 3 2 │ │ │ 2x x + x + x │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ o4 : QQ[a ..a ] │ │ │ 0,0 1,1 │ │ │ │ │ │ i5 : w = chowForm C; │ │ │ │ │ │ i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s)) │ │ │ - -- used 0.108209s (cpu); 0.0419396s (thread); 0s (gc) │ │ │ + -- used 0.101117s (cpu); 0.033817s (thread); 0s (gc) │ │ │ │ │ │ 3 2 3 2 │ │ │ o6 = {ideal(- x x + x x x - x - 3x x x + 2x x + │ │ │ 0,3 1,2 0,2 1,2 1,3 0,2 0,2 0,3 1,2 0,2 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 3 2 │ │ │ x x x - x x + x ), ideal(x x - 2x x x x + │ │ │ @@ -130,14 +130,14 @@ │ │ │ 2 3 2 │ │ │ 2x x - x + x )} │ │ │ 0,0 1,1 1,1 1,0 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : time apply(U,u->dim singularLocus u) │ │ │ - -- used 0.115818s (cpu); 0.0462516s (thread); 0s (gc) │ │ │ + -- used 0.111088s (cpu); 0.0420129s (thread); 0s (gc) │ │ │ │ │ │ o7 = {2, 2, 2, 2, 2, 2} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ + -p + -p p + 7p p + 6p p + -p p + --p ) │ │ │ 4 3 9 0 4 1 4 2 4 9 3 4 10 4 │ │ │ │ │ │ o1 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ i2 : time hurwitzForm Q │ │ │ - -- used 0.125887s (cpu); 0.0563778s (thread); 0s (gc) │ │ │ + -- used 0.112866s (cpu); 0.0507715s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o2 = 11966535p + 14645610p p + 11354175p + 1666980p p + │ │ │ 0,1 0,1 0,2 0,2 0,1 1,2 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 │ │ │ 4456620p p + 1127196p + 54176850p p + 20326950p p + │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out │ │ │ @@ -26,15 +26,15 @@ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ o1 : -------------------------------------- │ │ │ p p - p p + p p │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i2 : time isCoisotropic w │ │ │ - -- used 0.00877231s (cpu); 0.00877035s (thread); 0s (gc) │ │ │ + -- used 0.00786174s (cpu); 0.00786149s (thread); 0s (gc) │ │ │ │ │ │ o2 = true │ │ │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ 3 2 3 7 2 5 7 2 10 │ │ │ @@ -56,12 +56,12 @@ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ o3 : -------------------------------------- │ │ │ p p - p p + p p │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i4 : time isCoisotropic w' │ │ │ - -- used 0.00769242s (cpu); 0.00769265s (thread); 0s (gc) │ │ │ + -- used 0.00682201s (cpu); 0.00682168s (thread); 0s (gc) │ │ │ │ │ │ o4 = false │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out │ │ │ @@ -31,12 +31,12 @@ │ │ │ 4 5 │ │ │ │ │ │ ZZ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ 33331 0 5 │ │ │ │ │ │ i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I)) │ │ │ - -- used 0.0181849s (cpu); 0.0181858s (thread); 0s (gc) │ │ │ + -- used 0.0171715s (cpu); 0.0171706s (thread); 0s (gc) │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out │ │ │ @@ -13,15 +13,15 @@ │ │ │ 2 2 2 3 │ │ │ c x x x + c x x + c x x + c x x + c x } │ │ │ 4 0 1 2 7 1 2 5 0 2 8 1 2 9 2 │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : time (D,D') = macaulayFormula F │ │ │ - -- used 0.00373056s (cpu); 0.00372864s (thread); 0s (gc) │ │ │ + -- used 0.00343268s (cpu); 0.00343168s (thread); 0s (gc) │ │ │ │ │ │ o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0 0 0 0 0 0 0 0 0 0 0 │ │ │ | 0 a_0 0 a_1 a_2 0 a_3 a_4 a_5 0 0 0 0 0 0 0 0 │ │ │ | 0 0 a_0 0 a_1 a_2 0 a_3 a_4 a_5 0 0 0 0 0 0 0 │ │ │ | 0 0 0 a_0 0 0 a_1 a_2 0 0 a_3 a_4 a_5 0 0 0 0 │ │ │ | 0 0 0 0 a_0 0 0 a_1 a_2 0 0 a_3 a_4 a_5 0 0 0 │ │ │ | 0 0 0 0 0 a_0 0 0 a_1 a_2 0 0 a_3 a_4 a_5 0 0 │ │ │ @@ -78,15 +78,15 @@ │ │ │ 10 2 7 2 5 3 │ │ │ --p p + -p p + -p } │ │ │ 9 0 2 8 1 2 6 2 │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : time (D,D') = macaulayFormula F │ │ │ - -- used 0.00252047s (cpu); 0.00251818s (thread); 0s (gc) │ │ │ + -- used 0.00224073s (cpu); 0.00223913s (thread); 0s (gc) │ │ │ │ │ │ o4 = (| 9/2 9/4 3/4 7/4 7/9 7/10 0 0 0 0 0 0 0 0 0 │ │ │ | 0 9/2 0 9/4 3/4 0 7/4 7/9 7/10 0 0 0 0 0 0 │ │ │ | 0 0 9/2 0 9/4 3/4 0 7/4 7/9 7/10 0 0 0 0 0 │ │ │ | 0 0 0 9/2 0 0 9/4 3/4 0 0 7/4 7/9 7/10 0 0 │ │ │ | 0 0 0 0 9/2 0 0 9/4 3/4 0 0 7/4 7/9 7/10 0 │ │ │ | 0 0 0 0 0 9/2 0 0 9/4 3/4 0 0 7/4 7/9 7/10 │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out │ │ │ @@ -9,29 +9,29 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ 664x ) │ │ │ 4 │ │ │ │ │ │ o3 : Ideal of P4 │ │ │ │ │ │ i4 : time p = plucker L │ │ │ - -- used 0.00473681s (cpu); 0.00473472s (thread); 0s (gc) │ │ │ + -- used 0.00461257s (cpu); 0.00461104s (thread); 0s (gc) │ │ │ │ │ │ o4 = ideal (x + 8480x , x - 6727x , x + 15777x , x + │ │ │ 2,4 3,4 1,4 3,4 0,4 3,4 2,3 │ │ │ ------------------------------------------------------------------------ │ │ │ 11656x , x - 14853x , x + 664x , x + 13522x , x + │ │ │ 3,4 1,3 3,4 0,3 3,4 1,2 3,4 0,2 │ │ │ ------------------------------------------------------------------------ │ │ │ 11804x , x + 14854x ) │ │ │ 3,4 0,1 3,4 │ │ │ │ │ │ o4 : Ideal of G'1'4 │ │ │ │ │ │ i5 : time L' = plucker p │ │ │ - -- used 0.121088s (cpu); 0.0718194s (thread); 0s (gc) │ │ │ + -- used 0.105445s (cpu); 0.043898s (thread); 0s (gc) │ │ │ │ │ │ o5 = ideal (x + 8480x - 11656x , x - 6727x + 14853x , x + 15777x - │ │ │ 2 3 4 1 3 4 0 3 │ │ │ ------------------------------------------------------------------------ │ │ │ 664x ) │ │ │ 4 │ │ │ │ │ │ @@ -40,25 +40,25 @@ │ │ │ i6 : assert(L' == L) │ │ │ │ │ │ i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve │ │ │ │ │ │ o7 : Ideal of G'1'4 │ │ │ │ │ │ i8 : time W = plucker Y; -- surface swept out by the lines of Y │ │ │ - -- used 0.148823s (cpu); 0.0922017s (thread); 0s (gc) │ │ │ + -- used 0.0991924s (cpu); 0.0464414s (thread); 0s (gc) │ │ │ │ │ │ o8 : Ideal of P4 │ │ │ │ │ │ i9 : (codim W,degree W) │ │ │ │ │ │ o9 = (2, 5) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : time Y' = plucker(W,1); -- variety of lines contained in W │ │ │ - -- used 0.244058s (cpu); 0.244015s (thread); 0s (gc) │ │ │ + -- used 0.142696s (cpu); 0.142698s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of G'1'4 │ │ │ │ │ │ i11 : assert(Y' == Y) │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out │ │ │ @@ -35,15 +35,15 @@ │ │ │ 3 2 9 7 2 9 3 1 8 4 │ │ │ -b)y*w + (-a + -b)z*w + (-a + 2b)w , 2x + -y + -z + -w} │ │ │ 4 8 8 7 4 3 5 │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time resultant(F,Algorithm=>"Poisson2") │ │ │ - -- used 0.562085s (cpu); 0.289584s (thread); 0s (gc) │ │ │ + -- used 0.489603s (cpu); 0.213305s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o3 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ │ @@ -56,15 +56,15 @@ │ │ │ 1146977327343523453866040839029 4 194441910898734675845094443 5 │ │ │ -------------------------------a*b - ---------------------------b │ │ │ 1119954511872000000000 895963609497600000 │ │ │ │ │ │ o3 : QQ[a..b] │ │ │ │ │ │ i4 : time resultant(F,Algorithm=>"Macaulay2") │ │ │ - -- used 0.273183s (cpu); 0.131953s (thread); 0s (gc) │ │ │ + -- used 0.300703s (cpu); 0.146374s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o4 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ │ @@ -77,15 +77,15 @@ │ │ │ 1146977327343523453866040839029 4 194441910898734675845094443 5 │ │ │ -------------------------------a*b - ---------------------------b │ │ │ 1119954511872000000000 895963609497600000 │ │ │ │ │ │ o4 : QQ[a..b] │ │ │ │ │ │ i5 : time resultant(F,Algorithm=>"Poisson") │ │ │ - -- used 0.302008s (cpu); 0.301983s (thread); 0s (gc) │ │ │ + -- used 0.301989s (cpu); 0.301989s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o5 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ │ @@ -98,15 +98,15 @@ │ │ │ 1146977327343523453866040839029 4 194441910898734675845094443 5 │ │ │ -------------------------------a*b - ---------------------------b │ │ │ 1119954511872000000000 895963609497600000 │ │ │ │ │ │ o5 : QQ[a..b] │ │ │ │ │ │ i6 : time resultant(F,Algorithm=>"Macaulay") │ │ │ - -- used 0.625765s (cpu); 0.559323s (thread); 0s (gc) │ │ │ + -- used 0.638822s (cpu); 0.560047s (thread); 0s (gc) │ │ │ │ │ │ 21002161660529014459938925799 5 2085933800619238998825958079203 4 │ │ │ o6 = - -----------------------------a - -------------------------------a b - │ │ │ 2222549728809984000000 12700284164628480000000 │ │ │ ------------------------------------------------------------------------ │ │ │ 348237304382147063838108483692249 3 2 │ │ │ ---------------------------------a b - │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ │ │ 2 2 3 2 4 │ │ │ o2 = {x + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y + t*x y*z + u*z } │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time resultant F │ │ │ - -- used 0.123723s (cpu); 0.0492639s (thread); 0s (gc) │ │ │ + -- used 0.100779s (cpu); 0.0373295s (thread); 0s (gc) │ │ │ │ │ │ 12 11 2 10 3 9 4 8 5 7 6 │ │ │ o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u │ │ │ ------------------------------------------------------------------------ │ │ │ 6 7 5 8 11 10 2 9 3 │ │ │ - 413343t u - 177147t u + 567t u + 10206t u + 76545t u + │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -64,15 +64,15 @@ │ │ │ 3 │ │ │ + c x } │ │ │ 9 2 │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : time resultant F │ │ │ - -- used 3.07394s (cpu); 2.19358s (thread); 0s (gc) │ │ │ + -- used 2.28559s (cpu); 1.74838s (thread); 0s (gc) │ │ │ │ │ │ 6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 2 2 │ │ │ o5 = a b c - 3a a b b c + 3a a b b c - a a b c + 3a a b b c - │ │ │ 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 5 0 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 3 2 4 2 2 2 4 2 2 2 5 2 2 6 3 2 │ │ │ 6a a b b b c + 3a a b b c + 3a a b b c - 3a a b b c + a b c - │ │ │ @@ -1690,12 +1690,12 @@ │ │ │ 2 2 2 2 │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : time # terms resultant F │ │ │ - -- used 0.644433s (cpu); 0.456975s (thread); 0s (gc) │ │ │ + -- used 0.483304s (cpu); 0.355059s (thread); 0s (gc) │ │ │ │ │ │ o7 = 21894 │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out │ │ │ @@ -8,15 +8,15 @@ │ │ │ 1 2 0 3 1 3 0 4 3 2 4 │ │ │ │ │ │ o2 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form) │ │ │ time tangentialChowForm(S,0) │ │ │ - -- used 0.0398049s (cpu); 0.039807s (thread); 0s (gc) │ │ │ + -- used 0.0383661s (cpu); 0.0383665s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o3 = p p - p p p - p p p + p p p + p p + │ │ │ 1,3 2,3 1,2 1,3 2,4 0,3 1,3 2,4 0,2 1,4 2,4 1,2 3,4 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 │ │ │ p p - 2p p p - p p p │ │ │ @@ -26,15 +26,15 @@ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 │ │ │ o3 : ---------------------------------------------------------------------------------------------------------------------------------------------------------------- │ │ │ (p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p ) │ │ │ 2,3 1,4 1,3 2,4 1,2 3,4 2,3 0,4 0,3 2,4 0,2 3,4 1,3 0,4 0,3 1,4 0,1 3,4 1,2 0,4 0,2 1,4 0,1 2,4 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i4 : -- 1-th associated hypersurface of S in G(2,4) │ │ │ time tangentialChowForm(S,1) │ │ │ - -- used 0.16271s (cpu); 0.0952426s (thread); 0s (gc) │ │ │ + -- used 0.136019s (cpu); 0.0794229s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 3 2 2 │ │ │ o4 = p p + p p - 2p p + p p - │ │ │ 1,2,3 1,2,4 0,2,4 1,2,4 0,2,3 1,2,4 0,2,4 0,3,4 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 3 3 │ │ │ 4p p - 4p p - 2p p + │ │ │ @@ -68,32 +68,32 @@ │ │ │ 0,1,2 0,1,3 0,2,3 1,2,3 0,1,4 0,2,4 1,2,4 0,3,4 1,3,4 2,3,4 │ │ │ o4 : ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- │ │ │ (p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + p p ) │ │ │ 1,2,4 0,3,4 0,2,4 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4 0,2,3 1,2,4 0,1,2 2,3,4 1,2,3 0,1,4 0,1,3 1,2,4 0,1,2 1,3,4 0,2,3 0,1,4 0,1,3 0,2,4 0,1,2 0,3,4 │ │ │ │ │ │ i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S) │ │ │ time tangentialChowForm(S,2) │ │ │ - -- used 0.117742s (cpu); 0.0528496s (thread); 0s (gc) │ │ │ + -- used 0.103017s (cpu); 0.0501147s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o5 = p p - p p p + p p │ │ │ 0,1,3,4 0,2,3,4 0,1,2,4 0,2,3,4 1,2,3,4 0,1,2,3 1,2,3,4 │ │ │ │ │ │ o5 : QQ[p ..p , p , p , p ] │ │ │ 0,1,2,3 0,1,2,4 0,1,3,4 0,2,3,4 1,2,3,4 │ │ │ │ │ │ i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing │ │ │ time S' = ideal dualize tangentialChowForm(S,2) │ │ │ - -- used 0.130131s (cpu); 0.0621371s (thread); 0s (gc) │ │ │ + -- used 0.112314s (cpu); 0.0555025s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o6 = ideal(p p - p p p + p p ) │ │ │ 1 2 0 1 3 0 4 │ │ │ │ │ │ o6 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ i7 : -- we then can recover S │ │ │ time assert(dualize tangentialChowForm(S',3) == S) │ │ │ - -- used 0.190994s (cpu); 0.117998s (thread); 0s (gc) │ │ │ + -- used 0.164298s (cpu); 0.106233s (thread); 0s (gc) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ @@ -91,24 +91,24 @@ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.187761s (cpu); 0.10659s (thread); 0s (gc)
    │ │ │ + -- used 0.11934s (cpu); 0.0621734s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1)
│ │ │ - -- used 0.297065s (cpu); 0.151907s (thread); 0s (gc)
│ │ │ + -- used 0.201418s (cpu); 0.08334s (thread); 0s (gc)
│ │ │  
│ │ │                                                                             
│ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                              
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ @@ -135,26 +135,26 @@
│ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,20 +38,20 @@ │ │ │ │ │ │ │ │ o2 = ideal(x x - x x ) │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ │ 0 3 │ │ │ │ i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2); │ │ │ │ - -- used 0.187761s (cpu); 0.10659s (thread); 0s (gc) │ │ │ │ + -- used 0.11934s (cpu); 0.0621734s (thread); 0s (gc) │ │ │ │ In the next example, we calculate the defining ideal of $\mathbb │ │ │ │ {P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$ and that of its │ │ │ │ dual variety. │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ │ - -- used 0.297065s (cpu); 0.151907s (thread); 0s (gc) │ │ │ │ + -- used 0.201418s (cpu); 0.08334s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ o4 = (ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ 0,3 1,2 0,2 1,3 1,0 1,1 1,2 1,3 0,3 1,1 0,1 1,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ @@ -73,18 +73,18 @@ │ │ │ │ 4x x x x - 2x x x x + x x )) │ │ │ │ 0,0 0,1 1,2 1,3 0,2 0,3 1,2 1,3 0,2 1,3 │ │ │ │ │ │ │ │ o4 : Sequence │ │ │ │ If the option Duality is set to true, then the method applies the so-called │ │ │ │ "dual Cayley trick". │ │ │ │ i5 : time cayleyTrick(P1xP1,1,Duality=>true); │ │ │ │ - -- used 0.288499s (cpu); 0.151777s (thread); 0s (gc) │ │ │ │ + -- used 0.224121s (cpu); 0.109781s (thread); 0s (gc) │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ │ - -- used 0.340521s (cpu); 0.183633s (thread); 0s (gc) │ │ │ │ + -- used 0.263383s (cpu); 0.11857s (thread); 0s (gc) │ │ │ │ i8 : assert(oo == (P1xP1xP2,P1xP1xP2')) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_u_a_l_V_a_r_i_e_t_y -- projective dual variety │ │ │ │ ********** WWaayyss ttoo uussee ccaayylleeyyTTrriicckk:: ********** │ │ │ │ * cayleyTrick(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_a_y_l_e_y_T_r_i_c_k is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o2 : Ideal of P3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.288499s (cpu); 0.151777s (thread); 0s (gc)
    │ │ │ + -- used 0.224121s (cpu); 0.109781s (thread); 0s (gc) │ │ │
    │ │ │ 
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    │ │ │
    │ │ │ 
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.340521s (cpu); 0.183633s (thread); 0s (gc)
    │ │ │ + -- used 0.263383s (cpu); 0.11857s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    │ │ │
    │ │ │ 
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.0407572s (cpu); 0.040758s (thread); 0s (gc)
│ │ │ + -- used 0.0423502s (cpu); 0.0423495s (thread); 0s (gc)
│ │ │  
│ │ │               2 2    2 2    2 2    4                2      2 2   2      
│ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
│ │ │ @@ -167,15 +167,15 @@
│ │ │  o5 : Ideal of P3
    │ │ │
    │ │ │ 
    i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.0401433s (cpu); 0.0401425s (thread); 0s (gc)
│ │ │ + -- used 0.0389318s (cpu); 0.0389348s (thread); 0s (gc)
│ │ │  
│ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2 
│ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
│ │ │       ------------------------------------------------------------------------
│ │ │            2      3 2   3        3 2   4         2         2 2       3    
│ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ @@ -227,30 +227,30 @@
│ │ │  o9 : Ideal of P3
    │ │ │
    │ │ │ 
    i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.160178s (cpu); 0.106293s (thread); 0s (gc)
│ │ │ + -- used 0.164697s (cpu); 0.104009s (thread); 0s (gc)
│ │ │  
│ │ │                       2                              2
│ │ │  o10 = (x x  + x x , x    - 4x   x    + 2x   x    + x   , x     x      +
│ │ │          0 1    2 3   0,1     0,2 1,3     0,1 2,3    2,3   0,1,2 0,1,3  
│ │ │        -----------------------------------------------------------------------
│ │ │        x     x     )
│ │ │         0,2,3 1,2,3
│ │ │  
│ │ │  o10 : Sequence
    │ │ │
    │ │ │ 
    i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
│ │ │ - -- used 0.0594214s (cpu); 0.0594176s (thread); 0s (gc)
│ │ │ + -- used 0.0602164s (cpu); 0.0602231s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,15 +28,15 @@ │ │ │ │ 2 2 2 2 │ │ │ │ o2 = ideal (x + x + x + x , x x + x x + x x ) │ │ │ │ 0 1 2 3 0 1 1 2 2 3 │ │ │ │ │ │ │ │ o2 : Ideal of P3 │ │ │ │ i3 : -- Chow equations of C │ │ │ │ time eqsC = chowEquations chowForm C │ │ │ │ - -- used 0.0407572s (cpu); 0.040758s (thread); 0s (gc) │ │ │ │ + -- used 0.0423502s (cpu); 0.0423495s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 4 2 2 2 2 │ │ │ │ o3 = ideal (x x + x x + x x + x , x x x x + x x x + x x , x x x + │ │ │ │ 0 3 1 3 2 3 3 0 1 2 3 1 2 3 2 3 0 2 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 2 3 3 2 2 2 2 │ │ │ │ x x x + x x - 2x x x - 2x x x - x x , x x + 2x x x - x x x + x x │ │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ 2 3 2 2 │ │ │ │ o5 = ideal (x - x x , x - x x x , x x - x x ) │ │ │ │ 1 0 2 2 0 1 3 1 2 0 3 │ │ │ │ │ │ │ │ o5 : Ideal of P3 │ │ │ │ i6 : -- Chow equations of D │ │ │ │ time eqsD = chowEquations chowForm D │ │ │ │ - -- used 0.0401433s (cpu); 0.0401425s (thread); 0s (gc) │ │ │ │ + -- used 0.0389318s (cpu); 0.0389348s (thread); 0s (gc) │ │ │ │ │ │ │ │ 4 3 2 3 2 2 3 2 2 2 2 2 2 │ │ │ │ o6 = ideal (x x - x x , x x x - x x x , x x x - x x x , x x x - x x x , │ │ │ │ 2 3 1 3 1 2 3 0 1 3 0 2 3 0 1 3 1 2 3 0 1 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 3 3 2 4 2 2 2 3 │ │ │ │ x x x x - x x , x x x - x x , x x - 4x x x x + 3x x x , x x x - │ │ │ │ @@ -135,27 +135,27 @@ │ │ │ │ o9 = ideal(x x + x x ) │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o9 : Ideal of P3 │ │ │ │ i10 : -- tangential Chow forms of Q │ │ │ │ time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm │ │ │ │ (Q,1),tangentialChowForm(Q,2)) │ │ │ │ - -- used 0.160178s (cpu); 0.106293s (thread); 0s (gc) │ │ │ │ + -- used 0.164697s (cpu); 0.104009s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o10 = (x x + x x , x - 4x x + 2x x + x , x x + │ │ │ │ 0 1 2 3 0,1 0,2 1,3 0,1 2,3 2,3 0,1,2 0,1,3 │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ x x ) │ │ │ │ 0,2,3 1,2,3 │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ i11 : time (Q,Q,Q) == (chowEquations(W0,0),chowEquations(W1,1),chowEquations │ │ │ │ (W2,2)) │ │ │ │ - -- used 0.0594214s (cpu); 0.0594176s (thread); 0s (gc) │ │ │ │ + -- used 0.0602164s (cpu); 0.0602231s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = true │ │ │ │ Note that chowEquations(W,0) is not the same as chowEquations W. │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwEEqquuaattiioonnss:: ********** │ │ │ │ * chowEquations(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_o_w_E_q_u_a_t_i_o_n_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ 3331 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian)
│ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ - -- used 5.50368s (cpu); 4.92753s (thread); 0s (gc)
│ │ │ + -- used 4.50303s (cpu); 4.15467s (thread); 0s (gc)
│ │ │  
│ │ │        4               2              2     2               2            
│ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5  
│ │ │       ------------------------------------------------------------------------
│ │ │        2     2              2                                       
│ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ @@ -232,22 +232,22 @@
│ │ │         2,3,5 1,4,5    1,3,5 2,4,5    1,2,5 3,4,5   2,3,4 1,4,5    1,3,4 2,4,5    1,2,4 3,4,5   2,3,5 0,4,5    0,3,5 2,4,5    0,2,5 3,4,5   1,3,5 0,4,5    0,3,5 1,4,5    0,1,5 3,4,5   1,2,5 0,4,5    0,2,5 1,4,5    0,1,5 2,4,5   2,3,4 0,4,5    0,3,4 2,4,5    0,2,4 3,4,5   1,3,4 0,4,5    0,3,4 1,4,5    0,1,4 3,4,5   1,2,4 0,4,5    0,2,4 1,4,5    0,1,4 2,4,5   1,2,3 0,4,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 1,3,5    1,3,4 2,3,5    1,2,3 3,4,5   1,2,5 0,3,5    0,2,5 1,3,5    0,1,5 2,3,5   2,3,4 0,3,5    0,3,4 2,3,5    0,2,3 3,4,5   1,3,4 0,3,5    0,3,4 1,3,5    0,1,3 3,4,5   1,2,4 0,3,5    0,2,4 1,3,5    0,1,4 2,3,5    0,1,2 3,4,5   1,2,3 0,3,5    0,2,3 1,3,5    0,1,3 2,3,5   2,3,4 1,2,5    1,2,4 2,3,5    1,2,3 2,4,5   1,3,4 1,2,5    1,2,4 1,3,5    1,2,3 1,4,5   0,3,4 1,2,5    0,2,4 1,3,5    0,1,4 2,3,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 0,2,5    0,2,4 2,3,5    0,2,3 2,4,5   1,3,4 0,2,5    0,2,4 1,3,5    0,2,3 1,4,5    0,1,2 3,4,5   0,3,4 0,2,5    0,2,4 0,3,5    0,2,3 0,4,5   1,2,4 0,2,5    0,2,4 1,2,5    0,1,2 2,4,5   1,2,3 0,2,5    0,2,3 1,2,5    0,1,2 2,3,5   2,3,4 0,1,5    0,1,4 2,3,5    0,1,3 2,4,5    0,1,2 3,4,5   1,3,4 0,1,5    0,1,4 1,3,5    0,1,3 1,4,5   0,3,4 0,1,5    0,1,4 0,3,5    0,1,3 0,4,5   1,2,4 0,1,5    0,1,4 1,2,5    0,1,2 1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5    0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : -- equivalently (but faster)...
│ │ │       time assert(ChowV === chowForm f)
│ │ │ - -- used 1.1406s (cpu); 0.999558s (thread); 0s (gc)
    │ │ │ + -- used 0.95788s (cpu); 0.885679s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.325978s (cpu); 0.196045s (thread); 0s (gc)
    │ │ │ + -- used 0.266935s (cpu); 0.193022s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │ @@ -262,15 +262,15 @@
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.341473s (cpu); 0.215038s (thread); 0s (gc)
    │ │ │ + -- used 0.294847s (cpu); 0.218844s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring Xres))
    │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -41,15 +41,15 @@ │ │ │ │ │ │ │ │ ZZ │ │ │ │ o2 : Ideal of ----[x ..x ] │ │ │ │ 3331 0 5 │ │ │ │ i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an │ │ │ │ affine chart of the Grassmannian) │ │ │ │ time ChowV = chowForm(V,AffineChartGrass=>{1,2,3}) │ │ │ │ - -- used 5.50368s (cpu); 4.92753s (thread); 0s (gc) │ │ │ │ + -- used 4.50303s (cpu); 4.15467s (thread); 0s (gc) │ │ │ │ │ │ │ │ 4 2 2 2 2 │ │ │ │ o3 = x + 2x x x + x x - 2x x x + │ │ │ │ 1,2,4 0,2,4 1,2,4 2,3,4 0,2,4 2,3,4 1,2,3 1,2,4 1,2,5 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 │ │ │ │ x x - x x x + x x x x + │ │ │ │ @@ -234,33 +234,33 @@ │ │ │ │ 1,4,5 0,2,4 0,1,5 0,1,4 0,2,5 0,1,2 0,4,5 1,2,3 0,1,5 0,1,3 1,2,5 │ │ │ │ 0,1,2 1,3,5 0,2,3 0,1,5 0,1,3 0,2,5 0,1,2 0,3,5 1,2,4 0,3,4 0,2,4 │ │ │ │ 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 0,2,3 1,3,4 0,1,3 2,3,4 1,2,3 0,2,4 │ │ │ │ 0,2,3 1,2,4 0,1,2 2,3,4 1,2,3 0,1,4 0,1,3 1,2,4 0,1,2 1,3,4 0,2,3 │ │ │ │ 0,1,4 0,1,3 0,2,4 0,1,2 0,3,4 │ │ │ │ i4 : -- equivalently (but faster)... │ │ │ │ time assert(ChowV === chowForm f) │ │ │ │ - -- used 1.1406s (cpu); 0.999558s (thread); 0s (gc) │ │ │ │ + -- used 0.95788s (cpu); 0.885679s (thread); 0s (gc) │ │ │ │ i5 : -- X-resultant of V │ │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ │ - -- used 0.325978s (cpu); 0.196045s (thread); 0s (gc) │ │ │ │ + -- used 0.266935s (cpu); 0.193022s (thread); 0s (gc) │ │ │ │ i6 : -- three generic ternary quadrics │ │ │ │ F = genericPolynomials({2,2,2},ZZ/3331) │ │ │ │ │ │ │ │ 2 2 2 2 2 │ │ │ │ o6 = {a x + a x x + a x + a x x + a x x + a x , b x + b x x + b x + │ │ │ │ 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 2 │ │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : -- resultant of the three forms │ │ │ │ time resF = resultant F; │ │ │ │ - -- used 0.341473s (cpu); 0.215038s (thread); 0s (gc) │ │ │ │ + -- used 0.294847s (cpu); 0.218844s (thread); 0s (gc) │ │ │ │ i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring │ │ │ │ Xres)) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _h_u_r_w_i_t_z_F_o_r_m -- Hurwitz form of a projective variety │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwFFoorrmm:: ********** │ │ │ │ * chowForm(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ │ │ o2 : ZZ[a..c][x..y] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time discriminant F
│ │ │ - -- used 0.00924086s (cpu); 0.00924015s (thread); 0s (gc)
│ │ │ + -- used 0.0104301s (cpu); 0.0104054s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
    │ │ │ │ │ │ │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o5 : ZZ[a..d][x..y] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time discriminant F
│ │ │ - -- used 0.0098778s (cpu); 0.0098795s (thread); 0s (gc)
│ │ │ + -- used 0.0108438s (cpu); 0.0108453s (thread); 0s (gc)
│ │ │  
│ │ │          2 2       3     3                   2 2
│ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │  
│ │ │  o6 : ZZ[a..d]
    │ │ │ │ │ │ │ │ │ @@ -170,15 +170,15 @@ │ │ │ │ │ │ o12 : R' │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time D=discriminant pencil
│ │ │ - -- used 0.44332s (cpu); 0.443247s (thread); 0s (gc)
│ │ │ + -- used 0.371582s (cpu); 0.371418s (thread); 0s (gc)
│ │ │  
│ │ │             108      106 2       102 6      100 8       98 10       96 12  
│ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │             0        0   1       0   1      0   1       0  1        0  1   
│ │ │        -----------------------------------------------------------------------
│ │ │            94 14       92 16      90 18      88 20      86 22       84 24  
│ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,28 +23,28 @@
│ │ │ │  i1 : ZZ[a,b,c][x,y]; F = a*x^2+b*x*y+c*y^2
│ │ │ │  
│ │ │ │          2              2
│ │ │ │  o2 = a*x  + b*x*y + c*y
│ │ │ │  
│ │ │ │  o2 : ZZ[a..c][x..y]
│ │ │ │  i3 : time discriminant F
│ │ │ │ - -- used 0.00924086s (cpu); 0.00924015s (thread); 0s (gc)
│ │ │ │ + -- used 0.0104301s (cpu); 0.0104054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2
│ │ │ │  o3 = - b  + 4a*c
│ │ │ │  
│ │ │ │  o3 : ZZ[a..c]
│ │ │ │  i4 : ZZ[a,b,c,d][x,y]; F = a*x^3+b*x^2*y+c*x*y^2+d*y^3
│ │ │ │  
│ │ │ │          3      2         2      3
│ │ │ │  o5 = a*x  + b*x y + c*x*y  + d*y
│ │ │ │  
│ │ │ │  o5 : ZZ[a..d][x..y]
│ │ │ │  i6 : time discriminant F
│ │ │ │ - -- used 0.0098778s (cpu); 0.0098795s (thread); 0s (gc)
│ │ │ │ + -- used 0.0108438s (cpu); 0.0108453s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2       3     3                   2 2
│ │ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │ │  
│ │ │ │  o6 : ZZ[a..d]
│ │ │ │  The next example illustrates how computing the intersection of a pencil
│ │ │ │  generated by two degree $d$ forms $F(x_0,\ldots,x_n), G(x_0,\ldots,x_n)$ with
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  
│ │ │ │                  4        3      4             4        3      4
│ │ │ │  o12 = (t  + t )x  - t x x  + t x  + (t  - t )x  + t x x  + t x
│ │ │ │          0    1  0    1 0 1    0 1     0    1  2    1 2 3    0 3
│ │ │ │  
│ │ │ │  o12 : R'
│ │ │ │  i13 : time D=discriminant pencil
│ │ │ │ - -- used 0.44332s (cpu); 0.443247s (thread); 0s (gc)
│ │ │ │ + -- used 0.371582s (cpu); 0.371418s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             108      106 2       102 6      100 8       98 10       96 12
│ │ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │ │             0        0   1       0   1      0   1       0  1        0  1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            94 14       92 16      90 18      88 20      86 22       84 24
│ │ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html
│ │ │ @@ -95,28 +95,28 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time V' = dualVariety V
│ │ │ - -- used 0.202374s (cpu); 0.134305s (thread); 0s (gc)
│ │ │ + -- used 0.178788s (cpu); 0.111587s (thread); 0s (gc)
│ │ │  
│ │ │              2                 2    2
│ │ │  o2 = ideal(x x  - x x x  + x x  + x x  - 4x x x )
│ │ │              2 3    1 2 4    0 4    1 5     0 3 5
│ │ │  
│ │ │  o2 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time V == dualVariety V'
│ │ │ - -- used 0.310587s (cpu); 0.159166s (thread); 0s (gc)
│ │ │ + -- used 0.271897s (cpu); 0.148364s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │

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

    │ │ │ │ │ │ @@ -136,25 +136,25 @@ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ 3331 0 9 0 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -31,24 +31,24 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ 0 3 │ │ │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ │ 0 5 │ │ │ │ i2 : time V' = dualVariety V │ │ │ │ - -- used 0.202374s (cpu); 0.134305s (thread); 0s (gc) │ │ │ │ + -- used 0.178788s (cpu); 0.111587s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 │ │ │ │ o2 = ideal(x x - x x x + x x + x x - 4x x x ) │ │ │ │ 2 3 1 2 4 0 4 1 5 0 3 5 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ │ 0 5 │ │ │ │ i3 : time V == dualVariety V' │ │ │ │ - -- used 0.310587s (cpu); 0.159166s (thread); 0s (gc) │ │ │ │ + -- used 0.271897s (cpu); 0.148364s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ In the next example, we verify that the discriminant of a generic ternary cubic │ │ │ │ form coincides with the dual variety of the 3-th Veronese embedding of the │ │ │ │ plane, which is a hypersurface of degree 12 in $\mathbb{P}^9$ │ │ │ │ i4 : F = first genericPolynomials({3,-1,-1},ZZ/3331) │ │ │ │ │ │ │ │ @@ -60,21 +60,21 @@ │ │ │ │ a x x + a x │ │ │ │ 8 1 2 9 2 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ │ 3331 0 9 0 2 │ │ │ │ i5 : time discF = ideal discriminant F; │ │ │ │ - -- used 0.162855s (cpu); 0.0858796s (thread); 0s (gc) │ │ │ │ + -- used 0.137864s (cpu); 0.0738357s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ │ 3331 0 9 │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ │ - -- used 0.745903s (cpu); 0.673493s (thread); 0s (gc) │ │ │ │ + -- used 0.60157s (cpu); 0.53061s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o6 : Ideal of ----[x ..x ] │ │ │ │ 3331 0 9 │ │ │ │ i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z) │ │ │ │ │ │ │ │ o7 = true │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ @@ -90,15 +90,15 @@ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.162855s (cpu); 0.0858796s (thread); 0s (gc)
│ │ │ + -- used 0.137864s (cpu); 0.0738357s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.745903s (cpu); 0.673493s (thread); 0s (gc)
│ │ │ + -- used 0.60157s (cpu); 0.53061s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.141834s (cpu); 0.0752672s (thread); 0s (gc)
│ │ │ + -- used 0.119929s (cpu); 0.0565095s (thread); 0s (gc)
│ │ │  
│ │ │          3   3          2   2              2       2          2   3    
│ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        2       2               2   2                                   
│ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ @@ -143,15 +143,15 @@
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
    │ │ │
    │ │ │ 
    i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.177092s (cpu); 0.069066s (thread); 0s (gc)
│ │ │ + -- used 0.219055s (cpu); 0.0776059s (thread); 0s (gc)
│ │ │  
│ │ │              3          2                         2                        
│ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │             2      3      2
│ │ │       2x   x    + x    + x
│ │ │ @@ -184,15 +184,15 @@
│ │ │
    		│ │ │ 
    i5 : w = chowForm C;
    │ │ │
    │ │ │ 
    i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.108209s (cpu); 0.0419396s (thread); 0s (gc)
│ │ │ + -- used 0.101117s (cpu); 0.033817s (thread); 0s (gc)
│ │ │  
│ │ │                     3          2          3                       2        
│ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2      2            2   3               2        
│ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ @@ -232,15 +232,15 @@
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.115818s (cpu); 0.0462516s (thread); 0s (gc)
│ │ │ + -- used 0.111088s (cpu); 0.0420129s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {2, 2, 2, 2, 2, 2}
│ │ │  
│ │ │  o7 : List
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -26,15 +26,15 @@ │ │ │ │ 2 2 │ │ │ │ o1 = ideal (x - x x , x x - x x , x - x x ) │ │ │ │ 2 1 3 1 2 0 3 1 0 2 │ │ │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ │ 0 3 │ │ │ │ i2 : time fromPluckerToStiefel dualize chowForm C │ │ │ │ - -- used 0.141834s (cpu); 0.0752672s (thread); 0s (gc) │ │ │ │ + -- used 0.119929s (cpu); 0.0565095s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 3 2 2 2 2 2 3 │ │ │ │ o2 = - x x + x x x x - x x x x + x x x - │ │ │ │ 0,3 1,0 0,2 0,3 1,0 1,1 0,1 0,3 1,0 1,1 0,0 0,3 1,1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 2 │ │ │ │ x x x x + 2x x x x + x x x x x x - │ │ │ │ @@ -75,15 +75,15 @@ │ │ │ │ 2 2 2 2 2 2 3 3 │ │ │ │ x x x x - 2x x x x - x x x x + x x │ │ │ │ 0,0 0,1 1,1 1,3 0,0 0,2 1,1 1,3 0,0 0,1 1,2 1,3 0,0 1,3 │ │ │ │ │ │ │ │ o2 : QQ[x ..x ] │ │ │ │ 0,0 1,3 │ │ │ │ i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1}) │ │ │ │ - -- used 0.177092s (cpu); 0.069066s (thread); 0s (gc) │ │ │ │ + -- used 0.219055s (cpu); 0.0776059s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 2 2 │ │ │ │ o3 = - x x + x x x - x x x + x x + 3x x x - │ │ │ │ 0,3 1,2 0,2 1,2 1,3 0,2 0,3 1,2 0,2 1,3 0,3 1,2 1,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 │ │ │ │ 2x x + x + x │ │ │ │ @@ -105,15 +105,15 @@ │ │ │ │ o4 : QQ[a ..a ] │ │ │ │ 0,0 1,1 │ │ │ │ As another application, we check that the singular locus of the Chow form of │ │ │ │ the twisted cubic has dimension 2 (on each standard chart). │ │ │ │ i5 : w = chowForm C; │ │ │ │ i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel │ │ │ │ (w,AffineChartGrass=>s)) │ │ │ │ - -- used 0.108209s (cpu); 0.0419396s (thread); 0s (gc) │ │ │ │ + -- used 0.101117s (cpu); 0.033817s (thread); 0s (gc) │ │ │ │ │ │ │ │ 3 2 3 2 │ │ │ │ o6 = {ideal(- x x + x x x - x - 3x x x + 2x x + │ │ │ │ 0,3 1,2 0,2 1,2 1,3 0,2 0,2 0,3 1,2 0,2 1,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 3 2 │ │ │ │ x x x - x x + x ), ideal(x x - 2x x x x + │ │ │ │ @@ -149,15 +149,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 3 2 │ │ │ │ 2x x - x + x )} │ │ │ │ 0,0 1,1 1,1 1,0 │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : time apply(U,u->dim singularLocus u) │ │ │ │ - -- used 0.115818s (cpu); 0.0462516s (thread); 0s (gc) │ │ │ │ + -- used 0.111088s (cpu); 0.0420129s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {2, 2, 2, 2, 2, 2} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** WWaayyss ttoo uussee ffrroommPPlluucckkeerrTTooSSttiieeffeell:: ********** │ │ │ │ * fromPluckerToStiefel(Ideal) │ │ │ │ * fromPluckerToStiefel(Matrix) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ o1 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time hurwitzForm Q
│ │ │ - -- used 0.125887s (cpu); 0.0563778s (thread); 0s (gc)
│ │ │ + -- used 0.112866s (cpu); 0.0507715s (thread); 0s (gc)
│ │ │  
│ │ │                2                                 2                      
│ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │                0,1            0,1 0,2            0,2           0,1 1,2  
│ │ │       ------------------------------------------------------------------------
│ │ │                                 2                                          
│ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │         5 2   7                       2        3 2
│ │ │ │       + -p  + -p p  + 7p p  + 6p p  + -p p  + --p )
│ │ │ │         4 3   9 0 4     1 4     2 4   9 3 4   10 4
│ │ │ │  
│ │ │ │  o1 : Ideal of QQ[p ..p ]
│ │ │ │                    0   4
│ │ │ │  i2 : time hurwitzForm Q
│ │ │ │ - -- used 0.125887s (cpu); 0.0563778s (thread); 0s (gc)
│ │ │ │ + -- used 0.112866s (cpu); 0.0507715s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2                                 2
│ │ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │ │                0,1            0,1 0,2            0,2           0,1 1,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                                 2
│ │ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time isCoisotropic w
│ │ │ - -- used 0.00877231s (cpu); 0.00877035s (thread); 0s (gc)
│ │ │ + -- used 0.00786174s (cpu); 0.00786149s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- random quadric in G(1,3)
│ │ │ @@ -145,15 +145,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time isCoisotropic w'
│ │ │ - -- used 0.00769242s (cpu); 0.00769265s (thread); 0s (gc)
│ │ │ + -- used 0.00682201s (cpu); 0.00682168s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = false
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -44,15 +44,15 @@ │ │ │ │ │ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ │ o1 : -------------------------------------- │ │ │ │ p p - p p + p p │ │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ i2 : time isCoisotropic w │ │ │ │ - -- used 0.00877231s (cpu); 0.00877035s (thread); 0s (gc) │ │ │ │ + -- used 0.00786174s (cpu); 0.00786149s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = true │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ │ │ 3 2 3 7 2 5 7 2 10 │ │ │ │ o3 = --p + -p p + -p + 5p p + -p p + -p + --p p │ │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ │ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ │ o3 : -------------------------------------- │ │ │ │ p p - p p + p p │ │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ i4 : time isCoisotropic w' │ │ │ │ - -- used 0.00769242s (cpu); 0.00769265s (thread); 0s (gc) │ │ │ │ + -- used 0.00682201s (cpu); 0.00682168s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = false │ │ │ │ ********** WWaayyss ttoo uussee iissCCooiissoottrrooppiicc:: ********** │ │ │ │ * isCoisotropic(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_i_s_o_t_r_o_p_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html │ │ │ @@ -122,15 +122,15 @@ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ 33331 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0181849s (cpu); 0.0181858s (thread); 0s (gc)
│ │ │ + -- used 0.0171715s (cpu); 0.0171706s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2380x + 9482x ) │ │ │ │ 4 5 │ │ │ │ │ │ │ │ ZZ │ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ │ 33331 0 5 │ │ │ │ i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I)) │ │ │ │ - -- used 0.0181849s (cpu); 0.0181858s (thread); 0s (gc) │ │ │ │ + -- used 0.0171715s (cpu); 0.0171706s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _p_l_u_c_k_e_r -- get the Plücker coordinates of a linear subspace │ │ │ │ ********** WWaayyss ttoo uussee iissIInnCCooiissoottrrooppiicc:: ********** │ │ │ │ * isInCoisotropic(Ideal,Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ @@ -92,15 +92,15 @@ │ │ │ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00373056s (cpu); 0.00372864s (thread); 0s (gc)
│ │ │ + -- used 0.00343268s (cpu); 0.00343168s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
│ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
│ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
│ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
│ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
│ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
│ │ │ @@ -163,15 +163,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00252047s (cpu); 0.00251818s (thread); 0s (gc)
│ │ │ + -- used 0.00224073s (cpu); 0.00223913s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0   
│ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0   
│ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0   
│ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0   
│ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0   
│ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                     2          2        2      3
│ │ │ │       c x x x  + c x x  + c x x  + c x x  + c x }
│ │ │ │        4 0 1 2    7 1 2    5 0 2    8 1 2    9 2
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  i2 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00373056s (cpu); 0.00372864s (thread); 0s (gc)
│ │ │ │ + -- used 0.00343268s (cpu); 0.00343168s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0
│ │ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0
│ │ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0
│ │ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0
│ │ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0
│ │ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       10   2   7   2   5 3
│ │ │ │       --p p  + -p p  + -p }
│ │ │ │        9 0 2   8 1 2   6 2
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : time (D,D') = macaulayFormula F
│ │ │ │ - -- used 0.00252047s (cpu); 0.00251818s (thread); 0s (gc)
│ │ │ │ + -- used 0.00224073s (cpu); 0.00223913s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0
│ │ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0
│ │ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0
│ │ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0
│ │ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0
│ │ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : Ideal of P4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time p = plucker L
│ │ │ - -- used 0.00473681s (cpu); 0.00473472s (thread); 0s (gc)
│ │ │ + -- used 0.00461257s (cpu); 0.00461104s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o4 : Ideal of G'1'4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time L' = plucker p
│ │ │ - -- used 0.121088s (cpu); 0.0718194s (thread); 0s (gc)
│ │ │ + -- used 0.105445s (cpu); 0.043898s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o7 : Ideal of G'1'4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.148823s (cpu); 0.0922017s (thread); 0s (gc)
│ │ │ + -- used 0.0991924s (cpu); 0.0464414s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : (codim W,degree W)
│ │ │ @@ -163,15 +163,15 @@
│ │ │            
│ │ │          
│ │ │          
    In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ - -- used 0.244058s (cpu); 0.244015s (thread); 0s (gc)
│ │ │ + -- used 0.142696s (cpu); 0.142698s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of G'1'4
    
│ │ │              
    
│ │ │                
    i11 : assert(Y' == Y)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,28 +28,28 @@
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │  o3 : Ideal of P4
│ │ │ │  i4 : time p = plucker L
│ │ │ │ - -- used 0.00473681s (cpu); 0.00473472s (thread); 0s (gc)
│ │ │ │ + -- used 0.00461257s (cpu); 0.00461104s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       11804x   , x    + 14854x   )
│ │ │ │             3,4   0,1         3,4
│ │ │ │  
│ │ │ │  o4 : Ideal of G'1'4
│ │ │ │  i5 : time L' = plucker p
│ │ │ │ - -- used 0.121088s (cpu); 0.0718194s (thread); 0s (gc)
│ │ │ │ + -- used 0.105445s (cpu); 0.043898s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │ │               2        3         4   1        3         4   0         3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       664x )
│ │ │ │           4
│ │ │ │  
│ │ │ │ @@ -60,26 +60,26 @@
│ │ │ │  $W\subset\mathbb{P}^n$ swept out by the linear spaces corresponding to points
│ │ │ │  of $Y$. As an example, we now compute a surface scroll $W\subset\mathbb{P}^4$
│ │ │ │  over an elliptic curve $Y\subset\mathbb{G}(1,\mathbb{P}^4)$.
│ │ │ │  i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve
│ │ │ │  
│ │ │ │  o7 : Ideal of G'1'4
│ │ │ │  i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ │ - -- used 0.148823s (cpu); 0.0922017s (thread); 0s (gc)
│ │ │ │ + -- used 0.0991924s (cpu); 0.0464414s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of P4
│ │ │ │  i9 : (codim W,degree W)
│ │ │ │  
│ │ │ │  o9 = (2, 5)
│ │ │ │  
│ │ │ │  o9 : Sequence
│ │ │ │  In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb
│ │ │ │  {P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
│ │ │ │  i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ │ - -- used 0.244058s (cpu); 0.244015s (thread); 0s (gc)
│ │ │ │ + -- used 0.142696s (cpu); 0.142698s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of G'1'4
│ │ │ │  i11 : assert(Y' == Y)
│ │ │ │  WWaarrnniinngg: Notice that, by default, the computation is done on a randomly chosen
│ │ │ │  affine chart on the Grassmannian. To change this behavior, you can use the
│ │ │ │  _A_f_f_i_n_e_C_h_a_r_t_G_r_a_s_s option.
│ │ │ │  ********** WWaayyss ttoo uussee pplluucckkeerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o2 : List
│ │ │              
    
│ │ │                
    i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.562085s (cpu); 0.289584s (thread); 0s (gc)
│ │ │ + -- used 0.489603s (cpu); 0.213305s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o3 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.273183s (cpu); 0.131953s (thread); 0s (gc)
│ │ │ + -- used 0.300703s (cpu); 0.146374s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -161,15 +161,15 @@
│ │ │  
│ │ │  o4 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.302008s (cpu); 0.301983s (thread); 0s (gc)
│ │ │ + -- used 0.301989s (cpu); 0.301989s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o5 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.625765s (cpu); 0.559323s (thread); 0s (gc)
│ │ │ + -- used 0.638822s (cpu); 0.560047s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ ├── html2text {}
│ │ │ │ @@ -58,15 +58,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       3     2    9    7     2    9        3       1    8    4
│ │ │ │       -b)y*w  + (-a + -b)z*w  + (-a + 2b)w , 2x + -y + -z + -w}
│ │ │ │       4          8    8          7                4    3    5
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ │ - -- used 0.562085s (cpu); 0.289584s (thread); 0s (gc)
│ │ │ │ + -- used 0.489603s (cpu); 0.213305s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -78,15 +78,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o3 : QQ[a..b]
│ │ │ │  i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ │ - -- used 0.273183s (cpu); 0.131953s (thread); 0s (gc)
│ │ │ │ + -- used 0.300703s (cpu); 0.146374s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o4 : QQ[a..b]
│ │ │ │  i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ │ - -- used 0.302008s (cpu); 0.301983s (thread); 0s (gc)
│ │ │ │ + -- used 0.301989s (cpu); 0.301989s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ │ │ @@ -118,15 +118,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1146977327343523453866040839029   4   194441910898734675845094443 5
│ │ │ │       -------------------------------a*b  - ---------------------------b
│ │ │ │            1119954511872000000000                895963609497600000
│ │ │ │  
│ │ │ │  o5 : QQ[a..b]
│ │ │ │  i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ │ - -- used 0.625765s (cpu); 0.559323s (thread); 0s (gc)
│ │ │ │ + -- used 0.638822s (cpu); 0.560047s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time resultant F
│ │ │ - -- used 0.123723s (cpu); 0.0492639s (thread); 0s (gc)
│ │ │ + -- used 0.100779s (cpu); 0.0373295s (thread); 0s (gc)
│ │ │  
│ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
│ │ │       ------------------------------------------------------------------------
│ │ │                6 7          5 8       11          10 2         9 3  
│ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time resultant F
│ │ │ - -- used 3.07394s (cpu); 2.19358s (thread); 0s (gc)
│ │ │ + -- used 2.28559s (cpu); 1.74838s (thread); 0s (gc)
│ │ │  
│ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
│ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
│ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ @@ -1795,15 +1795,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time # terms resultant F
│ │ │ - -- used 0.644433s (cpu); 0.456975s (thread); 0s (gc)
│ │ │ + -- used 0.483304s (cpu); 0.355059s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = 21894
    
│ │ │              
    
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -32,15 +32,15 @@ │ │ │ │ i2 : F = {x^2+3*t*y*z-u*z^2,(t+3*u-1)*x-y,-t*x*y^3+t*x^2*y*z+u*z^4} │ │ │ │ │ │ │ │ 2 2 3 2 4 │ │ │ │ o2 = {x + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y + t*x y*z + u*z } │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : time resultant F │ │ │ │ - -- used 0.123723s (cpu); 0.0492639s (thread); 0s (gc) │ │ │ │ + -- used 0.100779s (cpu); 0.0373295s (thread); 0s (gc) │ │ │ │ │ │ │ │ 12 11 2 10 3 9 4 8 5 7 6 │ │ │ │ o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 6 7 5 8 11 10 2 9 3 │ │ │ │ - 413343t u - 177147t u + 567t u + 10206t u + 76545t u + │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 │ │ │ │ + c x } │ │ │ │ 9 2 │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : time resultant F │ │ │ │ - -- used 3.07394s (cpu); 2.19358s (thread); 0s (gc) │ │ │ │ + -- used 2.28559s (cpu); 1.74838s (thread); 0s (gc) │ │ │ │ │ │ │ │ 6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 2 2 │ │ │ │ o5 = a b c - 3a a b b c + 3a a b b c - a a b c + 3a a b b c - │ │ │ │ 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 5 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 3 2 4 2 2 2 4 2 2 2 5 2 2 6 3 2 │ │ │ │ 6a a b b b c + 3a a b b c + 3a a b b c - 3a a b b c + a b c - │ │ │ │ @@ -1712,15 +1712,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 2 2 2 │ │ │ │ b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x } │ │ │ │ 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : time # terms resultant F │ │ │ │ - -- used 0.644433s (cpu); 0.456975s (thread); 0s (gc) │ │ │ │ + -- used 0.483304s (cpu); 0.355059s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 21894 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_h_o_w_F_o_r_m -- Chow form of a projective variety │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_R_i_n_g_E_l_e_m_e_n_t_) │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * resultant(List) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ 0 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.0398049s (cpu); 0.039807s (thread); 0s (gc)
│ │ │ + -- used 0.0383661s (cpu); 0.0383665s (thread); 0s (gc)
│ │ │  
│ │ │        2                                                       2        
│ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │        2
│ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ @@ -123,15 +123,15 @@
│ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │       time tangentialChowForm(S,1)
│ │ │ - -- used 0.16271s (cpu); 0.0952426s (thread); 0s (gc)
│ │ │ + -- used 0.136019s (cpu); 0.0794229s (thread); 0s (gc)
│ │ │  
│ │ │        2     2        2     2               3        2     2      
│ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │               3         3               3            
│ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ @@ -168,43 +168,43 @@
│ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S)
│ │ │       time tangentialChowForm(S,2)
│ │ │ - -- used 0.117742s (cpu); 0.0528496s (thread); 0s (gc)
│ │ │ + -- used 0.103017s (cpu); 0.0501147s (thread); 0s (gc)
│ │ │  
│ │ │                2                                             2
│ │ │  o5 = p       p        - p       p       p        + p       p
│ │ │        0,1,3,4 0,2,3,4    0,1,2,4 0,2,3,4 1,2,3,4    0,1,2,3 1,2,3,4
│ │ │  
│ │ │  o5 : QQ[p       ..p       , p       , p       , p       ]
│ │ │           0,1,2,3   0,1,2,4   0,1,3,4   0,2,3,4   1,2,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing
│ │ │       time S' = ideal dualize tangentialChowForm(S,2)
│ │ │ - -- used 0.130131s (cpu); 0.0621371s (thread); 0s (gc)
│ │ │ + -- used 0.112314s (cpu); 0.0555025s (thread); 0s (gc)
│ │ │  
│ │ │              2               2
│ │ │  o6 = ideal(p p  - p p p  + p p )
│ │ │              1 2    0 1 3    0 4
│ │ │  
│ │ │  o6 : Ideal of QQ[p ..p ]
│ │ │                    0   4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : -- we then can recover S
│ │ │       time assert(dualize tangentialChowForm(S',3) == S)
│ │ │ - -- used 0.190994s (cpu); 0.117998s (thread); 0s (gc)
    │ │ │ + -- used 0.164298s (cpu); 0.106233s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -63,15 +63,15 @@ │ │ │ │ o2 = ideal (- p p + p p , - p p + p p , - p + p p ) │ │ │ │ 1 2 0 3 1 3 0 4 3 2 4 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[p ..p ] │ │ │ │ 0 4 │ │ │ │ i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form) │ │ │ │ time tangentialChowForm(S,0) │ │ │ │ - -- used 0.0398049s (cpu); 0.039807s (thread); 0s (gc) │ │ │ │ + -- used 0.0383661s (cpu); 0.0383665s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o3 = p p - p p p - p p p + p p p + p p + │ │ │ │ 1,3 2,3 1,2 1,3 2,4 0,3 1,3 2,4 0,2 1,4 2,4 1,2 3,4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 │ │ │ │ p p - 2p p p - p p p │ │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ - p p + p p , p p - p p + p p , p p - p p + p │ │ │ │ p ) │ │ │ │ 2,3 1,4 1,3 2,4 1,2 3,4 2,3 0,4 0,3 2,4 0,2 3,4 1,3 0,4 │ │ │ │ 0,3 1,4 0,1 3,4 1,2 0,4 0,2 1,4 0,1 2,4 1,2 0,3 0,2 1,3 0,1 │ │ │ │ 2,3 │ │ │ │ i4 : -- 1-th associated hypersurface of S in G(2,4) │ │ │ │ time tangentialChowForm(S,1) │ │ │ │ - -- used 0.16271s (cpu); 0.0952426s (thread); 0s (gc) │ │ │ │ + -- used 0.136019s (cpu); 0.0794229s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 3 2 2 │ │ │ │ o4 = p p + p p - 2p p + p p - │ │ │ │ 1,2,3 1,2,4 0,2,4 1,2,4 0,2,3 1,2,4 0,2,4 0,3,4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 3 3 │ │ │ │ 4p p - 4p p - 2p p + │ │ │ │ @@ -138,35 +138,35 @@ │ │ │ │ p + p p , p p - p p + p p ) │ │ │ │ 1,2,4 0,3,4 0,2,4 1,3,4 0,1,4 2,3,4 1,2,3 0,3,4 0,2,3 1,3,4 │ │ │ │ 0,1,3 2,3,4 1,2,3 0,2,4 0,2,3 1,2,4 0,1,2 2,3,4 1,2,3 0,1,4 0,1,3 │ │ │ │ 1,2,4 0,1,2 1,3,4 0,2,3 0,1,4 0,1,3 0,2,4 0,1,2 0,3,4 │ │ │ │ i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent │ │ │ │ hyperplanes to S) │ │ │ │ time tangentialChowForm(S,2) │ │ │ │ - -- used 0.117742s (cpu); 0.0528496s (thread); 0s (gc) │ │ │ │ + -- used 0.103017s (cpu); 0.0501147s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o5 = p p - p p p + p p │ │ │ │ 0,1,3,4 0,2,3,4 0,1,2,4 0,2,3,4 1,2,3,4 0,1,2,3 1,2,3,4 │ │ │ │ │ │ │ │ o5 : QQ[p ..p , p , p , p ] │ │ │ │ 0,1,2,3 0,1,2,4 0,1,3,4 0,2,3,4 1,2,3,4 │ │ │ │ i6 : -- we get the dual hypersurface of S in G(0,4) by dualizing │ │ │ │ time S' = ideal dualize tangentialChowForm(S,2) │ │ │ │ - -- used 0.130131s (cpu); 0.0621371s (thread); 0s (gc) │ │ │ │ + -- used 0.112314s (cpu); 0.0555025s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o6 = ideal(p p - p p p + p p ) │ │ │ │ 1 2 0 1 3 0 4 │ │ │ │ │ │ │ │ o6 : Ideal of QQ[p ..p ] │ │ │ │ 0 4 │ │ │ │ i7 : -- we then can recover S │ │ │ │ time assert(dualize tangentialChowForm(S',3) == S) │ │ │ │ - -- used 0.190994s (cpu); 0.117998s (thread); 0s (gc) │ │ │ │ + -- used 0.164298s (cpu); 0.106233s (thread); 0s (gc) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _i_s_C_o_i_s_o_t_r_o_p_i_c -- whether a hypersurface of a Grassmannian is a tangential │ │ │ │ Chow form │ │ │ │ * _c_h_o_w_F_o_r_m -- Chow form of a projective variety │ │ │ │ ********** WWaayyss ttoo uussee ttaannggeennttiiaallCChhoowwFFoorrmm:: ********** │ │ │ │ * tangentialChowForm(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ aXNFeHRlcm5hbE0yUGFyZW50 │ │ │ #:len=1043 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW5kaWNhdGUgaWYgdGhpcyBwcm9jZXNz │ │ │ IGlzIGEgcGFyZW50IHByb2Nlc3Mgb3Igbm90IiwgImxpbmVudW0iID0+IDg5NCwgImZpbGVuYW1l │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out │ │ │ @@ -4,15 +4,15 @@ │ │ │ time(seconds) 700 │ │ │ file(blocks) unlimited │ │ │ data(kbytes) unlimited │ │ │ stack(kbytes) 8192 │ │ │ coredump(blocks) unlimited │ │ │ memory(kbytes) 850000 │ │ │ locked memory(kbytes) 8192 │ │ │ -process 63815 │ │ │ +process 63520 │ │ │ nofiles 512 │ │ │ vmemory(kbytes) unlimited │ │ │ locks unlimited │ │ │ rtprio 0 │ │ │ │ │ │ o1 = 0 │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ @@ -1,129 +1,129 @@ │ │ │ -- -*- M2-comint -*- hash: 2927978066455787395 │ │ │ │ │ │ i1 : fn=temporaryFileName()|".m2" │ │ │ │ │ │ -o1 = /tmp/M2-24375-0/0.m2 │ │ │ +o1 = /tmp/M2-31226-0/0.m2 │ │ │ │ │ │ i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-24375-0/1.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31226-0/1.m2" >"/tmp/M2-31226-0/1.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i7 : h │ │ │ │ │ │ o7 = HashTable{"answer file" => null} │ │ │ "exit code" => 0 │ │ │ "output file" => null │ │ │ "return code" => 0 │ │ │ "statistics" => null │ │ │ - "time used" => 5 │ │ │ + "time used" => 3 │ │ │ value => 16 │ │ │ │ │ │ o7 : HashTable │ │ │ │ │ │ i8 : h#value===4^2 │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : h#"exit code"===0 │ │ │ │ │ │ o9 = true │ │ │ │ │ │ i10 : h=runExternalM2(fn,"justexit",()); │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-24375-0/2.m2" >"/tmp/M2-24375-0/2.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31226-0/2.m2" >"/tmp/M2-31226-0/2.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i11 : h │ │ │ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24375-0/2.ans} │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-31226-0/2.ans} │ │ │ "exit code" => 27 │ │ │ - "output file" => /tmp/M2-24375-0/2.out │ │ │ + "output file" => /tmp/M2-31226-0/2.out │ │ │ "return code" => 6912 │ │ │ "statistics" => null │ │ │ - "time used" => 5 │ │ │ + "time used" => 3 │ │ │ value => null │ │ │ │ │ │ o11 : HashTable │ │ │ │ │ │ i12 : fileExists(h#"output file") │ │ │ │ │ │ o12 = true │ │ │ │ │ │ i13 : fileExists(h#"answer file") │ │ │ │ │ │ o13 = false │ │ │ │ │ │ i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2"); │ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-24375-0/3.m2" >"/tmp/M2-24375-0/3.out" 2>&1 )) │ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31226-0/3.m2" >"/tmp/M2-31226-0/3.out" 2>&1 )) │ │ │ Killed │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i15 : h │ │ │ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24375-0/3.ans} │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-31226-0/3.ans} │ │ │ "exit code" => 0 │ │ │ - "output file" => /tmp/M2-24375-0/3.out │ │ │ + "output file" => /tmp/M2-31226-0/3.out │ │ │ "return code" => 9 │ │ │ "statistics" => null │ │ │ - "time used" => 2 │ │ │ + "time used" => 1 │ │ │ value => null │ │ │ │ │ │ o15 : HashTable │ │ │ │ │ │ i16 : if h#"output file" =!= null and fileExists(h#"output file") then get(h#"output file") │ │ │ │ │ │ o16 = │ │ │ │ │ │ i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get(h#"answer file") │ │ │ │ │ │ i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true); │ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1') >"/tmp/M2-24375-0/4.stat" 2>&1 )) │ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31226-0/4.m2" >"/tmp/M2-31226-0/4.out" 2>&1') >"/tmp/M2-31226-0/4.stat" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i19 : h#"statistics" │ │ │ │ │ │ -o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1" │ │ │ - User time (seconds): 7.49 │ │ │ - System time (seconds): 0.10 │ │ │ - Percent of CPU this job got: 93% │ │ │ - Elapsed (wall clock) time (h:mm:ss or m:ss): 0:08.15 │ │ │ +o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31226-0/4.m2" >"/tmp/M2-31226-0/4.out" 2>&1" │ │ │ + User time (seconds): 7.07 │ │ │ + System time (seconds): 0.30 │ │ │ + Percent of CPU this job got: 119% │ │ │ + Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.17 │ │ │ Average shared text size (kbytes): 0 │ │ │ Average unshared data size (kbytes): 0 │ │ │ Average stack size (kbytes): 0 │ │ │ Average total size (kbytes): 0 │ │ │ - Maximum resident set size (kbytes): 255852 │ │ │ + Maximum resident set size (kbytes): 339212 │ │ │ Average resident set size (kbytes): 0 │ │ │ Major (requiring I/O) page faults: 0 │ │ │ - Minor (reclaiming a frame) page faults: 10158 │ │ │ - Voluntary context switches: 2973 │ │ │ - Involuntary context switches: 2382 │ │ │ + Minor (reclaiming a frame) page faults: 13081 │ │ │ + Voluntary context switches: 18679 │ │ │ + Involuntary context switches: 3144 │ │ │ Swaps: 0 │ │ │ File system inputs: 0 │ │ │ - File system outputs: 0 │ │ │ + File system outputs: 16 │ │ │ Socket messages sent: 0 │ │ │ Socket messages received: 0 │ │ │ Signals delivered: 0 │ │ │ Page size (bytes): 4096 │ │ │ Exit status: 0 │ │ │ │ │ │ │ │ │ i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///; │ │ │ │ │ │ i21 : (runExternalM2(fn,identity,v))#value===v │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-24375-0/6.m2" >"/tmp/M2-24375-0/6.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31226-0/6.m2" >"/tmp/M2-31226-0/6.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ o21 = true │ │ │ │ │ │ i22 : R=QQ[x,y]; │ │ │ │ │ │ i23 : v=coker random(R^2,R^{3:-1}) │ │ │ @@ -131,54 +131,54 @@ │ │ │ o23 = cokernel | 9/2x+9/4y 7/9x+7/10y 7x+3/7y | │ │ │ | 3/4x+7/4y 7/10x+7/3y 6/7x+6y | │ │ │ │ │ │ 2 │ │ │ o23 : R-module, quotient of R │ │ │ │ │ │ i24 : h=runExternalM2(fn,identity,v) │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-24375-0/7.m2" >"/tmp/M2-24375-0/7.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31226-0/7.m2" >"/tmp/M2-31226-0/7.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24375-0/7.ans} │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-31226-0/7.ans} │ │ │ "exit code" => 1 │ │ │ - "output file" => /tmp/M2-24375-0/7.out │ │ │ + "output file" => /tmp/M2-31226-0/7.out │ │ │ "return code" => 256 │ │ │ "statistics" => null │ │ │ - "time used" => 4 │ │ │ + "time used" => 3 │ │ │ value => null │ │ │ │ │ │ o24 : HashTable │ │ │ │ │ │ i25 : get(h#"output file") │ │ │ │ │ │ o25 = │ │ │ - i1 : -- Script /tmp/M2-24375-0/7.m2 automatically generated by RunExternalM2 │ │ │ + i1 : -- Script /tmp/M2-31226-0/7.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-24375-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-31226-0/0.m2"; │ │ │ │ │ │ - i3 : runExternalM2ReturnAnswer("/tmp/M2-24375-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}})))); │ │ │ + i3 : runExternalM2ReturnAnswer("/tmp/M2-31226-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}})))); │ │ │ stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to objects: │ │ │ R (of class Symbol) │ │ │ ^ 2 (of class ZZ) │ │ │ │ │ │ │ │ │ i26 : fn<"/tmp/M2-24375-0/8.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31226-0/8.m2" >"/tmp/M2-31226-0/8.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ o27 = true │ │ │ │ │ │ i28 : v=R; │ │ │ │ │ │ i29 : h=runExternalM2(fn,identity,v); │ │ │ -Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-24375-0/9.m2" >"/tmp/M2-24375-0/9.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31226-0/9.m2" >"/tmp/M2-31226-0/9.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i30 : h#value │ │ │ │ │ │ o30 = QQ[x..y] │ │ │ │ │ │ o30 : PolynomialRing │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ @@ -80,15 +80,15 @@ │ │ │ time(seconds) 700 │ │ │ file(blocks) unlimited │ │ │ data(kbytes) unlimited │ │ │ stack(kbytes) 8192 │ │ │ coredump(blocks) unlimited │ │ │ memory(kbytes) 850000 │ │ │ locked memory(kbytes) 8192 │ │ │ -process 63815 │ │ │ +process 63520 │ │ │ nofiles 512 │ │ │ vmemory(kbytes) unlimited │ │ │ locks unlimited │ │ │ rtprio 0 │ │ │ │ │ │ o1 = 0 │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -35,15 +35,15 @@ │ │ │ │ time(seconds) 700 │ │ │ │ file(blocks) unlimited │ │ │ │ data(kbytes) unlimited │ │ │ │ stack(kbytes) 8192 │ │ │ │ coredump(blocks) unlimited │ │ │ │ memory(kbytes) 850000 │ │ │ │ locked memory(kbytes) 8192 │ │ │ │ -process 63815 │ │ │ │ +process 63520 │ │ │ │ nofiles 512 │ │ │ │ vmemory(kbytes) unlimited │ │ │ │ locks unlimited │ │ │ │ rtprio 0 │ │ │ │ │ │ │ │ o1 = 0 │ │ │ │ This starts a new shell and executes the command given, which in this case │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_run__External__M2.html │ │ │ @@ -89,15 +89,15 @@ │ │ │

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

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

    and then call them:

    │ │ │
    │ │ │
    │ │ │ 
    i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-24375-0/0.m2
    │ │ │ +o1 = /tmp/M2-31226-0/0.m2 │ │ │
    │ │ │ 
    i2 : fn<</// square = (x) -> (stderr<<"Running"<<endl; sleep(1); x^2); ///<<endl;
    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : h=runExternalM2(fn,"square",(4));
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/1.m2" >"/tmp/M2-24375-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31226-0/1.m2" >"/tmp/M2-31226-0/1.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │
    │ │ │ 
    i7 : h
│ │ │  
│ │ │  o7 = HashTable{"answer file" => null}
│ │ │                 "exit code" => 0
│ │ │                 "output file" => null
│ │ │                 "return code" => 0
│ │ │                 "statistics" => null
│ │ │ -               "time used" => 5
│ │ │ +               "time used" => 3
│ │ │                 value => 16
│ │ │  
│ │ │  o7 : HashTable
    │ │ │
    │ │ │ @@ -162,29 +162,29 @@ │ │ │

    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i10 : h=runExternalM2(fn,"justexit",());
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/2.m2" >"/tmp/M2-24375-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31226-0/2.m2" >"/tmp/M2-31226-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    │ │ │
    │ │ │ 
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24375-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-31226-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-24375-0/2.out
│ │ │ +                "output file" => /tmp/M2-31226-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 5
│ │ │ +                "time used" => 3
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
    │ │ │
    │ │ │ @@ -204,30 +204,30 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/3.m2" >"/tmp/M2-24375-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31226-0/3.m2" >"/tmp/M2-31226-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    │ │ │
    │ │ │ 
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24375-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-31226-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-24375-0/3.out
│ │ │ +                "output file" => /tmp/M2-31226-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 2
│ │ │ +                "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o15 : HashTable
    │ │ │
    │ │ │ @@ -246,40 +246,40 @@ │ │ │

    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -293,15 +293,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1') >"/tmp/M2-24375-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31226-0/4.m2" >"/tmp/M2-31226-0/4.out" 2>&1') >"/tmp/M2-31226-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    │ │ │
    │ │ │ 
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 7.49
│ │ │ -              System time (seconds): 0.10
│ │ │ -              Percent of CPU this job got: 93%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:08.15
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31226-0/4.m2" >"/tmp/M2-31226-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 7.07
│ │ │ +              System time (seconds): 0.30
│ │ │ +              Percent of CPU this job got: 119%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.17
│ │ │                Average shared text size (kbytes): 0
│ │ │                Average unshared data size (kbytes): 0
│ │ │                Average stack size (kbytes): 0
│ │ │                Average total size (kbytes): 0
│ │ │ -              Maximum resident set size (kbytes): 255852
│ │ │ +              Maximum resident set size (kbytes): 339212
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 10158
│ │ │ -              Voluntary context switches: 2973
│ │ │ -              Involuntary context switches: 2382
│ │ │ +              Minor (reclaiming a frame) page faults: 13081
│ │ │ +              Voluntary context switches: 18679
│ │ │ +              Involuntary context switches: 3144
│ │ │                Swaps: 0
│ │ │                File system inputs: 0
│ │ │ -              File system outputs: 0
│ │ │ +              File system outputs: 16
│ │ │                Socket messages sent: 0
│ │ │                Socket messages received: 0
│ │ │                Signals delivered: 0
│ │ │                Page size (bytes): 4096
│ │ │                Exit status: 0
    │ │ │
    │ │ │ 
    i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
    │ │ │
    │ │ │ 
    i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/6.m2" >"/tmp/M2-24375-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31226-0/6.m2" >"/tmp/M2-31226-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    │ │ │
    │ │ │
    │ │ │ @@ -323,24 +323,24 @@ │ │ │ 2 │ │ │ o23 : R-module, quotient of R │ │ │
    │ │ │ 
    i24 : h=runExternalM2(fn,identity,v)
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/7.m2" >"/tmp/M2-24375-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31226-0/7.m2" >"/tmp/M2-31226-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24375-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-31226-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-24375-0/7.out
│ │ │ +                "output file" => /tmp/M2-31226-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 4
│ │ │ +                "time used" => 3
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    │ │ │
    │ │ │
    │ │ │ @@ -348,20 +348,20 @@ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-24375-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-31226-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-24375-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-31226-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-24375-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-31226-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    │ │ │
    │ │ │
    │ │ │ @@ -372,15 +372,15 @@ │ │ │
    		│ │ │ 
    i26 : fn<<///R=QQ[x,y];///<<endl<<flush;
    │ │ │
    │ │ │ 
    i27 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/8.m2" >"/tmp/M2-24375-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31226-0/8.m2" >"/tmp/M2-31226-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    │ │ │
    │ │ │
    │ │ │ @@ -392,15 +392,15 @@ │ │ │
    		│ │ │ 
    i28 : v=R;
    │ │ │
    │ │ │ 
    i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/9.m2" >"/tmp/M2-24375-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31226-0/9.m2" >"/tmp/M2-31226-0/9.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │
    │ │ │ 
    i30 : h#value
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,193 +45,193 @@
│ │ │ │  the output file (unless it was deleted), the name of the answer file (unless it
│ │ │ │  was deleted), any statistics recorded about the resource usage, and the value
│ │ │ │  returned by the function func. If the child process terminates abnormally, then
│ │ │ │  usually the exit code is nonzero and the value returned is _n_u_l_l.
│ │ │ │  For example, we can write a few functions to a temporary file:
│ │ │ │  i1 : fn=temporaryFileName()|".m2"
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-24375-0/0.m2
│ │ │ │ +o1 = /tmp/M2-31226-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-24375-0/1.out" 2>&1 ))
│ │ │ │ +M2-31226-0/1.m2" >"/tmp/M2-31226-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │                 "statistics" => null
│ │ │ │ -               "time used" => 5
│ │ │ │ +               "time used" => 3
│ │ │ │                 value => 16
│ │ │ │  
│ │ │ │  o7 : HashTable
│ │ │ │  i8 : h#value===4^2
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : h#"exit code"===0
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  An abnormal program exit will have a nonzero exit code; also, the value will be
│ │ │ │  null, the output file should exist, but the answer file may not exist unless
│ │ │ │  the routine finished successfully.
│ │ │ │  i10 : h=runExternalM2(fn,"justexit",());
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24375-0/2.m2" >"/tmp/M2-24375-0/2.out" 2>&1 ))
│ │ │ │ +M2-31226-0/2.m2" >"/tmp/M2-31226-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24375-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-31226-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-24375-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-31226-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 5
│ │ │ │ +                "time used" => 3
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  i12 : fileExists(h#"output file")
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : fileExists(h#"answer file")
│ │ │ │  
│ │ │ │  o13 = false
│ │ │ │  Here, we use _r_e_s_o_u_r_c_e_ _l_i_m_i_t_s to limit the routine to 2 seconds of computational
│ │ │ │  time, while the system is asked to use 10 seconds of computational time:
│ │ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ │  Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -
│ │ │ │ -q  <"/tmp/M2-24375-0/3.m2" >"/tmp/M2-24375-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-31226-0/3.m2" >"/tmp/M2-31226-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24375-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-31226-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-24375-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-31226-0/3.out
│ │ │ │                  "return code" => 9
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 2
│ │ │ │ +                "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o15 : HashTable
│ │ │ │  i16 : if h#"output file" =!= null and fileExists(h#"output file") then get
│ │ │ │  (h#"output file")
│ │ │ │  
│ │ │ │  o16 =
│ │ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get
│ │ │ │  (h#"answer file")
│ │ │ │  We can get quite a lot of detail on the resources used with the _K_e_e_p_S_t_a_t_i_s_t_i_c_s
│ │ │ │  command:
│ │ │ │  i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ │  Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop -
│ │ │ │ --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-24375-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-31226-0/4.m2" >"/tmp/M2-31226-0/4.out" 2>&1')
│ │ │ │ +>"/tmp/M2-31226-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 7.49
│ │ │ │ -              System time (seconds): 0.10
│ │ │ │ -              Percent of CPU this job got: 93%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:08.15
│ │ │ │ +--silent  -q  <"/tmp/M2-31226-0/4.m2" >"/tmp/M2-31226-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 7.07
│ │ │ │ +              System time (seconds): 0.30
│ │ │ │ +              Percent of CPU this job got: 119%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:06.17
│ │ │ │                Average shared text size (kbytes): 0
│ │ │ │                Average unshared data size (kbytes): 0
│ │ │ │                Average stack size (kbytes): 0
│ │ │ │                Average total size (kbytes): 0
│ │ │ │ -              Maximum resident set size (kbytes): 255852
│ │ │ │ +              Maximum resident set size (kbytes): 339212
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 10158
│ │ │ │ -              Voluntary context switches: 2973
│ │ │ │ -              Involuntary context switches: 2382
│ │ │ │ +              Minor (reclaiming a frame) page faults: 13081
│ │ │ │ +              Voluntary context switches: 18679
│ │ │ │ +              Involuntary context switches: 3144
│ │ │ │                Swaps: 0
│ │ │ │                File system inputs: 0
│ │ │ │ -              File system outputs: 0
│ │ │ │ +              File system outputs: 16
│ │ │ │                Socket messages sent: 0
│ │ │ │                Socket messages received: 0
│ │ │ │                Signals delivered: 0
│ │ │ │                Page size (bytes): 4096
│ │ │ │                Exit status: 0
│ │ │ │  We can handle most kinds of objects as return values, although _M_u_t_a_b_l_e_M_a_t_r_i_x
│ │ │ │  does not work. Here, we use the built-in _i_d_e_n_t_i_t_y function:
│ │ │ │  i20 : v=/// A complicated string^%&C@#CERQVASDFQ#BQBSDH"' ewrjwklsf///;
│ │ │ │  i21 : (runExternalM2(fn,identity,v))#value===v
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24375-0/6.m2" >"/tmp/M2-24375-0/6.out" 2>&1 ))
│ │ │ │ +M2-31226-0/6.m2" >"/tmp/M2-31226-0/6.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  Some care is required, however:
│ │ │ │  i22 : R=QQ[x,y];
│ │ │ │  i23 : v=coker random(R^2,R^{3:-1})
│ │ │ │  
│ │ │ │  o23 = cokernel | 9/2x+9/4y 7/9x+7/10y 7x+3/7y |
│ │ │ │                 | 3/4x+7/4y 7/10x+7/3y 6/7x+6y |
│ │ │ │  
│ │ │ │                               2
│ │ │ │  o23 : R-module, quotient of R
│ │ │ │  i24 : h=runExternalM2(fn,identity,v)
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24375-0/7.m2" >"/tmp/M2-24375-0/7.out" 2>&1 ))
│ │ │ │ +M2-31226-0/7.m2" >"/tmp/M2-31226-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24375-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-31226-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-24375-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-31226-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 4
│ │ │ │ +                "time used" => 3
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-24375-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-31226-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-24375-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-31226-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-24375-0/7.ans",identity (cokernel
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-31226-0/7.ans",identity (cokernel
│ │ │ │  (map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/
│ │ │ │  4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ │        stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to
│ │ │ │  objects:
│ │ │ │                    R (of class Symbol)
│ │ │ │              ^     2 (of class ZZ)
│ │ │ │  Keep in mind that the object you are passing must make sense in the context of
│ │ │ │  the file containing your function! For instance, here we need to define the
│ │ │ │  ring:
│ │ │ │  i26 : fn<"/tmp/M2-24375-0/8.out" 2>&1 ))
│ │ │ │ +M2-31226-0/8.m2" >"/tmp/M2-31226-0/8.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  
│ │ │ │  o27 = true
│ │ │ │  This problem can be avoided by following some _s_u_g_g_e_s_t_i_o_n_s_ _f_o_r_ _u_s_i_n_g
│ │ │ │  _R_u_n_E_x_t_e_r_n_a_l_M_2.
│ │ │ │  The objects may unavoidably lose some internal references, though:
│ │ │ │  i28 : v=R;
│ │ │ │  i29 : h=runExternalM2(fn,identity,v);
│ │ │ │  Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/
│ │ │ │ -M2-24375-0/9.m2" >"/tmp/M2-24375-0/9.out" 2>&1 ))
│ │ │ │ +M2-31226-0/9.m2" >"/tmp/M2-31226-0/9.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i30 : h#value
│ │ │ │  
│ │ │ │  o30 = QQ[x..y]
│ │ │ │  
│ │ │ │  o30 : PolynomialRing
│ │ │ │  i31 : v===h#value
│ │ ├── ./usr/share/doc/Macaulay2/SCMAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  Y2Fub25pY2FsTW9kdWxlKElkZWFsKQ==
│ │ │  #:len=266
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjYW5vbmljYWxNb2R1bGUsSWRlYWwpLCJjYW5vbmlj
│ │ ├── ./usr/share/doc/Macaulay2/SCSCP/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  UmVtb3RlT2JqZWN0IGFuZCBSZW1vdGVPYmplY3Q=
│ │ │  #:len=217
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMwLCAidW5kb2N1bWVudGVkIiA9PiB0
│ │ │  cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wg
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  dHJhbnNwb3NlKEdhdGVNYXRyaXgp
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjU1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0cmFuc3Bvc2UsR2F0ZU1hdHJpeCksInRyYW5zcG9z
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out
│ │ │ @@ -30,23 +30,23 @@
│ │ │                                              )
│ │ │  
│ │ │                            "variable positions" => {-1}
│ │ │  
│ │ │  o5 : InterpretedSLProgram
│ │ │  
│ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000243396s (cpu); 0.000238547s (thread); 0s (gc)
│ │ │ + -- used 0.000220875s (cpu); 0.000215759s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.61088s (cpu); 3.68025s (thread); 0s (gc)
│ │ │ + -- used 4.06622s (cpu); 2.80027s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
│ │ │ @@ -109,29 +109,29 @@
│ │ │  
│ │ │  o5 : InterpretedSLProgram
    │ │ │
    │ │ │ 
    i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000243396s (cpu); 0.000238547s (thread); 0s (gc)
│ │ │ + -- used 0.000220875s (cpu); 0.000215759s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    │ │ │
    │ │ │ 
    i7 : ZZ[y];
    │ │ │
    │ │ │ 
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.61088s (cpu); 3.68025s (thread); 0s (gc)
│ │ │ + -- used 4.06622s (cpu); 2.80027s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ │ ├── html2text {}
│ │ │ │ @@ -38,21 +38,21 @@
│ │ │ │                                              output nodes: 1
│ │ │ │                                              )
│ │ │ │  
│ │ │ │                            "variable positions" => {-1}
│ │ │ │  
│ │ │ │  o5 : InterpretedSLProgram
│ │ │ │  i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ │ - -- used 0.000243396s (cpu); 0.000238547s (thread); 0s (gc)
│ │ │ │ + -- used 0.000220875s (cpu); 0.000215759s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 5.61088s (cpu); 3.68025s (thread); 0s (gc)
│ │ │ │ + -- used 4.06622s (cpu); 2.80027s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c2xFcXVpdmFyaWFudFZlY3RvckJ1bmRsZQ==
│ │ │  #:len=4962
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBTTC1lcXVpdmFyaWFu
│ │ │  dCB2ZWN0b3IgYnVuZGxlIG92ZXIgc29tZSBwcm9qZWN0aXZlIHNwYWNlIiwgImxpbmVudW0iID0+
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  RXhhbXBsZSBmaXJzdCBvcmRlciBkZWZvcm1hdGlvbg==
│ │ │  #:len=3174
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXhhbXBsZSBhY2Nlc3NpbmcgdGhlIGRh
│ │ │  dGEgc3RvcmVkIGluIGEgZmlyc3Qgb3JkZXIgZGVmb3JtYXRpb24uIiwgImxpbmVudW0iID0+IDQy
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/example-output/___Co__Complex.out
│ │ │ @@ -56,15 +56,15 @@
│ │ │  
│ │ │  o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i8 : cC=complement C
│ │ │  
│ │ │  o8 = 2: x x x  x x x  x x x  x x x  x x x  x x x  x x x  x x x  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i9 : dualize cC
│ │ │  
│ │ │  o9 = 2: v v v  v v v  v v v  v v v  v v v  v v v  v v v  v v v
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/html/___Co__Complex.html
│ │ │ @@ -163,15 +163,15 @@
│ │ │
    │ │ │ 
    i8 : cC=complement C
│ │ │  
│ │ │  o8 = 2: x x x  x x x  x x x  x x x  x x x  x x x  x x x  x x x  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -82,15 +82,15 @@ │ │ │ │ 0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 │ │ │ │ │ │ │ │ o7 : co-complex of dim 2 embedded in dim 5 (printing facets) │ │ │ │ equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1 │ │ │ │ i8 : cC=complement C │ │ │ │ │ │ │ │ o8 = 2: x x x x x x x x x x x x x x x x x x x x x x x x │ │ │ │ - 3 4 5 1 4 5 2 1 5 2 3 4 3 0 4 1 0 4 2 3 1 2 1 0 │ │ │ │ + 4 5 3 1 4 5 1 5 2 4 2 3 0 4 3 1 0 4 1 2 3 1 0 2 │ │ │ │ │ │ │ │ o8 : co-complex of dim 2 embedded in dim 5 (printing facets) │ │ │ │ equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1 │ │ │ │ i9 : dualize cC │ │ │ │ │ │ │ │ o9 = 2: v v v v v v v v v v v v v v v v v v v v v v v v │ │ │ │ 3 4 5 2 3 5 1 2 5 0 4 5 0 3 4 0 2 3 0 1 5 0 1 2 │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ cHJvamVjdFRvQ29tcGxleChDb21wbGV4LEhhc2hUYWJsZSk= │ │ │ #:len=298 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE4NCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocHJvamVjdFRvQ29tcGxleCxDb21wbGV4LEhhc2hU │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ i3 : r={5,11,3,2} │ │ │ │ │ │ o3 = {5, 11, 3, 2} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>4) │ │ │ - -- .0570737s elapsed │ │ │ + -- .0198887s elapsed │ │ │ │ │ │ 6 19 19 7 3 │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o4 : Complex │ │ │ @@ -41,15 +41,15 @@ │ │ │ 53 53 53 53 53 │ │ │ │ │ │ -1 0 1 2 3 │ │ │ │ │ │ o6 : Complex │ │ │ │ │ │ i7 : elapsedTime (h,U)=SVDComplex CR; │ │ │ - -- .00246826s elapsed │ │ │ + -- .00226706s elapsed │ │ │ │ │ │ i8 : h │ │ │ │ │ │ o8 = HashTable{-1 => 1} │ │ │ 0 => 3 │ │ │ 1 => 5 │ │ │ 2 => 2 │ │ │ @@ -85,15 +85,15 @@ │ │ │ i12 : maximalEntry complex errors │ │ │ │ │ │ o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian); │ │ │ - -- .00494602s elapsed │ │ │ + -- .00477556s elapsed │ │ │ │ │ │ i14 : hL === h │ │ │ │ │ │ o14 = true │ │ │ │ │ │ i15 : SigmaL =source U; │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out │ │ │ @@ -15,15 +15,15 @@ │ │ │ i3 : r={4,3,3} │ │ │ │ │ │ o3 = {4, 3, 3} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ - -- .0717096s elapsed │ │ │ + -- .0195031s elapsed │ │ │ │ │ │ 5 10 11 5 │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o4 : Complex │ │ │ @@ -41,25 +41,25 @@ │ │ │ 53 53 53 53 │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o6 : Complex │ │ │ │ │ │ i7 : elapsedTime (h,h1)=SVDHomology CR │ │ │ - -- .000541143s elapsed │ │ │ + -- .000524014s elapsed │ │ │ │ │ │ o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, ) }) │ │ │ 1 => 3 2 => (37.9214, 30.3707, 1.61954e-14) │ │ │ 2 => 5 3 => (14.972, 8.57847, 3.90646e-15) │ │ │ 3 => 2 │ │ │ │ │ │ o7 : Sequence │ │ │ │ │ │ i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian) │ │ │ - -- .00894283s elapsed │ │ │ + -- .00120428s elapsed │ │ │ │ │ │ o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14) }) │ │ │ 1 => 3 1 => (1.71747, 922.381, 2.51496e-13) │ │ │ 2 => 5 2 => (922.381, 73.5901, 1.81323e-13) │ │ │ 3 => 2 3 => (73.5901, , 2.82914e-13) │ │ │ │ │ │ o8 : Sequence │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out │ │ │ @@ -18,15 +18,15 @@ │ │ │ i4 : r={4,3,5} │ │ │ │ │ │ o4 = {4, 3, 5} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true) │ │ │ - -- .0727341s elapsed │ │ │ + -- .0313919s elapsed │ │ │ │ │ │ 6 10 13 8 │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ @@ -18,15 +18,15 @@ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ - -- .0562647s elapsed │ │ │ + -- .018874s elapsed │ │ │ │ │ │ 6 10 11 5 │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out │ │ │ @@ -18,15 +18,15 @@ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ - -- .00316014s elapsed │ │ │ + -- .00277459s elapsed │ │ │ │ │ │ 6 10 11 5 │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ @@ -110,15 +110,15 @@ │ │ │ │ │ │ o3 : List │ │ │
    │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .0570737s elapsed
│ │ │ + -- .0198887s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : Complex
    │ │ │ @@ -145,15 +145,15 @@ │ │ │ │ │ │ o6 : Complex │ │ │
    │ │ │ 
    i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00246826s elapsed
    │ │ │ + -- .00226706s elapsed │ │ │
    │ │ │ 
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │ @@ -207,15 +207,15 @@
│ │ │  
│ │ │  o12 : List
    │ │ │
    │ │ │ 
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00494602s elapsed
    │ │ │ + -- .00477556s elapsed │ │ │
    │ │ │ 
    i14 : hL === h
│ │ │  
│ │ │  o14 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -37,15 +37,15 @@ │ │ │ │ o2 : List │ │ │ │ i3 : r={5,11,3,2} │ │ │ │ │ │ │ │ o3 = {5, 11, 3, 2} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>4) │ │ │ │ - -- .0570737s elapsed │ │ │ │ + -- .0198887s elapsed │ │ │ │ │ │ │ │ 6 19 19 7 3 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o4 : Complex │ │ │ │ @@ -60,15 +60,15 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 53 │ │ │ │ │ │ │ │ -1 0 1 2 3 │ │ │ │ │ │ │ │ o6 : Complex │ │ │ │ i7 : elapsedTime (h,U)=SVDComplex CR; │ │ │ │ - -- .00246826s elapsed │ │ │ │ + -- .00226706s elapsed │ │ │ │ i8 : h │ │ │ │ │ │ │ │ o8 = HashTable{-1 => 1} │ │ │ │ 0 => 3 │ │ │ │ 1 => 5 │ │ │ │ 2 => 2 │ │ │ │ 3 => 1 │ │ │ │ @@ -99,15 +99,15 @@ │ │ │ │ 1)*Sigma.dd_ell*transpose U_ell); │ │ │ │ i12 : maximalEntry complex errors │ │ │ │ │ │ │ │ o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian); │ │ │ │ - -- .00494602s elapsed │ │ │ │ + -- .00477556s elapsed │ │ │ │ i14 : hL === h │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : SigmaL =source U; │ │ │ │ i16 : for i from min CR+1 to max CR list maximalEntry(SigmaL.dd_i -Sigma.dd_i) │ │ │ │ │ │ │ │ o16 = {1.77636e-14, 6.39488e-14, 8.52651e-14, 3.55271e-15} │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ @@ -112,15 +112,15 @@ │ │ │ │ │ │ o3 : List │ │ │
    │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0717096s elapsed
│ │ │ + -- .0195031s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : Complex
    │ │ │ @@ -147,28 +147,28 @@ │ │ │ │ │ │ o6 : Complex │ │ │
    │ │ │ 
    i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000541143s elapsed
│ │ │ + -- .000524014s elapsed
│ │ │  
│ │ │  o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, )           })
│ │ │                  1 => 3             2 => (37.9214, 30.3707, 1.61954e-14)
│ │ │                  2 => 5             3 => (14.972, 8.57847, 3.90646e-15)
│ │ │                  3 => 2
│ │ │  
│ │ │  o7 : Sequence
    │ │ │
    │ │ │ 
    i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
│ │ │ - -- .00894283s elapsed
│ │ │ + -- .00120428s elapsed
│ │ │  
│ │ │  o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14)      })
│ │ │                  1 => 3             1 => (1.71747, 922.381, 2.51496e-13)
│ │ │                  2 => 5             2 => (922.381, 73.5901, 1.81323e-13)
│ │ │                  3 => 2             3 => (73.5901, , 2.82914e-13)
│ │ │  
│ │ │  o8 : Sequence
    │ │ │ ├── html2text {} │ │ │ │ @@ -40,15 +40,15 @@ │ │ │ │ o2 : List │ │ │ │ i3 : r={4,3,3} │ │ │ │ │ │ │ │ o3 = {4, 3, 3} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .0717096s elapsed │ │ │ │ + -- .0195031s elapsed │ │ │ │ │ │ │ │ 5 10 11 5 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o4 : Complex │ │ │ │ @@ -63,24 +63,24 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o6 : Complex │ │ │ │ i7 : elapsedTime (h,h1)=SVDHomology CR │ │ │ │ - -- .000541143s elapsed │ │ │ │ + -- .000524014s elapsed │ │ │ │ │ │ │ │ o7 = (HashTable{0 => 1}, HashTable{1 => (7.87842, 1.31052, ) }) │ │ │ │ 1 => 3 2 => (37.9214, 30.3707, 1.61954e-14) │ │ │ │ 2 => 5 3 => (14.972, 8.57847, 3.90646e-15) │ │ │ │ 3 => 2 │ │ │ │ │ │ │ │ o7 : Sequence │ │ │ │ i8 : elapsedTime (hL,hL1)=SVDHomology(CR,Strategy=>Laplacian) │ │ │ │ - -- .00894283s elapsed │ │ │ │ + -- .00120428s elapsed │ │ │ │ │ │ │ │ o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14) }) │ │ │ │ 1 => 3 1 => (1.71747, 922.381, 2.51496e-13) │ │ │ │ 2 => 5 2 => (922.381, 73.5901, 1.81323e-13) │ │ │ │ 3 => 2 3 => (73.5901, , 2.82914e-13) │ │ │ │ │ │ │ │ o8 : Sequence │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ @@ -115,15 +115,15 @@ │ │ │ │ │ │ o4 : List │ │ │
    │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .0727341s elapsed
│ │ │ + -- .0313919s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -34,15 +34,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,5} │ │ │ │ │ │ │ │ o4 = {4, 3, 5} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true) │ │ │ │ - -- .0727341s elapsed │ │ │ │ + -- .0313919s elapsed │ │ │ │ │ │ │ │ 6 10 13 8 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o4 : List │ │ │
    │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0562647s elapsed
│ │ │ + -- .018874s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .0562647s elapsed │ │ │ │ + -- .018874s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o4 : List │ │ │
    │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00316014s elapsed
│ │ │ + -- .00277459s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .00316014s elapsed │ │ │ │ + -- .00277459s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SagbiGbDetection/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ d2VpZ2h0VmVjdG9yc1JlYWxpemluZ1NBR0JJ │ │ │ #:len=1819 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIG1haW4gZnVuY3Rpb24gZm9yIGRl │ │ │ dGVjdGluZyBTQUdCSSBiYXNlcyIsICJsaW5lbnVtIiA9PiAyMTUsIElucHV0cyA9PiB7U1BBTntU │ │ ├── ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ c2F0dXJhdGUoSWRlYWwsTGlzdCk= │ │ │ #:len=258 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzYXR1cmF0ZSxJZGVhbCxMaXN0KSwic2F0dXJhdGUo │ │ ├── ./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out │ │ │ @@ -37,33 +37,33 @@ │ │ │ o5 : Ideal of S │ │ │ │ │ │ i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5)); │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ i7 : time quotient(I^3, J^2, Strategy => Iterate); │ │ │ - -- used 0.412639s (cpu); 0.315292s (thread); 0s (gc) │ │ │ + -- used 0.291013s (cpu); 0.290751s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ - -- used 0.660447s (cpu); 0.585082s (thread); 0s (gc) │ │ │ + -- used 0.570089s (cpu); 0.497805s (thread); 0s (gc) │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ i9 : S = ZZ/101[vars(0..4)]; │ │ │ │ │ │ i10 : I = ideal vars S; │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ │ │ i11 : time quotient(I^5, I^3, Strategy => Iterate); │ │ │ - -- used 0.021082s (cpu); 0.021086s (thread); 0s (gc) │ │ │ + -- used 0.0205427s (cpu); 0.020544s (thread); 0s (gc) │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ - -- used 0.00785722s (cpu); 0.00785869s (thread); 0s (gc) │ │ │ + -- used 0.0071302s (cpu); 0.00713333s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ @@ -129,23 +129,23 @@ │ │ │ │ │ │ o6 : Ideal of S │ │ │
    │ │ │ 
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.412639s (cpu); 0.315292s (thread); 0s (gc)
│ │ │ + -- used 0.291013s (cpu); 0.290751s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    │ │ │
    │ │ │ 
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.660447s (cpu); 0.585082s (thread); 0s (gc)
│ │ │ + -- used 0.570089s (cpu); 0.497805s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    │ │ │
    │ │ │
    │ │ │

    Strategy => Quotient is faster in other cases:

    │ │ │ @@ -162,23 +162,23 @@ │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.021082s (cpu); 0.021086s (thread); 0s (gc)
│ │ │ + -- used 0.0205427s (cpu); 0.020544s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00785722s (cpu); 0.00785869s (thread); 0s (gc)
│ │ │ + -- used 0.0071302s (cpu); 0.00713333s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -52,32 +52,32 @@ │ │ │ │ i5 : I = monomialCurveIdeal(S, 1..n-1); │ │ │ │ │ │ │ │ o5 : Ideal of S │ │ │ │ i6 : J = ideal(map(S^1, S^n, (p, q) -> S_q^5)); │ │ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ i7 : time quotient(I^3, J^2, Strategy => Iterate); │ │ │ │ - -- used 0.412639s (cpu); 0.315292s (thread); 0s (gc) │ │ │ │ + -- used 0.291013s (cpu); 0.290751s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ │ - -- used 0.660447s (cpu); 0.585082s (thread); 0s (gc) │ │ │ │ + -- used 0.570089s (cpu); 0.497805s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ Strategy => Quotient is faster in other cases: │ │ │ │ i9 : S = ZZ/101[vars(0..4)]; │ │ │ │ i10 : I = ideal vars S; │ │ │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ i11 : time quotient(I^5, I^3, Strategy => Iterate); │ │ │ │ - -- used 0.021082s (cpu); 0.021086s (thread); 0s (gc) │ │ │ │ + -- used 0.0205427s (cpu); 0.020544s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ │ - -- used 0.00785722s (cpu); 0.00785869s (thread); 0s (gc) │ │ │ │ + -- used 0.0071302s (cpu); 0.00713333s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ ********** RReeffeerreenncceess ********** │ │ │ │ For further information see for example Exercise 15.41 in Eisenbud's │ │ │ │ Commutative Algebra with a View Towards Algebraic Geometry. │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd SSttrraatteeggyy:: ********** │ │ │ │ * addHook(...,Strategy=>...) -- see _a_d_d_H_o_o_k -- add a hook function to an │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ aW5jbHVzaW9uKC4uLixTdWJEaW1lbnNpb249Pi4uLik= │ │ │ #:len=260 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg1Nywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaW5jbHVzaW9uLFN1YkRpbWVuc2lvbl0sImluY2x1 │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ @@ -40,23 +40,23 @@ │ │ │ ) │ │ │ │ │ │ o6 = f │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ i7 : for n from 2 to 10 list time f n │ │ │ - -- used 0.00475218s (cpu); 0.00474919s (thread); 0s (gc) │ │ │ - -- used 0.00626054s (cpu); 0.00626186s (thread); 0s (gc) │ │ │ - -- used 0.0101612s (cpu); 0.010164s (thread); 0s (gc) │ │ │ - -- used 0.0184984s (cpu); 0.018504s (thread); 0s (gc) │ │ │ - -- used 0.0345749s (cpu); 0.0345811s (thread); 0s (gc) │ │ │ - -- used 0.0689669s (cpu); 0.0689727s (thread); 0s (gc) │ │ │ - -- used 0.104701s (cpu); 0.104707s (thread); 0s (gc) │ │ │ - -- used 0.307541s (cpu); 0.188629s (thread); 0s (gc) │ │ │ - -- used 0.386656s (cpu); 0.275611s (thread); 0s (gc) │ │ │ + -- used 0.00614237s (cpu); 0.00614108s (thread); 0s (gc) │ │ │ + -- used 0.00687526s (cpu); 0.00688183s (thread); 0s (gc) │ │ │ + -- used 0.0100654s (cpu); 0.0100728s (thread); 0s (gc) │ │ │ + -- used 0.0189899s (cpu); 0.0190002s (thread); 0s (gc) │ │ │ + -- used 0.0350112s (cpu); 0.0350299s (thread); 0s (gc) │ │ │ + -- used 0.066128s (cpu); 0.0661347s (thread); 0s (gc) │ │ │ + -- used 0.1004s (cpu); 0.100406s (thread); 0s (gc) │ │ │ + -- used 0.14747s (cpu); 0.147477s (thread); 0s (gc) │ │ │ + -- used 0.350371s (cpu); 0.250323s (thread); 0s (gc) │ │ │ │ │ │ o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, │ │ │ ------------------------------------------------------------------------ │ │ │ 289139638632755625, 520764738758073845321} │ │ │ │ │ │ o7 : List │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ @@ -131,23 +131,23 @@ │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00475218s (cpu); 0.00474919s (thread); 0s (gc)
│ │ │ - -- used 0.00626054s (cpu); 0.00626186s (thread); 0s (gc)
│ │ │ - -- used 0.0101612s (cpu); 0.010164s (thread); 0s (gc)
│ │ │ - -- used 0.0184984s (cpu); 0.018504s (thread); 0s (gc)
│ │ │ - -- used 0.0345749s (cpu); 0.0345811s (thread); 0s (gc)
│ │ │ - -- used 0.0689669s (cpu); 0.0689727s (thread); 0s (gc)
│ │ │ - -- used 0.104701s (cpu); 0.104707s (thread); 0s (gc)
│ │ │ - -- used 0.307541s (cpu); 0.188629s (thread); 0s (gc)
│ │ │ - -- used 0.386656s (cpu); 0.275611s (thread); 0s (gc)
│ │ │ + -- used 0.00614237s (cpu); 0.00614108s (thread); 0s (gc)
│ │ │ + -- used 0.00687526s (cpu); 0.00688183s (thread); 0s (gc)
│ │ │ + -- used 0.0100654s (cpu); 0.0100728s (thread); 0s (gc)
│ │ │ + -- used 0.0189899s (cpu); 0.0190002s (thread); 0s (gc)
│ │ │ + -- used 0.0350112s (cpu); 0.0350299s (thread); 0s (gc)
│ │ │ + -- used 0.066128s (cpu); 0.0661347s (thread); 0s (gc)
│ │ │ + -- used 0.1004s (cpu); 0.100406s (thread); 0s (gc)
│ │ │ + -- used 0.14747s (cpu); 0.147477s (thread); 0s (gc)
│ │ │ + -- used 0.350371s (cpu); 0.250323s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │       ------------------------------------------------------------------------
│ │ │       289139638632755625, 520764738758073845321}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -56,23 +56,23 @@ │ │ │ │ integral chern symmetricPower_(2*n-3) last bundles G │ │ │ │ ) │ │ │ │ │ │ │ │ o6 = f │ │ │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ i7 : for n from 2 to 10 list time f n │ │ │ │ - -- used 0.00475218s (cpu); 0.00474919s (thread); 0s (gc) │ │ │ │ - -- used 0.00626054s (cpu); 0.00626186s (thread); 0s (gc) │ │ │ │ - -- used 0.0101612s (cpu); 0.010164s (thread); 0s (gc) │ │ │ │ - -- used 0.0184984s (cpu); 0.018504s (thread); 0s (gc) │ │ │ │ - -- used 0.0345749s (cpu); 0.0345811s (thread); 0s (gc) │ │ │ │ - -- used 0.0689669s (cpu); 0.0689727s (thread); 0s (gc) │ │ │ │ - -- used 0.104701s (cpu); 0.104707s (thread); 0s (gc) │ │ │ │ - -- used 0.307541s (cpu); 0.188629s (thread); 0s (gc) │ │ │ │ - -- used 0.386656s (cpu); 0.275611s (thread); 0s (gc) │ │ │ │ + -- used 0.00614237s (cpu); 0.00614108s (thread); 0s (gc) │ │ │ │ + -- used 0.00687526s (cpu); 0.00688183s (thread); 0s (gc) │ │ │ │ + -- used 0.0100654s (cpu); 0.0100728s (thread); 0s (gc) │ │ │ │ + -- used 0.0189899s (cpu); 0.0190002s (thread); 0s (gc) │ │ │ │ + -- used 0.0350112s (cpu); 0.0350299s (thread); 0s (gc) │ │ │ │ + -- used 0.066128s (cpu); 0.0661347s (thread); 0s (gc) │ │ │ │ + -- used 0.1004s (cpu); 0.100406s (thread); 0s (gc) │ │ │ │ + -- used 0.14747s (cpu); 0.147477s (thread); 0s (gc) │ │ │ │ + -- used 0.350371s (cpu); 0.250323s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 289139638632755625, 520764738758073845321} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ Note: in characteristic zero, using Bertini's theorem, the numbers computed can │ │ ├── ./usr/share/doc/Macaulay2/SchurComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ c2NodXJDb21wbGV4 │ │ │ #:len=3519 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NodXIgZnVuY3RvcnMgb2YgY2hhaW4g │ │ │ Y29tcGxleGVzIiwgImxpbmVudW0iID0+IDU5NywgSW5wdXRzID0+IHtTUEFOe1RUeyJsYW1iZGEi │ │ ├── ./usr/share/doc/Macaulay2/SchurFunctors/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ aXNXZXlsU3RhbmRhcmQoV2V5bEZpbGxpbmcp │ │ │ #:len=281 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTcwMywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNXZXlsU3RhbmRhcmQsV2V5bEZpbGxpbmcpLCJp │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ c2NodXJSZXNvbHV0aW9uKFJpbmdFbGVtZW50LExpc3Qp │ │ │ #:len=286 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjU4Miwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2NodXJSZXNvbHV0aW9uLFJpbmdFbGVtZW50LExp │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/___Eor__H.out │ │ │ @@ -33,13 +33,13 @@ │ │ │ i6 : toS fe == toS fh │ │ │ │ │ │ o6 = true │ │ │ │ │ │ i7 : R = symmetricRing(QQ,12); │ │ │ │ │ │ i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false); │ │ │ - -- .00389276s elapsed │ │ │ + -- .00308612s elapsed │ │ │ │ │ │ i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false); │ │ │ - -- .000092754s elapsed │ │ │ + -- .000115838s elapsed │ │ │ │ │ │ i10 : │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/___Memoize.out │ │ │ @@ -1,23 +1,23 @@ │ │ │ -- -*- M2-comint -*- hash: 9113092561535051477 │ │ │ │ │ │ i1 : R = symmetricRing(QQ, 10); │ │ │ │ │ │ i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ - -- .000475089s elapsed │ │ │ + -- .000479285s elapsed │ │ │ │ │ │ i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ - -- .000014267s elapsed │ │ │ + -- .00001862s elapsed │ │ │ │ │ │ i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ - -- .000441827s elapsed │ │ │ + -- .000497843s elapsed │ │ │ │ │ │ i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ - -- .000013796s elapsed │ │ │ + -- .000021314s elapsed │ │ │ │ │ │ i6 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ - -- .000013405s elapsed │ │ │ + -- .000013494s elapsed │ │ │ │ │ │ i7 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ - -- .000012954s elapsed │ │ │ + -- .000012171s elapsed │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi.out │ │ │ @@ -50,18 +50,18 @@ │ │ │ i11 : toS fe == toS fh │ │ │ │ │ │ o11 = true │ │ │ │ │ │ i12 : R = symmetricRing(QQ,6); │ │ │ │ │ │ i13 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ - -- .000476681s elapsed │ │ │ + -- .000371998s elapsed │ │ │ │ │ │ i14 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ - -- .000014307s elapsed │ │ │ + -- .000015763s elapsed │ │ │ │ │ │ i15 : R = symmetricRing(QQ,5); │ │ │ │ │ │ i16 : S = schurRing R; │ │ │ │ │ │ i17 : jacobiTrudi({3,2,1},R) == toSymm(S_{3,2,1}) │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.out │ │ │ @@ -3,16 +3,16 @@ │ │ │ i1 : R = symmetricRing(QQ,6); │ │ │ │ │ │ i2 : jacobiTrudi({4,3,2,1},R,Memoize => true) == jacobiTrudi({4,3,2,1},R,Memoize => false) │ │ │ │ │ │ o2 = true │ │ │ │ │ │ i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ - -- .000459589s elapsed │ │ │ + -- .000405202s elapsed │ │ │ │ │ │ i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ - -- .00001618s elapsed │ │ │ + -- .000015965s elapsed │ │ │ │ │ │ i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false); │ │ │ - -- .000383086s elapsed │ │ │ + -- .000344644s elapsed │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/___Eor__H.html │ │ │ @@ -143,21 +143,21 @@ │ │ │ │ │ │ 
    i7 : R = symmetricRing(QQ,12);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false);
│ │ │ - -- .00389276s elapsed
    │ │ │ + -- .00308612s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false);
│ │ │ - -- .000092754s elapsed
    │ │ │ + -- .000115838s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Functions with optional argument named EorH:

    │ │ │ ├── html2text {} │ │ │ │ @@ -46,17 +46,17 @@ │ │ │ │ o6 = true │ │ │ │ When the conjugate partition is much longer than lambda itself, the "H"-branch │ │ │ │ requires a smaller determinant and runs measurably faster. For example on │ │ │ │ lambda = (10) the conjugate is $(1^{10})$, so "H" only sets up a 1x1 │ │ │ │ determinant: │ │ │ │ i7 : R = symmetricRing(QQ,12); │ │ │ │ i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false); │ │ │ │ - -- .00389276s elapsed │ │ │ │ + -- .00308612s elapsed │ │ │ │ i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false); │ │ │ │ - -- .000092754s elapsed │ │ │ │ + -- .000115838s elapsed │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd EEoorrHH:: ********** │ │ │ │ * jacobiTrudi(...,EorH=>...) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _E_o_r_H is a _s_y_m_b_o_l. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/SchurRings.m2:6153:0. │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/___Memoize.html │ │ │ @@ -66,59 +66,59 @@ │ │ │ │ │ │ 
    i1 : R = symmetricRing(QQ, 10);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000475089s elapsed
    │ │ │ + -- .000479285s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000014267s elapsed
    │ │ │ + -- .00001862s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    │ │ │

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

    │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000441827s elapsed
    │ │ │ + -- .000497843s elapsed │ │ │
    │ │ │ 
    i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000013796s elapsed
    │ │ │ + -- .000021314s elapsed │ │ │
    │ │ │
    │ │ │

    │ │ │

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

    │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000013405s elapsed
    │ │ │ + -- .000013494s elapsed │ │ │
    │ │ │ 
    i7 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000012954s elapsed
    │ │ │ + -- .000012171s elapsed │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -7,31 +7,31 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ This is an optional argument for the _j_a_c_o_b_i_T_r_u_d_i function, allowing one to │ │ │ │ store its values in order to speed up computations. When Memoize => true, every │ │ │ │ computed value is cached in a hash table on the symmetric ring, so repeated │ │ │ │ calls on the same partition return the cached value in constant time. │ │ │ │ i1 : R = symmetricRing(QQ, 10); │ │ │ │ i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ │ - -- .000475089s elapsed │ │ │ │ + -- .000479285s elapsed │ │ │ │ i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ │ - -- .000014267s elapsed │ │ │ │ + -- .00001862s elapsed │ │ │ │ The cache is attached to the ring R. After one partition is memoized, │ │ │ │ subsequent calls with a different partition perform the full Jacobi-Trudi │ │ │ │ determinant expansion, then cache it as well: │ │ │ │ i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ │ - -- .000441827s elapsed │ │ │ │ + -- .000497843s elapsed │ │ │ │ i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ │ - -- .000013796s elapsed │ │ │ │ + -- .000021314s elapsed │ │ │ │ Without Memoize => true, each call recomputes the determinant from scratch; for │ │ │ │ large partitions this can be substantially more expensive than a single cached │ │ │ │ lookup. │ │ │ │ i6 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ │ - -- .000013405s elapsed │ │ │ │ + -- .000013494s elapsed │ │ │ │ i7 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ │ - -- .000012954s elapsed │ │ │ │ + -- .000012171s elapsed │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _j_a_c_o_b_i_T_r_u_d_i -- Jacobi-Trudi determinant │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd MMeemmooiizzee:: ********** │ │ │ │ * _j_a_c_o_b_i_T_r_u_d_i_(_._._._,_M_e_m_o_i_z_e_=_>_._._._) -- Store values of the jacobiTrudi │ │ │ │ function. │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _M_e_m_o_i_z_e is a _s_y_m_b_o_l. │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi.html │ │ │ @@ -186,21 +186,21 @@ │ │ │ │ │ │ 
      i12 : R = symmetricRing(QQ,6);
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i13 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000476681s elapsed
      │ │ │ + -- .000371998s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i14 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000014307s elapsed
      │ │ │ + -- .000015763s elapsed │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │

      │ │ │

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

      │ │ │

      │ │ │ ├── html2text {} │ │ │ │ @@ -67,17 +67,17 @@ │ │ │ │ │ │ │ │ o11 = true │ │ │ │ The routine caches intermediate subdeterminants on the ring via _j_a_c_o_b_i_T_r_u_d_i │ │ │ │ _(_._._._,_M_e_m_o_i_z_e_=_>_._._._), so a second call on a large partition returns almost │ │ │ │ instantly: │ │ │ │ i12 : R = symmetricRing(QQ,6); │ │ │ │ i13 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ │ - -- .000476681s elapsed │ │ │ │ + -- .000371998s elapsed │ │ │ │ i14 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ │ - -- .000014307s elapsed │ │ │ │ + -- .000015763s elapsed │ │ │ │ Passing a partition through _t_o_S_y_m_m applied to the corresponding Schur label │ │ │ │ reproduces the Jacobi-Trudi output: │ │ │ │ i15 : R = symmetricRing(QQ,5); │ │ │ │ i16 : S = schurRing R; │ │ │ │ i17 : jacobiTrudi({3,2,1},R) == toSymm(S_{3,2,1}) │ │ │ │ │ │ │ │ o17 = true │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.html │ │ │ @@ -85,27 +85,27 @@ │ │ │ │ │ │ o2 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .000459589s elapsed
      │ │ │ + -- .000405202s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .00001618s elapsed
      │ │ │ + -- .000015965s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false);
│ │ │ - -- .000383086s elapsed
      │ │ │ + -- .000344644s elapsed │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │
      │ │ │
      │ │ │

      Functions with optional argument named Memoize:

      │ │ │ ├── html2text {} │ │ │ │ @@ -17,19 +17,19 @@ │ │ │ │ extra memory in the ring. │ │ │ │ i1 : R = symmetricRing(QQ,6); │ │ │ │ i2 : jacobiTrudi({4,3,2,1},R,Memoize => true) == jacobiTrudi( │ │ │ │ {4,3,2,1},R,Memoize => false) │ │ │ │ │ │ │ │ o2 = true │ │ │ │ i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ │ - -- .000459589s elapsed │ │ │ │ + -- .000405202s elapsed │ │ │ │ i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ │ - -- .00001618s elapsed │ │ │ │ + -- .000015965s elapsed │ │ │ │ i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false); │ │ │ │ - -- .000383086s elapsed │ │ │ │ + -- .000344644s elapsed │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd MMeemmooiizzee:: ********** │ │ │ │ * _j_a_c_o_b_i_T_r_u_d_i_(_._._._,_M_e_m_o_i_z_e_=_>_._._._) -- Store values of the jacobiTrudi │ │ │ │ function. │ │ │ │ ********** FFuurrtthheerr iinnffoorrmmaattiioonn ********** │ │ │ │ * Default value: _t_r_u_e │ │ │ │ * Function: _j_a_c_o_b_i_T_r_u_d_i -- Jacobi-Trudi determinant │ │ │ │ * Option key: _M_e_m_o_i_z_e -- Option to record values of the jacobiTrudi │ │ ├── ./usr/share/doc/Macaulay2/SchurVeronese/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ bWFrZUJldHRpVGFsbHk= │ │ │ #:len=1334 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydHMgYSBoYXNoIHRhYmxlIHJl │ │ │ cHJlc2VudGluZyBhIEJldHRpIHRhYmxlIHRvIGEgQmV0dGkgdGFsbHkiLCAibGluZW51bSIgPT4g │ │ ├── ./usr/share/doc/Macaulay2/SectionRing/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ bVJlZ3VsYXI= │ │ │ #:len=1136 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibVJlZ3VsYXIoRixHKSBjb21wdXRlcyB0 │ │ │ aGUgcmVndWxhcml0eSBvZiBGIHdpdGggcmVzcGVjdCB0byBHIChnbG9iYWxseSBnZW5lcmF0ZWQp │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ bWFrZVByb2R1Y3RSaW5nKFJpbmcsTGlzdCk= │ │ │ #:len=277 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlUHJvZHVjdFJpbmcsUmluZyxMaXN0KSwibWFr │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out │ │ │ @@ -53,15 +53,15 @@ │ │ │ o9 : Ideal of R │ │ │ │ │ │ i10 : X=((W)*ideal(y)+ideal(f)); │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ i11 : time isComponentContained(X,Y) │ │ │ - -- used 4.54047s (cpu); 3.61426s (thread); 0s (gc) │ │ │ + -- used 5.81931s (cpu); 3.15124s (thread); 0s (gc) │ │ │ │ │ │ o11 = true │ │ │ │ │ │ i12 : print "we could confirm this with the computation:" │ │ │ we could confirm this with the computation: │ │ │ │ │ │ i13 : B=ideal(x)*ideal(y)*ideal(z) │ │ │ @@ -71,12 +71,12 @@ │ │ │ b*d*g, b*d*h, b*d*i, b*e*g, b*e*h, b*e*i, b*f*g, b*f*h, b*f*i, c*d*g, │ │ │ ----------------------------------------------------------------------- │ │ │ c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i) │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ │ │ i14 : time isSubset(saturate(Y,B),saturate(X,B)) │ │ │ - -- used 54.8103s (cpu); 50.9154s (thread); 0s (gc) │ │ │ + -- used 57.0039s (cpu); 52.7537s (thread); 0s (gc) │ │ │ │ │ │ o14 = true │ │ │ │ │ │ i15 : │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out │ │ │ @@ -23,24 +23,24 @@ │ │ │ i5 : A = makeChowRing(R) │ │ │ │ │ │ o5 = A │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ │ │ i6 : time s = segreDimX(X,Y,A) │ │ │ - -- used 0.261889s (cpu); 0.158635s (thread); 0s (gc) │ │ │ + -- used 1.38501s (cpu); 0.292149s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o6 = 2H + 4H H + 2H │ │ │ 1 1 2 2 │ │ │ │ │ │ o6 : A │ │ │ │ │ │ i7 : time segre(X,Y,A) │ │ │ - -- used 0.158315s (cpu); 0.114099s (thread); 0s (gc) │ │ │ + -- used 0.203979s (cpu); 0.138184s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o7 = 12H H - 6H H - 6H H + 2H + 4H H + 2H │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ o7 : A │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ @@ -167,15 +167,15 @@ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.54047s (cpu); 3.61426s (thread); 0s (gc)
│ │ │ + -- used 5.81931s (cpu); 3.15124s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i12 : print "we could confirm this with the computation:"
│ │ │ @@ -194,15 +194,15 @@
│ │ │  
│ │ │  o13 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 54.8103s (cpu); 50.9154s (thread); 0s (gc)
│ │ │ + -- used 57.0039s (cpu); 52.7537s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
      │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -68,30 +68,30 @@ │ │ │ │ i9 : Y=ideal (z_0*W_0-z_1*W_1)+ideal(f); │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : X=((W)*ideal(y)+ideal(f)); │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : time isComponentContained(X,Y) │ │ │ │ - -- used 4.54047s (cpu); 3.61426s (thread); 0s (gc) │ │ │ │ + -- used 5.81931s (cpu); 3.15124s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = true │ │ │ │ i12 : print "we could confirm this with the computation:" │ │ │ │ we could confirm this with the computation: │ │ │ │ i13 : B=ideal(x)*ideal(y)*ideal(z) │ │ │ │ │ │ │ │ o13 = ideal (a*d*g, a*d*h, a*d*i, a*e*g, a*e*h, a*e*i, a*f*g, a*f*h, a*f*i, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ b*d*g, b*d*h, b*d*i, b*e*g, b*e*h, b*e*i, b*f*g, b*f*h, b*f*i, c*d*g, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ c*d*h, c*d*i, c*e*g, c*e*h, c*e*i, c*f*g, c*f*h, c*f*i) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time isSubset(saturate(Y,B),saturate(X,B)) │ │ │ │ - -- used 54.8103s (cpu); 50.9154s (thread); 0s (gc) │ │ │ │ + -- used 57.0039s (cpu); 52.7537s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ ********** WWaayyss ttoo uussee iissCCoommppoonneennttCCoonnttaaiinneedd:: ********** │ │ │ │ * isComponentContained(Ideal,Ideal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_m_p_o_n_e_n_t_C_o_n_t_a_i_n_e_d is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ @@ -123,27 +123,27 @@ │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.261889s (cpu); 0.158635s (thread); 0s (gc)
│ │ │ + -- used 1.38501s (cpu); 0.292149s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i7 : time segre(X,Y,A)
│ │ │ - -- used 0.158315s (cpu); 0.114099s (thread); 0s (gc)
│ │ │ + -- used 0.203979s (cpu); 0.138184s (thread); 0s (gc)
│ │ │  
│ │ │          2 2     2         2     2             2
│ │ │  o7 = 12H H  - 6H H  - 6H H  + 2H  + 4H H  + 2H
│ │ │          1 2     1 2     1 2     1     1 2     2
│ │ │  
│ │ │  o7 : A
      │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -48,23 +48,23 @@ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : A = makeChowRing(R) │ │ │ │ │ │ │ │ o5 = A │ │ │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ i6 : time s = segreDimX(X,Y,A) │ │ │ │ - -- used 0.261889s (cpu); 0.158635s (thread); 0s (gc) │ │ │ │ + -- used 1.38501s (cpu); 0.292149s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o6 = 2H + 4H H + 2H │ │ │ │ 1 1 2 2 │ │ │ │ │ │ │ │ o6 : A │ │ │ │ i7 : time segre(X,Y,A) │ │ │ │ - -- used 0.158315s (cpu); 0.114099s (thread); 0s (gc) │ │ │ │ + -- used 0.203979s (cpu); 0.138184s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ │ o7 = 12H H - 6H H - 6H H + 2H + 4H H + 2H │ │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ │ │ o7 : A │ │ │ │ ********** WWaayyss ttoo uussee sseeggrreeDDiimmXX:: ********** │ │ ├── ./usr/share/doc/Macaulay2/SemidefiniteProgramming/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=28 │ │ │ b3B0aW1pemUoLi4uLFZlcmJvc2l0eT0+Li4uKQ== │ │ │ #:len=300 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tvcHRpbWl6ZSxWZXJib3NpdHldLCJvcHRpbWl6ZSgu │ │ ├── ./usr/share/doc/Macaulay2/Seminormalization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ YmV0dGVyTm9ybWFsaXphdGlvbk1hcA== │ │ │ #:len=1762 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibm9ybWFsaXplcyBub24gZG9tYWlucyIs │ │ │ ICJsaW5lbnVtIiA9PiA4ODEsIElucHV0cyA9PiB7U1BBTntUVHsiUyJ9LCIsICIsU1BBTnsiYSAi │ │ ├── ./usr/share/doc/Macaulay2/Serialization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=13 │ │ │ U2VyaWFsaXphdGlvbg== │ │ │ #:len=554 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV2ZXJzaWJsZSBjb252ZXJzaW9uIG9m │ │ │ IGFsbCBNYWNhdWxheTIgb2JqZWN0cyB0byBzdHJpbmdzIiwgRGVzY3JpcHRpb24gPT4gMTooRElW │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ YXJYaXYoU3RyaW5nLFN0cmluZyk= │ │ │ #:len=236 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzA5LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhclhpdixTdHJpbmcsU3RyaW5nKSwiYXJYaXYoU3Ry │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_test__Example.out │ │ │ @@ -1,7 +1,7 @@ │ │ │ -- -*- M2-comint -*- hash: 1331702921222 │ │ │ │ │ │ i1 : check SimpleDoc │ │ │ - -- capturing check(0, "SimpleDoc") -- .36758s elapsed │ │ │ - -- capturing check(1, "SimpleDoc") -- .401467s elapsed │ │ │ + -- capturing check(0, "SimpleDoc") -- .206247s elapsed │ │ │ + -- capturing check(1, "SimpleDoc") -- .167673s elapsed │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html │ │ │ @@ -79,16 +79,16 @@ │ │ │
      │ │ │

      The check method executes all package tests defined this way.

      │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ 
      i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .36758s elapsed
│ │ │ - -- capturing check(1, "SimpleDoc")           -- .401467s elapsed
      │ │ │ + -- capturing check(0, "SimpleDoc") -- .206247s elapsed │ │ │ + -- capturing check(1, "SimpleDoc") -- .167673s elapsed │ │ │
      │ │ │
      │ │ │
      │ │ │

      See also

      │ │ │
        │ │ │ ├── html2text {} │ │ │ │ @@ -10,16 +10,16 @@ │ │ │ │ The variable testExample is a _S_t_r_i_n_g containing an example of the use of _T_E_S_T │ │ │ │ to write a test case. │ │ │ │ TEST /// -* test for simpleDocFrob *- │ │ │ │ assert(simpleDocFrob(2,matrix{{1,2}}) == matrix{{1,2,0,0},{0,0,1,2}}) │ │ │ │ /// │ │ │ │ The _c_h_e_c_k method executes all package tests defined this way. │ │ │ │ i1 : check SimpleDoc │ │ │ │ - -- capturing check(0, "SimpleDoc") -- .36758s elapsed │ │ │ │ - -- capturing check(1, "SimpleDoc") -- .401467s elapsed │ │ │ │ + -- capturing check(0, "SimpleDoc") -- .206247s elapsed │ │ │ │ + -- capturing check(1, "SimpleDoc") -- .167673s elapsed │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _T_E_S_T -- add a test for a package │ │ │ │ * _c_h_e_c_k -- perform tests of a package │ │ │ │ * _p_a_c_k_a_g_e_T_e_m_p_l_a_t_e -- a template for a package │ │ │ │ * _d_o_c_E_x_a_m_p_l_e -- an example of a documentation string │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _t_e_s_t_E_x_a_m_p_l_e is a _s_t_r_i_n_g. │ │ ├── ./usr/share/doc/Macaulay2/SimplicialComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ ZWxlbWVudGFyeUNvbGxhcHNl │ │ │ #:len=365 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU1Miwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiZWxlbWVudGFyeUNvbGxhcHNlIiwiZWxlbWVudGFy │ │ ├── ./usr/share/doc/Macaulay2/SimplicialDecomposability/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ c2hlbGxpbmdPcmRlciguLi4sUmFuZG9tPT4uLi4p │ │ │ #:len=311 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODkzLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzaGVsbGluZ09yZGVyLFJhbmRvbV0sInNoZWxsaW5n │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=44 │ │ │ bm9ybWFsaXplKFNpbXBsaWNpYWxNb2R1bGUsWlosQ2hlY2tTdW09Pi4uLik= │ │ │ #:len=375 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg0Nywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbKG5vcm1hbGl6ZSxTaW1wbGljaWFsTW9kdWxlLFpa │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/___Simplicial__Modules.out │ │ │ @@ -111,25 +111,25 @@ │ │ │ o7 = R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o7 : SimplicialModule │ │ │ │ │ │ i8 : elapsedTime simplicialModule(K,6) --specify top degree 6 │ │ │ - -- .0671636s elapsed │ │ │ + -- .0684915s elapsed │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ o8 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o8 : SimplicialModule │ │ │ │ │ │ i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true) │ │ │ - -- .195242s elapsed │ │ │ + -- .140675s elapsed │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ o9 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o9 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_ext__Power.out │ │ │ @@ -12,15 +12,15 @@ │ │ │ o2 = Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 │ │ │ │ │ │ o2 : Complex │ │ │ │ │ │ i3 : w3K = elapsedTime prune extPower(3, K) │ │ │ - -- 7.53161s elapsed │ │ │ + -- 5.09979s elapsed │ │ │ │ │ │ 1 18 63 91 60 15 │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 1 2 3 4 5 6 │ │ │ │ │ │ o3 : Complex │ │ │ @@ -93,15 +93,15 @@ │ │ │ 1 2 │ │ │ 2 : Q <--------------- Q : 2 │ │ │ {2} | 0 b | │ │ │ │ │ │ o7 : ComplexMap │ │ │ │ │ │ i8 : f = elapsedTime prune extPower(2, phi) │ │ │ - -- .604017s elapsed │ │ │ + -- .360082s elapsed │ │ │ │ │ │ 3 6 │ │ │ o8 = 1 : Q <------------------------- Q : 1 │ │ │ {1} | a b 0 0 0 0 | │ │ │ {1} | 0 0 0 b 0 0 | │ │ │ {2} | 0 0 0 0 ab b2 | │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_exterior__Inclusion.out │ │ │ @@ -12,15 +12,15 @@ │ │ │ o2 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o2 : Complex │ │ │ │ │ │ i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ - -- .210615s elapsed │ │ │ + -- .186876s elapsed │ │ │ │ │ │ i4 : isWellDefined phi │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : isCommutative phi │ │ │ │ │ │ @@ -160,15 +160,15 @@ │ │ │ o10 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o10 : Complex │ │ │ │ │ │ i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ - -- .21521s elapsed │ │ │ + -- .182241s elapsed │ │ │ │ │ │ i12 : isWellDefined phi │ │ │ │ │ │ o12 = true │ │ │ │ │ │ i13 : isCommutative phi │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_forget__Degeneracy_lp__Simplicial__Module_rp.out │ │ │ @@ -21,15 +21,15 @@ │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ │ │ i4 : elapsedTime S**S │ │ │ - -- .487245s elapsed │ │ │ + -- .328986s elapsed │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ o4 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o4 : SimplicialModule │ │ │ @@ -40,15 +40,15 @@ │ │ │ o5 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o5 : SimplicialModule │ │ │ │ │ │ i6 : elapsedTime fS**fS --faster when degeneracy is ignored │ │ │ - -- .463629s elapsed │ │ │ + -- .260303s elapsed │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ o6 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o6 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_normalize_lp__Simplicial__Module_cm__Z__Z_rp.out │ │ │ @@ -114,35 +114,35 @@ │ │ │ o10 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ o10 : SimplicialModule │ │ │ │ │ │ i11 : elapsedTime prune normalize S10 │ │ │ - -- 4.94576s elapsed │ │ │ + -- 3.29347s elapsed │ │ │ │ │ │ 10 30 30 10 │ │ │ o11 = R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o11 : Complex │ │ │ │ │ │ i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times slower; becomes significant for larger ranks │ │ │ - -- 8.29592s elapsed │ │ │ + -- 6.06738s elapsed │ │ │ │ │ │ 10 30 30 10 │ │ │ o12 = R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o12 : Complex │ │ │ │ │ │ i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER! │ │ │ - -- .0571616s elapsed │ │ │ + -- .0332799s elapsed │ │ │ │ │ │ 10 30 30 10 │ │ │ o13 = R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o13 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_schur__Map.out │ │ │ @@ -14,25 +14,25 @@ │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ │ │ i4 : S2S = elapsedTime schurMap({2}, S) │ │ │ - -- .142323s elapsed │ │ │ + -- .119469s elapsed │ │ │ │ │ │ 1 6 21 55 120 │ │ │ o4 = Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o4 : SimplicialModule │ │ │ │ │ │ i5 : elapsedTime schurMap({2}, S, Degeneracy => true) │ │ │ - -- .309043s elapsed │ │ │ + -- .260382s elapsed │ │ │ │ │ │ 1 6 21 55 120 │ │ │ o5 = Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o5 : SimplicialModule │ │ │ @@ -78,15 +78,15 @@ │ │ │ o10 = Q │ │ │ │ │ │ o10 : PolynomialRing │ │ │ │ │ │ i11 : K = koszulComplex vars Q; │ │ │ │ │ │ i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4) │ │ │ - -- 1.00439s elapsed │ │ │ + -- .769551s elapsed │ │ │ │ │ │ 1 9 36 74 81 │ │ │ o12 = Q <-- Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o12 : Complex │ │ │ @@ -273,15 +273,15 @@ │ │ │ 1 2 │ │ │ 2 : Q <--------------- Q : 2 │ │ │ {2} | 0 b | │ │ │ │ │ │ o30 : ComplexMap │ │ │ │ │ │ i31 : f = elapsedTime prune schurMap({2}, phi) │ │ │ - -- .912527s elapsed │ │ │ + -- .821635s elapsed │ │ │ │ │ │ 1 1 │ │ │ o31 = 0 : Q <--------- Q : 0 │ │ │ | 1 | │ │ │ │ │ │ 5 9 │ │ │ 1 : Q <----------------------------------- Q : 1 │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_tensor__L__E__S.out │ │ │ @@ -62,15 +62,15 @@ │ │ │ o7 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o7 : Complex │ │ │ │ │ │ i8 : hL = elapsedTime prune tensorLES(L,4) │ │ │ - -- 2.28436s elapsed │ │ │ + -- 1.54337s elapsed │ │ │ │ │ │ 25 50 25 │ │ │ o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b | <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0 | <-- cokernel | 0 0 b a 0 0 0 0 | <-- cokernel | ab2 a2b b2 a2 | <-- cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q <-- Q <-- Q │ │ │ | 0 b a | | 0 b a | | 0 0 b a | | 0 0 b a 0 0 0 0 | | 0 0 0 0 b a 0 0 | │ │ │ 0 1 2 3 | 0 0 0 0 b a 0 0 | | ab2 a2b 0 0 0 0 b2 a2 | 9 10 11 12 13 14 │ │ │ 4 5 6 | 0 0 0 0 0 0 b a | │ │ │ 8 │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_ext__Power.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ │ │ │ o2 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : w3K = elapsedTime prune extPower(3, K)
│ │ │ - -- 7.53161s elapsed
│ │ │ + -- 5.09979s elapsed
│ │ │  
│ │ │        1      18      63      91      60      15
│ │ │  o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q
│ │ │                                               
│ │ │       1      2       3       4       5       6
│ │ │  
│ │ │  o3 : Complex
        │ │ │ @@ -203,15 +203,15 @@ │ │ │ │ │ │ o7 : ComplexMap │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i8 : f = elapsedTime prune extPower(2, phi)
│ │ │ - -- .604017s elapsed
│ │ │ + -- .360082s elapsed
│ │ │  
│ │ │            3                             6
│ │ │  o8 = 1 : Q  <------------------------- Q  : 1
│ │ │                 {1} | a b 0 0 0  0  |
│ │ │                 {1} | 0 0 0 b 0  0  |
│ │ │                 {2} | 0 0 0 0 ab b2 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │        1      2      1
│ │ │ │  o2 = Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │       0      1      2
│ │ │ │  
│ │ │ │  o2 : Complex
│ │ │ │  i3 : w3K = elapsedTime prune extPower(3, K)
│ │ │ │ - -- 7.53161s elapsed
│ │ │ │ + -- 5.09979s elapsed
│ │ │ │  
│ │ │ │        1      18      63      91      60      15
│ │ │ │  o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q
│ │ │ │  
│ │ │ │       1      2       3       4       5       6
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │ @@ -115,15 +115,15 @@
│ │ │ │  
│ │ │ │            1                   2
│ │ │ │       2 : Q  <--------------- Q  : 2
│ │ │ │                 {2} | 0 b |
│ │ │ │  
│ │ │ │  o7 : ComplexMap
│ │ │ │  i8 : f = elapsedTime prune extPower(2, phi)
│ │ │ │ - -- .604017s elapsed
│ │ │ │ + -- .360082s elapsed
│ │ │ │  
│ │ │ │            3                             6
│ │ │ │  o8 = 1 : Q  <------------------------- Q  : 1
│ │ │ │                 {1} | a b 0 0 0  0  |
│ │ │ │                 {1} | 0 0 0 b 0  0  |
│ │ │ │                 {2} | 0 0 0 0 ab b2 |
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_exterior__Inclusion.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 : Complex
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .210615s elapsed
        │ │ │ + -- .186876s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : isWellDefined phi
│ │ │  
│ │ │  o4 = true
        │ │ │ @@ -278,15 +278,15 @@ │ │ │ │ │ │ o10 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .21521s elapsed
        │ │ │ + -- .182241s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i12 : isWellDefined phi
│ │ │  
│ │ │  o12 = true
        │ │ │ ├── html2text {} │ │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ 1 3 3 1 │ │ │ │ o2 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o2 : Complex │ │ │ │ i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ │ - -- .210615s elapsed │ │ │ │ + -- .186876s elapsed │ │ │ │ i4 : isWellDefined phi │ │ │ │ │ │ │ │ o4 = true │ │ │ │ i5 : isCommutative phi │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : prune coker phi │ │ │ │ @@ -176,15 +176,15 @@ │ │ │ │ 1 3 3 1 │ │ │ │ o10 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o10 : Complex │ │ │ │ i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ │ - -- .21521s elapsed │ │ │ │ + -- .182241s elapsed │ │ │ │ i12 : isWellDefined phi │ │ │ │ │ │ │ │ o12 = true │ │ │ │ i13 : isCommutative phi │ │ │ │ │ │ │ │ o13 = true │ │ │ │ i14 : for i to 2 list prune HH_i source phi │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Degeneracy_lp__Simplicial__Module_rp.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : elapsedTime S**S
│ │ │ - -- .487245s elapsed
│ │ │ + -- .328986s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o4 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o4 : SimplicialModule
        │ │ │ @@ -134,15 +134,15 @@ │ │ │ │ │ │ o5 : SimplicialModule │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i6 : elapsedTime fS**fS --faster when degeneracy is ignored
│ │ │ - -- .463629s elapsed
│ │ │ + -- .260303s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o6 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o6 : SimplicialModule
        │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 1 5 15 35 70 126 210 │ │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ i4 : elapsedTime S**S │ │ │ │ - -- .487245s elapsed │ │ │ │ + -- .328986s elapsed │ │ │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ │ o4 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o4 : SimplicialModule │ │ │ │ @@ -53,15 +53,15 @@ │ │ │ │ 1 5 15 35 70 126 210 │ │ │ │ o5 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o5 : SimplicialModule │ │ │ │ i6 : elapsedTime fS**fS --faster when degeneracy is ignored │ │ │ │ - -- .463629s elapsed │ │ │ │ + -- .260303s elapsed │ │ │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ │ o6 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o6 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_normalize_lp__Simplicial__Module_cm__Z__Z_rp.html │ │ │ @@ -237,28 +237,28 @@ │ │ │ │ │ │ o10 : SimplicialModule │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i11 : elapsedTime prune normalize S10
│ │ │ - -- 4.94576s elapsed
│ │ │ + -- 3.29347s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o11 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o11 : Complex
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times slower; becomes significant for larger ranks
│ │ │ - -- 8.29592s elapsed
│ │ │ + -- 6.06738s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o12 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o12 : Complex
        │ │ │ @@ -268,15 +268,15 @@ │ │ │
        │ │ │

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

        │ │ │
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
        │ │ │ 
        i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER!
│ │ │ - -- .0571616s elapsed
│ │ │ + -- .0332799s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o13 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o13 : Complex
        │ │ │ ├── html2text {} │ │ │ │ @@ -156,38 +156,38 @@ │ │ │ │ │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 │ │ │ │ 9 10 │ │ │ │ │ │ │ │ o10 : SimplicialModule │ │ │ │ i11 : elapsedTime prune normalize S10 │ │ │ │ - -- 4.94576s elapsed │ │ │ │ + -- 3.29347s elapsed │ │ │ │ │ │ │ │ 10 30 30 10 │ │ │ │ o11 = R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o11 : Complex │ │ │ │ i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times │ │ │ │ slower; becomes significant for larger ranks │ │ │ │ - -- 8.29592s elapsed │ │ │ │ + -- 6.06738s elapsed │ │ │ │ │ │ │ │ 10 30 30 10 │ │ │ │ o12 = R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o12 : Complex │ │ │ │ The user may also specify the top homological degree to compute the │ │ │ │ normalization up to. Note that this can help speed up computational time; if │ │ │ │ the user knows the normalization should have a shorter length, then they should │ │ │ │ specify this upper bound in the syntax: │ │ │ │ i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER! │ │ │ │ - -- .0571616s elapsed │ │ │ │ + -- .0332799s elapsed │ │ │ │ │ │ │ │ 10 30 30 10 │ │ │ │ o13 = R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o13 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_schur__Map.html │ │ │ @@ -110,28 +110,28 @@ │ │ │ │ │ │ o3 : SimplicialModule │ │ │
        │ │ │ 
        i4 : S2S = elapsedTime schurMap({2}, S)
│ │ │ - -- .142323s elapsed
│ │ │ + -- .119469s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o4 : SimplicialModule
        │ │ │
        │ │ │ 
        i5 : elapsedTime schurMap({2}, S, Degeneracy => true)
│ │ │ - -- .309043s elapsed
│ │ │ + -- .260382s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o5 : SimplicialModule
        │ │ │ @@ -208,15 +208,15 @@ │ │ │         		│ │ │ 
        i11 : K = koszulComplex vars Q;
        │ │ │
        │ │ │ 
        i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)
│ │ │ - -- 1.00439s elapsed
│ │ │ + -- .769551s elapsed
│ │ │  
│ │ │         1      9      36      74      81
│ │ │  o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o12 : Complex
        │ │ │ @@ -470,15 +470,15 @@ │ │ │ │ │ │ o30 : ComplexMap │ │ │
        │ │ │ 
        i31 : f = elapsedTime prune schurMap({2}, phi)
│ │ │ - -- .912527s elapsed
│ │ │ + -- .821635s elapsed
│ │ │  
│ │ │             1             1
│ │ │  o31 = 0 : Q  <--------- Q  : 0
│ │ │                  | 1 |
│ │ │  
│ │ │             5                                       9
│ │ │        1 : Q  <----------------------------------- Q  : 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,24 +42,24 @@
│ │ │ │        1      3      6      10      15
│ │ │ │  o3 = Q  <-- Q  <-- Q  <-- Q   <-- Q  <-- ...
│ │ │ │  
│ │ │ │       0      1      2      3       4
│ │ │ │  
│ │ │ │  o3 : SimplicialModule
│ │ │ │  i4 : S2S = elapsedTime schurMap({2}, S)
│ │ │ │ - -- .142323s elapsed
│ │ │ │ + -- .119469s elapsed
│ │ │ │  
│ │ │ │        1      6      21      55      120
│ │ │ │  o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4
│ │ │ │  
│ │ │ │  o4 : SimplicialModule
│ │ │ │  i5 : elapsedTime schurMap({2}, S, Degeneracy => true)
│ │ │ │ - -- .309043s elapsed
│ │ │ │ + -- .260382s elapsed
│ │ │ │  
│ │ │ │        1      6      21      55      120
│ │ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4
│ │ │ │  
│ │ │ │  o5 : SimplicialModule
│ │ │ │ @@ -106,15 +106,15 @@
│ │ │ │  i10 : Q = ZZ/101[a..c]
│ │ │ │  
│ │ │ │  o10 = Q
│ │ │ │  
│ │ │ │  o10 : PolynomialRing
│ │ │ │  i11 : K = koszulComplex vars Q;
│ │ │ │  i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)
│ │ │ │ - -- 1.00439s elapsed
│ │ │ │ + -- .769551s elapsed
│ │ │ │  
│ │ │ │         1      9      36      74      81
│ │ │ │  o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q
│ │ │ │  
│ │ │ │        0      1      2       3       4
│ │ │ │  
│ │ │ │  o12 : Complex
│ │ │ │ @@ -299,15 +299,15 @@
│ │ │ │  
│ │ │ │             1                   2
│ │ │ │        2 : Q  <--------------- Q  : 2
│ │ │ │                  {2} | 0 b |
│ │ │ │  
│ │ │ │  o30 : ComplexMap
│ │ │ │  i31 : f = elapsedTime prune schurMap({2}, phi)
│ │ │ │ - -- .912527s elapsed
│ │ │ │ + -- .821635s elapsed
│ │ │ │  
│ │ │ │             1             1
│ │ │ │  o31 = 0 : Q  <--------- Q  : 0
│ │ │ │                  | 1 |
│ │ │ │  
│ │ │ │             5                                       9
│ │ │ │        1 : Q  <----------------------------------- Q  : 1
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_tensor__L__E__S.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o7 : Complex
        │ │ │
        │ │ │ 
        i8 : hL = elapsedTime prune tensorLES(L,4)
│ │ │ - -- 2.28436s elapsed
│ │ │ + -- 1.54337s elapsed
│ │ │  
│ │ │                                                                                                                                                                                                                                                                                                     25      50      25
│ │ │  o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b | <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0 | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <-- cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q
│ │ │                                                                      | 0 b a |              | 0 b a |              | 0 0 b a |              | 0 0 b a 0 0 0 0 |              | 0   0   0 0 b a 0  0  |                                                                                                              
│ │ │       0                    1                    2     3                                                                                     | 0 0 0 0 b a 0 0 |              | ab2 a2b 0 0 0 0 b2 a2 |     9                              10                        11                             12      13      14
│ │ │                                                             4                      5                      6                                 | 0 0 0 0 0 0 b a |      
│ │ │                                                                                                                                                                     8
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,15 +114,15 @@
│ │ │ │        1      2      2      1
│ │ │ │  o7 = Q  <-- Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │       0      1      2      3
│ │ │ │  
│ │ │ │  o7 : Complex
│ │ │ │  i8 : hL = elapsedTime prune tensorLES(L,4)
│ │ │ │ - -- 2.28436s elapsed
│ │ │ │ + -- 1.54337s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  25      50      25
│ │ │ │  o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b |
│ │ │ │  <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0
│ │ │ │  | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <-
│ │ │ │  - cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/index.html
│ │ │ @@ -211,28 +211,28 @@
│ │ │  
│ │ │  o7 : SimplicialModule
        │ │ │
        │ │ │ 
        i8 : elapsedTime simplicialModule(K,6) --specify top degree 6
│ │ │ - -- .0671636s elapsed
│ │ │ + -- .0684915s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o8 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o8 : SimplicialModule
        │ │ │
        │ │ │ 
        i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true)
│ │ │ - -- .195242s elapsed
│ │ │ + -- .140675s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o9 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o9 : SimplicialModule
        │ │ │ ├── html2text {} │ │ │ │ @@ -135,24 +135,24 @@ │ │ │ │ 1 4 10 20 │ │ │ │ o7 = R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o7 : SimplicialModule │ │ │ │ i8 : elapsedTime simplicialModule(K,6) --specify top degree 6 │ │ │ │ - -- .0671636s elapsed │ │ │ │ + -- .0684915s elapsed │ │ │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ │ o8 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o8 : SimplicialModule │ │ │ │ i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true) │ │ │ │ - -- .195242s elapsed │ │ │ │ + -- .140675s elapsed │ │ │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ │ o9 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o9 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialPosets/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ aXNCb29sZWFu │ │ │ #:len=1246 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRGV0ZXJtaW5lIGlmIGEgcG9zZXQgaXMg │ │ │ YSBib29sZWFuIGFsZ2VicmEuIiwgImxpbmVudW0iID0+IDMzNywgSW5wdXRzID0+IHtTUEFOe1RU │ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ dW5pdmVyc2FsSWRlYWw= │ │ │ #:len=2182 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIHVuaXZlcnNhbCBy │ │ │ ZWFsaXphdGlvbiBpZGVhbCBvZiBhIG1hdHJvaWQiLCAibGluZW51bSIgPT4gMjE0MiwgSW5wdXRz │ │ ├── ./usr/share/doc/Macaulay2/SpaceCurves/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ aXNTbW9vdGhBQ01CZXR0aQ== │ │ │ #:len=902 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIHdoZXRoZXIgYSBCZXR0aSB0 │ │ │ YWJsZSBpcyB0aGF0IG9mIGEgc21vb3RoIEFDTSBjdXJ2ZSIsICJsaW5lbnVtIiA9PiAxMjY5LCBJ │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=47 │ │ │ TXVsdGlkaW1lbnNpb25hbE1hdHJpeCAqIE11bHRpZGltZW5zaW9uYWxNYXRyaXg= │ │ │ #:len=1779 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJvZHVjdCBvZiBtdWx0aWRpbWVuc2lv │ │ │ bmFsIG1hdHJpY2VzIiwgImxpbmVudW0iID0+IDE0MjUsIElucHV0cyA9PiB7U1BBTntUVHsiTSJ9 │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out │ │ │ @@ -3,15 +3,15 @@ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ o1 : List │ │ │ │ │ │ i2 : time degreeDeterminant n │ │ │ - -- used 9.3134e-05s (cpu); 8.2705e-05s (thread); 0s (gc) │ │ │ + -- used 6.5316e-05s (cpu); 5.9067e-05s (thread); 0s (gc) │ │ │ │ │ │ o2 = 6 │ │ │ │ │ │ i3 : M = genericMultidimensionalMatrix n; │ │ │ warning: clearing value of symbol x2 to allow access to subscripted variables based on it │ │ │ : debug with expression debug 1368 or with command line option --debug 1368 │ │ │ warning: clearing value of symbol x1 to allow access to subscripted variables based on it │ │ │ @@ -19,14 +19,14 @@ │ │ │ warning: clearing value of symbol x0 to allow access to subscripted variables based on it │ │ │ : debug with expression debug 6010 or with command line option --debug 6010 │ │ │ │ │ │ o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a ..a ] │ │ │ 0,0,0 1,2,1 │ │ │ │ │ │ i4 : time degree determinant M │ │ │ - -- used 0.0324525s (cpu); 0.0316018s (thread); 0s (gc) │ │ │ + -- used 0.195399s (cpu); 0.0620636s (thread); 0s (gc) │ │ │ │ │ │ o4 = {6} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out │ │ │ @@ -1,13 +1,13 @@ │ │ │ -- -*- M2-comint -*- hash: 17130321902108223178 │ │ │ │ │ │ i1 : (d,n) := (2,3); │ │ │ │ │ │ i2 : time Disc = denseDiscriminant(d,n) │ │ │ - -- used 0.44512s (cpu); 0.234802s (thread); 0s (gc) │ │ │ + -- used 0.462113s (cpu); 0.276856s (thread); 0s (gc) │ │ │ │ │ │ o2 = Disc │ │ │ │ │ │ o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |) │ │ │ | 0 0 0 1 1 2 0 0 1 0 | │ │ │ | 0 1 2 0 1 0 0 1 0 0 | │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out │ │ │ @@ -9,18 +9,18 @@ │ │ │ 2 │ │ │ c x x + c x + c x + c x + c ) │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ o1 : Sequence │ │ │ │ │ │ i2 : time denseResultant(f0,f1,f2); -- using Poisson formula │ │ │ - -- used 0.0911372s (cpu); 0.0911356s (thread); 0s (gc) │ │ │ + -- used 0.0982456s (cpu); 0.0982455s (thread); 0s (gc) │ │ │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula │ │ │ - -- used 0.298973s (cpu); 0.23937s (thread); 0s (gc) │ │ │ + -- used 0.387662s (cpu); 0.287586s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ - -- used 0.382249s (cpu); 0.323056s (thread); 0s (gc) │ │ │ + -- used 0.356397s (cpu); 0.307338s (thread); 0s (gc) │ │ │ │ │ │ i5 : assert(o2 == o3 and o3 == o4) │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2, │ │ │ ------------------------------------------------------------------------ │ │ │ 3}}}} │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ │ │ i2 : time det M │ │ │ - -- used 0.0864658s (cpu); 0.0851857s (thread); 0s (gc) │ │ │ + -- used 0.435044s (cpu); 0.146788s (thread); 0s (gc) │ │ │ │ │ │ o2 = 9698337990421512192 │ │ │ │ │ │ i3 : M = randomMultidimensionalMatrix(2,2,2,2,5) │ │ │ │ │ │ o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5, │ │ │ ------------------------------------------------------------------------ │ │ │ @@ -24,13 +24,13 @@ │ │ │ 7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6, │ │ │ ------------------------------------------------------------------------ │ │ │ 4, 8, 4, 2}}}}} │ │ │ │ │ │ o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ │ │ │ │ │ │ i4 : time det M │ │ │ - -- used 0.778868s (cpu); 0.689399s (thread); 0s (gc) │ │ │ + -- used 0.482915s (cpu); 0.482914s (thread); 0s (gc) │ │ │ │ │ │ o4 = 912984499996938980479447727885644530753184525786986940737407301278806287 │ │ │ 9257139493926586400187927813888 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Discriminant.out │ │ │ @@ -11,15 +11,15 @@ │ │ │ a x y z + a x y z + a x y z │ │ │ 1,1,1 1 1 1 1,2,0 1 2 0 1,2,1 1 2 1 │ │ │ │ │ │ o1 : ZZ[a ..a ][x ..x , y ..y , z ..z ] │ │ │ 0,0,0 1,2,1 0 1 0 2 0 1 │ │ │ │ │ │ i2 : time sparseDiscriminant f │ │ │ - -- used 3.23258s (cpu); 2.80646s (thread); 0s (gc) │ │ │ + -- used 2.70971s (cpu); 2.27788s (thread); 0s (gc) │ │ │ │ │ │ 2 │ │ │ o2 = a a a a a a - a a a a a - │ │ │ 0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0 0,1,0 0,2,1 1,0,0 1,0,1 1,1,0 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 2 │ │ │ a a a a + a a a a a - │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out │ │ │ @@ -1,11 +1,11 @@ │ │ │ -- -*- M2-comint -*- hash: 16228363821945730064 │ │ │ │ │ │ i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}}) │ │ │ - -- used 0.477656s (cpu); 0.408514s (thread); 0s (gc) │ │ │ + -- used 0.682852s (cpu); 0.476812s (thread); 0s (gc) │ │ │ │ │ │ o1 = Res │ │ │ │ │ │ o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |}) │ │ │ | 0 0 1 1 | | 1 0 1 2 | | 0 1 0 1 | │ │ │ │ │ │ i2 : QQ[c_(1,1)..c_(3,4)][x,y]; │ │ │ @@ -18,15 +18,15 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ c x*y + c x + c y + c ) │ │ │ 3,3 3,4 3,2 3,1 │ │ │ │ │ │ o3 : Sequence │ │ │ │ │ │ i4 : time Res(f,g,h) │ │ │ - -- used 0.00964993s (cpu); 0.00964869s (thread); 0s (gc) │ │ │ + -- used 0.011705s (cpu); 0.0117014s (thread); 0s (gc) │ │ │ │ │ │ 2 4 2 2 4 │ │ │ o4 = - c c c c c c c + c c c c c c + │ │ │ 1,2 1,3 1,4 2,1 2,2 2,3 3,1 1,2 1,3 2,1 2,2 2,4 3,1 │ │ │ ------------------------------------------------------------------------ │ │ │ 3 2 3 2 3 │ │ │ c c c c c c - 2c c c c c c c c + │ │ │ @@ -730,30 +730,30 @@ │ │ │ │ │ │ o4 : QQ[c ..c ] │ │ │ 1,1 3,4 │ │ │ │ │ │ i5 : assert(Res(f,g,h) == sparseResultant(f,g,h)) │ │ │ │ │ │ i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331); │ │ │ - -- used 0.0310848s (cpu); 0.0302486s (thread); 0s (gc) │ │ │ + -- used 0.272997s (cpu); 0.0714601s (thread); 0s (gc) │ │ │ │ │ │ o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331) │ │ │ | 0 1 0 1 | │ │ │ │ │ │ i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y]; │ │ │ │ │ │ i8 : (f,g,h) = (a_0 + a_1*x + a_2*y + a_3*x*y, b_0 + b_1*x + b_2*y + b_3*x*y, c_0 + c_1*x + c_2*y + c_3*x*y) │ │ │ │ │ │ o8 = (a x*y + a x + a y + a , b x*y + b x + b y + b , c x*y + c x + c y + c ) │ │ │ 3 1 2 0 3 1 2 0 3 1 2 0 │ │ │ │ │ │ o8 : Sequence │ │ │ │ │ │ i9 : time Res(f,g,h) │ │ │ - -- used 0.00326741s (cpu); 0.00326931s (thread); 0s (gc) │ │ │ + -- used 0.00397003s (cpu); 0.00396874s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 2 │ │ │ o9 = a b b c - a a b b c - a a b b c + a a b c - a b b c c - │ │ │ 3 1 2 0 2 3 1 3 0 1 3 2 3 0 1 2 3 0 3 0 2 0 1 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 2 │ │ │ a a b b c c + a a b c c + a a b b c c + a b b c c - a a b b c c + │ │ │ @@ -822,15 +822,15 @@ │ │ │ 2 │ │ │ c x x + c x + c x + c x + c ) │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ o11 : Sequence │ │ │ │ │ │ i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true)); │ │ │ - -- used 0.277694s (cpu); 0.202525s (thread); 0s (gc) │ │ │ + -- used 0.411406s (cpu); 0.213775s (thread); 0s (gc) │ │ │ │ │ │ i13 : quotientRemainder(UnmixedRes,MixedRes) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o13 = (b c - b b c c + b b c + b c c - 2b b c c - b b c c + b c , 0) │ │ │ 5 2 4 5 2 4 2 5 4 4 2 5 2 5 2 5 2 4 4 5 2 5 │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ │ │ │ o1 : List │ │ │
        │ │ │ 
        i2 : time degreeDeterminant n
│ │ │ - -- used 9.3134e-05s (cpu); 8.2705e-05s (thread); 0s (gc)
│ │ │ + -- used 6.5316e-05s (cpu); 5.9067e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
        │ │ │
        │ │ │ 
        i3 : M = genericMultidimensionalMatrix n;
│ │ │ @@ -103,15 +103,15 @@
│ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │                                                        0,0,0   1,2,1
        │ │ │
        │ │ │ 
        i4 : time degree determinant M
│ │ │ - -- used 0.0324525s (cpu); 0.0316018s (thread); 0s (gc)
│ │ │ + -- used 0.195399s (cpu); 0.0620636s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = {6}
│ │ │  
│ │ │  o4 : List
        │ │ │
        │ │ │ ├── html2text {} │ │ │ │ @@ -15,15 +15,15 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : n = {2,3,2} │ │ │ │ │ │ │ │ o1 = {2, 3, 2} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ i2 : time degreeDeterminant n │ │ │ │ - -- used 9.3134e-05s (cpu); 8.2705e-05s (thread); 0s (gc) │ │ │ │ + -- used 6.5316e-05s (cpu); 5.9067e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 6 │ │ │ │ i3 : M = genericMultidimensionalMatrix n; │ │ │ │ warning: clearing value of symbol x2 to allow access to subscripted variables │ │ │ │ based on it │ │ │ │ : debug with expression debug 1368 or with command line option -- │ │ │ │ debug 1368 │ │ │ │ @@ -35,15 +35,15 @@ │ │ │ │ based on it │ │ │ │ : debug with expression debug 6010 or with command line option -- │ │ │ │ debug 6010 │ │ │ │ │ │ │ │ o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a ..a ] │ │ │ │ 0,0,0 1,2,1 │ │ │ │ i4 : time degree determinant M │ │ │ │ - -- used 0.0324525s (cpu); 0.0316018s (thread); 0s (gc) │ │ │ │ + -- used 0.195399s (cpu); 0.0620636s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = {6} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_t_e_r_m_i_n_a_n_t_(_M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x_) -- hyperdeterminant of a │ │ │ │ multidimensional matrix │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ 
        i1 : (d,n) := (2,3);
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.44512s (cpu); 0.234802s (thread); 0s (gc)
│ │ │ + -- used 0.462113s (cpu); 0.276856s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = Disc
│ │ │  
│ │ │  o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |)
│ │ │                                                             | 0 0 0 1 1 2 0 0 1 0 |
│ │ │                                                             | 0 1 2 0 1 0 0 1 0 0 |
        │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -18,15 +18,15 @@ │ │ │ │ * Outputs: │ │ │ │ o for (d,n), this is the same as _s_p_a_r_s_e_D_i_s_c_r_i_m_i_n_a_n_t _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x │ │ │ │ ""ggeenneerriicc ppoollyynnoommiiaall ooff ddeeggrreeee dd iinn nn vvaarriiaabblleess"";; │ │ │ │ o for f, this is the same as _a_f_f_i_n_e_D_i_s_c_r_i_m_i_n_a_n_t(f). │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : (d,n) := (2,3); │ │ │ │ i2 : time Disc = denseDiscriminant(d,n) │ │ │ │ - -- used 0.44512s (cpu); 0.234802s (thread); 0s (gc) │ │ │ │ + -- used 0.462113s (cpu); 0.276856s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = Disc │ │ │ │ │ │ │ │ o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 │ │ │ │ 2 |) │ │ │ │ | 0 0 0 1 1 2 0 0 1 │ │ │ │ 0 | │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ @@ -95,27 +95,27 @@ │ │ │ │ │ │ o1 : Sequence │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.0911372s (cpu); 0.0911356s (thread); 0s (gc)
        │ │ │ + -- used 0.0982456s (cpu); 0.0982455s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.298973s (cpu); 0.23937s (thread); 0s (gc)
        │ │ │ + -- used 0.387662s (cpu); 0.287586s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.382249s (cpu); 0.323056s (thread); 0s (gc)
        │ │ │ + -- used 0.356397s (cpu); 0.307338s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i5 : assert(o2 == o3 and o3 == o4)
        │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -28,20 +28,20 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 │ │ │ │ c x x + c x + c x + c x + c ) │ │ │ │ 4 1 2 2 2 3 1 1 2 0 │ │ │ │ │ │ │ │ o1 : Sequence │ │ │ │ i2 : time denseResultant(f0,f1,f2); -- using Poisson formula │ │ │ │ - -- used 0.0911372s (cpu); 0.0911356s (thread); 0s (gc) │ │ │ │ + -- used 0.0982456s (cpu); 0.0982455s (thread); 0s (gc) │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay │ │ │ │ formula │ │ │ │ - -- used 0.298973s (cpu); 0.23937s (thread); 0s (gc) │ │ │ │ + -- used 0.387662s (cpu); 0.287586s (thread); 0s (gc) │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ │ - -- used 0.382249s (cpu); 0.323056s (thread); 0s (gc) │ │ │ │ + -- used 0.356397s (cpu); 0.307338s (thread); 0s (gc) │ │ │ │ i5 : assert(o2 == o3 and o3 == o4) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_p_a_r_s_e_R_e_s_u_l_t_a_n_t -- sparse resultant (A-resultant) │ │ │ │ * _a_f_f_i_n_e_R_e_s_u_l_t_a_n_t -- affine resultant │ │ │ │ * _d_e_n_s_e_D_i_s_c_r_i_m_i_n_a_n_t -- dense discriminant (classical discriminant) │ │ │ │ * _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x -- exponents in one or more polynomials │ │ │ │ * _g_e_n_e_r_i_c_L_a_u_r_e_n_t_P_o_l_y_n_o_m_i_a_l_s -- generic (Laurent) polynomials │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ @@ -89,15 +89,15 @@ │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i2 : time det M
│ │ │ - -- used 0.0864658s (cpu); 0.0851857s (thread); 0s (gc)
│ │ │ + -- used 0.435044s (cpu); 0.146788s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : time det M
│ │ │ - -- used 0.778868s (cpu); 0.689399s (thread); 0s (gc)
│ │ │ + -- used 0.482915s (cpu); 0.482914s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 912984499996938980479447727885644530753184525786986940737407301278806287
│ │ │       9257139493926586400187927813888
        │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -25,15 +25,15 @@ │ │ │ │ │ │ │ │ o1 = {{{{8, 1}, {3, 7}}, {{8, 3}, {3, 7}}}, {{{8, 8}, {5, 7}}, {{8, 5}, {2, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3}}}} │ │ │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ i2 : time det M │ │ │ │ - -- used 0.0864658s (cpu); 0.0851857s (thread); 0s (gc) │ │ │ │ + -- used 0.435044s (cpu); 0.146788s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = 9698337990421512192 │ │ │ │ i3 : M = randomMultidimensionalMatrix(2,2,2,2,5) │ │ │ │ │ │ │ │ o3 = {{{{{6, 3, 6, 8, 6}, {9, 3, 7, 6, 9}}, {{6, 2, 6, 0, 2}, {6, 9, 3, 5, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 6}}}, {{{3, 5, 7, 7, 9}, {4, 5, 0, 4, 3}}, {{1, 8, 9, 1, 2}, {9, 6, 6, │ │ │ │ @@ -42,15 +42,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 7, 4, 5}}}, {{{4, 0, 1, 4, 4}, {2, 6, 1, 1, 4}}, {{5, 4, 9, 7, 4}, {6, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 4, 8, 4, 2}}}}} │ │ │ │ │ │ │ │ o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ │ │ │ │ i4 : time det M │ │ │ │ - -- used 0.778868s (cpu); 0.689399s (thread); 0s (gc) │ │ │ │ + -- used 0.482915s (cpu); 0.482914s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 912984499996938980479447727885644530753184525786986940737407301278806287 │ │ │ │ 9257139493926586400187927813888 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x -- the class of all multidimensional matrices │ │ │ │ * _d_e_g_r_e_e_D_e_t_e_r_m_i_n_a_n_t -- degree of the hyperdeterminant of a generic │ │ │ │ multidimensional matrix │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Discriminant.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o1 : ZZ[a ..a ][x ..x , y ..y , z ..z ] │ │ │ 0,0,0 1,2,1 0 1 0 2 0 1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : time sparseDiscriminant f
│ │ │ - -- used 3.23258s (cpu); 2.80646s (thread); 0s (gc)
│ │ │ + -- used 2.70971s (cpu); 2.27788s (thread); 0s (gc)
│ │ │  
│ │ │                                                     2                        
│ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │              2     2                                2            
│ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       a     x y z  + a     x y z  + a     x y z
│ │ │ │        1,1,1 1 1 1    1,2,0 1 2 0    1,2,1 1 2 1
│ │ │ │  
│ │ │ │  o1 : ZZ[a     ..a     ][x ..x , y ..y , z ..z ]
│ │ │ │           0,0,0   1,2,1   0   1   0   2   0   1
│ │ │ │  i2 : time sparseDiscriminant f
│ │ │ │ - -- used 3.23258s (cpu); 2.80646s (thread); 0s (gc)
│ │ │ │ + -- used 2.70971s (cpu); 2.27788s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                     2
│ │ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2     2                                2
│ │ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │          
      Description
      
│ │ │          
      Alternatively, one can apply the method directly to the list of Laurent polynomials $f_0,\ldots,f_n$. In this case, the matrices $A_0,\ldots,A_n$ are automatically determined by exponentsMatrix. If you want require that $A_0=\cdots=A_n$, then use the option Unmixed=>true (this could be faster). Below we consider some examples.
      
│ │ │          
      In the first example, we calculate the sparse (mixed) resultant associated to the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$. Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{(1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{(2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o3 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
      
│ │ │                
      i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ - -- used 0.477656s (cpu); 0.408514s (thread); 0s (gc)
│ │ │ + -- used 0.682852s (cpu); 0.476812s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = Res
│ │ │  
│ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1 2 2 |, | 0 0 1 1 |})
│ │ │                                                              | 0 0 1 1 |  | 1 0 1 2 |  | 0 1 0 1 |
      
│ │ │              
      
│ │ │                
      i4 : time Res(f,g,h)
│ │ │ - -- used 0.00964993s (cpu); 0.00964869s (thread); 0s (gc)
│ │ │ + -- used 0.011705s (cpu); 0.0117014s (thread); 0s (gc)
│ │ │  
│ │ │          2                       4      2   2               4    
│ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        3       2       3               2                   3        
│ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ @@ -830,15 +830,15 @@
│ │ │            
      
│ │ │          
      In the second example, we calculate the sparse unmixed resultant associated to the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0 + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over $\mathbb{Z}/3331$.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,15 +68,15 @@
│ │ │ │  
│ │ │ │  o4 = | 0 1 |
│ │ │ │       | 2 3 |
│ │ │ │       | 4 |
│ │ │ │  
│ │ │ │  o4 : YoungTableau
│ │ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ │ - -- used 0.00150749s (cpu); 0.00150181s (thread); 0s (gc)
│ │ │ │ + -- used 0.00146204s (cpu); 0.00145852s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 2
│ │ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │ @@ -88,15 +88,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o5 : R
│ │ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ │ - -- used 0.0012075s (cpu); 0.00120561s (thread); 0s (gc)
│ │ │ │ + -- used 0.00133065s (cpu); 0.00133051s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        3 2          2 3      3     2        3 2    3   2      2   3
│ │ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3 2        3 2        2 3        2 3        3   2        3 2
│ │ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ │ @@ -108,15 +108,15 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │ │  
│ │ │ │  o6 : R
│ │ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ │ - -- used 0.00201494s (cpu); 0.00201272s (thread); 0s (gc)
│ │ │ │ + -- used 0.00217066s (cpu); 0.0021715s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )
│ │ │ │  (x )(x ))
│ │ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4
│ │ │ │  2   0
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  2*x_2*x_4-x_3*x_4-
│ │ │ │  2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-
│ │ │ │  3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i3 : time toGrass X
│ │ │ │ - -- used 4.10188s (cpu); 2.6702s (thread); 0s (gc)
│ │ │ │ + -- used 4.25112s (cpu); 2.43041s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces
│ │ │ │  of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │ │  i3 : time toGrass X
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 4.22961s (cpu); 2.84603s (thread); 0s (gc)
│ │ │ │ + -- used 4.40074s (cpu); 2.56707s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0310848s (cpu); 0.0302486s (thread); 0s (gc)
│ │ │ + -- used 0.272997s (cpu); 0.0714601s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
      
│ │ │              
      
│ │ │ @@ -854,15 +854,15 @@
│ │ │  
│ │ │  o8 : Sequence
│ │ │              
      
│ │ │                
      i9 : time Res(f,g,h)
│ │ │ - -- used 0.00326741s (cpu); 0.00326931s (thread); 0s (gc)
│ │ │ + -- used 0.00397003s (cpu); 0.00396874s (thread); 0s (gc)
│ │ │  
│ │ │        2     2            2            2        2 2    2          
│ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2                       2                         
│ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ @@ -943,15 +943,15 @@
│ │ │  
│ │ │  o11 : Sequence
      
│ │ │              
      
│ │ │                
      i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.277694s (cpu); 0.202525s (thread); 0s (gc)
      
│ │ │ + -- used 0.411406s (cpu); 0.213775s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │  
│ │ │          2 2                   2    2                               2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  In the first example, we calculate the sparse (mixed) resultant associated to
│ │ │ │  the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$.
│ │ │ │  Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{
│ │ │ │  (1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{
│ │ │ │  (2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
│ │ │ │  i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},
│ │ │ │  {1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ │ - -- used 0.477656s (cpu); 0.408514s (thread); 0s (gc)
│ │ │ │ + -- used 0.682852s (cpu); 0.476812s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = Res
│ │ │ │  
│ │ │ │  o1 : SparseResultant (sparse mixed resultant associated to {| 0 1 1 2 |, | 0 1
│ │ │ │  2 2 |, | 0 0 1 1 |})
│ │ │ │                                                              | 0 0 1 1 |  | 1 0
│ │ │ │  1 2 |  | 0 1 0 1 |
│ │ │ │ @@ -55,15 +55,15 @@
│ │ │ │         1,3       1,2       1,4     1,1   2,2        2,3       2,4     2,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       c   x*y + c   x + c   y + c   )
│ │ │ │        3,3       3,4     3,2     3,1
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : time Res(f,g,h)
│ │ │ │ - -- used 0.00964993s (cpu); 0.00964869s (thread); 0s (gc)
│ │ │ │ + -- used 0.011705s (cpu); 0.0117014s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2                       4      2   2               4
│ │ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        3       2       3               2                   3
│ │ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ │ @@ -771,29 +771,29 @@
│ │ │ │  In the second example, we calculate the sparse unmixed resultant associated to
│ │ │ │  the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials
│ │ │ │  $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0
│ │ │ │  + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over
│ │ │ │  $\mathbb{Z}/3331$.
│ │ │ │  i6 : time Res = sparseResultant(matrix{{0,0,1,1},
│ │ │ │  {0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ │ - -- used 0.0310848s (cpu); 0.0302486s (thread); 0s (gc)
│ │ │ │ + -- used 0.272997s (cpu); 0.0714601s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over
│ │ │ │  ZZ/3331)
│ │ │ │                                                               | 0 1 0 1 |
│ │ │ │  i7 : ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
│ │ │ │  i8 : (f,g,h) = (a_0 + a_1*x + a_2*y + a_3*x*y, b_0 + b_1*x + b_2*y + b_3*x*y,
│ │ │ │  c_0 + c_1*x + c_2*y + c_3*x*y)
│ │ │ │  
│ │ │ │  o8 = (a x*y + a x + a y + a , b x*y + b x + b y + b , c x*y + c x + c y + c )
│ │ │ │         3       1     2     0   3       1     2     0   3       1     2     0
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : time Res(f,g,h)
│ │ │ │ - -- used 0.00326741s (cpu); 0.00326931s (thread); 0s (gc)
│ │ │ │ + -- used 0.00397003s (cpu); 0.00396874s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        2     2            2            2        2 2    2
│ │ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                           2                       2
│ │ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ │ @@ -863,15 +863,15 @@
│ │ │ │                    2
│ │ │ │        c x x  + c x  + c x  + c x  + c )
│ │ │ │         4 1 2    2 2    3 1    1 2    0
│ │ │ │  
│ │ │ │  o11 : Sequence
│ │ │ │  i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant
│ │ │ │  (f,g,h,Unmixed=>true));
│ │ │ │ - -- used 0.277694s (cpu); 0.202525s (thread); 0s (gc)
│ │ │ │ + -- used 0.411406s (cpu); 0.213775s (thread); 0s (gc)
│ │ │ │  i13 : quotientRemainder(UnmixedRes,MixedRes)
│ │ │ │  
│ │ │ │          2 2                   2    2                               2 2
│ │ │ │  o13 = (b c  - b b c c  + b b c  + b c c  - 2b b c c  - b b c c  + b c , 0)
│ │ │ │          5 2    4 5 2 4    2 5 4    4 2 5     2 5 2 5    2 4 4 5    2 5
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  c2NodXJQb2x5bm9taWFsKC4uLixBc0V4cHJlc3Npb249Pi4uLik=
│ │ │  #:len=287
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzMwNSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2NodXJQb2x5bm9taWFsLEFzRXhwcmVzc2lvbl0s
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o4 = | 0 1 |
│ │ │       | 2 3 |
│ │ │       | 4 |
│ │ │  
│ │ │  o4 : YoungTableau
│ │ │  
│ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00150749s (cpu); 0.00150181s (thread); 0s (gc)
│ │ │ + -- used 0.00146204s (cpu); 0.00145852s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -46,15 +46,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o5 : R
│ │ │  
│ │ │  i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0012075s (cpu); 0.00120561s (thread); 0s (gc)
│ │ │ + -- used 0.00133065s (cpu); 0.00133051s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -67,15 +67,15 @@
│ │ │          2   3    2     3    2     3      2   3        2 3        2 3
│ │ │       x x x x  - x x x x  - x x x x  + x x x x  - x x x x  + x x x x
│ │ │        0 1 3 4    0 2 3 4    1 2 3 4    0 2 3 4    0 1 3 4    1 2 3 4
│ │ │  
│ │ │  o6 : R
│ │ │  
│ │ │  i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00201494s (cpu); 0.00201272s (thread); 0s (gc)
│ │ │ + -- used 0.00217066s (cpu); 0.0021715s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : tal := tally apply (H,h->conjugacyClass h);
│ │ │  
│ │ │  i4 : partis = partitions 6;
│ │ │  
│ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.425094s (cpu); 0.317058s (thread); 0s (gc)
│ │ │ + -- used 0.426326s (cpu); 0.346572s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out
│ │ │ @@ -20,15 +20,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.815181s (cpu); 0.539564s (thread); 0s (gc)
│ │ │ + -- used 0.784693s (cpu); 0.575895s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │  o4 = HashTable{{0, 1, 2, 3, 4, 5} => HashTable{0 => - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        }
│ │ │                                                        3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6
│ │ │                                                      2 3 2 2     4 2 3 2     2 2 2 3     4 3 2   2   2 2 3   2   2 3   2 2   2   3 2 2   2 2   3 2   4   2 3 2   2 2 2   3   4 2   2 3   2   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     1 3 2 2     1 2 3 2     2 3 2 2     1 3 2 2     2 2 3 2     1 2 3 2     1 3 2 2     2 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 3 2     2 2 2 3     4 2 2 3     2 2 2 3     2 2 2 3     4 2 2 3     2 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 3 2   2   1 2 3   2   2 3 2   2   1 3 2   2   2 2 3   2   1 2 3   2   1 3 2   2   2 3 2   2   1 3 2   2   1 2 3   2   2 2 3   2   1 2 3   2   2 3   2 2   2   3 2 2   4 3   2 2   2 3   2 2   4   3 2 2   2   3 2 2   2 3   2 2   4 3   2 2   2 3   2 2   2   3 2 2   4   3 2 2   2   3 2 2   1 2   3 2   1   2 3 2   2 2   3 2   1 2   3 2   2   2 3 2   1   2 3 2   1 2   3 2   2 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   2 2 2   3   4 2 2   3   2 2 2   3   2 2 2   3   4 2 2   3   2 2 2   3   1 2   2 3   1   2 2 3   2 2   2 3   1 2   2 3   2   2 2 3   1   2 2 3   1 2   2 3   2 2   2 3   1 2   2 3   1   2 2 3   2   2 2 3   1   2 2 3
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html
│ │ │ @@ -130,15 +130,15 @@
│ │ │  
│ │ │  o4 : YoungTableau
      
│ │ │              
      
│ │ │                
      i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00150749s (cpu); 0.00150181s (thread); 0s (gc)
│ │ │ + -- used 0.00146204s (cpu); 0.00145852s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -154,15 +154,15 @@
│ │ │  
│ │ │  o5 : R
      
│ │ │              
      
│ │ │                
      i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0012075s (cpu); 0.00120561s (thread); 0s (gc)
│ │ │ + -- used 0.00133065s (cpu); 0.00133051s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o6 : R
      
│ │ │              
      
│ │ │                
      i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00201494s (cpu); 0.00201272s (thread); 0s (gc)
│ │ │ + -- used 0.00217066s (cpu); 0.0021715s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )(x )(x ))
│ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4   2   0
│ │ │  
│ │ │  o7 : Expression of class Product
      
│ │ │              
      
│ │ │                
      i4 : partis = partitions 6;
      
│ │ │              
      
│ │ │                
      i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity(tal,p))
│ │ │ - -- used 0.425094s (cpu); 0.317058s (thread); 0s (gc)
│ │ │ + -- used 0.426326s (cpu); 0.346572s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │                 Partition{2, 2, 2} => 1
│ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │                 Partition{3, 2, 1} => 0
│ │ │ ├── html2text {}
│ │ │ │ @@ -63,15 +63,15 @@
│ │ │ │  representations of $H$ in each irreducible representation of $S_6$. We take
│ │ │ │  into account that there are multiple copies of each representation by
│ │ │ │  multiplying the values with the number of copies which is given by the
│ │ │ │  hookLengthFormula.
│ │ │ │  i4 : partis = partitions 6;
│ │ │ │  i5 : time multi = hashTable apply (partis, p-> p=> representationMultiplicity
│ │ │ │  (tal,p))
│ │ │ │ - -- used 0.425094s (cpu); 0.317058s (thread); 0s (gc)
│ │ │ │ + -- used 0.426326s (cpu); 0.346572s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │ │                 Partition{2, 2, 2} => 1
│ │ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.815181s (cpu); 0.539564s (thread); 0s (gc)
      
│ │ │ + -- used 0.784693s (cpu); 0.575895s (thread); 0s (gc)
│ │ │              
      
│ │ │                
      i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -56,15 +56,15 @@
│ │ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ │ - -- used 0.815181s (cpu); 0.539564s (thread); 0s (gc)
│ │ │ │ + -- used 0.784693s (cpu); 0.575895s (thread); 0s (gc)
│ │ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │ │  
│ │ │ │                                                        2 2 2       4 2   2     2
│ │ │ │  2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2
│ │ │ │  1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2
│ │ │ │  1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2
│ │ │ │  2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  ZGV0ZWN0Q29uZ3J1ZW5jZSguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=296
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTUsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2RldGVjdENvbmdydWVuY2UsVmVyYm9zZV0sImRldGVj
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out
│ │ │ @@ -45,15 +45,15 @@
│ │ │    -- top 1, degrees: 1^1 2^3 3^3 
│ │ │    -- top 2, degrees: 2^4 3^3 
│ │ │    -- top 3, degrees: 2^3 3^4 
│ │ │    -- top 4, degrees: 2^3 3^3 4^1 
│ │ │    -- top 5, degrees: 2^3 3^3 5^1 
│ │ │    -- top 6, degrees: 2^3 3^3 6^1 
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Complete intersection of 3 quadrics in PP^7
│ │ │       of discriminant 31 = det| 8 1 |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Cubic__Fourfold_rp.out
│ │ │ @@ -42,15 +42,15 @@
│ │ │  -- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
│ │ │    -- top 1, degrees: 1^4 2^1 
│ │ │    -- top 2, degrees: 1^3 2^2 
│ │ │    -- top 3, degrees: 1^3 2^1 3^1 
│ │ │    -- top 4, degrees: 1^3 2^1 4^1 
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated K3 successfully completed in 1 second (cpu: 1 second)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Special cubic fourfold of discriminant 14
│ │ │       containing a rational surface of degree 4 and sectional genus 0
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -48,15 +48,15 @@
│ │ │    -- top 3, degrees: 1^1 2^4 3^2 
│ │ │    -- top 4, degrees: 1^1 2^4 4^2 
│ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │  -- computing normalization of the surface image
│ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ + ✦ associated K3 successfully completed in 4 seconds (cpu: 6 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface of degree 2 and sectional genus 0
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -8,28 +8,28 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.79687s (cpu); 2.14089s (thread); 0s (gc)
│ │ │ + -- used 2.83976s (cpu); 1.75497s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │  
│ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │  
│ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │  
│ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.325742s (cpu); 0.325011s (thread); 0s (gc)
│ │ │ + -- used 0.385969s (cpu); 0.279928s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │       containing a surface of degree 9 and sectional genus 2
│ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 15.0623s (cpu); 7.94045s (thread); 0s (gc)
│ │ │ + -- used 15.4087s (cpu); 6.63493s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │  
│ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ @@ -29,15 +29,15 @@
│ │ │  i5 : p := point Y -- random point on Y
│ │ │  
│ │ │  o5 = point of coordinates [14360, -1933, -494, -6471, -10457, -2246, -11879, -12725, 1]
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │  
│ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.828054s (cpu); 0.576108s (thread); 0s (gc)
│ │ │ + -- used 0.642364s (cpu); 0.358576s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │  
│ │ │  i7 : S = surface X;
│ │ │  
│ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.427896s (cpu); 0.253136s (thread); 0s (gc)
│ │ │ + -- used 0.380407s (cpu); 0.208817s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Cubic__Fourfold_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = cubicFourfold V;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.7594s (cpu); 0.447905s (thread); 0s (gc)
│ │ │ + -- used 0.618736s (cpu); 0.375246s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │  
│ │ │  i3 : X = gushelMukaiFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 4.53348s (cpu); 2.63125s (thread); 0s (gc)
│ │ │ + -- used 3.19399s (cpu); 2.27114s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i3 : ? X
│ │ │  
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
│ │ │  
│ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 0.850872s (cpu); 0.559168s (thread); 0s (gc)
│ │ │ + -- used 0.879995s (cpu); 0.557429s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_polarized__K3surface.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- surface was already irreducible
│ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 21 seconds (cpu: 23 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 11 seconds (cpu: 16 seconds)
│ │ │  
│ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -108,16 +108,16 @@
│ │ │  -- computing p2^*(H_PP^2)
│ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │    -- computing image on K3 surface...
│ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │  -- constructing lattice polarization...
│ │ │  -- verifying self-intersection of the curve...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ --- total time (K3 surface + polarization): 25 seconds (cpu: 26 seconds)
│ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 4 seconds)
│ │ │ +-- total time (K3 surface + polarization): 14 seconds (cpu: 20 seconds)
│ │ │  
│ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice intersection matrix on Ũ: | 14 7 |
│ │ │ @@ -205,15 +205,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- removing 1 components of degrees {3}
│ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 7 seconds (cpu: 7 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 4 seconds (cpu: 8 seconds)
│ │ │  
│ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -226,15 +226,15 @@
│ │ │  -- available strategies: "SpecialCurve", "MapFromW", "MapFromU", "MapFromW-Virtual", "MapFromU-Virtual"
│ │ │  -- special curves already detected on U
│ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1 
│ │ │  -- constructing lattice polarization...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ --- total time (K3 surface + polarization): 7 seconds (cpu: 7 seconds)
│ │ │ +-- total time (K3 surface + polarization): 4 seconds (cpu: 8 seconds)
│ │ │  
│ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice intersection matrix on Ũ: | 6 5  |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : x := gens ring PP_(ZZ/33331)^8;
│ │ │  
│ │ │  i2 : X = gushelMukaiFourfold(ideal(x_6-x_7, x_5, x_3-x_4, x_1, x_0-x_4, x_2*x_7-x_4*x_8), ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │ - -- used 4.10188s (cpu); 2.6702s (thread); 0s (gc)
│ │ │ + -- used 4.25112s (cpu); 2.43041s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : X = projectiveVariety ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8);
│ │ │  
│ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.22961s (cpu); 2.84603s (thread); 0s (gc)
│ │ │ + -- used 4.40074s (cpu); 2.56707s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = cubicFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time f = unirationalParametrization X;
│ │ │ - -- used 0.856935s (cpu); 0.482912s (thread); 0s (gc)
│ │ │ + -- used 0.944796s (cpu); 0.595987s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │  
│ │ │  i5 : degreeSequence f
│ │ │  
│ │ │  o5 = {[10]}
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │    -- top 1, degrees: 1^1 2^3 3^3 
│ │ │    -- top 2, degrees: 2^4 3^3 
│ │ │    -- top 3, degrees: 2^3 3^4 
│ │ │    -- top 4, degrees: 2^3 3^3 4^1 
│ │ │    -- top 5, degrees: 2^3 3^3 5^1 
│ │ │    -- top 6, degrees: 2^3 3^3 6^1 
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │    -- top 1, degrees: 1^1 2^3 3^3
│ │ │ │    -- top 2, degrees: 2^4 3^3
│ │ │ │    -- top 3, degrees: 2^3 3^4
│ │ │ │    -- top 4, degrees: 2^3 3^3 4^1
│ │ │ │    -- top 5, degrees: 2^3 3^3 5^1
│ │ │ │    -- top 6, degrees: 2^3 3^3 6^1
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Complete intersection of 3 quadrics in PP^7
│ │ │ │       of discriminant 31 = det| 8 1 |
│ │ │ │                               | 1 4 |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Cubic__Fourfold_rp.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  -- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
│ │ │    -- top 1, degrees: 1^4 2^1 
│ │ │    -- top 2, degrees: 1^3 2^2 
│ │ │    -- top 3, degrees: 1^3 2^1 3^1 
│ │ │    -- top 4, degrees: 1^3 2^1 4^1 
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated K3 successfully completed in 1 second (cpu: 1 second)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │  interpolation
│ │ │ │    -- top 1, degrees: 1^4 2^1
│ │ │ │    -- top 2, degrees: 1^3 2^2
│ │ │ │    -- top 3, degrees: 1^3 2^1 3^1
│ │ │ │    -- top 4, degrees: 1^3 2^1 4^1
│ │ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ │ + ✦ associated K3 successfully completed in 1 second (cpu: 1 second)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Special cubic fourfold of discriminant 14
│ │ │ │       containing a rational surface of degree 4 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degree 2
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │    -- top 3, degrees: 1^1 2^4 3^2 
│ │ │    -- top 4, degrees: 1^1 2^4 4^2 
│ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │  -- computing normalization of the surface image
│ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ + ✦ associated K3 successfully completed in 4 seconds (cpu: 6 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -84,15 +84,15 @@
│ │ │ │    -- top 3, degrees: 1^1 2^4 3^2
│ │ │ │    -- top 4, degrees: 1^1 2^4 4^2
│ │ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │ │  -- computing normalization of the surface image
│ │ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ │ + ✦ associated K3 successfully completed in 4 seconds (cpu: 6 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees 1^5 2^1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
      
│ │ │              
      
│ │ │                
      i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.79687s (cpu); 2.14089s (thread); 0s (gc)
│ │ │ + -- used 2.83976s (cpu); 1.75497s (thread); 0s (gc)
│ │ │  number lines contained in the image of the cubic map and passing through a general point: 8
│ │ │  number 2-secant lines = 7
│ │ │  number 5-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
      
│ │ │              
      
│ │ │                
      i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ - -- used 0.325742s (cpu); 0.325011s (thread); 0s (gc)
│ │ │ + -- used 0.385969s (cpu); 0.279928s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
      
│ │ │              
      
│ │ │                
      i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 3.79687s (cpu); 2.14089s (thread); 0s (gc)
│ │ │ │ + -- used 2.83976s (cpu); 1.75497s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the cubic map and passing through a
│ │ │ │  general point: 8
│ │ │ │  number 2-secant lines = 7
│ │ │ │  number 5-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 5-secant conics to surface in PP^5
│ │ │ │  i4 : p := point ambient X -- random point on P^5
│ │ │ │  
│ │ │ │  o4 = point of coordinates [15092, -9738, -3620, -15181, 12688, 1]
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │ │  i5 : time C = f p; -- 5-secant conic to the surface
│ │ │ │ - -- used 0.325742s (cpu); 0.325011s (thread); 0s (gc)
│ │ │ │ + -- used 0.385969s (cpu); 0.279928s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree
│ │ │ │  (C * surface X) == 5 and isSubset(p, C))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_t_e_c_t_C_o_n_g_r_u_e_n_c_e_(_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_,_Z_Z_) -- detect and return a
│ │ │ │        congruence of (2e-1)-secant curves of degree e inside a del Pezzo
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │              
      
│ │ │                
      i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 15.0623s (cpu); 7.94045s (thread); 0s (gc)
│ │ │ + -- used 15.4087s (cpu); 6.63493s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
      
│ │ │              
      
│ │ │                
      i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.828054s (cpu); 0.576108s (thread); 0s (gc)
│ │ │ + -- used 0.642364s (cpu); 0.358576s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
      
│ │ │              
      
│ │ │                
      i7 : S = surface X;
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 20
│ │ │ │       containing a surface of degree 9 and sectional genus 2
│ │ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 15.0623s (cpu); 7.94045s (thread); 0s (gc)
│ │ │ │ + -- used 15.4087s (cpu); 6.63493s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the quadratic map and passing through a
│ │ │ │  general point: 7
│ │ │ │  number 1-secant lines = 6
│ │ │ │  number 3-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ │ @@ -52,15 +52,15 @@
│ │ │ │  i5 : p := point Y -- random point on Y
│ │ │ │  
│ │ │ │  o5 = point of coordinates [14360, -1933, -494, -6471, -10457, -2246, -11879, -
│ │ │ │  12725, 1]
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ │ - -- used 0.828054s (cpu); 0.576108s (thread); 0s (gc)
│ │ │ │ + -- used 0.642364s (cpu); 0.358576s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │ │  i7 : S = surface X;
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ │ │  i8 : assert(dim C == 1 and degree C == 2 and dim(C*S) == 0 and degree(C*S) == 3
│ │ │ │  and isSubset(p,C) and isSubset(C,Y))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.427896s (cpu); 0.253136s (thread); 0s (gc)
│ │ │ + -- used 0.380407s (cpu); 0.208817s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │  i1 : K = ZZ/33331; S = PP_K^(1,5);
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ │ - -- used 0.427896s (cpu); 0.253136s (thread); 0s (gc)
│ │ │ │ + -- used 0.380407s (cpu); 0.208817s (thread); 0s (gc)
│ │ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3
│ │ │ │  hypersurfaces of degree 2
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 32
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Cubic__Fourfold_rp.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.7594s (cpu); 0.447905s (thread); 0s (gc)
│ │ │ + -- used 0.618736s (cpu); 0.375246s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = cubicFourfold V;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 0.7594s (cpu); 0.447905s (thread); 0s (gc)
│ │ │ │ + -- used 0.618736s (cpu); 0.375246s (thread); 0s (gc)
│ │ │ │  S: Veronese surface in PP^5
│ │ │ │  X: smooth cubic hypersurface in PP^5
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 27
│ │ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 4.53348s (cpu); 2.63125s (thread); 0s (gc)
│ │ │ + -- used 3.19399s (cpu); 2.27114s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │ │  i3 : X = gushelMukaiFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 4.53348s (cpu); 2.63125s (thread); 0s (gc)
│ │ │ │ + -- used 3.19399s (cpu); 2.27114s (thread); 0s (gc)
│ │ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │ │  X: GM fourfold containing S
│ │ │ │  Y: del Pezzo fivefold containing X
│ │ │ │  h^1(N_{S,Y}) = 0
│ │ │ │  h^0(N_{S,Y}) = 11
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
      
│ │ │              
      
│ │ │                
      i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 0.850872s (cpu); 0.559168s (thread); 0s (gc)
│ │ │ + -- used 0.879995s (cpu); 0.557429s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 4-dimensional subvariety of PP^9
│ │ │ │  i3 : ? X
│ │ │ │  
│ │ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │ │       1^2 2^5
│ │ │ │  i4 : time parametrizeFanoFourfold X
│ │ │ │ - -- used 0.850872s (cpu); 0.559168s (thread); 0s (gc)
│ │ │ │ + -- used 0.879995s (cpu); 0.557429s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7
│ │ │ │  hypersurfaces of degrees 1^2 2^5
│ │ │ │       dominance: true
│ │ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html
│ │ │ @@ -176,15 +176,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- surface was already irreducible
│ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 21 seconds (cpu: 23 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 11 seconds (cpu: 16 seconds)
│ │ │  
│ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -215,16 +215,16 @@
│ │ │  -- computing p2^*(H_PP^2)
│ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │    -- computing image on K3 surface...
│ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │  -- constructing lattice polarization...
│ │ │  -- verifying self-intersection of the curve...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ --- total time (K3 surface + polarization): 25 seconds (cpu: 26 seconds)
│ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 4 seconds)
│ │ │ +-- total time (K3 surface + polarization): 14 seconds (cpu: 20 seconds)
│ │ │  
│ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice intersection matrix on Ũ: | 14 7 |
│ │ │ @@ -327,15 +327,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- removing 1 components of degrees {3}
│ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 7 seconds (cpu: 7 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 4 seconds (cpu: 8 seconds)
│ │ │  
│ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -351,15 +351,15 @@
│ │ │  -- available strategies: "SpecialCurve", "MapFromW", "MapFromU", "MapFromW-Virtual", "MapFromU-Virtual"
│ │ │  -- special curves already detected on U
│ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1 
│ │ │  -- constructing lattice polarization...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ --- total time (K3 surface + polarization): 7 seconds (cpu: 7 seconds)
│ │ │ +-- total time (K3 surface + polarization): 4 seconds (cpu: 8 seconds)
│ │ │  
│ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice intersection matrix on Ũ: | 6 5  |
│ │ │ ├── html2text {}
│ │ │ │ @@ -119,15 +119,15 @@
│ │ │ │  -- analyzing the base locus of the inverse map...
│ │ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │ │  -- projecting to PP^3 for surface decomposition
│ │ │ │    -- surface was already irreducible
│ │ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ underlying K3 successfully completed in 21 seconds (cpu: 23 seconds)
│ │ │ │ + ✦ underlying K3 successfully completed in 11 seconds (cpu: 16 seconds)
│ │ │ │  
│ │ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       No exceptional curves
│ │ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by
│ │ │ │ @@ -160,16 +160,16 @@
│ │ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of
│ │ │ │  degrees 1^3 2^8
│ │ │ │    -- computing image on K3 surface...
│ │ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8
│ │ │ │  -- constructing lattice polarization...
│ │ │ │  -- verifying self-intersection of the curve...
│ │ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ │ --- total time (K3 surface + polarization): 25 seconds (cpu: 26 seconds)
│ │ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 4 seconds)
│ │ │ │ +-- total time (K3 surface + polarization): 14 seconds (cpu: 20 seconds)
│ │ │ │  
│ │ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       No exceptional curves
│ │ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by
│ │ │ │ @@ -267,15 +267,15 @@
│ │ │ │  -- analyzing the base locus of the inverse map...
│ │ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │ │  -- projecting to PP^3 for surface decomposition
│ │ │ │    -- removing 1 components of degrees {3}
│ │ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1
│ │ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ underlying K3 successfully completed in 7 seconds (cpu: 7 seconds)
│ │ │ │ + ✦ underlying K3 successfully completed in 4 seconds (cpu: 8 seconds)
│ │ │ │  
│ │ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │ │        Mirror fourfold: PP^4
│ │ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2
│ │ │ │  hypersurfaces of degrees 2^1 3^1
│ │ │ │        No exceptional curves
│ │ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by
│ │ │ │ @@ -292,15 +292,15 @@
│ │ │ │  Virtual", "MapFromU-Virtual"
│ │ │ │  -- special curves already detected on U
│ │ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1
│ │ │ │  -- constructing lattice polarization...
│ │ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ │ --- total time (K3 surface + polarization): 7 seconds (cpu: 7 seconds)
│ │ │ │ +-- total time (K3 surface + polarization): 4 seconds (cpu: 8 seconds)
│ │ │ │  
│ │ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │ │        Mirror fourfold: PP^4
│ │ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2
│ │ │ │  hypersurfaces of degrees 2^1 3^1
│ │ │ │        No exceptional curves
│ │ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i3 : time toGrass X
│ │ │ - -- used 4.10188s (cpu); 2.6702s (thread); 0s (gc)
│ │ │ + -- used 4.25112s (cpu); 2.43041s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
      
│ │ │              
      
│ │ │                
      i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.22961s (cpu); 2.84603s (thread); 0s (gc)
│ │ │ + -- used 4.40074s (cpu); 2.56707s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
      
│ │ │              
      
│ │ │                
      i4 : time f = unirationalParametrization X;
│ │ │ - -- used 0.856935s (cpu); 0.482912s (thread); 0s (gc)
│ │ │ + -- used 0.944796s (cpu); 0.595987s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
      
│ │ │              
      
│ │ │                
      i5 : degreeSequence f
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = cubicFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time f = unirationalParametrization X;
│ │ │ │ - -- used 0.856935s (cpu); 0.482912s (thread); 0s (gc)
│ │ │ │ + -- used 0.944796s (cpu); 0.595987s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │ │  i5 : degreeSequence f
│ │ │ │  
│ │ │ │  o5 = {[10]}
│ │ │ │  
│ │ │ │  o5 : List
│ │ ├── ./usr/share/doc/Macaulay2/SpectralSequences/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  U3BlY3RyYWxTZXF1ZW5jZSBeIEluZmluaXRlTnVtYmVy
│ │ │  #:len=967
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGluZmluaXR5IHBhZ2Ugb2YgYSBz
│ │ │  cGVjdHJhbCBzZXF1ZW5jZSIsICJsaW5lbnVtIiA9PiAzNDExLCBJbnB1dHMgPT4ge1NQQU57VFR7
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  aXNMb29wbGVzcyhEaWdyYXBoKQ==
│ │ │  #:len=247
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0xvb3BsZXNzLERpZ3JhcGgpLCJpc0xvb3BsZXNz
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/example-output/_graph_lp__Mixed__Graph_rp.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │                                b => {a, c}
│ │ │                                c => {b}
│ │ │  
│ │ │  o2 : HashTable
│ │ │  
│ │ │  i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │  
│ │ │  o4 = true
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/example-output/_to__String_lp__Mixed__Graph_rp.out
│ │ │ @@ -11,12 +11,12 @@
│ │ │                  Graph => Graph{1 => {3}}
│ │ │                                 3 => {1}
│ │ │  
│ │ │  o1 : MixedGraph
│ │ │  
│ │ │  i2 : toString G
│ │ │  
│ │ │ -o2 = new HashTable from {Graph => graph ({3, 1}, {{1, 3}}), Bigraph =>
│ │ │ -     bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Digraph => digraph ({1, 2, 3},
│ │ │ -     {{1, 2}, {2, 3}})}
│ │ │ +o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}),
│ │ │ +     Graph => graph ({3, 1}, {{1, 3}}), Bigraph => bigraph ({3, 4, 2}, {{4,
│ │ │ +     3}, {4, 2}})}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_graph_lp__Mixed__Graph_rp.html
│ │ │ @@ -121,15 +121,15 @@
│ │ │  o2 : HashTable
      
│ │ │              
      
│ │ │                
      i3 : keys (graph G)
│ │ │  
│ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │  
│ │ │  o3 : List
      
│ │ │              
      
│ │ │                
      i4 : (graph G)#Bigraph === bigraph G
│ │ │ ├── html2text {}
│ │ │ │ @@ -46,15 +46,15 @@
│ │ │ │                 Graph => Graph{a => {b}   }
│ │ │ │                                b => {a, c}
│ │ │ │                                c => {b}
│ │ │ │  
│ │ │ │  o2 : HashTable
│ │ │ │  i3 : keys (graph G)
│ │ │ │  
│ │ │ │ -o3 = {Graph, Bigraph, Digraph}
│ │ │ │ +o3 = {Digraph, Graph, Bigraph}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (graph G)#Bigraph === bigraph G
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_i_x_e_d_G_r_a_p_h -- a graph that has undirected, directed and bidirected edges
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_to__String_lp__Mixed__Graph_rp.html
│ │ │ @@ -93,17 +93,17 @@
│ │ │  o1 : MixedGraph
      
│ │ │              
      
│ │ │                
      i2 : toString G
│ │ │  
│ │ │ -o2 = new HashTable from {Graph => graph ({3, 1}, {{1, 3}}), Bigraph =>
│ │ │ -     bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Digraph => digraph ({1, 2, 3},
│ │ │ -     {{1, 2}, {2, 3}})}
      
│ │ │ +o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}),
│ │ │ +     Graph => graph ({3, 1}, {{1, 3}}), Bigraph => bigraph ({3, 4, 2}, {{4,
│ │ │ +     3}, {4, 2}})}
│ │ │              
      
│ │ │
      │ │ │
      │ │ │

      See also

      │ │ │
        │ │ │ ├── html2text {} │ │ │ │ @@ -23,17 +23,17 @@ │ │ │ │ 3 => {} │ │ │ │ Graph => Graph{1 => {3}} │ │ │ │ 3 => {1} │ │ │ │ │ │ │ │ o1 : MixedGraph │ │ │ │ i2 : toString G │ │ │ │ │ │ │ │ -o2 = new HashTable from {Graph => graph ({3, 1}, {{1, 3}}), Bigraph => │ │ │ │ - bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Digraph => digraph ({1, 2, 3}, │ │ │ │ - {{1, 2}, {2, 3}})} │ │ │ │ +o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}), │ │ │ │ + Graph => graph ({3, 1}, {{1, 3}}), Bigraph => bigraph ({3, 4, 2}, {{4, │ │ │ │ + 3}, {4, 2}})} │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _M_i_x_e_d_G_r_a_p_h -- a graph that has undirected, directed and bidirected edges │ │ │ │ * _n_e_t_(_M_i_x_e_d_G_r_a_p_h_) -- print a mixed graph as a net │ │ │ │ * _S_t_r_i_n_g -- the class of all strings │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _t_o_S_t_r_i_n_g_(_M_i_x_e_d_G_r_a_p_h_) -- print a mixed graph as a string │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/StatePolytope/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=13 │ │ │ aW5pdGlhbElkZWFscw== │ │ │ #:len=950 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2FsbHMgZ2ZhbiBhbmQgcmV0dXJucyB0 │ │ │ aGUgbGlzdCBvZiBpbml0aWFsIGlkZWFscyIsICJsaW5lbnVtIiA9PiAxNDAsIElucHV0cyA9PiB7 │ │ ├── ./usr/share/doc/Macaulay2/StronglyStableIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=34 │ │ │ bWFjYXVsYXlEZWNvbXBvc2l0aW9uKFJpbmdFbGVtZW50KQ== │ │ │ #:len=329 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODUxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWNhdWxheURlY29tcG9zaXRpb24sUmluZ0VsZW1l │ │ ├── ./usr/share/doc/Macaulay2/Style/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=39 │ │ │ Z2VuZXJhdGVHcmFtbWFyKFN0cmluZyxTdHJpbmcsRnVuY3Rpb24p │ │ │ #:len=282 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnZW5lcmF0ZUdyYW1tYXIsU3RyaW5nLFN0cmluZyxG │ │ ├── ./usr/share/doc/Macaulay2/Style/example-output/_generate__Grammar.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 3455701143666534588 │ │ │ │ │ │ i1 : outfile = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10829-0/0 │ │ │ +o1 = /tmp/M2-10929-0/0 │ │ │ │ │ │ i2 : template = outfile | ".in" │ │ │ │ │ │ -o2 = /tmp/M2-10829-0/0.in │ │ │ +o2 = /tmp/M2-10929-0/0.in │ │ │ │ │ │ i3 : template << "@M2BANNER@" << endl << endl; │ │ │ │ │ │ i4 : template << "This is an example file for the generateGrammar method!"; │ │ │ │ │ │ i5 : template << endl; │ │ │ │ │ │ @@ -30,15 +30,15 @@ │ │ │ String regex: @M2STRINGS@ │ │ │ List of keywords: { │ │ │ @M2KEYWORDS@ │ │ │ } │ │ │ │ │ │ │ │ │ i11 : generateGrammar(template, outfile, x -> demark(",\n ", x)) │ │ │ - -- generating /tmp/M2-10829-0/0 │ │ │ + -- generating /tmp/M2-10929-0/0 │ │ │ │ │ │ i12 : get outfile │ │ │ │ │ │ o12 = Auto-generated for Macaulay2-1.26.06. Do not modify this file manually. │ │ │ │ │ │ This is an example file for the generateGrammar method! │ │ │ String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///" │ │ ├── ./usr/share/doc/Macaulay2/Style/html/_generate__Grammar.html │ │ │ @@ -87,22 +87,22 @@ │ │ │

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

        │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -148,15 +148,15 @@ │ │ │ @M2KEYWORDS@ │ │ │ } │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ 
      i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10829-0/0
      │ │ │ +o1 = /tmp/M2-10929-0/0 │ │ │
      │ │ │ 
      i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10829-0/0.in
      │ │ │ +o2 = /tmp/M2-10929-0/0.in │ │ │
      │ │ │ 
      i3 : template << "@M2BANNER@" << endl << endl;
      │ │ │
      │ │ │ 
      i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10829-0/0
      │ │ │ + -- generating /tmp/M2-10929-0/0 │ │ │
      │ │ │ 
      i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.26.06. Do not modify this file manually.
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,18 +26,18 @@
│ │ │ │      * @M2CONSTANTS@, for a list of Macaulay2 symbols and packages.
│ │ │ │      * @M2STRINGS@, for a regular expression that matches Macaulay2 strings.
│ │ │ │  The function demarkf indicates how the elements of each of the lists will be
│ │ │ │  demarked in the resulting file. The file outfile will then be generated,
│ │ │ │  replacing each of these strings as indicated above.
│ │ │ │  i1 : outfile = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-10829-0/0
│ │ │ │ +o1 = /tmp/M2-10929-0/0
│ │ │ │  i2 : template = outfile | ".in"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10829-0/0.in
│ │ │ │ +o2 = /tmp/M2-10929-0/0.in
│ │ │ │  i3 : template << "@M2BANNER@" << endl << endl;
│ │ │ │  i4 : template << "This is an example file for the generateGrammar method!";
│ │ │ │  i5 : template << endl;
│ │ │ │  i6 : template << "String regex: @M2STRINGS@" << endl;
│ │ │ │  i7 : template << "List of keywords: {" << endl;
│ │ │ │  i8 : template << "    @M2KEYWORDS@" << endl;
│ │ │ │  i9 : template << "}" << endl << close;
│ │ │ │ @@ -47,15 +47,15 @@
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: @M2STRINGS@
│ │ │ │        List of keywords: {
│ │ │ │            @M2KEYWORDS@
│ │ │ │        }
│ │ │ │  i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ │ - -- generating /tmp/M2-10829-0/0
│ │ │ │ + -- generating /tmp/M2-10929-0/0
│ │ │ │  i12 : get outfile
│ │ │ │  
│ │ │ │  o12 = Auto-generated for Macaulay2-1.26.06. Do not modify this file manually.
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ │ │        List of keywords: {
│ │ ├── ./usr/share/doc/Macaulay2/SubalgebraBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  c2FnYmkoLi4uLFN0cmF0ZWd5PT4uLi4p
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU3OSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2FnYmksU3RyYXRlZ3ldLCJzYWdiaSguLi4sU3Ry
│ │ ├── ./usr/share/doc/Macaulay2/SumsOfSquares/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  bG93ZXJCb3VuZCguLi4sVmVyYm9zaXR5PT4uLi4p
│ │ │  #:len=302
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tsb3dlckJvdW5kLFZlcmJvc2l0eV0sImxvd2VyQm91
│ │ ├── ./usr/share/doc/Macaulay2/SuperLinearAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  cGFyaXR5
│ │ │  #:len=1709
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicGFyaXR5IG9mIGFuIGVsZW1lbnQgb2Yg
│ │ │  YSBzdXBlciByaW5nLiIsICJsaW5lbnVtIiA9PiA2NDcsIElucHV0cyA9PiB7U1BBTntUVHsiZiJ9
│ │ ├── ./usr/share/doc/Macaulay2/SwitchingFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  ZmllbGRCYXNlQ2hhbmdlKFJpbmcsR2Fsb2lzRmllbGQp
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmaWVsZEJhc2VDaGFuZ2UsUmluZyxHYWxvaXNGaWVs
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  bm9QYWNrZWRBbGxTdWJz
│ │ │  #:len=1151
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZHMgYWxsIHN1YnN0aXR1dGlvbnMg
│ │ │  b2YgdmFyaWFibGVzIGJ5IDEgYW5kL29yIDAgZm9yIHdoaWNoIGlkZWFsIGlzIG5vdCBLb25pZy4i
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out
│ │ │ @@ -31,15 +31,15 @@
│ │ │  o5 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i6 : isHomogeneous P
│ │ │  
│ │ │  o6 = false
│ │ │  
│ │ │  i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.441308s (cpu); 0.240277s (thread); 0s (gc)
│ │ │ + -- used 0.447811s (cpu); 0.203361s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │  
│ │ │               2         3         2     2
│ │ │  o8 = ideal (y  - x*z, x  - y*z, x y - z )
│ │ │ @@ -47,12 +47,12 @@
│ │ │  o8 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i9 : isHomogeneous Q
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.115068s (cpu); 0.0574305s (thread); 0s (gc)
│ │ │ + -- used 0.11174s (cpu); 0.0529114s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │  
│ │ │  o6 = false
      │ │ │
      │ │ │ 
      i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.441308s (cpu); 0.240277s (thread); 0s (gc)
│ │ │ + -- used 0.447811s (cpu); 0.203361s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
      │ │ │
      │ │ │ 
      i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o9 = true
      │ │ │
      │ │ │ 
      i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.115068s (cpu); 0.0574305s (thread); 0s (gc)
│ │ │ + -- used 0.11174s (cpu); 0.0529114s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
      │ │ │
      │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -59,28 +59,28 @@ │ │ │ │ o5 = ideal (y - x*z, x y - z , x - y*z) │ │ │ │ │ │ │ │ o5 : Ideal of QQ[x..z] │ │ │ │ i6 : isHomogeneous P │ │ │ │ │ │ │ │ o6 = false │ │ │ │ i7 : time symbolicPower(P,4); │ │ │ │ - -- used 0.441308s (cpu); 0.240277s (thread); 0s (gc) │ │ │ │ + -- used 0.447811s (cpu); 0.203361s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of QQ[x..z] │ │ │ │ i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5}) │ │ │ │ │ │ │ │ 2 3 2 2 │ │ │ │ o8 = ideal (y - x*z, x - y*z, x y - z ) │ │ │ │ │ │ │ │ o8 : Ideal of QQ[x..z] │ │ │ │ i9 : isHomogeneous Q │ │ │ │ │ │ │ │ o9 = true │ │ │ │ i10 : time symbolicPower(Q,4); │ │ │ │ - -- used 0.115068s (cpu); 0.0574305s (thread); 0s (gc) │ │ │ │ + -- used 0.11174s (cpu); 0.0529114s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 : Ideal of QQ[x..z] │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_y_m_b_P_o_w_e_r_P_r_i_m_e_P_o_s_C_h_a_r │ │ │ │ ********** WWaayyss ttoo uussee ssyymmbboolliiccPPoowweerr:: ********** │ │ │ │ * symbolicPower(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/SymmetricPolynomials/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ YnVpbGRTeW1tZXRyaWNHQihQb2x5bm9taWFsUmluZyk= │ │ │ #:len=1047 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiR3JvZWJuZXIgYmFzaXMgb2YgZWxlbWVu │ │ │ dGFyeSBzeW1tZXRyaWMgcG9seW5vbWlhbHMgYWxnZWJyYSIsICJsaW5lbnVtIiA9PiAxNzksIElu │ │ ├── ./usr/share/doc/Macaulay2/TSpreadIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ Y291bnRUTGV4TW9uKC4uLixGaXhlZE1heD0+Li4uKQ== │ │ │ #:len=268 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTQwMSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY291bnRUTGV4TW9uLEZpeGVkTWF4XSwiY291bnRU │ │ ├── ./usr/share/doc/Macaulay2/Tableaux/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ aXNDb3JuZXI= │ │ │ #:len=1388 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIGlmIGEgYm94IGlzIGEgY29y │ │ │ bmVyIG9mIGEgdGFibGVhdSIsICJsaW5lbnVtIiA9PiA2MTEsIElucHV0cyA9PiB7U1BBTntUVHsi │ │ ├── ./usr/share/doc/Macaulay2/TangentCone/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=11 │ │ │ VGFuZ2VudENvbmU= │ │ │ #:len=312 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGFuZ2VudCBjb25lcyIsIERlc2NyaXB0 │ │ │ aW9uID0+IDE6KCJUaGlzIHBhY2thZ2UgcHJvdmlkZXMgYSBzaW5nbGUgZnVuY3Rpb24gdGhhdCBj │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=52 │ │ │ cHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyguLi4sQ29lZmZpY2llbnRGaWVsZD0+Li4uKQ== │ │ │ #:len=347 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDM3Mywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbcHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyxDb2Vm │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ o3 = 0 <-- E <-- 0 │ │ │ │ │ │ -1 0 1 │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : time T=tateExtension W; │ │ │ - -- used 1.08449s (cpu); 0.725679s (thread); 0s (gc) │ │ │ + -- used 0.917849s (cpu); 0.605884s (thread); 0s (gc) │ │ │ │ │ │ i5 : cohomologyMatrix(T,-{3,3},{3,3}) │ │ │ │ │ │ o5 = | 8h 4h 0 4 8 12 16 | │ │ │ | 6h 3h 0 3 6 9 12 | │ │ │ | 4h 2h 0 2 4 6 8 | │ │ │ | 2h h 0 1 2 3 4 | │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time T=tateExtension W;
│ │ │ - -- used 1.08449s (cpu); 0.725679s (thread); 0s (gc)
    │ │ │ + -- used 0.917849s (cpu); 0.605884s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │               1
│ │ │ │  o3 = 0  <-- E  <-- 0
│ │ │ │  
│ │ │ │       -1     0      1
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  i4 : time T=tateExtension W;
│ │ │ │ - -- used 1.08449s (cpu); 0.725679s (thread); 0s (gc)
│ │ │ │ + -- used 0.917849s (cpu); 0.605884s (thread); 0s (gc)
│ │ │ │  i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │ │  
│ │ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ │       | 6h  3h  0 3  6  9  12 |
│ │ │ │       | 4h  2h  0 2  4  6  8  |
│ │ │ │       | 2h  h   0 1  2  3  4  |
│ │ │ │       | 0   0   0 0  0  0  0  |
│ │ ├── ./usr/share/doc/Macaulay2/TensorComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  bWlub3JzTWFwKE1hdHJpeCxMYWJlbGVkTW9kdWxlKQ==
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkzOCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWlub3JzTWFwLE1hdHJpeCxMYWJlbGVkTW9kdWxl
│ │ ├── ./usr/share/doc/Macaulay2/TerraciniLoci/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  dGVycmFjaW5pTG9jdXMoWlosUmluZ01hcCk=
│ │ │  #:len=278
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjM1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0ZXJyYWNpbmlMb2N1cyxaWixSaW5nTWFwKSwidGVy
│ │ ├── ./usr/share/doc/Macaulay2/TestAudit/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  dGVzdEF1ZGl0KFN0cmluZyk=
│ │ │  #:len=238
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0ZXN0QXVkaXQsU3RyaW5nKSwidGVzdEF1ZGl0KFN0
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  ZnJvYmVuaXVzUHJlaW1hZ2U=
│ │ │  #:len=934
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZHMgdGhlIGlkZWFsIG9mIGVsZW1l
│ │ │  bnRzIG1hcHBlZCBpbnRvIGEgZ2l2ZW4gaWRlYWwsIHVuZGVyIGFsbCAkcF57LWV9JC1saW5lYXIg
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out
│ │ │ @@ -63,20 +63,20 @@
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 1.06996s (cpu); 0.723329s (thread); 0s (gc)
│ │ │ + -- used 1.07017s (cpu); 0.691447s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.70336s (cpu); 2.192s (thread); 0s (gc)
│ │ │ + -- used 2.55895s (cpu); 2.08234s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
│ │ │  
│ │ │  i19 : J1 == J2
│ │ │  
│ │ │  o19 = true
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3});
│ │ │  
│ │ │  o3 : RingMap T <-- S
│ │ │  
│ │ │  i4 : R = S/(ker g);
│ │ │  
│ │ │  i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00348147s (cpu); 0.00347735s (thread); 0s (gc)
│ │ │ + -- used 0.00288977s (cpu); 0.00287445s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00548896s (cpu); 0.00549078s (thread); 0s (gc)
│ │ │ + -- used 0.00415003s (cpu); 0.00415379s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │  
│ │ │  i8 : isCohenMacaulay(R)
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out
│ │ │ @@ -60,49 +60,49 @@
│ │ │  i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3)
│ │ │  
│ │ │  o19 = R
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │  
│ │ │  i20 : time isFInjective(R)
│ │ │ - -- used 0.0276842s (cpu); 0.027688s (thread); 0s (gc)
│ │ │ + -- used 0.0268528s (cpu); 0.0268523s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.0387s (cpu); 1.62786s (thread); 0s (gc)
│ │ │ + -- used 1.6212s (cpu); 1.17784s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
│ │ │  
│ │ │  i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
│ │ │  
│ │ │  i23 : time isFInjective(R)
│ │ │ - -- used 0.109724s (cpu); 0.109728s (thread); 0s (gc)
│ │ │ + -- used 0.0645249s (cpu); 0.0645284s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.109777s (cpu); 0.109789s (thread); 0s (gc)
│ │ │ + -- used 0.0663085s (cpu); 0.0663155s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
│ │ │  
│ │ │  i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2);
│ │ │  
│ │ │  i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});
│ │ │  
│ │ │  o27 : RingMap EP1 <-- S
│ │ │  
│ │ │  i28 : R = S/(ker f);
│ │ │  
│ │ │  i29 : time isFInjective(R)
│ │ │ - -- used 0.802488s (cpu); 0.65363s (thread); 0s (gc)
│ │ │ + -- used 0.673708s (cpu); 0.60719s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.1693s (cpu); 0.169306s (thread); 0s (gc)
│ │ │ + -- used 0.291671s (cpu); 0.22174s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out
│ │ │ @@ -80,19 +80,19 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │  
│ │ │  i26 : debugLevel = 1;
│ │ │  
│ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.170186s (cpu); 0.0965721s (thread); 0s (gc)
│ │ │ + -- used 0.144943s (cpu); 0.0780549s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.142596s (cpu); 0.142603s (thread); 0s (gc)
│ │ │ + -- used 0.133106s (cpu); 0.133114s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
│ │ │  
│ │ │  i29 : debugLevel = 0;
│ │ │  
│ │ │  i30 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out
│ │ │ @@ -81,21 +81,21 @@
│ │ │  i22 : testIdeal({3/4, 2/3, 3/5}, L)
│ │ │  
│ │ │  o22 = ideal (y, x)
│ │ │  
│ │ │  o22 : Ideal of R
│ │ │  
│ │ │  i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.332881s (cpu); 0.257332s (thread); 0s (gc)
│ │ │ + -- used 0.236572s (cpu); 0.160965s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.45624s (cpu); 0.377858s (thread); 0s (gc)
│ │ │ + -- used 0.284792s (cpu); 0.218044s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
│ │ │ @@ -231,23 +231,23 @@
│ │ │  
│ │ │  o16 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 1.06996s (cpu); 0.723329s (thread); 0s (gc)
│ │ │ + -- used 1.07017s (cpu); 0.691447s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.70336s (cpu); 2.192s (thread); 0s (gc)
│ │ │ + -- used 2.55895s (cpu); 2.08234s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i19 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  i15 : I2 = ideal(x^20*y^100, x + z^100);
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ │ - -- used 1.06996s (cpu); 0.723329s (thread); 0s (gc)
│ │ │ │ + -- used 1.07017s (cpu); 0.691447s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.70336s (cpu); 2.192s (thread); 0s (gc)
│ │ │ │ + -- used 2.55895s (cpu); 2.08234s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 : Ideal of R
│ │ │ │  i19 : J1 == J2
│ │ │ │  
│ │ │ │  o19 = true
│ │ │ │  For legacy reasons, the last ideal in the list can be specified separately,
│ │ │ │  using frobeniusRoot(e, \{a_1,\ldots,a_n\}, \{I_1,\ldots,I_n\}, I). The last
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html
│ │ │ @@ -101,23 +101,23 @@
│ │ │              
│ │ │                
    i4 : R = S/(ker g);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00348147s (cpu); 0.00347735s (thread); 0s (gc)
│ │ │ + -- used 0.00288977s (cpu); 0.00287445s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00548896s (cpu); 0.00549078s (thread); 0s (gc)
│ │ │ + -- used 0.00415003s (cpu); 0.00415379s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,19 +23,19 @@
│ │ │ │  i1 : T = ZZ/5[x,y];
│ │ │ │  i2 : S = ZZ/5[a,b,c,d];
│ │ │ │  i3 : g = map(T, S, {x^3, x^2*y, x*y^2, y^3});
│ │ │ │  
│ │ │ │  o3 : RingMap T <-- S
│ │ │ │  i4 : R = S/(ker g);
│ │ │ │  i5 : time isCohenMacaulay(R)
│ │ │ │ - -- used 0.00348147s (cpu); 0.00347735s (thread); 0s (gc)
│ │ │ │ + -- used 0.00288977s (cpu); 0.00287445s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ │ - -- used 0.00548896s (cpu); 0.00549078s (thread); 0s (gc)
│ │ │ │ + -- used 0.00415003s (cpu); 0.00415379s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │ │  i8 : isCohenMacaulay(R)
│ │ │ │  
│ │ │ │  o8 = false
│ │ │ │  The function isCohenMacaulay considers $R$ as a quotient of a polynomial ring,
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html
│ │ │ @@ -219,23 +219,23 @@
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i20 : time isFInjective(R)
│ │ │ - -- used 0.0276842s (cpu); 0.027688s (thread); 0s (gc)
│ │ │ + -- used 0.0268528s (cpu); 0.0268523s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
    
│ │ │              
    
│ │ │                
    i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.0387s (cpu); 1.62786s (thread); 0s (gc)
│ │ │ + -- used 1.6212s (cpu); 1.17784s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If the option AtOrigin (default value false) is set to true, isFInjective will only check $F$-injectivity at the origin. Otherwise, it will check $F$-injectivity globally. Note that checking $F$-injectivity at the origin can be slower than checking it globally. Consider the following example of a non-$F$-injective ring.
    
│ │ │ @@ -245,23 +245,23 @@
│ │ │              
│ │ │                
    i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time isFInjective(R)
│ │ │ - -- used 0.109724s (cpu); 0.109728s (thread); 0s (gc)
│ │ │ + -- used 0.0645249s (cpu); 0.0645284s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.109777s (cpu); 0.109789s (thread); 0s (gc)
│ │ │ + -- used 0.0663085s (cpu); 0.0663155s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If the option AssumeCM (default value false) is set to true, then isFInjective only checks the Frobenius action on top cohomology (which is typically much faster). Note that it can give an incorrect answer if the non-injective Frobenius occurs in a lower degree.  Consider the example of the cone over a supersingular elliptic curve times $\mathbb{P}^1$.
    
│ │ │ @@ -288,23 +288,23 @@
│ │ │              
│ │ │                
    i28 : R = S/(ker f);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : time isFInjective(R)
│ │ │ - -- used 0.802488s (cpu); 0.65363s (thread); 0s (gc)
│ │ │ + -- used 0.673708s (cpu); 0.60719s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.1693s (cpu); 0.169306s (thread); 0s (gc)
│ │ │ + -- used 0.291671s (cpu); 0.22174s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If the option AssumedReduced is set to true (its default behavior), then the bottom local cohomology is avoided (this means the Frobenius action on the top potentially nonzero Ext is not computed).
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -81,52 +81,52 @@
│ │ │ │  much faster.
│ │ │ │  i19 : R = ZZ/5[x,y,z]/(y^2*z + x*y*z-x^3)
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : QuotientRing
│ │ │ │  i20 : time isFInjective(R)
│ │ │ │ - -- used 0.0276842s (cpu); 0.027688s (thread); 0s (gc)
│ │ │ │ + -- used 0.0268528s (cpu); 0.0268523s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = true
│ │ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ │ - -- used 2.0387s (cpu); 1.62786s (thread); 0s (gc)
│ │ │ │ + -- used 1.6212s (cpu); 1.17784s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = true
│ │ │ │  If the option AtOrigin (default value false) is set to true, isFInjective will
│ │ │ │  only check $F$-injectivity at the origin. Otherwise, it will check $F$-
│ │ │ │  injectivity globally. Note that checking $F$-injectivity at the origin can be
│ │ │ │  slower than checking it globally. Consider the following example of a non-$F$-
│ │ │ │  injective ring.
│ │ │ │  i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
│ │ │ │  i23 : time isFInjective(R)
│ │ │ │ - -- used 0.109724s (cpu); 0.109728s (thread); 0s (gc)
│ │ │ │ + -- used 0.0645249s (cpu); 0.0645284s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ │ - -- used 0.109777s (cpu); 0.109789s (thread); 0s (gc)
│ │ │ │ + -- used 0.0663085s (cpu); 0.0663155s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  If the option AssumeCM (default value false) is set to true, then isFInjective
│ │ │ │  only checks the Frobenius action on top cohomology (which is typically much
│ │ │ │  faster). Note that it can give an incorrect answer if the non-injective
│ │ │ │  Frobenius occurs in a lower degree. Consider the example of the cone over a
│ │ │ │  supersingular elliptic curve times $\mathbb{P}^1$.
│ │ │ │  i25 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
│ │ │ │  i26 : EP1 = ZZ/3[x,y,z,s,t]/(x^3 + y^2*z - x*z^2);
│ │ │ │  i27 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});
│ │ │ │  
│ │ │ │  o27 : RingMap EP1 <-- S
│ │ │ │  i28 : R = S/(ker f);
│ │ │ │  i29 : time isFInjective(R)
│ │ │ │ - -- used 0.802488s (cpu); 0.65363s (thread); 0s (gc)
│ │ │ │ + -- used 0.673708s (cpu); 0.60719s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = false
│ │ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ │ - -- used 0.1693s (cpu); 0.169306s (thread); 0s (gc)
│ │ │ │ + -- used 0.291671s (cpu); 0.22174s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  If the option AssumedReduced is set to true (its default behavior), then the
│ │ │ │  bottom local cohomology is avoided (this means the Frobenius action on the top
│ │ │ │  potentially nonzero Ext is not computed).
│ │ │ │  If the option AssumeNormal (default value false) is set to true, then the
│ │ │ │  bottom two local cohomology modules (or, rather, their duals) need not be
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html
│ │ │ @@ -278,23 +278,23 @@
│ │ │                
    i26 : debugLevel = 1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.170186s (cpu); 0.0965721s (thread); 0s (gc)
│ │ │ + -- used 0.144943s (cpu); 0.0780549s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.142596s (cpu); 0.142603s (thread); 0s (gc)
│ │ │ + -- used 0.133106s (cpu); 0.133114s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : debugLevel = 0;
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  i25 : I = minors(2, matrix {{x, y, z}, {u, v, w}});
│ │ │ │  
│ │ │ │  o25 : Ideal of S
│ │ │ │  i26 : debugLevel = 1;
│ │ │ │  i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing
│ │ │ │  DepthOfSearch will let the function search more deeply.
│ │ │ │ - -- used 0.170186s (cpu); 0.0965721s (thread); 0s (gc)
│ │ │ │ + -- used 0.144943s (cpu); 0.0780549s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = false
│ │ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ │ - -- used 0.142596s (cpu); 0.142603s (thread); 0s (gc)
│ │ │ │ + -- used 0.133106s (cpu); 0.133114s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : debugLevel = 0;
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_e_s_t_I_d_e_a_l -- compute a test ideal in a Q-Gorenstein ring
│ │ │ │      * _i_s_F_R_a_t_i_o_n_a_l -- whether a ring is F-rational
│ │ │ │  ********** WWaayyss ttoo uussee iissFFRReegguullaarr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html
│ │ │ @@ -260,25 +260,25 @@
│ │ │          
    
│ │ │            
    It is often more efficient to pass a list, as opposed to finding a common denominator and passing a single element, since testIdeal can do things in a more intelligent way for such a list.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.332881s (cpu); 0.257332s (thread); 0s (gc)
│ │ │ + -- used 0.236572s (cpu); 0.160965s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
    
│ │ │                
    i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.45624s (cpu); 0.377858s (thread); 0s (gc)
│ │ │ + -- used 0.284792s (cpu); 0.218044s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -100,21 +100,21 @@
│ │ │ │  o22 = ideal (y, x)
│ │ │ │  
│ │ │ │  o22 : Ideal of R
│ │ │ │  It is often more efficient to pass a list, as opposed to finding a common
│ │ │ │  denominator and passing a single element, since testIdeal can do things in a
│ │ │ │  more intelligent way for such a list.
│ │ │ │  i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ │ - -- used 0.332881s (cpu); 0.257332s (thread); 0s (gc)
│ │ │ │ + -- used 0.236572s (cpu); 0.160965s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = ideal (y, x)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ │ - -- used 0.45624s (cpu); 0.377858s (thread); 0s (gc)
│ │ │ │ + -- used 0.284792s (cpu); 0.218044s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = ideal (y, x)
│ │ │ │  
│ │ │ │  o24 : Ideal of R
│ │ │ │  The option AssumeDomain (default value false) is used when finding a test
│ │ │ │  element. The option FrobeniusRootStrategy (default value Substitution) is
│ │ │ │  passed to internal _f_r_o_b_e_n_i_u_s_R_o_o_t calls.
│ │ ├── ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  bmV3IFRPSCBmcm9tIFRoaW5n
│ │ │  #:len=219
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODQwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhOZXdGcm9tTWV0aG9kLFRPSCxUaGluZyksIm5ldyBU
│ │ ├── ./usr/share/doc/Macaulay2/ThinSincereQuivers/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  Y2hhaW5RdWl2ZXIoLi4uLEhlaWdodD0+Li4uKQ==
│ │ │  #:len=283
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI5MSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY2hhaW5RdWl2ZXIsSGVpZ2h0XSwiY2hhaW5RdWl2
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  dGdiKExpc3Qp
│ │ │  #:len=213
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0Z2IsTGlzdCksInRnYihMaXN0KSIsIlRocmVhZGVk
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out
│ │ │ @@ -2,16 +2,16 @@
│ │ │  
│ │ │  i1 : S = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │ -o3 = LineageTable{((0, 2), 0) => null}
│ │ │ -                  ((0, 2), 1) => null
│ │ │ +o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ +                  ((0, 1), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out
│ │ │ @@ -2,17 +2,15 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ -                  ((0, 1), 1) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out
│ │ │ @@ -2,18 +2,16 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -24,16 +22,15 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out
│ │ │ @@ -6,34 +6,48 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : allowableThreads  = 4;
│ │ │  
│ │ │  i4 : H = tgb I
│ │ │  
│ │ │ +                                          2 4
│ │ │ +o4 = LineageTable{(((0, 2), 1), 3) => -46y z         }
│ │ │ +                                    3 7      2 10
│ │ │ +                  ((0, 2), 1) => 20y z  - 47y z
│ │ │ +                                      4 4      2 7
│ │ │ +                  ((0, 2), 3) => - 10y z  + 23y z
│ │ │ +                                         3 5      2 4
│ │ │ +                  ((0, 3), (0, 2)) => 25y z  - 22y z
│ │ │ +                                      3 5     2 13
│ │ │ +                  ((0, 3), 1) => - 11y z  - 5y z
│ │ │                                     4 4      2 6
│ │ │ -o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z        }
│ │ │ -                                    2 6      2 5
│ │ │ -                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ -                                           4 4     2 4
│ │ │ -                  ((0, 2), (0, 1)) => - 13y z  + 9y z
│ │ │ -                                    3 4      2 4
│ │ │ -                  ((0, 2), 1) => 18y z  - 33y z
│ │ │ -                                    3 4      2 5
│ │ │ -                  ((0, 2), 2) => 37y z  - 45y z
│ │ │ -                                    2 5
│ │ │ -                  ((0, 2), 3) => 16y z
│ │ │ -                                   2 4
│ │ │ -                  ((0, 3), 2) => 5y z
│ │ │ -                                 5 2      3 4
│ │ │ -                  (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ +                  ((0, 3), 2) => 9y z  + 23y z
│ │ │ +                                         2 6     2 5
│ │ │ +                  ((1, 2), (0, 3)) => 22y z  + 2y z
│ │ │ +                                      2 6      2 5
│ │ │ +                  ((1, 2), 3) => - 16y z  - 29y z
│ │ │ +                                        2 4
│ │ │ +                  ((1, 3), (1, 2)) => 5y z
│ │ │ +                                    2 6    2 4
│ │ │ +                  ((1, 3), 1) => 36y z  - y z
│ │ │ +                                      2 5      2 4
│ │ │ +                  ((1, 3), 2) => - 34y z  - 49y z
│ │ │ +                                   3 4      2 5
│ │ │ +                  ((1, 3), 3) => 4y z  - 30y z
│ │ │                                 5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ +                  (0, 1) => 39y z - 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
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │                    2 => 5x*y z  + 9x*z
│ │ │                             3         3
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html
│ │ │ @@ -78,16 +78,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │ -o3 = LineageTable{((0, 2), 0) => null}
│ │ │ -                  ((0, 2), 1) => null
│ │ │ +o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ +                  ((0, 1), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,16 +12,16 @@
│ │ │ │  Gröbner basis is minimized. Lineages of non-minimal Gröbner basis elements
│ │ │ │  that were added to the basis during the distributed computation are saved, with
│ │ │ │  the corresponding entry in the table being null.
│ │ │ │  i1 : S = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │ │  
│ │ │ │ -o3 = LineageTable{((0, 2), 0) => null}
│ │ │ │ -                  ((0, 2), 1) => null
│ │ │ │ +o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ +                  ((0, 1), 0) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html
│ │ │ @@ -91,17 +91,15 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ -                  ((0, 1), 1) => null
│ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,17 +19,15 @@
│ │ │ │  This simple function just returns the Gröbner basis computed with threaded
│ │ │ │  Gröbner basis function _t_g_b in the expected Macaulay2 format, so that further
│ │ │ │  computation are one step easier to set up.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │ │  
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ -                  ((0, 1), 1) => null
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html
│ │ │ @@ -87,18 +87,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ +                                   3
│ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │                                     2
│ │ │                    ((1, 2), 0) => -c
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │ @@ -112,16 +110,15 @@
│ │ │  o3 : LineageTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,18 +20,16 @@
│ │ │ │  remainder on the division by the remaining values H.
│ │ │ │  If values H constitute a Gröbner basis of the ideal they generate, this method
│ │ │ │  returns a reduced Gröbner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │ -                                     2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │ +                                   3
│ │ │ │ +o3 = LineageTable{((0, 2), 0) => -c      }
│ │ │ │                                     2
│ │ │ │                    ((1, 2), 0) => -c
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │ @@ -41,16 +39,15 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : reduce T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │ +o4 = LineageTable{((0, 2), 0) => null}
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html
│ │ │ @@ -100,34 +100,48 @@
│ │ │                
    i3 : allowableThreads  = 4;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : H = tgb I
│ │ │  
│ │ │ +                                          2 4
│ │ │ +o4 = LineageTable{(((0, 2), 1), 3) => -46y z         }
│ │ │ +                                    3 7      2 10
│ │ │ +                  ((0, 2), 1) => 20y z  - 47y z
│ │ │ +                                      4 4      2 7
│ │ │ +                  ((0, 2), 3) => - 10y z  + 23y z
│ │ │ +                                         3 5      2 4
│ │ │ +                  ((0, 3), (0, 2)) => 25y z  - 22y z
│ │ │ +                                      3 5     2 13
│ │ │ +                  ((0, 3), 1) => - 11y z  - 5y z
│ │ │                                     4 4      2 6
│ │ │ -o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z        }
│ │ │ -                                    2 6      2 5
│ │ │ -                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ -                                           4 4     2 4
│ │ │ -                  ((0, 2), (0, 1)) => - 13y z  + 9y z
│ │ │ -                                    3 4      2 4
│ │ │ -                  ((0, 2), 1) => 18y z  - 33y z
│ │ │ -                                    3 4      2 5
│ │ │ -                  ((0, 2), 2) => 37y z  - 45y z
│ │ │ -                                    2 5
│ │ │ -                  ((0, 2), 3) => 16y z
│ │ │ -                                   2 4
│ │ │ -                  ((0, 3), 2) => 5y z
│ │ │ -                                 5 2      3 4
│ │ │ -                  (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ +                  ((0, 3), 2) => 9y z  + 23y z
│ │ │ +                                         2 6     2 5
│ │ │ +                  ((1, 2), (0, 3)) => 22y z  + 2y z
│ │ │ +                                      2 6      2 5
│ │ │ +                  ((1, 2), 3) => - 16y z  - 29y z
│ │ │ +                                        2 4
│ │ │ +                  ((1, 3), (1, 2)) => 5y z
│ │ │ +                                    2 6    2 4
│ │ │ +                  ((1, 3), 1) => 36y z  - y z
│ │ │ +                                      2 5      2 4
│ │ │ +                  ((1, 3), 2) => - 34y z  - 49y z
│ │ │ +                                   3 4      2 5
│ │ │ +                  ((1, 3), 3) => 4y z  - 30y z
│ │ │                                 5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ +                  (0, 1) => 39y z - 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
│ │ │                                 2
│ │ │                    0 => 2x + 10y z
│ │ │                           2           3
│ │ │                    1 => 8x y + 10x*y*z
│ │ │                             3 2       3
│ │ │                    2 => 5x*y z  + 9x*z
│ │ │                             3         3
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,34 +26,48 @@
│ │ │ │  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 4
│ │ │ │ +o4 = LineageTable{(((0, 2), 1), 3) => -46y z         }
│ │ │ │ +                                    3 7      2 10
│ │ │ │ +                  ((0, 2), 1) => 20y z  - 47y z
│ │ │ │ +                                      4 4      2 7
│ │ │ │ +                  ((0, 2), 3) => - 10y z  + 23y z
│ │ │ │ +                                         3 5      2 4
│ │ │ │ +                  ((0, 3), (0, 2)) => 25y z  - 22y z
│ │ │ │ +                                      3 5     2 13
│ │ │ │ +                  ((0, 3), 1) => - 11y z  - 5y z
│ │ │ │                                     4 4      2 6
│ │ │ │ -o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z        }
│ │ │ │ -                                    2 6      2 5
│ │ │ │ -                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ │ -                                           4 4     2 4
│ │ │ │ -                  ((0, 2), (0, 1)) => - 13y z  + 9y z
│ │ │ │ -                                    3 4      2 4
│ │ │ │ -                  ((0, 2), 1) => 18y z  - 33y z
│ │ │ │ -                                    3 4      2 5
│ │ │ │ -                  ((0, 2), 2) => 37y z  - 45y z
│ │ │ │ -                                    2 5
│ │ │ │ -                  ((0, 2), 3) => 16y z
│ │ │ │ -                                   2 4
│ │ │ │ -                  ((0, 3), 2) => 5y z
│ │ │ │ -                                 5 2      3 4
│ │ │ │ -                  (0, 1) => - 25y z  - 19y z
│ │ │ │ -                                 3 5     2 4
│ │ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ │ +                  ((0, 3), 2) => 9y z  + 23y z
│ │ │ │ +                                         2 6     2 5
│ │ │ │ +                  ((1, 2), (0, 3)) => 22y z  + 2y z
│ │ │ │ +                                      2 6      2 5
│ │ │ │ +                  ((1, 2), 3) => - 16y z  - 29y z
│ │ │ │ +                                        2 4
│ │ │ │ +                  ((1, 3), (1, 2)) => 5y z
│ │ │ │ +                                    2 6    2 4
│ │ │ │ +                  ((1, 3), 1) => 36y z  - y z
│ │ │ │ +                                      2 5      2 4
│ │ │ │ +                  ((1, 3), 2) => - 34y z  - 49y z
│ │ │ │ +                                   3 4      2 5
│ │ │ │ +                  ((1, 3), 3) => 4y z  - 30y z
│ │ │ │                                 5       3 4
│ │ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ │ +                  (0, 1) => 39y z - 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
│ │ │ │                                 2
│ │ │ │                    0 => 2x + 10y z
│ │ │ │                           2           3
│ │ │ │                    1 => 8x y + 10x*y*z
│ │ │ │                             3 2       3
│ │ │ │                    2 => 5x*y z  + 9x*z
│ │ │ │                             3         3
│ │ ├── ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Y2hpcm90b3BlU3RyaW5n
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│ │ ├── ./usr/share/doc/Macaulay2/TorAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
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│ │ ├── ./usr/share/doc/Macaulay2/ToricHigherDirectImages/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
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│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  ZWREZWc=
│ │ │  #:len=2501
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIChnZW5lcmljKSBF
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│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/example-output/_ed__Deg.out
│ │ │ @@ -40,15 +40,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.33817s (cpu); 0.918817s (thread); 0s (gc)
│ │ │ + -- used 1.03121s (cpu); 0.686379s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
│ │ │  
│ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │  
│ │ │ @@ -56,14 +56,14 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 4.78144s (cpu); 3.03944s (thread); 0s (gc)
│ │ │ + -- used 4.25204s (cpu); 2.71076s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -136,15 +136,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.33817s (cpu); 0.918817s (thread); 0s (gc)
│ │ │ + -- used 1.03121s (cpu); 0.686379s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -155,15 +155,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 4.78144s (cpu); 3.03944s (thread); 0s (gc)
│ │ │ + -- used 4.25204s (cpu); 2.71076s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,30 +66,30 @@
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │ - -- used 1.33817s (cpu); 0.918817s (thread); 0s (gc)
│ │ │ │ + -- used 1.03121s (cpu); 0.686379s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 252
│ │ │ │  
│ │ │ │  o4 : QQ
│ │ │ │  i5 : time edDeg(A,ForceAmat=>true)
│ │ │ │  
│ │ │ │  The toric variety has degree = 28
│ │ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │ │  ED Degree = 252
│ │ │ │  
│ │ │ │                         5      4      3      2
│ │ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ │ - -- used 4.78144s (cpu); 3.03944s (thread); 0s (gc)
│ │ │ │ + -- used 4.25204s (cpu); 2.71076s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 252
│ │ │ │  
│ │ │ │  o5 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee eeddDDeegg:: **********
│ │ │ │      * edDeg(Matrix)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  YmV0dGkoTm9ybWFsVG9yaWNWYXJpZXR5KQ==
│ │ │  #:len=272
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDc0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhiZXR0aSxOb3JtYWxUb3JpY1ZhcmlldHkpLCJiZXR0
│ │ ├── ./usr/share/doc/Macaulay2/ToricVectorBundles/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  ZHVhbChUb3JpY1ZlY3RvckJ1bmRsZSk=
│ │ │  #:len=1504
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiIHRoZSBkdWFsIGJ1bmRsZSBvZiBhIHRv
│ │ │  cmljIHZlY3RvciBidW5kbGUiLCAibGluZW51bSIgPT4gMjgwOCwgSW5wdXRzID0+IHtTUEFOe1RU
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  Y2hlY2tJbnRlcmZhY2U=
│ │ │  #:len=792
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAid2hldGhlciB0aGUgTWFwbGUgaW50ZXJm
│ │ │  YWNlIGlzIHdvcmtpbmcgKGZvciBkZXZlbG9wZXJzKSIsICJsaW5lbnVtIiA9PiAzMzUsIElucHV0
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=50
│ │ │  cmVndWxhclRyaWFuZ3VsYXRpb25XZWlnaHRzKC4uLixEZWdyZWVNYXRyaXg9Pi4uLik=
│ │ │  #:len=347
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgzNCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbcmVndWxhclRyaWFuZ3VsYXRpb25XZWlnaHRzLERl
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out
│ │ │ @@ -17,15 +17,15 @@
│ │ │       | -1 1  2  -1 -1 1 -1 1 0 0 |
│ │ │       | 1  0  -1 0  0  0 0  0 0 0 |
│ │ │  
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .116686s elapsed
│ │ │ + -- .0960999s elapsed
│ │ │  
│ │ │  i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │       ------------------------------------------------------------------------
│ │ │       1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -50,14 +50,14 @@
│ │ │  i7 : T = regularFineTriangulation A
│ │ │  
│ │ │  o7 = triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, 1, 6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8}, {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6, 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}}
│ │ │  
│ │ │  o7 : Triangulation
│ │ │  
│ │ │  i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 2.08042s elapsed
│ │ │ + -- 1.00514s elapsed
│ │ │  
│ │ │  i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/index.html
│ │ │ @@ -168,15 +168,15 @@
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .116686s elapsed
    
│ │ │ + -- .0960999s elapsed
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : select(Ts, T -> isStar T)
│ │ │  
│ │ │  o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
│ │ │ @@ -216,15 +216,15 @@
│ │ │  
│ │ │  o7 : Triangulation
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 2.08042s elapsed
    
│ │ │ + -- 1.00514s elapsed
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ | 0 0 0 1 0 0 -1 0 0 0 | │ │ │ │ | -1 1 2 -1 -1 1 -1 1 0 0 | │ │ │ │ | 1 0 -1 0 0 0 0 0 0 0 | │ │ │ │ │ │ │ │ 4 10 │ │ │ │ o2 : Matrix ZZ <-- ZZ │ │ │ │ i3 : elapsedTime Ts = allTriangulations(A, Fine => true); │ │ │ │ - -- .116686s elapsed │ │ │ │ + -- .0960999s elapsed │ │ │ │ i4 : select(Ts, T -> isStar T) │ │ │ │ │ │ │ │ o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 5, 7, 9}, {0, 3, 5, 8, 9}, {0, 4, 6, 8, 9}, {0, 5, 6, 7, 9}, {0, 5, 6, │ │ │ │ @@ -120,15 +120,15 @@ │ │ │ │ 6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8}, │ │ │ │ {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6, │ │ │ │ 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, │ │ │ │ 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}} │ │ │ │ │ │ │ │ o7 : Triangulation │ │ │ │ i8 : elapsedTime Ts2 = generateTriangulations T; │ │ │ │ - -- 2.08042s elapsed │ │ │ │ + -- 1.00514s elapsed │ │ │ │ i9 : #Ts2 == #Ts │ │ │ │ │ │ │ │ o9 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _P_o_l_y_h_e_d_r_a -- convex polyhedra │ │ │ │ * _T_o_p_c_o_m -- interface to the topcom software package which in particular │ │ │ │ computes triangulations │ │ ├── ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=7 │ │ │ cm90Rm9ydw== │ │ │ #:len=229 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODM2LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJyb3RGb3J3Iiwicm90Rm9ydyIsIlRyaXBsZXRzIn0s │ │ ├── ./usr/share/doc/Macaulay2/Tropical/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ aXNCYWxhbmNlZA== │ │ │ #:len=1097 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIHdoZXRoZXIgYSB0cm9waWNh │ │ │ bCBjeWNsZSBpcyBiYWxhbmNlZCIsICJsaW5lbnVtIiA9PiA5NTEsIElucHV0cyA9PiB7U1BBTntU │ │ ├── ./usr/share/doc/Macaulay2/TropicalToric/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=43 │ │ │ Y2xhc3NGcm9tVHJvcGljYWwoTm9ybWFsVG9yaWNWYXJpZXR5LElkZWFsKQ== │ │ │ #:len=335 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM1LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjbGFzc0Zyb21Ucm9waWNhbCxOb3JtYWxUb3JpY1Zh │ │ ├── ./usr/share/doc/Macaulay2/Truncations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ dHJ1bmNhdGUoSW5maW5pdGVOdW1iZXIsVGhpbmcp │ │ │ #:len=222 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAsICJ1bmRvY3VtZW50ZWQiID0+IHRy │ │ │ dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KHRydW5jYXRl │ │ ├── ./usr/share/doc/Macaulay2/Units/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=5 │ │ │ VW5pdHM= │ │ │ #:len=298 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidW5pdHMgY29udmVyc2lvbiBhbmQgcGh5 │ │ │ c2ljYWwgY29uc3RhbnRzIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7VEVYeyJUaGlzIHBh │ │ ├── ./usr/share/doc/Macaulay2/VNumber/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ c3RhYmxlTWF4 │ │ │ #:len=1353 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSB0aGUgc2V0IG9mIHN0YWJs │ │ │ ZSBwcmltZXMgb2YgYSBtb25vbWlhbCBpZGVhbCB0aGF0IGFyZSBtYXhpbWFsIHdpdGggcmVzcGVj │ │ ├── ./usr/share/doc/Macaulay2/Valuations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ VmFsdWF0aW9u │ │ │ #:len=1226 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIHR5cGUgb2YgYWxsIHZhbHVhdGlv │ │ │ bnMiLCAibGluZW51bSIgPT4gNzk4LCBTZWVBbHNvID0+IERJVntIRUFERVIyeyJTZWUgYWxzbyJ9 │ │ ├── ./usr/share/doc/Macaulay2/Varieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ PiBJbmZpbml0ZU51bWJlcg== │ │ │ #:len=253 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjk4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wgPixJbmZpbml0ZU51bWJlciksIj4gSW5m │ │ ├── ./usr/share/doc/Macaulay2/VectorFields/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ Y29tbXV0YXRvcg== │ │ │ #:len=3282 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNvbW11dGF0b3Igb2YgYSBjb2xs │ │ │ ZWN0aW9uIG9mIHZlY3RvciBmaWVsZHMiLCAibGluZW51bSIgPT4gMjE1NSwgSW5wdXRzID0+IHtT │ │ ├── ./usr/share/doc/Macaulay2/VectorGraphics/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ dGV4KFNWRyk= │ │ │ #:len=181 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg4OCwgInVuZG9jdW1lbnRlZCIgPT4g │ │ │ dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4LFNW │ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ bGlmdERlZm9ybWF0aW9uKC4uLixWZXJib3NlPT4uLi4p │ │ │ #:len=285 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA1MCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbbGlmdERlZm9ybWF0aW9uLFZlcmJvc2VdLCJsaWZ0 │ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ @@ -6,30 +6,30 @@ │ │ │ │ │ │ o2 = | xz yz z2 x3 | │ │ │ │ │ │ 1 4 │ │ │ o2 : Matrix S <-- S │ │ │ │ │ │ i3 : time (F,R,G,C)=localHilbertScheme(F0); │ │ │ - -- used 0.688076s (cpu); 0.53591s (thread); 0s (gc) │ │ │ + -- used 0.647333s (cpu); 0.520293s (thread); 0s (gc) │ │ │ │ │ │ i4 : T=ring first G; │ │ │ │ │ │ i5 : sum G │ │ │ │ │ │ o5 = | t_1t_16 | │ │ │ | t_9t_16 | │ │ │ | -t_4t_16 | │ │ │ | -2t_14t_16+t_15t_16 | │ │ │ │ │ │ 4 1 │ │ │ o5 : Matrix T <-- T │ │ │ │ │ │ i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false); │ │ │ - -- used 0.537824s (cpu); 0.386837s (thread); 0s (gc) │ │ │ + -- used 0.49421s (cpu); 0.373995s (thread); 0s (gc) │ │ │ │ │ │ i7 : sum G │ │ │ │ │ │ o7 = | t_1t_16 │ │ │ | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+ │ │ │ | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t │ │ │ | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2- │ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ │ │ │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -103,15 +103,15 @@ │ │ │ │ │ │
    │ │ │ 
    i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 0.688076s (cpu); 0.53591s (thread); 0s (gc)
    │ │ │ + -- used 0.647333s (cpu); 0.520293s (thread); 0s (gc) │ │ │
    │ │ │ 
    i4 : T=ring first G;
    │ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -26,15 +26,15 @@ │ │ │ │ o2 : List │ │ │ │ i3 : range1=makeRange(L,{4,6}) │ │ │ │ │ │ │ │ o3 = {4, 6, 8, 12} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range1) │ │ │ │ - -- .850657s elapsed │ │ │ │ + -- .431438s elapsed │ │ │ │ │ │ │ │ 3 2 4 8 2 2 2 6 3 2 │ │ │ │ o4 = (true, ideal (x - x x - x z - x z , x x - x - x z , x - x x + │ │ │ │ 0 1 3 0 0 0 1 3 0 1 0 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 4 6 8 │ │ │ │ x x z - x x z + x z ), {0}) │ │ │ │ @@ -46,23 +46,23 @@ │ │ │ │ i5 : range2=drop(makeRange(L,{1}),9) │ │ │ │ │ │ │ │ o5 = {10, 11, 12, 13, 14, 15} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range2,Verbose=>1) │ │ │ │ time to decompose J1 : │ │ │ │ - -- .000769529s elapsed │ │ │ │ + -- .0009533s elapsed │ │ │ │ │ │ │ │ component number = 0 │ │ │ │ deformation weights = {{10}} │ │ │ │ semigroup = {4, 5, 7} │ │ │ │ smoothing components numbers = {0} │ │ │ │ - -- .00146584s elapsed │ │ │ │ + -- .00102092s elapsed │ │ │ │ flat = true │ │ │ │ - -- .720102s elapsed │ │ │ │ + -- .361303s elapsed │ │ │ │ │ │ │ │ 3 2 2 10 3 2 10 │ │ │ │ o6 = (true, ideal (x - x x , x x - x - x z , x - x x - x z ), {0}) │ │ │ │ 0 1 3 0 1 3 0 1 0 3 1 │ │ │ │ │ │ │ │ o6 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_prune__Family.html │ │ │ @@ -132,15 +132,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 0.537824s (cpu); 0.386837s (thread); 0s (gc)
    │ │ │ + -- used 0.49421s (cpu); 0.373995s (thread); 0s (gc) │ │ │
    │ │ │ 
    i7 : sum G
│ │ │  
│ │ │  o7 = | t_1t_16
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,29 +18,29 @@
│ │ │ │  o2 = | xz yz z2 x3 |
│ │ │ │  
│ │ │ │               1      4
│ │ │ │  o2 : Matrix S  <-- S
│ │ │ │  With the default setting SmartLift=>true we get very nice equations for the
│ │ │ │  base space:
│ │ │ │  i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ │ - -- used 0.688076s (cpu); 0.53591s (thread); 0s (gc)
│ │ │ │ + -- used 0.647333s (cpu); 0.520293s (thread); 0s (gc)
│ │ │ │  i4 : T=ring first G;
│ │ │ │  i5 : sum G
│ │ │ │  
│ │ │ │  o5 = | t_1t_16             |
│ │ │ │       | t_9t_16             |
│ │ │ │       | -t_4t_16            |
│ │ │ │       | -2t_14t_16+t_15t_16 |
│ │ │ │  
│ │ │ │               4      1
│ │ │ │  o5 : Matrix T  <-- T
│ │ │ │  With the setting SmartLift=>false the calculation is faster, but the equations
│ │ │ │  are no longer homogeneous:
│ │ │ │  i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ │ - -- used 0.537824s (cpu); 0.386837s (thread); 0s (gc)
│ │ │ │ + -- used 0.49421s (cpu); 0.373995s (thread); 0s (gc)
│ │ │ │  i7 : sum G
│ │ │ │  
│ │ │ │  o7 = | t_1t_16
│ │ │ │       | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
│ │ │ │       | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
│ │ │ │       | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/VirtualResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=42
│ │ │  bXVsdGlncmFkZWRSZWd1bGFyaXR5KC4uLixMb3dlckxpbWl0PT4uLi4p
│ │ │  #:len=334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjAzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/Visualize/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  dmlzdWFsaXplKEdyYXBoLFZlcmJvc2U9Pi4uLik=
│ │ │  #:len=1329
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gdG8gdmlldyBjb21tdW5pY2F0aW9u
│ │ │  IGJldHdlZW4gTWFjYXVsYXkyIHNlcnZlciBhbmQgYnJvd3NlciAiLCAibGluZW51bSIgPT4gMjEw
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  Z2V0UmFuZ2VPZk9uZVBhcmFtZXRlckZhbWlseShJZGVhbCk=
│ │ │  #:len=349
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjMwMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2V0UmFuZ2VPZk9uZVBhcmFtZXRlckZhbWlseSxJ
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_append__Family.out
│ │ │ @@ -21,15 +21,15 @@
│ │ │  i5 : setRandomSeed("always successful");
│ │ │   -- setting random seed to 11500776554390917551191162798934277
│ │ │  
│ │ │  i6 : elapsedTime (smooth,J)=getSmoothingFamily(L,6,Verbose=>1)
│ │ │  number of components = 1, codimension of components = {0}
│ │ │  semigroup = {5, 7, 9}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .817637s elapsed
│ │ │ + -- .557727s elapsed
│ │ │  
│ │ │                      2             7      9     14   5      2    2 7    2   8
│ │ │  o6 = (true, ideal (x  - x x  - x z  - x z  - 2z  , x  - x x  - x z  - x x z 
│ │ │                      2    0 4    2      0            0    2 4    4      0 2  
│ │ │       ------------------------------------------------------------------------
│ │ │              9    3 10        11        13    2 15      16      18      20  
│ │ │       - x x z  - x z   - x x z   - x x z   - x z   - x z   - x z   - x z   -
│ │ │ @@ -43,15 +43,15 @@
│ │ │       x z   - x z   - z  ))
│ │ │        0       0
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : elapsedTime smoothnessWithReductions(J,Verbose=>1)
│ │ │  semigroup = {5, 7, 9}
│ │ │ - -- .0189065s elapsed
│ │ │ + -- .0203689s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : assert(flatten drop(degrees ring J,-1)==L)
│ │ │  
│ │ │  i9 : "appendFamily(L,J,X,Xdbm)";
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Smoothing__Family__With__Versal__Deformation.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  Calculating first order relations
│ │ │  Calculating standard expressions for obstructions
│ │ │  Starting lifting
│ │ │  Order 2
│ │ │  Order 3
│ │ │  Order 4
│ │ │  Solution is polynomial
│ │ │ - -- .253446s elapsed
│ │ │ + -- .150663s elapsed
│ │ │  
│ │ │                      3    2      2   4        5    3 6    2 7        9      10
│ │ │  o2 = (true, ideal (x  - x x  - x x z  - x x z  - x z  - x z  + x x z  - x z  
│ │ │                      2    0 1    0 2      0 1      0      2      0 2      1   
│ │ │       ------------------------------------------------------------------------
│ │ │          2 11      14       16     21   3      2        4    2   5        6  
│ │ │       - x z   + x z   - 3x z   - 2z  , x x  - x  - x x z  - x x z  - x x z  -
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Range.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : range1=makeRange(L,{4,6})
│ │ │  
│ │ │  o3 = {4, 6, 8, 12}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range1)
│ │ │ - -- .850657s elapsed
│ │ │ + -- .431438s elapsed
│ │ │  
│ │ │                      3           2 4      8     2    2    2 6   3    2    
│ │ │  o4 = (true, ideal (x  - x x  - x z  - x z , x x  - x  - x z , x  - x x  +
│ │ │                      0    1 3    0      0     0 1    3    0     1    0 3  
│ │ │       ------------------------------------------------------------------------
│ │ │            4        6      8
│ │ │       x x z  - x x z  + x z ), {0})
│ │ │ @@ -35,23 +35,23 @@
│ │ │  
│ │ │  o5 = {10, 11, 12, 13, 14, 15}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range2,Verbose=>1)
│ │ │  time to decompose J1 : 
│ │ │ - -- .000769529s elapsed
│ │ │ + -- .0009533s elapsed
│ │ │  
│ │ │  component number = 0
│ │ │  deformation weights = {{10}}
│ │ │  semigroup = {4, 5, 7}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .00146584s elapsed
│ │ │ + -- .00102092s elapsed
│ │ │  flat = true
│ │ │ - -- .720102s elapsed
│ │ │ + -- .361303s elapsed
│ │ │  
│ │ │                      3            2    2      10   3    2        10
│ │ │  o6 = (true, ideal (x  - x x , x x  - x  - x z  , x  - x x  - x z  ), {0})
│ │ │                      0    1 3   0 1    3    0      1    0 3    1
│ │ │  
│ │ │  o6 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_prune__Family.out
│ │ │ @@ -23,15 +23,15 @@
│ │ │  i6 : as=apply(numgens I,i->a=drop(support unfolding_{i},#L));
│ │ │  
│ │ │  i7 : rL=apply(#as,i->select(as_i,m->(degree m)_0> b));
│ │ │  
│ │ │  i8 : restrictionList=apply(flatten rL,m->sub(m,A));
│ │ │  
│ │ │  i9 : elapsedTime (J,family)=getFlatFamily(I,A,unfolding,restrictionList);
│ │ │ - -- .325212s elapsed
│ │ │ + -- .175525s elapsed
│ │ │  
│ │ │  i10 : leadTerm J
│ │ │  
│ │ │  o10 = ideal (a      a      , a      a      , a      a      , a      a      ,
│ │ │                {1, 4} {2, 6}   {2, 4} {1, 4}   {2, 3} {1, 4}   {2, 4} {1, 3} 
│ │ │        -----------------------------------------------------------------------
│ │ │        a      a      , a      a      )
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_smoothness__With__Reductions.out
│ │ │ @@ -27,12 +27,12 @@
│ │ │          0      3       1       0      0    1 3    0 3      3
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2)
│ │ │  semigroup = {5, 6, 8}
│ │ │  dim and degree singF = (0, 4)
│ │ │ - -- .0188212s elapsed
│ │ │ + -- .0203886s elapsed
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_append__Family.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │
    │ │ │ 
    i6 : elapsedTime (smooth,J)=getSmoothingFamily(L,6,Verbose=>1)
│ │ │  number of components = 1, codimension of components = {0}
│ │ │  semigroup = {5, 7, 9}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .817637s elapsed
│ │ │ + -- .557727s elapsed
│ │ │  
│ │ │                      2             7      9     14   5      2    2 7    2   8
│ │ │  o6 = (true, ideal (x  - x x  - x z  - x z  - 2z  , x  - x x  - x z  - x x z 
│ │ │                      2    0 4    2      0            0    2 4    4      0 2  
│ │ │       ------------------------------------------------------------------------
│ │ │              9    3 10        11        13    2 15      16      18      20  
│ │ │       - x x z  - x z   - x x z   - x x z   - x z   - x z   - x z   - x z   -
│ │ │ @@ -138,15 +138,15 @@
│ │ │  o6 : Sequence
    │ │ │
    │ │ │ 
    i7 : elapsedTime smoothnessWithReductions(J,Verbose=>1)
│ │ │  semigroup = {5, 7, 9}
│ │ │ - -- .0189065s elapsed
│ │ │ + -- .0203689s elapsed
│ │ │  
│ │ │  o7 = true
    │ │ │
    │ │ │ 
    i8 : assert(flatten drop(degrees ring J,-1)==L)
    │ │ │ ├── html2text {} │ │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ o4 = fam8.dbm │ │ │ │ i5 : setRandomSeed("always successful"); │ │ │ │ -- setting random seed to 11500776554390917551191162798934277 │ │ │ │ i6 : elapsedTime (smooth,J)=getSmoothingFamily(L,6,Verbose=>1) │ │ │ │ number of components = 1, codimension of components = {0} │ │ │ │ semigroup = {5, 7, 9} │ │ │ │ smoothing components numbers = {0} │ │ │ │ - -- .817637s elapsed │ │ │ │ + -- .557727s elapsed │ │ │ │ │ │ │ │ 2 7 9 14 5 2 2 7 2 8 │ │ │ │ o6 = (true, ideal (x - x x - x z - x z - 2z , x - x x - x z - x x z │ │ │ │ 2 0 4 2 0 0 2 4 4 0 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 9 3 10 11 13 2 15 16 18 20 │ │ │ │ - x x z - x z - x x z - x x z - x z - x z - x z - x z - │ │ │ │ @@ -51,15 +51,15 @@ │ │ │ │ 2 17 22 27 │ │ │ │ x z - x z - z )) │ │ │ │ 0 0 │ │ │ │ │ │ │ │ o6 : Sequence │ │ │ │ i7 : elapsedTime smoothnessWithReductions(J,Verbose=>1) │ │ │ │ semigroup = {5, 7, 9} │ │ │ │ - -- .0189065s elapsed │ │ │ │ + -- .0203689s elapsed │ │ │ │ │ │ │ │ o7 = true │ │ │ │ i8 : assert(flatten drop(degrees ring J,-1)==L) │ │ │ │ i9 : "appendFamily(L,J,X,Xdbm)"; │ │ │ │ Reading and writing to the disk does not work in the documentation. Hence we │ │ │ │ give the command in quotes. │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_get__Smoothing__Family__With__Versal__Deformation.html │ │ │ @@ -99,15 +99,15 @@ │ │ │ Calculating first order relations │ │ │ Calculating standard expressions for obstructions │ │ │ Starting lifting │ │ │ Order 2 │ │ │ Order 3 │ │ │ Order 4 │ │ │ Solution is polynomial │ │ │ - -- .253446s elapsed │ │ │ + -- .150663s elapsed │ │ │ │ │ │ 3 2 2 4 5 3 6 2 7 9 10 │ │ │ o2 = (true, ideal (x - x x - x x z - x x z - x z - x z + x x z - x z │ │ │ 2 0 1 0 2 0 1 0 2 0 2 1 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 11 14 16 21 3 2 4 2 5 6 │ │ │ - x z + x z - 3x z - 2z , x x - x - x x z - x x z - x x z - │ │ │ ├── html2text {} │ │ │ │ @@ -32,15 +32,15 @@ │ │ │ │ Calculating first order relations │ │ │ │ Calculating standard expressions for obstructions │ │ │ │ Starting lifting │ │ │ │ Order 2 │ │ │ │ Order 3 │ │ │ │ Order 4 │ │ │ │ Solution is polynomial │ │ │ │ - -- .253446s elapsed │ │ │ │ + -- .150663s elapsed │ │ │ │ │ │ │ │ 3 2 2 4 5 3 6 2 7 9 10 │ │ │ │ o2 = (true, ideal (x - x x - x x z - x x z - x z - x z + x x z - x z │ │ │ │ 2 0 1 0 2 0 1 0 2 0 2 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 11 14 16 21 3 2 4 2 5 6 │ │ │ │ - x z + x z - 3x z - 2z , x x - x - x x z - x x z - x x z - │ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_make__Range.html │ │ │ @@ -106,15 +106,15 @@ │ │ │ │ │ │ o3 : List │ │ │
    │ │ │ 
    i4 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range1)
│ │ │ - -- .850657s elapsed
│ │ │ + -- .431438s elapsed
│ │ │  
│ │ │                      3           2 4      8     2    2    2 6   3    2    
│ │ │  o4 = (true, ideal (x  - x x  - x z  - x z , x x  - x  - x z , x  - x x  +
│ │ │                      0    1 3    0      0     0 1    3    0     1    0 3  
│ │ │       ------------------------------------------------------------------------
│ │ │            4        6      8
│ │ │       x x z  - x x z  + x z ), {0})
│ │ │ @@ -137,23 +137,23 @@
│ │ │  o5 : List
    │ │ │
    │ │ │ 
    i6 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range2,Verbose=>1)
│ │ │  time to decompose J1 : 
│ │ │ - -- .000769529s elapsed
│ │ │ + -- .0009533s elapsed
│ │ │  
│ │ │  component number = 0
│ │ │  deformation weights = {{10}}
│ │ │  semigroup = {4, 5, 7}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .00146584s elapsed
│ │ │ + -- .00102092s elapsed
│ │ │  flat = true
│ │ │ - -- .720102s elapsed
│ │ │ + -- .361303s elapsed
│ │ │  
│ │ │                      3            2    2      10   3    2        10
│ │ │  o6 = (true, ideal (x  - x x , x x  - x  - x z  , x  - x x  - x z  ), {0})
│ │ │                      0    1 3   0 1    3    0      1    0 3    1
│ │ │  
│ │ │  o6 : Sequence
    │ │ │     		│ │ │ 
    i8 : restrictionList=apply(flatten rL,m->sub(m,A));
    │ │ │
    │ │ │ 
    i9 : elapsedTime (J,family)=getFlatFamily(I,A,unfolding,restrictionList);
│ │ │ - -- .325212s elapsed
    │ │ │ + -- .175525s elapsed │ │ │
    │ │ │ 
    i10 : leadTerm J
│ │ │  
│ │ │  o10 = ideal (a      a      , a      a      , a      a      , a      a      ,
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │                10007  0   1   3   4
│ │ │ │  i4 : (A,unfolding)=makeUnfolding I;
│ │ │ │  i5 : b = genus L-1;
│ │ │ │  i6 : as=apply(numgens I,i->a=drop(support unfolding_{i},#L));
│ │ │ │  i7 : rL=apply(#as,i->select(as_i,m->(degree m)_0> b));
│ │ │ │  i8 : restrictionList=apply(flatten rL,m->sub(m,A));
│ │ │ │  i9 : elapsedTime (J,family)=getFlatFamily(I,A,unfolding,restrictionList);
│ │ │ │ - -- .325212s elapsed
│ │ │ │ + -- .175525s elapsed
│ │ │ │  i10 : leadTerm J
│ │ │ │  
│ │ │ │  o10 = ideal (a      a      , a      a      , a      a      , a      a      ,
│ │ │ │                {1, 4} {2, 6}   {2, 4} {1, 4}   {2, 3} {1, 4}   {2, 4} {1, 3}
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        a      a      , a      a      )
│ │ │ │         {2, 3} {1, 3}   {1, 2} {2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_smoothness__With__Reductions.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │
    │ │ │ 
    i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2)
│ │ │  semigroup = {5, 6, 8}
│ │ │  dim and degree singF = (0, 4)
│ │ │ - -- .0188212s elapsed
│ │ │ + -- .0203886s elapsed
│ │ │  
│ │ │  o5 = true
    │ │ │
    │ │ │
    │ │ │

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

    │ │ │ ├── html2text {} │ │ │ │ @@ -49,15 +49,15 @@ │ │ │ │ - x z - x z - x z + x z , x - x x + x x z + x z - z ) │ │ │ │ 0 3 1 0 0 1 3 0 3 3 │ │ │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2) │ │ │ │ semigroup = {5, 6, 8} │ │ │ │ dim and degree singF = (0, 4) │ │ │ │ - -- .0188212s elapsed │ │ │ │ + -- .0203886s elapsed │ │ │ │ │ │ │ │ o5 = true │ │ │ │ The intermediate output dim and degree singF = (0, 4) says that after computing │ │ │ │ some minors of the jacobian matrix, we detect that the curve is smooth away │ │ │ │ from the zero dimensional scheme defined by singF of degree 4. │ │ │ │ The function checkSmoothness takes longer, some times much longer. │ │ │ │ ********** WWaayyss ttoo uussee ssmmooootthhnneessssWWiitthhRReedduuccttiioonnss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ Zmxvb3IoUldlaWxEaXZpc29yKQ== │ │ │ #:len=275 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU1Niwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxvb3IsUldlaWxEaXZpc29yKSwiZmxvb3IoUldl │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Basic__Divisor.out │ │ │ @@ -1,14 +1,14 @@ │ │ │ -- -*- M2-comint -*- hash: 18380296066161043289 │ │ │ │ │ │ i1 : R = QQ[x,y,z]; │ │ │ │ │ │ i2 : D = divisor(x*y^2*z^3) │ │ │ │ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : H = new HashTable from D │ │ │ │ │ │ o3 = HashTable{{x} => {1} } │ │ │ {y} => {2} │ │ │ @@ -16,18 +16,18 @@ │ │ │ cache => CacheTable{...1...} │ │ │ ring => R │ │ │ │ │ │ o3 : HashTable │ │ │ │ │ │ i4 : (2/3)*D │ │ │ │ │ │ -o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x) │ │ │ +o4 = 2/3*Div(x) + 2*Div(z) + 4/3*Div(y) │ │ │ │ │ │ o4 : QWeilDivisor on R │ │ │ │ │ │ i5 : 0.6*D │ │ │ │ │ │ -o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x) │ │ │ +o5 = .6*Div(x) + 1.8*Div(z) + 1.2*Div(y) │ │ │ │ │ │ o5 : RWeilDivisor on R │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Basic__Divisor_sp_pl_sp__Basic__Divisor.out │ │ │ @@ -1,32 +1,32 @@ │ │ │ -- -*- M2-comint -*- hash: 11085051886200177329 │ │ │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ │ │ i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)}) │ │ │ │ │ │ -o2 = 3*Div(y) + 2*Div(z) + Div(x) │ │ │ +o2 = Div(x) + 3*Div(y) + 2*Div(z) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)}) │ │ │ │ │ │ -o3 = -2*Div(z) + 3*Div(y) + -5*Div(x) │ │ │ +o3 = -5*Div(x) + -2*Div(z) + 3*Div(y) │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ │ │ i4 : D1 + D2 │ │ │ │ │ │ -o4 = 6*Div(y) + -4*Div(x) │ │ │ +o4 = -4*Div(x) + 6*Div(y) │ │ │ │ │ │ o4 : WeilDivisor on R │ │ │ │ │ │ i5 : D1 - D2 │ │ │ │ │ │ -o5 = 4*Div(z) + 6*Div(x) │ │ │ +o5 = 6*Div(x) + 4*Div(z) │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ │ │ i6 : R = QQ[x,y]; │ │ │ │ │ │ i7 : D1 = divisor({3, 1}, {ideal(x), ideal(y)}) │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Number_sp_st_sp__Basic__Divisor.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ │ │ │ o3 = 1/2*Div(x) + -5/3*Div(y) │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ │ │ i4 : F = divisor({1.5, 0, -3.2}, {ideal(x), ideal(y), ideal(x^2-y^3)}, CoefficientType=>RR) │ │ │ │ │ │ -o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2) │ │ │ +o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y) │ │ │ │ │ │ o4 : RWeilDivisor on R │ │ │ │ │ │ i5 : 8*D │ │ │ │ │ │ o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x) │ │ │ │ │ │ @@ -46,18 +46,18 @@ │ │ │ │ │ │ o9 = 2.35667*Div(y) + -.707*Div(x) │ │ │ │ │ │ o9 : RWeilDivisor on R │ │ │ │ │ │ i10 : 6*F │ │ │ │ │ │ -o10 = 9*Div(x) + -19.2*Div(-y^3+x^2) │ │ │ +o10 = -19.2*Div(-y^3+x^2) + 9*Div(x) │ │ │ │ │ │ o10 : RWeilDivisor on R │ │ │ │ │ │ i11 : (-3/2)*F │ │ │ │ │ │ -o11 = -2.25*Div(x) + 4.8*Div(-y^3+x^2) │ │ │ +o11 = 4.8*Div(-y^3+x^2) + -2.25*Div(x) │ │ │ │ │ │ o11 : RWeilDivisor on R │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_apply__To__Coefficients.out │ │ │ @@ -1,17 +1,17 @@ │ │ │ -- -*- M2-comint -*- hash: 14937934652040812889 │ │ │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ │ │ i2 : D = divisor(x*y^2/z) │ │ │ │ │ │ -o2 = Div(x) + -Div(z) + 2*Div(y) │ │ │ +o2 = -Div(z) + 2*Div(y) + Div(x) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : applyToCoefficients(D, u->5*u) │ │ │ │ │ │ -o3 = 5*Div(x) + 10*Div(y) + -5*Div(z) │ │ │ +o3 = 10*Div(y) + -5*Div(z) + 5*Div(x) │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clean__Support.out │ │ │ @@ -1,17 +1,17 @@ │ │ │ -- -*- M2-comint -*- hash: 2803818461661759459 │ │ │ │ │ │ i1 : R = QQ[x,y,z]; │ │ │ │ │ │ i2 : D = divisor({1,0,-2}, {ideal(x), ideal(y), ideal(z)}) │ │ │ │ │ │ -o2 = Div(x) + 0*Div(y) + -2*Div(z) │ │ │ +o2 = 0*Div(y) + -2*Div(z) + Div(x) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : cleanSupport(D) │ │ │ │ │ │ -o3 = Div(x) + -2*Div(z) │ │ │ +o3 = -2*Div(z) + Div(x) │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out │ │ │ @@ -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 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : E = divisor(x*y^2*z^3) │ │ │ │ │ │ -o3 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ +o3 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ │ │ i4 : F = divisor(ideal(x*y^2*z^3)) │ │ │ │ │ │ -o4 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ +o4 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ │ │ o4 : WeilDivisor on R │ │ │ │ │ │ i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}}) │ │ │ │ │ │ -o5 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ +o5 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ │ │ i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z) │ │ │ │ │ │ -o6 = Div(x) + 3*Div(z) + 2*Div(y) │ │ │ +o6 = 3*Div(z) + 2*Div(y) + Div(x) │ │ │ │ │ │ o6 : WeilDivisor on R │ │ │ │ │ │ i7 : R = QQ[x,y,z]/ideal(x^2-y*z); │ │ │ │ │ │ i8 : D = divisor({2}, {ideal(x,y)}) │ │ │ │ │ │ @@ -68,21 +68,21 @@ │ │ │ │ │ │ o15 : WeilDivisor on A │ │ │ │ │ │ i16 : R = ZZ/7[x,y]; │ │ │ │ │ │ i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ) │ │ │ │ │ │ -o17 = -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 0.0706325s (cpu); 0.0706342s (thread); 0s (gc) │ │ │ + -- used 0.0457191s (cpu); 0.0457183s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ │ │ i13 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ - -- used 0.491025s (cpu); 0.490974s (thread); 0s (gc) │ │ │ + -- used 0.382233s (cpu); 0.382241s (thread); 0s (gc) │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ │ │ i14 : time dualize(M, Strategy=>IdealStrategy); │ │ │ - -- used 0.528431s (cpu); 0.440429s (thread); 0s (gc) │ │ │ + -- used 0.523672s (cpu); 0.442914s (thread); 0s (gc) │ │ │ │ │ │ i15 : time dualize(M, Strategy=>ModuleStrategy); │ │ │ - -- used 0.00269572s (cpu); 0.00269697s (thread); 0s (gc) │ │ │ + -- used 0.00307887s (cpu); 0.00308346s (thread); 0s (gc) │ │ │ │ │ │ i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy); │ │ │ - -- used 0.00217526s (cpu); 0.00217625s (thread); 0s (gc) │ │ │ + -- used 0.00264852s (cpu); 0.00265352s (thread); 0s (gc) │ │ │ │ │ │ o16 : Ideal of R │ │ │ │ │ │ i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v); │ │ │ │ │ │ i18 : I = ideal(x,u); │ │ │ │ │ │ o18 : Ideal of R │ │ │ │ │ │ i19 : J = I^15; │ │ │ │ │ │ o19 : Ideal of R │ │ │ │ │ │ i20 : time dualize(J, Strategy=>IdealStrategy); │ │ │ - -- used 0.0637156s (cpu); 0.0637222s (thread); 0s (gc) │ │ │ + -- used 0.0735762s (cpu); 0.0735808s (thread); 0s (gc) │ │ │ │ │ │ o20 : Ideal of R │ │ │ │ │ │ i21 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ - -- used 0.00595766s (cpu); 0.00595829s (thread); 0s (gc) │ │ │ + -- used 0.00595823s (cpu); 0.00596222s (thread); 0s (gc) │ │ │ │ │ │ o21 : Ideal of R │ │ │ │ │ │ i22 : R = QQ[x,y]/ideal(x*y); │ │ │ │ │ │ i23 : J = ideal(x,y); │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Divisors.out │ │ │ @@ -1,20 +1,20 @@ │ │ │ -- -*- M2-comint -*- hash: 6816075660530398278 │ │ │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ │ │ i2 : D = divisor({-8, 2, 0}, {ideal(x), ideal(y), ideal(x^2+z)}) │ │ │ │ │ │ -o2 = -8*Div(x) + 2*Div(y) + 0*Div(x^2+z) │ │ │ +o2 = 0*Div(x^2+z) + -8*Div(x) + 2*Div(y) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : getPrimeDivisors( D ) │ │ │ │ │ │ -o3 = {Div(x), Div(y), Div(x^2+z)} │ │ │ +o3 = {Div(x^2+z), Div(x), Div(y)} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : getPrimeDivisors( cleanSupport D ) │ │ │ │ │ │ o4 = {Div(x), Div(y)} │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Cartier.out │ │ │ @@ -24,15 +24,15 @@ │ │ │ │ │ │ o6 = false │ │ │ │ │ │ i7 : R = QQ[x, y, z]; │ │ │ │ │ │ i8 : D = divisor({1, 2}, {ideal(x), ideal(y)}) │ │ │ │ │ │ -o8 = 2*Div(y) + Div(x) │ │ │ +o8 = Div(x) + 2*Div(y) │ │ │ │ │ │ o8 : WeilDivisor on R │ │ │ │ │ │ i9 : isCartier( D ) │ │ │ │ │ │ o9 = true │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Reduced.out │ │ │ @@ -1,20 +1,20 @@ │ │ │ -- -*- M2-comint -*- hash: 6263371580478090172 │ │ │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ │ │ i2 : D1 = divisor(x^2 * y^3 * z) │ │ │ │ │ │ -o2 = 3*Div(y) + Div(z) + 2*Div(x) │ │ │ +o2 = 2*Div(x) + 3*Div(y) + Div(z) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : D2 = divisor(x * y * z) │ │ │ │ │ │ -o3 = Div(y) + Div(z) + Div(x) │ │ │ +o3 = Div(x) + Div(y) + Div(z) │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ │ │ i4 : isReduced( D1 ) │ │ │ │ │ │ o4 = false │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__S__N__C.out │ │ │ @@ -36,39 +36,39 @@ │ │ │ │ │ │ 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 │ │ │ │ │ │ i13 : R = QQ[x, y]; │ │ │ │ │ │ i14 : D = divisor(x*y*(x+y)) │ │ │ │ │ │ -o14 = Div(y) + Div(x) + Div(x+y) │ │ │ +o14 = Div(x+y) + Div(y) + Div(x) │ │ │ │ │ │ o14 : WeilDivisor on R │ │ │ │ │ │ i15 : isSNC( D, IsGraded => true ) │ │ │ │ │ │ o15 = true │ │ │ │ │ │ i16 : R = QQ[x,y,z]; │ │ │ │ │ │ i17 : D = divisor(x*y*(x+y)) │ │ │ │ │ │ -o17 = Div(x) + Div(x+y) + Div(y) │ │ │ +o17 = Div(y) + Div(x) + Div(x+y) │ │ │ │ │ │ o17 : WeilDivisor on R │ │ │ │ │ │ i18 : isSNC( D, IsGraded => true) │ │ │ │ │ │ o18 = false │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Weil__Divisor.out │ │ │ @@ -1,20 +1,20 @@ │ │ │ -- -*- M2-comint -*- hash: 9088327857146195664 │ │ │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ │ │ i2 : D1 = divisor({1/1, 2/2, -6/3}, {ideal(x), ideal(y), ideal(z)}, CoefficientType=>QQ) │ │ │ │ │ │ -o2 = Div(y) + -2*Div(z) + Div(x) │ │ │ +o2 = Div(x) + Div(y) + -2*Div(z) │ │ │ │ │ │ o2 : QWeilDivisor on R │ │ │ │ │ │ i3 : D2 = divisor({1/2, 3/4, 5/6}, {ideal(y), ideal(z), ideal(x)}, CoefficientType=>QQ) │ │ │ │ │ │ -o3 = 1/2*Div(y) + 3/4*Div(z) + 5/6*Div(x) │ │ │ +o3 = 5/6*Div(x) + 1/2*Div(y) + 3/4*Div(z) │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ │ │ i4 : isWeilDivisor( D1 ) │ │ │ │ │ │ o4 = true │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_map__To__Projective__Space.out │ │ │ @@ -16,15 +16,15 @@ │ │ │ o3 : RingMap R <-- QQ[YY ..YY ] │ │ │ 1 2 │ │ │ │ │ │ i4 : R = ZZ/7[x,y,z]; │ │ │ │ │ │ i5 : D = divisor(x*y) │ │ │ │ │ │ -o5 = Div(x) + Div(y) │ │ │ +o5 = Div(y) + Div(x) │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ │ │ i6 : mapToProjectiveSpace(D, Variable=>"Z") │ │ │ │ │ │ ZZ 2 2 2 │ │ │ o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z }) │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out │ │ │ @@ -6,21 +6,21 @@ │ │ │ │ │ │ i3 : f = map(T, R, {a^3, a^2*b, a*b^2, b^3}); │ │ │ │ │ │ o3 : RingMap T <-- R │ │ │ │ │ │ i4 : D = divisor(y*z) │ │ │ │ │ │ -o4 = 3*Div(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) │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out │ │ │ @@ -103,104 +103,104 @@ │ │ │ o21 : Ideal of R │ │ │ │ │ │ i22 : J = I^21; │ │ │ │ │ │ o22 : Ideal of R │ │ │ │ │ │ i23 : time reflexify(J); │ │ │ - -- used 0.338433s (cpu); 0.253168s (thread); 0s (gc) │ │ │ + -- used 0.239803s (cpu); 0.156455s (thread); 0s (gc) │ │ │ │ │ │ o23 : Ideal of R │ │ │ │ │ │ i24 : time reflexify(J*R^1); │ │ │ - -- used 0.671713s (cpu); 0.510526s (thread); 0s (gc) │ │ │ + -- used 0.394678s (cpu); 0.320457s (thread); 0s (gc) │ │ │ │ │ │ i25 : R = ZZ/13[x,y,z]/ideal(x^3 + y^3-z^11*x*y); │ │ │ │ │ │ i26 : I = ideal(x-4*y, z); │ │ │ │ │ │ o26 : Ideal of R │ │ │ │ │ │ i27 : J = I^20; │ │ │ │ │ │ o27 : Ideal of R │ │ │ │ │ │ i28 : M = J*R^1; │ │ │ │ │ │ i29 : J1 = time reflexify( J, Strategy=>IdealStrategy ) │ │ │ - -- used 0.168829s (cpu); 0.168835s (thread); 0s (gc) │ │ │ + -- used 0.0895845s (cpu); 0.0895882s (thread); 0s (gc) │ │ │ │ │ │ 2 2 9 9 11 │ │ │ o29 = ideal (x + 5x*y + 3y , x*z - 4y*z , z + x - 4y) │ │ │ │ │ │ o29 : Ideal of R │ │ │ │ │ │ i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy ) │ │ │ - -- used 7.92394s (cpu); 5.32977s (thread); 0s (gc) │ │ │ + -- used 5.43945s (cpu); 3.95694s (thread); 0s (gc) │ │ │ │ │ │ 2 2 9 9 11 │ │ │ o30 = ideal (x + 5x*y + 3y , x*z - 4y*z , z + x - 4y) │ │ │ │ │ │ o30 : Ideal of R │ │ │ │ │ │ i31 : J1 == J2 │ │ │ │ │ │ o31 = true │ │ │ │ │ │ i32 : time reflexify( M, Strategy=>IdealStrategy ); │ │ │ - -- used 6.96991s (cpu); 5.49609s (thread); 0s (gc) │ │ │ + -- used 5.52999s (cpu); 4.16149s (thread); 0s (gc) │ │ │ │ │ │ i33 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ - -- used 0.831741s (cpu); 0.473654s (thread); 0s (gc) │ │ │ + -- used 0.504874s (cpu); 0.342262s (thread); 0s (gc) │ │ │ │ │ │ i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v); │ │ │ │ │ │ i35 : I = ideal(x,u); │ │ │ │ │ │ o35 : Ideal of R │ │ │ │ │ │ i36 : J = I^20; │ │ │ │ │ │ o36 : Ideal of R │ │ │ │ │ │ i37 : M = I^20*R^1; │ │ │ │ │ │ i38 : time reflexify( J, Strategy=>IdealStrategy ) │ │ │ - -- used 0.461613s (cpu); 0.267865s (thread); 0s (gc) │ │ │ + -- used 0.396157s (cpu); 0.229751s (thread); 0s (gc) │ │ │ │ │ │ 20 19 2 18 3 17 4 16 5 15 6 14 7 13 8 12 │ │ │ o38 = ideal (u , x*u , x u , x u , x u , x u , x u , x u , x u , │ │ │ ----------------------------------------------------------------------- │ │ │ 9 11 10 10 11 9 12 8 13 7 14 6 15 5 16 4 17 3 18 2 │ │ │ x u , x u , x u , x u , x u , x u , x u , x u , x u , x u , │ │ │ ----------------------------------------------------------------------- │ │ │ 19 20 │ │ │ x u, x ) │ │ │ │ │ │ o38 : Ideal of R │ │ │ │ │ │ i39 : time reflexify( J, Strategy=>ModuleStrategy ) │ │ │ - -- used 0.0135561s (cpu); 0.0135572s (thread); 0s (gc) │ │ │ + -- used 0.0124373s (cpu); 0.0124375s (thread); 0s (gc) │ │ │ │ │ │ 20 19 2 18 3 17 4 16 5 15 6 14 7 13 8 12 │ │ │ o39 = ideal (u , x*u , x u , x u , x u , x u , x u , x u , x u , │ │ │ ----------------------------------------------------------------------- │ │ │ 9 11 10 10 11 9 12 8 13 7 14 6 15 5 16 4 17 3 18 2 │ │ │ x u , x u , x u , x u , x u , x u , x u , x u , x u , x u , │ │ │ ----------------------------------------------------------------------- │ │ │ 19 20 │ │ │ x u, x ) │ │ │ │ │ │ o39 : Ideal of R │ │ │ │ │ │ i40 : time reflexify( M, Strategy=>IdealStrategy ); │ │ │ - -- used 0.0397151s (cpu); 0.0397234s (thread); 0s (gc) │ │ │ + -- used 0.037933s (cpu); 0.037937s (thread); 0s (gc) │ │ │ │ │ │ i41 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ - -- used 0.00682599s (cpu); 0.00682691s (thread); 0s (gc) │ │ │ + -- used 0.00612283s (cpu); 0.00613039s (thread); 0s (gc) │ │ │ │ │ │ i42 : R = QQ[x,y]/ideal(x*y); │ │ │ │ │ │ i43 : I = ideal(x,y); │ │ │ │ │ │ o43 : Ideal of R │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out │ │ │ @@ -23,44 +23,44 @@ │ │ │ i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3); │ │ │ │ │ │ i6 : I = ideal(x-z,y-2*z); │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ i7 : time J20a = reflexivePower(20, I); │ │ │ - -- used 0.0292441s (cpu); 0.0292465s (thread); 0s (gc) │ │ │ + -- used 0.029116s (cpu); 0.0291144s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ │ │ i8 : I20 = I^20; │ │ │ │ │ │ o8 : Ideal of R │ │ │ │ │ │ i9 : time J20b = reflexify(I20); │ │ │ - -- used 0.142968s (cpu); 0.14297s (thread); 0s (gc) │ │ │ + -- used 0.134859s (cpu); 0.13486s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ │ │ i10 : J20a == J20b │ │ │ │ │ │ o10 = true │ │ │ │ │ │ i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3); │ │ │ │ │ │ i12 : I = ideal(x-z,y-2*z); │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ │ │ i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy); │ │ │ - -- used 0.0299195s (cpu); 0.0299242s (thread); 0s (gc) │ │ │ + -- used 0.029618s (cpu); 0.0296198s (thread); 0s (gc) │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ │ │ i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy); │ │ │ - -- used 0.164355s (cpu); 0.0959583s (thread); 0s (gc) │ │ │ + -- used 0.0565307s (cpu); 0.0565374s (thread); 0s (gc) │ │ │ │ │ │ o14 : Ideal of R │ │ │ │ │ │ i15 : J1 == J2 │ │ │ │ │ │ o15 = true │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Basic__Divisor.html │ │ │ @@ -77,15 +77,15 @@ │ │ │ 
    i1 : R = QQ[x,y,z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : H = new HashTable from D
│ │ │ @@ -99,24 +99,24 @@
│ │ │  o3 : HashTable
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : (2/3)*D
│ │ │  
│ │ │ -o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x)
│ │ │ +o4 = 2/3*Div(x) + 2*Div(z) + 4/3*Div(y)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : 0.6*D
│ │ │  
│ │ │ -o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │ +o5 = .6*Div(x) + 1.8*Div(z) + 1.2*Div(y)
│ │ │  
│ │ │  o5 : RWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -15,34 +15,34 @@ │ │ │ │ specifies the ambient ring. Another key is cache which points to a CacheTable. │ │ │ │ The remaining keys are a Groebner basis $L$ for each prime ideal $P$ in the │ │ │ │ support with corresponding value a list with one entry {$n$} where $n$ is the │ │ │ │ coefficient of the height one prime. │ │ │ │ i1 : R = QQ[x,y,z]; │ │ │ │ i2 : D = divisor(x*y^2*z^3) │ │ │ │ │ │ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : H = new HashTable from D │ │ │ │ │ │ │ │ o3 = HashTable{{x} => {1} } │ │ │ │ {y} => {2} │ │ │ │ {z} => {3} │ │ │ │ cache => CacheTable{...1...} │ │ │ │ ring => R │ │ │ │ │ │ │ │ o3 : HashTable │ │ │ │ i4 : (2/3)*D │ │ │ │ │ │ │ │ -o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x) │ │ │ │ +o4 = 2/3*Div(x) + 2*Div(z) + 4/3*Div(y) │ │ │ │ │ │ │ │ o4 : QWeilDivisor on R │ │ │ │ i5 : 0.6*D │ │ │ │ │ │ │ │ -o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x) │ │ │ │ +o5 = .6*Div(x) + 1.8*Div(z) + 1.2*Div(y) │ │ │ │ │ │ │ │ o5 : RWeilDivisor on R │ │ │ │ ********** TTyyppeess ooff BBaassiiccDDiivviissoorr:: ********** │ │ │ │ * RWeilDivisor │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aann oobbjjeecctt ooff ccllaassss BBaassiiccDDiivviissoorr:: ********** │ │ │ │ * applyToCoefficients(BasicDivisor,Function) -- see _a_p_p_l_y_T_o_C_o_e_f_f_i_c_i_e_n_t_s - │ │ │ │ - apply a function to the coefficients of a divisor │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Basic__Divisor_sp_pl_sp__Basic__Divisor.html │ │ │ @@ -85,42 +85,42 @@ │ │ │ 
    i1 : R = QQ[x, y, z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = 3*Div(y) + 2*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 3*Div(y) + 2*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)})
│ │ │  
│ │ │ -o3 = -2*Div(z) + 3*Div(y) + -5*Div(x)
│ │ │ +o3 = -5*Div(x) + -2*Div(z) + 3*Div(y)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : D1 + D2
│ │ │  
│ │ │ -o4 = 6*Div(y) + -4*Div(x)
│ │ │ +o4 = -4*Div(x) + 6*Div(y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : D1 - D2
│ │ │  
│ │ │ -o5 = 4*Div(z) + 6*Div(x)
│ │ │ +o5 = 6*Div(x) + 4*Div(z)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    We can also add or subtract divisors with different coefficients.

    │ │ │ ├── html2text {} │ │ │ │ @@ -16,30 +16,30 @@ │ │ │ │ * Outputs: │ │ │ │ o an instance of the type _B_a_s_i_c_D_i_v_i_s_o_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ We can add or subtract two divisors: │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ i2 : D1 = divisor({1, 3, 2}, {ideal(x), ideal(y), ideal(z)}) │ │ │ │ │ │ │ │ -o2 = 3*Div(y) + 2*Div(z) + Div(x) │ │ │ │ +o2 = Div(x) + 3*Div(y) + 2*Div(z) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : D2 = divisor({-2, 3, -5}, {ideal(z), ideal(y), ideal(x)}) │ │ │ │ │ │ │ │ -o3 = -2*Div(z) + 3*Div(y) + -5*Div(x) │ │ │ │ +o3 = -5*Div(x) + -2*Div(z) + 3*Div(y) │ │ │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ i4 : D1 + D2 │ │ │ │ │ │ │ │ -o4 = 6*Div(y) + -4*Div(x) │ │ │ │ +o4 = -4*Div(x) + 6*Div(y) │ │ │ │ │ │ │ │ o4 : WeilDivisor on R │ │ │ │ i5 : D1 - D2 │ │ │ │ │ │ │ │ -o5 = 4*Div(z) + 6*Div(x) │ │ │ │ +o5 = 6*Div(x) + 4*Div(z) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ We can also add or subtract divisors with different coefficients. │ │ │ │ i6 : R = QQ[x,y]; │ │ │ │ i7 : D1 = divisor({3, 1}, {ideal(x), ideal(y)}) │ │ │ │ │ │ │ │ o7 = 3*Div(x) + Div(y) │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Number_sp_st_sp__Basic__Divisor.html │ │ │ @@ -99,15 +99,15 @@ │ │ │ o3 : QWeilDivisor on R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : F = divisor({1.5, 0, -3.2}, {ideal(x), ideal(y), ideal(x^2-y^3)}, CoefficientType=>RR)
│ │ │  
│ │ │ -o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2)
│ │ │ +o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y)
│ │ │  
│ │ │  o4 : RWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : 8*D
│ │ │ @@ -153,24 +153,24 @@
│ │ │  o9 : RWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : 6*F
│ │ │  
│ │ │ -o10 = 9*Div(x) + -19.2*Div(-y^3+x^2)
│ │ │ +o10 = -19.2*Div(-y^3+x^2) + 9*Div(x)
│ │ │  
│ │ │  o10 : RWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : (-3/2)*F
│ │ │  
│ │ │ -o11 = -2.25*Div(x) + 4.8*Div(-y^3+x^2)
│ │ │ +o11 = 4.8*Div(-y^3+x^2) + -2.25*Div(x)
│ │ │  
│ │ │  o11 : RWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,15 +22,15 @@ │ │ │ │ │ │ │ │ o3 = 1/2*Div(x) + -5/3*Div(y) │ │ │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ i4 : F = divisor({1.5, 0, -3.2}, {ideal(x), ideal(y), ideal(x^2-y^3)}, │ │ │ │ CoefficientType=>RR) │ │ │ │ │ │ │ │ -o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2) │ │ │ │ +o4 = -3.2*Div(-y^3+x^2) + 1.5*Div(x) + 0*Div(y) │ │ │ │ │ │ │ │ o4 : RWeilDivisor on R │ │ │ │ i5 : 8*D │ │ │ │ │ │ │ │ o5 = -8*Div(x+y) + 8*Div(y) + 16*Div(x) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ @@ -52,20 +52,20 @@ │ │ │ │ i9 : (-1.414)*E │ │ │ │ │ │ │ │ o9 = 2.35667*Div(y) + -.707*Div(x) │ │ │ │ │ │ │ │ o9 : RWeilDivisor on R │ │ │ │ i10 : 6*F │ │ │ │ │ │ │ │ -o10 = 9*Div(x) + -19.2*Div(-y^3+x^2) │ │ │ │ +o10 = -19.2*Div(-y^3+x^2) + 9*Div(x) │ │ │ │ │ │ │ │ o10 : RWeilDivisor on R │ │ │ │ i11 : (-3/2)*F │ │ │ │ │ │ │ │ -o11 = -2.25*Div(x) + 4.8*Div(-y^3+x^2) │ │ │ │ +o11 = 4.8*Div(-y^3+x^2) + -2.25*Div(x) │ │ │ │ │ │ │ │ o11 : RWeilDivisor on R │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _N_u_m_b_e_r_ _*_ _B_a_s_i_c_D_i_v_i_s_o_r -- multiply a divisor by a number │ │ │ │ * QQ * RWeilDivisor │ │ │ │ * QQ * WeilDivisor │ │ │ │ * RR * QWeilDivisor │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_apply__To__Coefficients.html │ │ │ @@ -87,24 +87,24 @@ │ │ │ 
    i1 : R = QQ[x, y, z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D = divisor(x*y^2/z)
│ │ │  
│ │ │ -o2 = Div(x) + -Div(z) + 2*Div(y)
│ │ │ +o2 = -Div(z) + 2*Div(y) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : applyToCoefficients(D, u->5*u)
│ │ │  
│ │ │ -o3 = 5*Div(x) + 10*Div(y) + -5*Div(z)
│ │ │ +o3 = 10*Div(y) + -5*Div(z) + 5*Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -25,20 +25,20 @@ │ │ │ │ the output D is the same as the class of the input D1 (WeilDivisor, │ │ │ │ QWeilDivisor, RWeilDivisor, BasicDivisor). If Safe is set to true (the default │ │ │ │ is false), then the function will check to make sure the output is a valid │ │ │ │ divisor. │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ i2 : D = divisor(x*y^2/z) │ │ │ │ │ │ │ │ -o2 = Div(x) + -Div(z) + 2*Div(y) │ │ │ │ +o2 = -Div(z) + 2*Div(y) + Div(x) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : applyToCoefficients(D, u->5*u) │ │ │ │ │ │ │ │ -o3 = 5*Div(x) + 10*Div(y) + -5*Div(z) │ │ │ │ +o3 = 10*Div(y) + -5*Div(z) + 5*Div(x) │ │ │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _f_l_o_o_r_(_R_W_e_i_l_D_i_v_i_s_o_r_) -- produce a WeilDivisor whose coefficients are │ │ │ │ ceilings or floors of the divisor │ │ │ │ * _c_e_i_l_i_n_g_(_R_W_e_i_l_D_i_v_i_s_o_r_) -- produce a WeilDivisor whose coefficients are │ │ │ │ ceilings or floors of the divisor │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_clean__Support.html │ │ │ @@ -81,24 +81,24 @@ │ │ │ 
    i1 : R = QQ[x,y,z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D = divisor({1,0,-2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = Div(x) + 0*Div(y) + -2*Div(z)
│ │ │ +o2 = 0*Div(y) + -2*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : cleanSupport(D)
│ │ │  
│ │ │ -o3 = Div(x) + -2*Div(z)
│ │ │ +o3 = -2*Div(z) + Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -13,20 +13,20 @@ │ │ │ │ o an instance of the type _B_a_s_i_c_D_i_v_i_s_o_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ This function returns a divisor where all entries with coefficient zero are │ │ │ │ removed. │ │ │ │ i1 : R = QQ[x,y,z]; │ │ │ │ i2 : D = divisor({1,0,-2}, {ideal(x), ideal(y), ideal(z)}) │ │ │ │ │ │ │ │ -o2 = Div(x) + 0*Div(y) + -2*Div(z) │ │ │ │ +o2 = 0*Div(y) + -2*Div(z) + Div(x) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : cleanSupport(D) │ │ │ │ │ │ │ │ -o3 = Div(x) + -2*Div(z) │ │ │ │ +o3 = -2*Div(z) + Div(x) │ │ │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ ********** WWaayyss ttoo uussee cclleeaannSSuuppppoorrtt:: ********** │ │ │ │ * cleanSupport(BasicDivisor) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_l_e_a_n_S_u_p_p_o_r_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_divisor.html │ │ │ @@ -100,51 +100,51 @@ │ │ │ 
    i1 : R = QQ[x,y,z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D = divisor({1,2,3}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : E = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o3 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o3 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : F = divisor(ideal(x*y^2*z^3))
│ │ │  
│ │ │ -o4 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o4 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}})
│ │ │  
│ │ │ -o5 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │ +o5 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │  
│ │ │ -o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │ +o6 = 3*Div(z) + 2*Div(y) + Div(x)
│ │ │  
│ │ │  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.

    │ │ │ @@ -228,24 +228,24 @@ │ │ │ 
    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 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ +o2 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : E = divisor(x*y^2*z^3) │ │ │ │ │ │ │ │ -o3 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ +o3 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ i4 : F = divisor(ideal(x*y^2*z^3)) │ │ │ │ │ │ │ │ -o4 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ +o4 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ │ │ │ │ o4 : WeilDivisor on R │ │ │ │ i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}}) │ │ │ │ │ │ │ │ -o5 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ +o5 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z) │ │ │ │ │ │ │ │ -o6 = Div(x) + 3*Div(z) + 2*Div(y) │ │ │ │ +o6 = 3*Div(z) + 2*Div(y) + Div(x) │ │ │ │ │ │ │ │ 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)}) │ │ │ │ @@ -103,20 +103,20 @@ │ │ │ │ │ │ │ │ o15 : WeilDivisor on A │ │ │ │ We can construct a Q-divisor as well. Here are two ways to do it (we work in │ │ │ │ $A^2$ this time). │ │ │ │ i16 : R = ZZ/7[x,y]; │ │ │ │ i17 : D = divisor({-1/2, 2/1}, {ideal(y^2-x^3), ideal(x)}, CoefficientType=>QQ) │ │ │ │ │ │ │ │ -o17 = -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 │ │ │ @@ -168,43 +168,43 @@ │ │ │ │ │ │ 
    i11 : M = J*R^1;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0706325s (cpu); 0.0706342s (thread); 0s (gc)
│ │ │ + -- used 0.0457191s (cpu); 0.0457183s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.491025s (cpu); 0.490974s (thread); 0s (gc)
│ │ │ + -- used 0.382233s (cpu); 0.382241s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.528431s (cpu); 0.440429s (thread); 0s (gc)
    │ │ │ + -- used 0.523672s (cpu); 0.442914s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00269572s (cpu); 0.00269697s (thread); 0s (gc)
    │ │ │ + -- used 0.00307887s (cpu); 0.00308346s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00217526s (cpu); 0.00217625s (thread); 0s (gc)
│ │ │ + -- used 0.00264852s (cpu); 0.00265352s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │ @@ -228,23 +228,23 @@ │ │ │ │ │ │ o19 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0637156s (cpu); 0.0637222s (thread); 0s (gc)
│ │ │ + -- used 0.0735762s (cpu); 0.0735808s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00595766s (cpu); 0.00595829s (thread); 0s (gc)
│ │ │ + -- used 0.00595823s (cpu); 0.00596222s (thread); 0s (gc)
│ │ │  
│ │ │  o21 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │ ├── html2text {} │ │ │ │ @@ -60,43 +60,43 @@ │ │ │ │ │ │ │ │ o9 : Ideal of R │ │ │ │ i10 : J = m^9; │ │ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ i11 : M = J*R^1; │ │ │ │ i12 : time dualize(J, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.0706325s (cpu); 0.0706342s (thread); 0s (gc) │ │ │ │ + -- used 0.0457191s (cpu); 0.0457183s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ i13 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.491025s (cpu); 0.490974s (thread); 0s (gc) │ │ │ │ + -- used 0.382233s (cpu); 0.382241s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time dualize(M, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.528431s (cpu); 0.440429s (thread); 0s (gc) │ │ │ │ + -- used 0.523672s (cpu); 0.442914s (thread); 0s (gc) │ │ │ │ i15 : time dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00269572s (cpu); 0.00269697s (thread); 0s (gc) │ │ │ │ + -- used 0.00307887s (cpu); 0.00308346s (thread); 0s (gc) │ │ │ │ i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00217526s (cpu); 0.00217625s (thread); 0s (gc) │ │ │ │ + -- used 0.00264852s (cpu); 0.00265352s (thread); 0s (gc) │ │ │ │ │ │ │ │ o16 : Ideal of R │ │ │ │ For monomial ideals in toric rings, frequently ModuleStrategy appears faster. │ │ │ │ i17 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v); │ │ │ │ i18 : I = ideal(x,u); │ │ │ │ │ │ │ │ o18 : Ideal of R │ │ │ │ i19 : J = I^15; │ │ │ │ │ │ │ │ o19 : Ideal of R │ │ │ │ i20 : time dualize(J, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.0637156s (cpu); 0.0637222s (thread); 0s (gc) │ │ │ │ + -- used 0.0735762s (cpu); 0.0735808s (thread); 0s (gc) │ │ │ │ │ │ │ │ o20 : Ideal of R │ │ │ │ i21 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00595766s (cpu); 0.00595829s (thread); 0s (gc) │ │ │ │ + -- used 0.00595823s (cpu); 0.00596222s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 : Ideal of R │ │ │ │ KnownDomain is an option for dualize. If it is false (default is true), then │ │ │ │ the computer will first check whether the ring is a domain, if it is not then │ │ │ │ it will revert to ModuleStrategy. If KnownDomain is set to true for a non- │ │ │ │ domain, then the function can return an incorrect answer. │ │ │ │ i22 : R = QQ[x,y]/ideal(x*y); │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_get__Prime__Divisors.html │ │ │ @@ -81,24 +81,24 @@ │ │ │ 
    i1 : R = QQ[x, y, z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D = divisor({-8, 2, 0}, {ideal(x), ideal(y), ideal(x^2+z)})
│ │ │  
│ │ │ -o2 = -8*Div(x) + 2*Div(y) + 0*Div(x^2+z)
│ │ │ +o2 = 0*Div(x^2+z) + -8*Div(x) + 2*Div(y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : getPrimeDivisors( D )
│ │ │  
│ │ │ -o3 = {Div(x), Div(y), Div(x^2+z)}
│ │ │ +o3 = {Div(x^2+z), Div(x), Div(y)}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : getPrimeDivisors( cleanSupport D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,20 +14,20 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns the list of prime divisors of a given divisor. The prime
│ │ │ │  divisors are all of the class WeilDivisor. If you do not call cleanSupport, you
│ │ │ │  may obtain divisors with zero coefficients.
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D = divisor({-8, 2, 0}, {ideal(x), ideal(y), ideal(x^2+z)})
│ │ │ │  
│ │ │ │ -o2 = -8*Div(x) + 2*Div(y) + 0*Div(x^2+z)
│ │ │ │ +o2 = 0*Div(x^2+z) + -8*Div(x) + 2*Div(y)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : getPrimeDivisors( D )
│ │ │ │  
│ │ │ │ -o3 = {Div(x), Div(y), Div(x^2+z)}
│ │ │ │ +o3 = {Div(x^2+z), Div(x), Div(y)}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : getPrimeDivisors( cleanSupport D )
│ │ │ │  
│ │ │ │  o4 = {Div(x), Div(y)}
│ │ │ │  
│ │ │ │  o4 : List
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Cartier.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │                
    i7 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : D = divisor({1, 2}, {ideal(x), ideal(y)})
│ │ │  
│ │ │ -o8 = 2*Div(y) + Div(x)
│ │ │ +o8 = Div(x) + 2*Div(y)
│ │ │  
│ │ │  o8 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : isCartier( D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  i6 : isCartier( D )
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  Of course the next divisor is Cartier.
│ │ │ │  i7 : R = QQ[x, y, z];
│ │ │ │  i8 : D = divisor({1, 2}, {ideal(x), ideal(y)})
│ │ │ │  
│ │ │ │ -o8 = 2*Div(y) + Div(x)
│ │ │ │ +o8 = Div(x) + 2*Div(y)
│ │ │ │  
│ │ │ │  o8 : WeilDivisor on R
│ │ │ │  i9 : isCartier( D )
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  If the option IsGraded is set to true (it is false by default), this will check
│ │ │ │  as if D is a divisor on the $Proj$ of the ambient graded ring.
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Reduced.html
│ │ │ @@ -81,24 +81,24 @@
│ │ │                
    i1 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D1 = divisor(x^2 * y^3 * z)
│ │ │  
│ │ │ -o2 = 3*Div(y) + Div(z) + 2*Div(x)
│ │ │ +o2 = 2*Div(x) + 3*Div(y) + Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : D2 = divisor(x * y * z)
│ │ │  
│ │ │ -o3 = Div(y) + Div(z) + Div(x)
│ │ │ +o3 = Div(x) + Div(y) + Div(z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : isReduced( D1 )
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,20 +12,20 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns true if the divisor is reduced (all coefficients equal to
│ │ │ │  1), otherwise it returns false.
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D1 = divisor(x^2 * y^3 * z)
│ │ │ │  
│ │ │ │ -o2 = 3*Div(y) + Div(z) + 2*Div(x)
│ │ │ │ +o2 = 2*Div(x) + 3*Div(y) + Div(z)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : D2 = divisor(x * y * z)
│ │ │ │  
│ │ │ │ -o3 = Div(y) + Div(z) + Div(x)
│ │ │ │ +o3 = Div(x) + Div(y) + Div(z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : isReduced( D1 )
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : isReduced( D2 )
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__S__N__C.html
│ │ │ @@ -157,15 +157,15 @@
│ │ │                
    i10 : R = QQ[x, y, z] / ideal(x * y - z^2 );
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : D = divisor({1, -2}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o11 = -2*Div(y, z) + Div(x, z)
│ │ │ +o11 = Div(x, z) + -2*Div(y, z)
│ │ │  
│ │ │  o11 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : isSNC( D, IsGraded => true )
│ │ │ @@ -180,15 +180,15 @@
│ │ │                
    i13 : R = QQ[x, y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │  
│ │ │  o14 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : isSNC( D, IsGraded => true )
│ │ │ @@ -203,15 +203,15 @@
│ │ │                
    i16 : R = QQ[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ +o17 = Div(y) + Div(x) + Div(x+y)
│ │ │  
│ │ │  o17 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : isSNC( D, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,33 +45,33 @@
│ │ │ │  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))
│ │ │ │  
│ │ │ │ -o14 = Div(y) + Div(x) + Div(x+y)
│ │ │ │ +o14 = Div(x+y) + Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o14 : WeilDivisor on R
│ │ │ │  i15 : isSNC( D, IsGraded => true )
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  i16 : R = QQ[x,y,z];
│ │ │ │  i17 : D = divisor(x*y*(x+y))
│ │ │ │  
│ │ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ │ +o17 = Div(y) + Div(x) + Div(x+y)
│ │ │ │  
│ │ │ │  o17 : WeilDivisor on R
│ │ │ │  i18 : isSNC( D, IsGraded => true)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  The output value of this function is stored in the divisor's cache with the
│ │ │ │  value of the last IsGraded option. If you change the IsGraded option, the value
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Weil__Divisor.html
│ │ │ @@ -81,24 +81,24 @@
│ │ │                
    i1 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D1 = divisor({1/1, 2/2, -6/3}, {ideal(x), ideal(y), ideal(z)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o2 = Div(y) + -2*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + Div(y) + -2*Div(z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : D2 = divisor({1/2, 3/4, 5/6}, {ideal(y), ideal(z), ideal(x)}, CoefficientType=>QQ)
│ │ │  
│ │ │ -o3 = 1/2*Div(y) + 3/4*Div(z) + 5/6*Div(x)
│ │ │ +o3 = 5/6*Div(x) + 1/2*Div(y) + 3/4*Div(z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : isWeilDivisor( D1 )
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,21 +13,21 @@
│ │ │ │            o a _B_o_o_l_e_a_n_ _v_a_l_u_e,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Check if a rational/real divisor is a Weil divisor
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D1 = divisor({1/1, 2/2, -6/3}, {ideal(x), ideal(y), ideal(z)},
│ │ │ │  CoefficientType=>QQ)
│ │ │ │  
│ │ │ │ -o2 = Div(y) + -2*Div(z) + Div(x)
│ │ │ │ +o2 = Div(x) + Div(y) + -2*Div(z)
│ │ │ │  
│ │ │ │  o2 : QWeilDivisor on R
│ │ │ │  i3 : D2 = divisor({1/2, 3/4, 5/6}, {ideal(y), ideal(z), ideal(x)},
│ │ │ │  CoefficientType=>QQ)
│ │ │ │  
│ │ │ │ -o3 = 1/2*Div(y) + 3/4*Div(z) + 5/6*Div(x)
│ │ │ │ +o3 = 5/6*Div(x) + 1/2*Div(y) + 3/4*Div(z)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : isWeilDivisor( D1 )
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : isWeilDivisor( D2 )
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_map__To__Projective__Space.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │                
    i4 : R = ZZ/7[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : D = divisor(x*y)
│ │ │  
│ │ │ -o5 = Div(x) + Div(y)
│ │ │ +o5 = Div(y) + Div(x)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,15 +36,15 @@
│ │ │ │  
│ │ │ │  o3 : RingMap R <-- QQ[YY ..YY ]
│ │ │ │                          1    2
│ │ │ │  The user may also specify the variable name of the new projective space.
│ │ │ │  i4 : R = ZZ/7[x,y,z];
│ │ │ │  i5 : D = divisor(x*y)
│ │ │ │  
│ │ │ │ -o5 = Div(x) + Div(y)
│ │ │ │ +o5 = Div(y) + Div(x)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : mapToProjectiveSpace(D, Variable=>"Z")
│ │ │ │  
│ │ │ │               ZZ            2             2        2
│ │ │ │  o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z })
│ │ │ │                7  1   6
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html
│ │ │ @@ -99,24 +99,24 @@
│ │ │  o3 : RingMap T <-- R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : D = divisor(y*z)
│ │ │  
│ │ │ -o4 = 3*Div(w, z, y) + 3*Div(z, y, x)
│ │ │ +o4 = 3*Div(z, y, x) + 3*Div(w, z, y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : pullback(f, D, Strategy=>Primes)
│ │ │  
│ │ │ -o5 = 3*Div(b) + 3*Div(a)
│ │ │ +o5 = 3*Div(a) + 3*Div(b)
│ │ │  
│ │ │  o5 : WeilDivisor on T
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : pullback(f, D, Strategy=>Sheaves)
│ │ │ ├── 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
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexify.html
│ │ │ @@ -272,23 +272,23 @@
│ │ │  
│ │ │  o22 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time reflexify(J);
│ │ │ - -- used 0.338433s (cpu); 0.253168s (thread); 0s (gc)
│ │ │ + -- used 0.239803s (cpu); 0.156455s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.671713s (cpu); 0.510526s (thread); 0s (gc)
    
│ │ │ + -- used 0.394678s (cpu); 0.320457s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Because of this, there are two strategies for computing a reflexification (at least if the module embeds as an ideal).
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -324,26 +324,26 @@
│ │ │              
│ │ │                
    i28 : M = J*R^1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.168829s (cpu); 0.168835s (thread); 0s (gc)
│ │ │ + -- used 0.0895845s (cpu); 0.0895882s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 7.92394s (cpu); 5.32977s (thread); 0s (gc)
│ │ │ + -- used 5.43945s (cpu); 3.95694s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │ @@ -353,21 +353,21 @@
│ │ │  
│ │ │  o31 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 6.96991s (cpu); 5.49609s (thread); 0s (gc)
    
│ │ │ + -- used 5.52999s (cpu); 4.16149s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.831741s (cpu); 0.473654s (thread); 0s (gc)
    
│ │ │ + -- used 0.504874s (cpu); 0.342262s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -394,15 +394,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : M = I^20*R^1;
    
│ │ │              
    
│ │ │                
    i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.461613s (cpu); 0.267865s (thread); 0s (gc)
│ │ │ + -- used 0.396157s (cpu); 0.229751s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -411,15 +411,15 @@
│ │ │  
│ │ │  o38 : Ideal of R
    
│ │ │              
    
│ │ │                
    i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.0135561s (cpu); 0.0135572s (thread); 0s (gc)
│ │ │ + -- used 0.0124373s (cpu); 0.0124375s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -428,21 +428,21 @@
│ │ │  
│ │ │  o39 : Ideal of R
    
│ │ │              
    
│ │ │                
    i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.0397151s (cpu); 0.0397234s (thread); 0s (gc)
    
│ │ │ + -- used 0.037933s (cpu); 0.037937s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00682599s (cpu); 0.00682691s (thread); 0s (gc)
    
│ │ │ + -- used 0.00612283s (cpu); 0.00613039s (thread); 0s (gc)
│ │ │              
    
│ │ │          
    
│ │ │            
    For ideals, if KnownDomain is false (default value is true), then the function will check whether it is a domain.  If it is a domain (or assumed to be a domain), it will reflexify using a strategy which can speed up computation, if not it will compute using a sometimes slower method which is essentially reflexifying it as a module.
    
│ │ │          
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,19 +114,19 @@
│ │ │ │  i21 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o21 : Ideal of R
│ │ │ │  i22 : J = I^21;
│ │ │ │  
│ │ │ │  o22 : Ideal of R
│ │ │ │  i23 : time reflexify(J);
│ │ │ │ - -- used 0.338433s (cpu); 0.253168s (thread); 0s (gc)
│ │ │ │ + -- used 0.239803s (cpu); 0.156455s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time reflexify(J*R^1);
│ │ │ │ - -- used 0.671713s (cpu); 0.510526s (thread); 0s (gc)
│ │ │ │ + -- used 0.394678s (cpu); 0.320457s (thread); 0s (gc)
│ │ │ │  Because of this, there are two strategies for computing a reflexification (at
│ │ │ │  least if the module embeds as an ideal).
│ │ │ │  IdealStrategy. In the case that $R$ is a domain, and our module is isomorphic
│ │ │ │  to an ideal $I$, then one can compute the reflexification by computing colons.
│ │ │ │  ModuleStrategy. This computes the reflexification simply by computing $Hom$
│ │ │ │  twice.
│ │ │ │  ModuleStrategy is the default strategy for modules, IdealStrategy is the
│ │ │ │ @@ -139,73 +139,73 @@
│ │ │ │  
│ │ │ │  o26 : Ideal of R
│ │ │ │  i27 : J = I^20;
│ │ │ │  
│ │ │ │  o27 : Ideal of R
│ │ │ │  i28 : M = J*R^1;
│ │ │ │  i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ │ - -- used 0.168829s (cpu); 0.168835s (thread); 0s (gc)
│ │ │ │ + -- used 0.0895845s (cpu); 0.0895882s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ │ - -- used 7.92394s (cpu); 5.32977s (thread); 0s (gc)
│ │ │ │ + -- used 5.43945s (cpu); 3.95694s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o30 : Ideal of R
│ │ │ │  i31 : J1 == J2
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ │ - -- used 6.96991s (cpu); 5.49609s (thread); 0s (gc)
│ │ │ │ + -- used 5.52999s (cpu); 4.16149s (thread); 0s (gc)
│ │ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.831741s (cpu); 0.473654s (thread); 0s (gc)
│ │ │ │ + -- used 0.504874s (cpu); 0.342262s (thread); 0s (gc)
│ │ │ │  However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
│ │ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i35 : I = ideal(x,u);
│ │ │ │  
│ │ │ │  o35 : Ideal of R
│ │ │ │  i36 : J = I^20;
│ │ │ │  
│ │ │ │  o36 : Ideal of R
│ │ │ │  i37 : M = I^20*R^1;
│ │ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ │ - -- used 0.461613s (cpu); 0.267865s (thread); 0s (gc)
│ │ │ │ + -- used 0.396157s (cpu); 0.229751s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12
│ │ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2
│ │ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         19    20
│ │ │ │        x  u, x  )
│ │ │ │  
│ │ │ │  o38 : Ideal of R
│ │ │ │  i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ │ - -- used 0.0135561s (cpu); 0.0135572s (thread); 0s (gc)
│ │ │ │ + -- used 0.0124373s (cpu); 0.0124375s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12
│ │ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2
│ │ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │         19    20
│ │ │ │        x  u, x  )
│ │ │ │  
│ │ │ │  o39 : Ideal of R
│ │ │ │  i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ │ - -- used 0.0397151s (cpu); 0.0397234s (thread); 0s (gc)
│ │ │ │ + -- used 0.037933s (cpu); 0.037937s (thread); 0s (gc)
│ │ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.00682599s (cpu); 0.00682691s (thread); 0s (gc)
│ │ │ │ + -- used 0.00612283s (cpu); 0.00613039s (thread); 0s (gc)
│ │ │ │  For ideals, if KnownDomain is false (default value is true), then the function
│ │ │ │  will check whether it is a domain. If it is a domain (or assumed to be a
│ │ │ │  domain), it will reflexify using a strategy which can speed up computation, if
│ │ │ │  not it will compute using a sometimes slower method which is essentially
│ │ │ │  reflexifying it as a module.
│ │ │ │  Consider the following example showing the importance of making the correct
│ │ │ │  assumption about the ring being a domain.
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexive__Power.html
│ │ │ @@ -129,30 +129,30 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0292441s (cpu); 0.0292465s (thread); 0s (gc)
│ │ │ + -- used 0.029116s (cpu); 0.0291144s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.142968s (cpu); 0.14297s (thread); 0s (gc)
│ │ │ + -- used 0.134859s (cpu); 0.13486s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : J20a == J20b
│ │ │ @@ -176,23 +176,23 @@
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0299195s (cpu); 0.0299242s (thread); 0s (gc)
│ │ │ + -- used 0.029618s (cpu); 0.0296198s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.164355s (cpu); 0.0959583s (thread); 0s (gc)
│ │ │ + -- used 0.0565307s (cpu); 0.0565374s (thread); 0s (gc)
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i15 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,39 +40,39 @@
│ │ │ │  of the generators of $I$. Consider the example of a cone over a point on an
│ │ │ │  elliptic curve.
│ │ │ │  i5 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │ │  i6 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : time J20a = reflexivePower(20, I);
│ │ │ │ - -- used 0.0292441s (cpu); 0.0292465s (thread); 0s (gc)
│ │ │ │ + -- used 0.029116s (cpu); 0.0291144s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : I20 = I^20;
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time J20b = reflexify(I20);
│ │ │ │ - -- used 0.142968s (cpu); 0.14297s (thread); 0s (gc)
│ │ │ │ + -- used 0.134859s (cpu); 0.13486s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : J20a == J20b
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  This passes the Strategy option to a reflexify call. Valid options are
│ │ │ │  IdealStrategy and ModuleStrategy.
│ │ │ │  i11 : R = QQ[x,y,z]/ideal(-y^2*z +x^3 + x^2*z + x*z^2+z^3);
│ │ │ │  i12 : I = ideal(x-z,y-2*z);
│ │ │ │  
│ │ │ │  o12 : Ideal of R
│ │ │ │  i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ │ - -- used 0.0299195s (cpu); 0.0299242s (thread); 0s (gc)
│ │ │ │ + -- used 0.029618s (cpu); 0.0296198s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.164355s (cpu); 0.0959583s (thread); 0s (gc)
│ │ │ │ + -- used 0.0565307s (cpu); 0.0565374s (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/WeylAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  ZXh0cmFjdFZhcnNBbGdlYnJh
│ │ │  #:len=1106
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidW5kZXJseWluZyBwb2x5bm9taWFsIHJp
│ │ │  bmcgaW4gdGhlIG9yZGluYXJ5IHZhcmlhYmxlcyBvZiBhIFdleWwgYWxnZWJyYSIsICJsaW5lbnVt
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  V2V5bEdyb3VwRWxlbWVudCAqIFJvb3Q=
│ │ │  #:len=994
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYXBwbHkgYW4gZWxlbWVudCBvZiBhIFdl
│ │ │  eWwgZ3JvdXAgdG8gYSByb290IiwgImxpbmVudW0iID0+IDI5NjMsIElucHV0cyA9PiB7U1BBTntU
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_above__Bruhat_lp__Basic__List_rp.out
│ │ │ @@ -36,34 +36,34 @@
│ │ │                                             |  2 |
│ │ │                                             | -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : aboveBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ -                                             |  3 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, |  1
│ │ │ +                                             |  1 |        |  2 |       |  1
│ │ │ +                                             |  2 |        | -1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ -     |                                            |  1 |        |  1 |      
│ │ │ -     |                                            | -2 |        |  1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            |  3 |        | -1 |      
│ │ │ +     |                                            | -1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |},
│ │ │ -     |  2 |                                            | -2 |        | -1 |  
│ │ │ -     | -1 |                                            |  1 |        |  2 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |},
│ │ │ +     | -1 |                                            | -2 |        |  1 |  
│ │ │ +     |  0 |                                            |  3 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ -         |  1 |                                            |  2 |        | -1
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1
│ │ │ +         | -1 |                                            |  1 |        |  1
│ │ │ +         |  0 |                                            | -2 |        |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0,
│ │ │ -     |       |  1 |                                            |  3 |       
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ +     |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1,
│ │ │ +     |       |  2 |                                            | -2 |       
│ │ │ +     |       | -1 |                                            |  1 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  0 |}, {2, |  2 |}}}}
│ │ │ -     | -1 |       | -1 |
│ │ │ -     |  2 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_eval_lp__Root__System_cm__Weight_cm__Root_rp.out
│ │ │ @@ -4,33 +4,33 @@
│ │ │  
│ │ │  o1 = RootSystem{...8...}
│ │ │  
│ │ │  o1 : RootSystem
│ │ │  
│ │ │  i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ -      |  2 |  | 1 |  | -1 |
│ │ │ +o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ +      | 1 |  | -1 |  |  2 |
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : v=weight(R,{1,2})
│ │ │  
│ │ │  o3 = | 1 |
│ │ │       | 2 |
│ │ │  
│ │ │         2
│ │ │  o3 : ZZ
│ │ │  
│ │ │  i4 : eval(R,v,L#0)
│ │ │  
│ │ │ -o4 = 2
│ │ │ +o4 = 3
│ │ │  
│ │ │  i5 : eval(R,v,L#1)
│ │ │  
│ │ │ -o5 = 3
│ │ │ +o5 = 1
│ │ │  
│ │ │  i6 : eval(R,v,L#2)
│ │ │  
│ │ │ -o6 = 1
│ │ │ +o6 = 2
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_eval_lp__Root__System_cm__Z__Z_cm__Root_rp.out
│ │ │ @@ -4,25 +4,25 @@
│ │ │  
│ │ │  o1 = RootSystem{...8...}
│ │ │  
│ │ │  o1 : RootSystem
│ │ │  
│ │ │  i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ -      |  2 |  | 1 |  | -1 |
│ │ │ +o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ +      | 1 |  | -1 |  |  2 |
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : eval(R,1,L#0)
│ │ │  
│ │ │ -o3 = 0
│ │ │ +o3 = 1
│ │ │  
│ │ │  i4 : eval(R,1,L#1)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : eval(R,1,L#2)
│ │ │  
│ │ │ -o5 = 1
│ │ │ +o5 = 0
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_hasse__Diagram__To__Graph_lp__Hasse__Diagram_rp.out
│ │ │ @@ -20,26 +20,26 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o3 : WeylGroupElement
│ │ │  
│ │ │  i4 : myInterval=intervalBruhat(w1,w2)
│ │ │  
│ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, |  1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  2 |       |  1 |       | 0 |       |  1 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        |  1 |       | -1 |       | -1 |                                              | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        | -1 |       | -1 |       | 1 |       |  1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  1 |       |  0 |       |  2 |                                              |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}, {2, |  1 |}, {4, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | -1 |}, {4, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  1 |       |  2 |       |  1 |       | 0 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        |  1 |       | -1 |       | -1 |       | 1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o4 : HasseDiagram
│ │ │  
│ │ │  i5 : hasseDiagramToGraph(myInterval)
│ │ │  
│ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}}, {, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 2}, {, 4}}}, {, {{, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │  
│ │ │  o5 : HasseGraph
│ │ │  
│ │ │  i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")
│ │ │  
│ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121, 1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232, 0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{232, 0}, {2, 1}, {121, 2}, {12321, 4}}}, {1232, {{12321, 2}, {2, 3}, {3, 4}}}, {1213, {{3, 0}, {232, 1}, {1, 3}}}}, {{121, {{2, 0}, {1, 3}}}, {213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │  
│ │ │  o6 : HasseGraph
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Left__Coset_cm__Weyl__Group__Left__Coset_rp.out
│ │ │ @@ -26,30 +26,30 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
│ │ │  
│ │ │  i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        |  1 |       | 0 |                                              | -1 |        |  2 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  1 |       | 1 |                                              |  3 |        | -1 |       |  0 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
│ │ │  
│ │ │  i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ -                                             |  3 |        |  1 |       |  1
│ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2
│ │ │ +                                             | -1 |        |  2 |       | -1
│ │ │ +                                             |  3 |        | -1 |       |  0
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  3 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2,
│ │ │ +     |                                            |  3 |        |  1 |      
│ │ │ +     |                                            | -2 |        | -1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}}
│ │ │ -     | -1 |
│ │ │ -     |  0 |
│ │ │ +     | -1 |}}}}
│ │ │ +     |  1 |
│ │ │ +     |  1 |
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Right__Coset_cm__Weyl__Group__Right__Coset_rp.out
│ │ │ @@ -26,30 +26,30 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
│ │ │  
│ │ │  i5 : myInterval=intervalBruhat(P % w1,P % w2)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            |  1 |        | 0 |       | -1 |                                              | -2 |        | -1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            |  1 |        | 1 |       |  2 |                                              |  3 |        |  0 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  0 |}, {2, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, |  0 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | -1 |       | 0 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        |  2 |       | 1 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  2 |       |  0 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            |  1 |        |  2 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
│ │ │  
│ │ │  i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1
│ │ │                                               | -3 |        |  1 |       |  2
│ │ │                                               |  1 |        |  1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1,
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1,
│ │ │       |                                            | -1 |        | -1 |      
│ │ │ -     |                                            |  2 |        |  2 |      
│ │ │ +     |                                            |  2 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, |  2 |}}}}
│ │ │ -     |  1 |       | -1 |
│ │ │ -     |  1 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_list__Weyl__Group__Elements_lp__Root__System_cm__Z__Z_rp.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 1330776363660
│ │ │  
│ │ │  i1 : listWeylGroupElements(rootSystemG2,4)
│ │ │  
│ │ │ -o1 = {WeylGroupElement{RootSystem{...8...}, |  4 |},
│ │ │ -                                            | -3 |  
│ │ │ +o1 = {WeylGroupElement{RootSystem{...8...}, | -5 |},
│ │ │ +                                            |  2 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     WeylGroupElement{RootSystem{...8...}, | -5 |}}
│ │ │ -                                           |  2 |
│ │ │ +     WeylGroupElement{RootSystem{...8...}, |  4 |}}
│ │ │ +                                           | -3 |
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_poincare__Series_lp__Hasse__Diagram_cm__Ring__Element_rp.out
│ │ │ @@ -4,16 +4,16 @@
│ │ │  
│ │ │  o1 = RootSystem{...8...}
│ │ │  
│ │ │  o1 : RootSystem
│ │ │  
│ │ │  i2 : H=intervalBruhat(neutralWeylGroupElement R, longestWeylGroupElement R)
│ │ │  
│ │ │ -o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ -                                                          | -1 |        | -1 |       |  2 |                                              |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       | 1 |                                              | -1 |        |  2 |                                            |  2 |        | -1 |                                              | 1 |
│ │ │ +o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ +                                                          | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |       | 1 |                                            |  1 |        | 1 |       |  2 |                                              | -1 |        |  2 |                                            |  2 |        | -1 |                                              | 1 |
│ │ │  
│ │ │  o2 : HasseDiagram
│ │ │  
│ │ │  i3 : ZZ[x]
│ │ │  
│ │ │  o3 = ZZ[x]
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_positive__Roots_lp__Root__System_cm__Parabolic_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  o2 = set {1, 2}
│ │ │  
│ │ │  o2 : Parabolic
│ │ │  
│ │ │  i3 : positiveRoots(R,P)
│ │ │  
│ │ │ -o3 = set {|  1 |, |  2 |, | -1 |}
│ │ │ -          |  1 |  | -1 |  |  2 |
│ │ │ -          | -1 |  |  0 |  | -1 |
│ │ │ +o3 = set {| -1 |, |  1 |, |  2 |}
│ │ │ +          |  2 |  |  1 |  | -1 |
│ │ │ +          | -1 |  | -1 |  |  0 |
│ │ │            |  0 |  |  0 |  |  0 |
│ │ │  
│ │ │  o3 : Set
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_positive__Roots_lp__Root__System_rp.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1330744940387
│ │ │  
│ │ │  i1 : positiveRoots(rootSystemA(3))
│ │ │  
│ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ +o1 = set {| 1 |, |  1 |, |  2 |, |  0 |, | -1 |, | -1 |}
│ │ │ +          | 0 |  |  1 |  | -1 |  | -1 |  |  1 |  |  2 |
│ │ │ +          | 1 |  | -1 |  |  0 |  |  2 |  |  1 |  | -1 |
│ │ │  
│ │ │  o1 : Set
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_under__Bruhat_lp__Basic__List_rp.out
│ │ │ @@ -36,34 +36,34 @@
│ │ │                                             | -2 |
│ │ │                                             | -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : underBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             |  2 |        |  1 |       | -1
│ │ │ -                                             | -1 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2, |  2
│ │ │ +                                             | -3 |        | -1 |       | -1
│ │ │ +                                             |  1 |        |  2 |       |  0
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  2 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ +     |                                            | -1 |        |  1 |      
│ │ │ +     |                                            | -2 |        |  1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |},
│ │ │ -     |  1 |                                            |  2 |        | -1 |  
│ │ │ -     | -1 |                                            | -3 |        |  2 |  
│ │ │ +     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |},
│ │ │ +     |  2 |                                            |  2 |        |  1 |  
│ │ │ +     | -1 |                                            | -1 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0
│ │ │ -         |  1 |                                            | -3 |        | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ +         | -1 |                                            | -1 |        |  2
│ │ │ +         |  0 |                                            |  2 |        | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ -     |       |  0 |                                            | -2 |       
│ │ │ +     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1,
│ │ │ +     |       |  1 |                                            |  2 |       
│ │ │ +     |       | -1 |                                            | -3 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ -     |  1 |       |  2 |
│ │ │ -     |  1 |       | -1 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_above__Bruhat_lp__Basic__List_rp.html
│ │ │ @@ -121,37 +121,37 @@
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : aboveBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ -                                             |  3 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, |  1
│ │ │ +                                             |  1 |        |  2 |       |  1
│ │ │ +                                             |  2 |        | -1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ -     |                                            |  1 |        |  1 |      
│ │ │ -     |                                            | -2 |        |  1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            |  3 |        | -1 |      
│ │ │ +     |                                            | -1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |},
│ │ │ -     |  2 |                                            | -2 |        | -1 |  
│ │ │ -     | -1 |                                            |  1 |        |  2 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |},
│ │ │ +     | -1 |                                            | -2 |        |  1 |  
│ │ │ +     |  0 |                                            |  3 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ -         |  1 |                                            |  2 |        | -1
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1
│ │ │ +         | -1 |                                            |  1 |        |  1
│ │ │ +         |  0 |                                            | -2 |        |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0,
│ │ │ -     |       |  1 |                                            |  3 |       
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ +     |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1,
│ │ │ +     |       |  2 |                                            | -2 |       
│ │ │ +     |       | -1 |                                            |  1 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  0 |}, {2, |  2 |}}}}
│ │ │ -     | -1 |       | -1 |
│ │ │ -     |  2 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,37 +48,37 @@ │ │ │ │ WeylGroupElement{RootSystem{...8...}, | 1 |}} │ │ │ │ | 2 | │ │ │ │ | -1 | │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : aboveBruhat(L1) │ │ │ │ │ │ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 1 |}, {1, | 2 │ │ │ │ - | -2 | | 1 | | -1 │ │ │ │ - | 3 | | -1 | | 0 │ │ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, | 1 │ │ │ │ + | 1 | | 2 | | 1 │ │ │ │ + | 2 | | -1 | | -1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 |}, {2, │ │ │ │ - | | 1 | | 1 | │ │ │ │ - | | -2 | | 1 | │ │ │ │ + |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | 0 |}, {2, │ │ │ │ + | | 3 | | -1 | │ │ │ │ + | | -1 | | 2 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{1, | 0 |}, │ │ │ │ - | 2 | | -2 | | -1 | │ │ │ │ - | -1 | | 1 | | 2 | │ │ │ │ + | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 1 |}, │ │ │ │ + | -1 | | -2 | | 1 | │ │ │ │ + | 0 | | 3 | | -1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 │ │ │ │ - | 1 | | 1 | | 2 │ │ │ │ - | 1 | | 2 | | -1 │ │ │ │ + {1, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 │ │ │ │ + | -1 | | 1 | | 1 │ │ │ │ + | 0 | | -2 | | 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, │ │ │ │ - | | 1 | | 3 | │ │ │ │ - | | -1 | | -1 | │ │ │ │ + |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{1, │ │ │ │ + | | 2 | | -2 | │ │ │ │ + | | -1 | | 1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | 0 |}, {2, | 2 |}}}} │ │ │ │ - | -1 | | -1 | │ │ │ │ - | 2 | | 0 | │ │ │ │ + | 0 |}, {2, | -1 |}}}} │ │ │ │ + | -1 | | 1 | │ │ │ │ + | 2 | | 1 | │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _a_b_o_v_e_B_r_u_h_a_t_(_B_a_s_i_c_L_i_s_t_) -- The Weyl group elements just under the ones in │ │ │ │ the list for the Bruhat order │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_eval_lp__Root__System_cm__Weight_cm__Root_rp.html │ │ │ @@ -85,16 +85,16 @@ │ │ │ o1 : RootSystem │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ -      |  2 |  | 1 |  | -1 |
│ │ │ +o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ +      | 1 |  | -1 |  |  2 |
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : v=weight(R,{1,2})
│ │ │ @@ -106,29 +106,29 @@
│ │ │  o3 : ZZ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : eval(R,v,L#0)
│ │ │  
│ │ │ -o4 = 2
    │ │ │ +o4 = 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : eval(R,v,L#1)
│ │ │  
│ │ │ -o5 = 3
    │ │ │ +o5 = 1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : eval(R,v,L#2)
│ │ │  
│ │ │ -o6 = 1
    │ │ │ +o6 = 2 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

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

    Ways to use this method:

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

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

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

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

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

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

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

    Description

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

    Description

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : positiveRoots(rootSystemA(3))
│ │ │  
│ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ +o1 = set {| 1 |, |  1 |, |  2 |, |  0 |, | -1 |, | -1 |}
│ │ │ +          | 0 |  |  1 |  | -1 |  | -1 |  |  1 |  |  2 |
│ │ │ +          | 1 |  | -1 |  |  0 |  |  2 |  |  1 |  | -1 |
│ │ │  
│ │ │  o1 : Set
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -10,17 +10,17 @@ │ │ │ │ * Inputs: │ │ │ │ o R, an instance of the type _R_o_o_t_S_y_s_t_e_m, │ │ │ │ * Outputs: │ │ │ │ o a _s_e_t, of all positive roots of R │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : positiveRoots(rootSystemA(3)) │ │ │ │ │ │ │ │ -o1 = set {| 0 |, | -1 |, | -1 |, | 1 |, | 1 |, | 2 |} │ │ │ │ - | -1 | | 1 | | 2 | | 0 | | 1 | | -1 | │ │ │ │ - | 2 | | 1 | | -1 | | 1 | | -1 | | 0 | │ │ │ │ +o1 = set {| 1 |, | 1 |, | 2 |, | 0 |, | -1 |, | -1 |} │ │ │ │ + | 0 | | 1 | | -1 | | -1 | | 1 | | 2 | │ │ │ │ + | 1 | | -1 | | 0 | | 2 | | 1 | | -1 | │ │ │ │ │ │ │ │ o1 : Set │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _p_o_s_i_t_i_v_e_R_o_o_t_s_(_R_o_o_t_S_y_s_t_e_m_) -- the set of all positive roots │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/WeylGroups.m2:2604:0. │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_under__Bruhat_lp__Basic__List_rp.html │ │ │ @@ -121,37 +121,37 @@ │ │ │ o3 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : underBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             |  2 |        |  1 |       | -1
│ │ │ -                                             | -1 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2, |  2
│ │ │ +                                             | -3 |        | -1 |       | -1
│ │ │ +                                             |  1 |        |  2 |       |  0
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  2 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ +     |                                            | -1 |        |  1 |      
│ │ │ +     |                                            | -2 |        |  1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |},
│ │ │ -     |  1 |                                            |  2 |        | -1 |  
│ │ │ -     | -1 |                                            | -3 |        |  2 |  
│ │ │ +     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |},
│ │ │ +     |  2 |                                            |  2 |        |  1 |  
│ │ │ +     | -1 |                                            | -1 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0
│ │ │ -         |  1 |                                            | -3 |        | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ +         | -1 |                                            | -1 |        |  2
│ │ │ +         |  0 |                                            |  2 |        | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ -     |       |  0 |                                            | -2 |       
│ │ │ +     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1,
│ │ │ +     |       |  1 |                                            |  2 |       
│ │ │ +     |       | -1 |                                            | -3 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ -     |  1 |       |  2 |
│ │ │ -     |  1 |       | -1 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,37 +48,37 @@ │ │ │ │ WeylGroupElement{RootSystem{...8...}, | 1 |}} │ │ │ │ | -2 | │ │ │ │ | -1 | │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : underBruhat(L1) │ │ │ │ │ │ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 |}, {1, | 2 │ │ │ │ - | 2 | | 1 | | -1 │ │ │ │ - | -1 | | -1 | | 0 │ │ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 0 |}, {2, | 2 │ │ │ │ + | -3 | | -1 | | -1 │ │ │ │ + | 1 | | 2 | | 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, │ │ │ │ - | | -1 | | 2 | │ │ │ │ - | | 2 | | -1 | │ │ │ │ + |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 |}, {2, │ │ │ │ + | | -1 | | 1 | │ │ │ │ + | | -2 | | 1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, | 0 |}, │ │ │ │ - | 1 | | 2 | | -1 | │ │ │ │ - | -1 | | -3 | | 2 | │ │ │ │ + | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 |}, │ │ │ │ + | 2 | | 2 | | 1 | │ │ │ │ + | -1 | | -1 | | -1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 0 │ │ │ │ - | 1 | | -3 | | -1 │ │ │ │ - | 1 | | 1 | | 2 │ │ │ │ + {1, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 │ │ │ │ + | -1 | | -1 | | 2 │ │ │ │ + | 0 | | 2 | | -1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}, {2, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, │ │ │ │ - | | -1 | | -1 | │ │ │ │ - | | 0 | | -2 | │ │ │ │ + |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, │ │ │ │ + | | 1 | | 2 | │ │ │ │ + | | -1 | | -3 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | -1 |}, {2, | -1 |}}}} │ │ │ │ - | 1 | | 2 | │ │ │ │ - | 1 | | -1 | │ │ │ │ + | 0 |}, {2, | -1 |}}}} │ │ │ │ + | -1 | | 1 | │ │ │ │ + | 2 | | 1 | │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _u_n_d_e_r_B_r_u_h_a_t_(_B_a_s_i_c_L_i_s_t_) -- Weyl group elements just under the ones in the │ │ │ │ list for the Bruhat order │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:32 2026 │ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ Y29ub3JtYWw= │ │ │ #:len=1038 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNvbm9ybWFsIHZh │ │ │ cmlldHkiLCAibGluZW51bSIgPT4gMTY2NCwgSW5wdXRzID0+IHtTUEFOe1RUeyJJIn0sIiwgIixT │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out │ │ │ @@ -122,15 +122,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 2.05196s (cpu); 1.19594s (thread); 0s (gc) │ │ │ + -- used 1.96839s (cpu); 1.12449s (thread); 0s (gc) │ │ │ │ │ │ o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o23 : List │ │ │ │ │ │ i24 : peek last ms │ │ │ │ │ │ @@ -142,15 +142,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 4.97937s (cpu); 2.77946s (thread); 0s (gc) │ │ │ + -- used 6.48536s (cpu); 2.6632s (thread); 0s (gc) │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o25 : List │ │ │ │ │ │ i26 : peek last ms │ │ │ │ │ │ @@ -162,15 +162,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 5.88833s (cpu); 3.18288s (thread); 0s (gc) │ │ │ + -- used 8.26795s (cpu); 3.05967s (thread); 0s (gc) │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o27 : List │ │ │ │ │ │ i28 : peek last ms │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html │ │ │ @@ -297,15 +297,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 2.05196s (cpu); 1.19594s (thread); 0s (gc) │ │ │ + -- used 1.96839s (cpu); 1.12449s (thread); 0s (gc) │ │ │ │ │ │ o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o23 : List │ │ │ │ │ │ │ │ │ │ │ │ @@ -323,15 +323,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 4.97937s (cpu); 2.77946s (thread); 0s (gc) │ │ │ + -- used 6.48536s (cpu); 2.6632s (thread); 0s (gc) │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o25 : List │ │ │ │ │ │ │ │ │ │ │ │ @@ -349,15 +349,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 5.88833s (cpu); 3.18288s (thread); 0s (gc) │ │ │ + -- used 8.26795s (cpu); 3.05967s (thread); 0s (gc) │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o27 : List │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -184,15 +184,15 @@ │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ - -- used 2.05196s (cpu); 1.19594s (thread); 0s (gc) │ │ │ │ + -- used 1.96839s (cpu); 1.12449s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o23 : List │ │ │ │ i24 : peek last ms │ │ │ │ │ │ │ │ o24 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ │ @@ -202,15 +202,15 @@ │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ - -- used 4.97937s (cpu); 2.77946s (thread); 0s (gc) │ │ │ │ + -- used 6.48536s (cpu); 2.6632s (thread); 0s (gc) │ │ │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o25 : List │ │ │ │ i26 : peek last ms │ │ │ │ │ │ │ │ o26 = MutableHashTable{0 => {ideal (P, M1)} } │ │ │ │ @@ -220,15 +220,15 @@ │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ - -- used 5.88833s (cpu); 3.18288s (thread); 0s (gc) │ │ │ │ + -- used 8.26795s (cpu); 3.05967s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o27 : List │ │ │ │ i28 : peek last ms │ │ │ │ │ │ │ │ o28 = MutableHashTable{0 => {ideal (P, M1)} } │ │ ├── ./usr/share/doc/Macaulay2/WittVectors/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=50 │ │ │ ZmluZEZyb2Jlbml1c0xpZnRDb25zdHJhaW50cyguLi4sSG9tb2dlbmVvdXM9Pi4uLik= │ │ │ #:len=354 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEyOCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbZmluZEZyb2Jlbml1c0xpZnRDb25zdHJhaW50cyxI │ │ ├── ./usr/share/doc/Macaulay2/XML/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=25 │ │ │ Z2V0QXR0cmlidXRlcyhMaWJ4bWxOb2RlKQ== │ │ │ #:len=465 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2V0IHRoZSBsaXN0IG9mIGF0dHJpYnV0 │ │ │ ZXMgb2YgYW4gWE1MIG5vZGUiLCBEZXNjcmlwdGlvbiA9PiAxOihUQUJMRXsiY2xhc3MiID0+ICJl │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ Z2ZhblBvbHlub21pYWxTZXRVbmlvbihMaXN0LExpc3Qp │ │ │ #:len=309 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzY0Mywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2ZhblBvbHlub21pYWxTZXRVbmlvbixMaXN0LExp │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/example-output/___Installation_spand_sp__Configuration_spof_spgfan__Interface.out │ │ │ @@ -19,15 +19,15 @@ │ │ │ i4 : prefixDirectory | currentLayout#"programs" │ │ │ │ │ │ o4 = /usr/x86_64-Linux- │ │ │ Debian-forky/libexec/Macaulay2/bin/ │ │ │ │ │ │ i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true); │ │ │ -- warning: reloading gfanInterface; recreate instances of types from this package │ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16342-0/172 │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20462-0/172 │ │ │ This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm. │ │ │ Options: │ │ │ -g: │ │ │ Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis. │ │ │ │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup. │ │ │ @@ -38,16 +38,16 @@ │ │ │ --disableSymmetryTest: │ │ │ When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal. │ │ │ │ │ │ --parameters value: │ │ │ With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters. │ │ │ --interrupt value: │ │ │ Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans). │ │ │ -using temporary file /tmp/M2-16342-0/172 │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16342-0/174 │ │ │ +using temporary file /tmp/M2-20462-0/172 │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20462-0/174 │ │ │ This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]). │ │ │ Options: │ │ │ -w: │ │ │ Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read. │ │ │ │ │ │ -r: │ │ │ Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used. │ │ │ @@ -56,69 +56,69 @@ │ │ │ Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored. │ │ │ │ │ │ -g: │ │ │ Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used. │ │ │ │ │ │ --parameters value: │ │ │ With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters. │ │ │ -using temporary file /tmp/M2-16342-0/174 │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16342-0/176 │ │ │ +using temporary file /tmp/M2-20462-0/174 │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20462-0/176 │ │ │ This program takes a marked Groebner basis of an ideal I and a set of polynomials on its input and tests if the polynomial set is contained in I by applying the division algorithm for each element. The output is 1 for true and 0 for false. │ │ │ Options: │ │ │ --remainder: │ │ │ Tell the program to output the remainders of the divisions rather than outputting 0 or 1. │ │ │ --multiplier: │ │ │ Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division. │ │ │ -using temporary file /tmp/M2-16342-0/176 │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16342-0/178 │ │ │ +using temporary file /tmp/M2-20462-0/176 │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20462-0/178 │ │ │ This program takes two polyhedral fans and computes their common refinement. │ │ │ Options: │ │ │ -i1 value: │ │ │ Specify the name of the first input file. │ │ │ -i2 value: │ │ │ Specify the name of the second input file. │ │ │ --stable: │ │ │ Compute the stable intersection. │ │ │ -using temporary file /tmp/M2-16342-0/178 │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16342-0/180 │ │ │ +using temporary file /tmp/M2-20462-0/178 │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20462-0/180 │ │ │ This program takes a polyhedral fan and a vector and computes the link of the polyhedral fan around that vertex. The link will have lineality space dimension equal to the dimension of the relative open polyhedral cone of the original fan containing the vector. │ │ │ Options: │ │ │ -i value: │ │ │ Specify the name of the input file. │ │ │ --symmetry: │ │ │ Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input. │ │ │ │ │ │ --star: │ │ │ Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector. │ │ │ -using temporary file /tmp/M2-16342-0/180 │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16342-0/182 │ │ │ +using temporary file /tmp/M2-20462-0/180 │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20462-0/182 │ │ │ This program takes two polyhedral fans and computes their product. │ │ │ Options: │ │ │ -i1 value: │ │ │ Specify the name of the first input file. │ │ │ -i2 value: │ │ │ Specify the name of the second input file. │ │ │ -using temporary file /tmp/M2-16342-0/182 │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16342-0/184 │ │ │ +using temporary file /tmp/M2-20462-0/182 │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20462-0/184 │ │ │ This program computes a Groebner cone. Three different cases are handled. The input may be a marked reduced Groebner basis in which case its Groebner cone is computed. The input may be just a marked minimal basis in which case the cone computed is not a Groebner cone in the usual sense but smaller. (These cones are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case is that the Groebner cone is possibly lower dimensional and given by a pair of Groebner bases as it is useful to do for tropical varieties, see option --pair. The facets of the cone can be read off in section FACETS and the equations in section IMPLIED_EQUATIONS. │ │ │ Options: │ │ │ --restrict: │ │ │ Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant. │ │ │ --pair: │ │ │ The Groebner cone is given by a pair of compatible Groebner bases. The first basis is for the initial ideal and the second for the ideal itself. See the tropical section of the manual. │ │ │ --asfan: │ │ │ Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed. │ │ │ --vectorinput: │ │ │ Compute a cone given list of inequalities rather than a Groebner cone. The input is an integer which specifies the dimension of the ambient space, a list of inequalities given as vectors and a list of equations. │ │ │ -using temporary file /tmp/M2-16342-0/184 │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16342-0/186 │ │ │ +using temporary file /tmp/M2-20462-0/184 │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20462-0/186 │ │ │ This program computes the homogeneity space of a list of polynomials - as a cone. Thus generators for the homogeneity space are found in the section LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first compute a set of homogeneous generators and call the program on these. A reduced Groebner basis will always suffice for this purpose. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/186 │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16342-0/188 │ │ │ +using temporary file /tmp/M2-20462-0/186 │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20462-0/188 │ │ │ This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree. │ │ │ Example: │ │ │ Input: │ │ │ Q[x,y]{y-1} │ │ │ z │ │ │ Output: │ │ │ Q[x,y,z]{y-z} │ │ │ @@ -126,30 +126,30 @@ │ │ │ -i: │ │ │ Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it. │ │ │ -w: │ │ │ Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials. │ │ │ │ │ │ -H: │ │ │ Let the name of the new variable be H rather than reading in a name from the input. │ │ │ -using temporary file /tmp/M2-16342-0/188 │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16342-0/190 │ │ │ +using temporary file /tmp/M2-20462-0/188 │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20462-0/190 │ │ │ This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list. │ │ │ Options: │ │ │ --ideal: │ │ │ Treat input as an ideal. This will make the program compute the initial ideal of the ideal generated by the input polynomials. The computation is done by computing a Groebner basis with respect to the given vector. The vector must be positive or the input polynomials must be homogeneous in a positive grading. None of these conditions are checked by the program. │ │ │ │ │ │ --pair: │ │ │ Produce a pair of polynomial lists. Used together with --ideal this option will also write a compatible reduced Groebner basis for the input ideal to the output. This is useful for finding the Groebner cone of a non-monomial initial ideal. │ │ │ │ │ │ --mark: │ │ │ If the --pair option is and the --ideal option is not used this option will still make sure that the second output basis is marked consistently with the vector. │ │ │ --list: │ │ │ Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output. │ │ │ -using temporary file /tmp/M2-16342-0/190 │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16342-0/192 │ │ │ +using temporary file /tmp/M2-20462-0/190 │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20462-0/192 │ │ │ This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program. │ │ │ Options: │ │ │ -L: │ │ │ Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi. │ │ │ │ │ │ -x: │ │ │ Exit immediately. │ │ │ @@ -164,57 +164,57 @@ │ │ │ Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis. │ │ │ │ │ │ -W: │ │ │ Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone. │ │ │ │ │ │ --tropical: │ │ │ Traverse a tropical variety interactively. │ │ │ -using temporary file /tmp/M2-16342-0/192 │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16342-0/194 │ │ │ +using temporary file /tmp/M2-20462-0/192 │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20462-0/194 │ │ │ This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/194 │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16342-0/196 │ │ │ +using temporary file /tmp/M2-20462-0/194 │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20462-0/196 │ │ │ Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis. │ │ │ Options: │ │ │ -g: │ │ │ Tell the program that the input is already a reduced Groebner basis. │ │ │ -using temporary file /tmp/M2-16342-0/196 │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16342-0/198 │ │ │ +using temporary file /tmp/M2-20462-0/196 │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20462-0/198 │ │ │ This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice. │ │ │ Options: │ │ │ -t: │ │ │ Compute the toric ideal of the matrix whose rows are given on the input instead. │ │ │ --convert: │ │ │ Does not do any computation, but just converts the vectors to binomials. │ │ │ -using temporary file /tmp/M2-16342-0/198 │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16342-0/200 │ │ │ +using temporary file /tmp/M2-20462-0/198 │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20462-0/200 │ │ │ This program converts a list of polynomials to a list of their leading terms. │ │ │ Options: │ │ │ -m: │ │ │ Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms. │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/200 │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16342-0/202 │ │ │ +using temporary file /tmp/M2-20462-0/200 │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20462-0/202 │ │ │ This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/202 │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16342-0/204 │ │ │ +using temporary file /tmp/M2-20462-0/202 │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20462-0/204 │ │ │ This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials. │ │ │ Options: │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup. │ │ │ │ │ │ --disableSymmetryTest: │ │ │ When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal. │ │ │ │ │ │ --nocones: │ │ │ Tell the program to not list cones in the output. │ │ │ -using temporary file /tmp/M2-16342-0/204 │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16342-0/206 │ │ │ +using temporary file /tmp/M2-20462-0/204 │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20462-0/206 │ │ │ This program will generate the r*r minors of a d*n matrix of indeterminates. │ │ │ Options: │ │ │ -r value: │ │ │ Specify r. │ │ │ -d value: │ │ │ Specify d. │ │ │ -n value: │ │ │ @@ -229,16 +229,16 @@ │ │ │ Do nothing but produce symmetry generators for the Pluecker ideal. │ │ │ --symmetry: │ │ │ Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names). │ │ │ --parametrize: │ │ │ Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials. │ │ │ --ultrametric: │ │ │ Produce tropical equations cutting out the ultrametrics. │ │ │ -using temporary file /tmp/M2-16342-0/206 │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16342-0/208 │ │ │ +using temporary file /tmp/M2-20462-0/206 │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20462-0/208 │ │ │ This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials. │ │ │ Options: │ │ │ --vectorinput: │ │ │ Read in a list of point configurations instead of a polynomial ring and a list of polynomials. │ │ │ --cyclic value: │ │ │ Use cyclic-n example instead of reading input. │ │ │ --noon value: │ │ │ @@ -251,47 +251,47 @@ │ │ │ Use Gaukwa-n example instead of reading input. │ │ │ --eco value: │ │ │ Use Eco-n example instead of reading input. │ │ │ -j value: │ │ │ Number of threads │ │ │ --resume: │ │ │ Continue enumeration from saved state... │ │ │ -using temporary file /tmp/M2-16342-0/208 │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16342-0/210 │ │ │ +using temporary file /tmp/M2-20462-0/208 │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20462-0/210 │ │ │ This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets. │ │ │ Options: │ │ │ -s: │ │ │ Sort output by degree. │ │ │ │ │ │ --twosetswithrings: │ │ │ Switch to mode where input is: ring polynomialset (same)ring polynomialset. │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/210 │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16342-0/212 │ │ │ +using temporary file /tmp/M2-20462-0/210 │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20462-0/212 │ │ │ This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables. │ │ │ Options: │ │ │ -L: │ │ │ Make the triangle larger so that the shape of the Groebner region appears. │ │ │ --shiftVariables value: │ │ │ Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0. │ │ │ -using temporary file /tmp/M2-16342-0/212 │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16342-0/214 │ │ │ +using temporary file /tmp/M2-20462-0/212 │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20462-0/214 │ │ │ This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables. │ │ │ Options: │ │ │ -m: │ │ │ Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn. │ │ │ │ │ │ -d value: │ │ │ Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8. │ │ │ │ │ │ -w value: │ │ │ Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5. │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/214 │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16342-0/216 │ │ │ +using temporary file /tmp/M2-20462-0/214 │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20462-0/216 │ │ │ This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution. │ │ │ Options: │ │ │ --codimension: │ │ │ Compute only the codimension of the resultant fan and return. │ │ │ │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry. │ │ │ @@ -304,25 +304,25 @@ │ │ │ │ │ │ --vectorinput: │ │ │ Read in a list of point configurations instead of a polynomial ring and a list of polynomials. │ │ │ │ │ │ --projection: │ │ │ Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates. │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/216 │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16342-0/218 │ │ │ +using temporary file /tmp/M2-20462-0/216 │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20462-0/218 │ │ │ This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous. │ │ │ Options: │ │ │ -h: │ │ │ Tell the program that the input is a homogeneous ideal (with homogeneous generators). │ │ │ │ │ │ --noideal: │ │ │ Do not treat input as an ideal but just factor out common monomial factors of the input polynomials. │ │ │ -using temporary file /tmp/M2-16342-0/218 │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16342-0/220 │ │ │ +using temporary file /tmp/M2-20462-0/218 │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20462-0/220 │ │ │ This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}. │ │ │ Options: │ │ │ --unimodular: │ │ │ Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan │ │ │ --scale value: │ │ │ Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future. │ │ │ --restrictingfan value: │ │ │ @@ -331,72 +331,72 @@ │ │ │ --symmetry: │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry. │ │ │ │ │ │ --nocones: │ │ │ Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used. │ │ │ --interrupt value: │ │ │ Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans). │ │ │ -using temporary file /tmp/M2-16342-0/220 │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16342-0/222 │ │ │ +using temporary file /tmp/M2-20462-0/220 │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20462-0/222 │ │ │ This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/222 │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16342-0/224 │ │ │ +using temporary file /tmp/M2-20462-0/222 │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20462-0/224 │ │ │ This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring. │ │ │ Example: │ │ │ Input: │ │ │ Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x] │ │ │ Output: │ │ │ Q[b,a,c,x]{2*b-3*a,c+x} │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/224 │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16342-0/226 │ │ │ +using temporary file /tmp/M2-20462-0/224 │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20462-0/226 │ │ │ This program converts ASCII math to TeX math. The data-type is specified by the options. │ │ │ Options: │ │ │ -h: │ │ │ Add a header to the output. Using this option the output will be LaTeXable right away. │ │ │ --polynomialset_: │ │ │ The data to be converted is a list of polynomials. │ │ │ --polynomialsetlist_: │ │ │ The data to be converted is a list of lists of polynomials. │ │ │ -using temporary file /tmp/M2-16342-0/226 │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16342-0/228 │ │ │ +using temporary file /tmp/M2-20462-0/226 │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20462-0/228 │ │ │ This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan. │ │ │ Options: │ │ │ --restrict: │ │ │ Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant. │ │ │ --generatedcones: │ │ │ Instead of a list of reduced Groebner bases, read in a list of closed generated cones given in modern format (like _cone). In this case --restrict is ignored. │ │ │ --symmetry: │ │ │ Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input. │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/228 │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16342-0/230 │ │ │ +using temporary file /tmp/M2-20462-0/228 │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20462-0/230 │ │ │ This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used. │ │ │ Options: │ │ │ -h: │ │ │ Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous. │ │ │ -using temporary file /tmp/M2-16342-0/230 │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16342-0/232 │ │ │ +using temporary file /tmp/M2-20462-0/230 │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20462-0/232 │ │ │ This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/232 │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16342-0/234 │ │ │ +using temporary file /tmp/M2-20462-0/232 │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20462-0/234 │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/234 │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16342-0/236 │ │ │ +using temporary file /tmp/M2-20462-0/234 │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20462-0/236 │ │ │ This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options: │ │ │ --exponents: │ │ │ Tell program to read a list of exponent vectors instead. │ │ │ -using temporary file /tmp/M2-16342-0/236 │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16342-0/238 │ │ │ +using temporary file /tmp/M2-20462-0/236 │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20462-0/238 │ │ │ This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options: │ │ │ -using temporary file /tmp/M2-16342-0/238 │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16342-0/240 │ │ │ +using temporary file /tmp/M2-20462-0/238 │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20462-0/240 │ │ │ This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces. │ │ │ Options: │ │ │ --tropicalbasistest: │ │ │ This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.) │ │ │ │ │ │ --tplane: │ │ │ This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed. │ │ │ @@ -408,16 +408,16 @@ │ │ │ Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used. │ │ │ --restrict: │ │ │ Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms. │ │ │ --stable: │ │ │ Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored. │ │ │ --parameters value: │ │ │ With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters. │ │ │ -using temporary file /tmp/M2-16342-0/240 │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16342-0/242 │ │ │ +using temporary file /tmp/M2-20462-0/240 │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20462-0/242 │ │ │ This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in: │ │ │ │ │ │ Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig: │ │ │ tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 │ │ │ │ │ │ See also │ │ │ │ │ │ @@ -442,48 +442,48 @@ │ │ │ Options: │ │ │ --noMult: │ │ │ Disable the multiplicity computation. │ │ │ -n value: │ │ │ Number of variables that should have negative weight. │ │ │ -c: │ │ │ Only output a list of vectors being the possible choices. │ │ │ -using temporary file /tmp/M2-16342-0/242 │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16342-0/244 │ │ │ +using temporary file /tmp/M2-20462-0/242 │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20462-0/244 │ │ │ This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input. │ │ │ Options: │ │ │ -d value: │ │ │ Specify d. │ │ │ -n value: │ │ │ Specify n. │ │ │ --trees: │ │ │ list the boundary trees (assumes d=3) │ │ │ -using temporary file /tmp/M2-16342-0/244 │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16342-0/246 │ │ │ +using temporary file /tmp/M2-20462-0/244 │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20462-0/246 │ │ │ This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/246 │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16342-0/248 │ │ │ +using temporary file /tmp/M2-20462-0/246 │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20462-0/248 │ │ │ This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM. │ │ │ Options: │ │ │ --kapranov: │ │ │ Compute Kapranov rank instead of tropical rank. │ │ │ --determinant: │ │ │ Compute the tropical determinant instead. │ │ │ -using temporary file /tmp/M2-16342-0/248 │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16342-0/250 │ │ │ +using temporary file /tmp/M2-20462-0/248 │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20462-0/250 │ │ │ This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex. │ │ │ Options: │ │ │ -g: │ │ │ Tell the program that the input is already a reduced Groebner basis. │ │ │ -d: │ │ │ Output dimension information to standard error. │ │ │ --stable: │ │ │ Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric. │ │ │ -using temporary file /tmp/M2-16342-0/250 │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16342-0/252 │ │ │ +using temporary file /tmp/M2-20462-0/250 │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20462-0/252 │ │ │ This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety. │ │ │ Options: │ │ │ --symmetry: │ │ │ Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster. │ │ │ --symsigns: │ │ │ Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix. │ │ │ --nocones: │ │ │ @@ -491,24 +491,24 @@ │ │ │ --disableSymmetryTest: │ │ │ When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal. │ │ │ │ │ │ --stable: │ │ │ Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric. │ │ │ --interrupt value: │ │ │ Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans). │ │ │ -using temporary file /tmp/M2-16342-0/252 │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16342-0/254 │ │ │ +using temporary file /tmp/M2-20462-0/252 │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20462-0/254 │ │ │ This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information. │ │ │ Options: │ │ │ -i1 value: │ │ │ Specify the name of the Polymake input file containing the k-cycle. │ │ │ -i2 value: │ │ │ Specify the name of the Polymake input file containing the piecewise linear function. │ │ │ -using temporary file /tmp/M2-16342-0/254 │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16342-0/256 │ │ │ +using temporary file /tmp/M2-20462-0/254 │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20462-0/256 │ │ │ This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n]. │ │ │ Several operations are supported by specifying the appropriate option: │ │ │ (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically), │ │ │ (2) computation of an initial ideal, │ │ │ (3) computation of the Groebner fan, │ │ │ (4) computation of a single Groebner cone. │ │ │ Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive. │ │ │ @@ -528,21 +528,21 @@ │ │ │ --groebnerCone: │ │ │ Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector. │ │ │ -m: │ │ │ For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list. │ │ │ -g: │ │ │ Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow. │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/256 │ │ │ +using temporary file /tmp/M2-20462-0/256 │ │ │ │ │ │ i6 : QQ[x,y]; │ │ │ │ │ │ i7 : gfan {x,y}; │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16342-0/258 │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20462-0/258 │ │ │ Q[x1,x2] │ │ │ {{ │ │ │ x2, │ │ │ x1} │ │ │ } │ │ │ -using temporary file /tmp/M2-16342-0/258 │ │ │ +using temporary file /tmp/M2-20462-0/258 │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html │ │ │ @@ -114,15 +114,15 @@ │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16342-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20462-0/172
│ │ │  This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │ @@ -133,16 +133,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16342-0/174
│ │ │ +using temporary file /tmp/M2-20462-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20462-0/174
│ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read.
│ │ │  
│ │ │  -r:
│ │ │   Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used.
│ │ │ @@ -151,69 +151,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16342-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16342-0/176
│ │ │ +using temporary file /tmp/M2-20462-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20462-0/176
│ │ │  This program takes a marked Groebner basis of an ideal I and a set of polynomials on its input and tests if the polynomial set is contained in I by applying the division algorithm for each element. The output is 1 for true and 0 for false.
│ │ │  Options:
│ │ │  --remainder:
│ │ │   Tell the program to output the remainders of the divisions rather than outputting 0 or 1.
│ │ │  --multiplier:
│ │ │   Reads in a polynomial that will be multiplied to the polynomial to be divided before doing the division.
│ │ │ -using temporary file /tmp/M2-16342-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16342-0/178
│ │ │ +using temporary file /tmp/M2-20462-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20462-0/178
│ │ │  This program takes two polyhedral fans and computes their common refinement.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │  --stable:
│ │ │   Compute the stable intersection.
│ │ │ -using temporary file /tmp/M2-16342-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16342-0/180
│ │ │ +using temporary file /tmp/M2-20462-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20462-0/180
│ │ │  This program takes a polyhedral fan and a vector and computes the link of the polyhedral fan around that vertex. The link will have lineality space dimension equal to the dimension of the relative open polyhedral cone of the original fan containing the vector.
│ │ │  Options:
│ │ │  -i value:
│ │ │   Specify the name of the input file.
│ │ │  --symmetry:
│ │ │   Reads in a fan stored with symmetry. The generators of the symmetry group must be given on the standard input.
│ │ │  
│ │ │  --star:
│ │ │   Computes the star instead. The star is defined as the smallest polyhedral fan containing all cones of the original fan containing the vector.
│ │ │ -using temporary file /tmp/M2-16342-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16342-0/182
│ │ │ +using temporary file /tmp/M2-20462-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20462-0/182
│ │ │  This program takes two polyhedral fans and computes their product.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the first input file.
│ │ │  -i2 value:
│ │ │   Specify the name of the second input file.
│ │ │ -using temporary file /tmp/M2-16342-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16342-0/184
│ │ │ +using temporary file /tmp/M2-20462-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20462-0/184
│ │ │  This program computes a Groebner cone. Three different cases are handled. The input may be a marked reduced Groebner basis in which case its Groebner cone is computed. The input may be just a marked minimal basis in which case the cone computed is not a Groebner cone in the usual sense but smaller. (These cones are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case is that the Groebner cone is possibly lower dimensional and given by a pair of Groebner bases as it is useful to do for tropical varieties, see option --pair. The facets of the cone can be read off in section FACETS and the equations in section IMPLIED_EQUATIONS.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cone is restricted to the non-negative orthant.
│ │ │  --pair:
│ │ │   The Groebner cone is given by a pair of compatible Groebner bases. The first basis is for the initial ideal and the second for the ideal itself. See the tropical section of the manual.
│ │ │  --asfan:
│ │ │   Writes the cone as a polyhedral fan with all its faces instead. In this way the extreme rays of the cone are also computed.
│ │ │  --vectorinput:
│ │ │   Compute a cone given list of inequalities rather than a Groebner cone. The input is an integer which specifies the dimension of the ambient space, a list of inequalities given as vectors and a list of equations.
│ │ │ -using temporary file /tmp/M2-16342-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16342-0/186
│ │ │ +using temporary file /tmp/M2-20462-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20462-0/186
│ │ │  This program computes the homogeneity space of a list of polynomials - as a cone. Thus generators for the homogeneity space are found in the section LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first compute a set of homogeneous generators and call the program on these. A reduced Groebner basis will always suffice for this purpose.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16342-0/188
│ │ │ +using temporary file /tmp/M2-20462-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20462-0/188
│ │ │  This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -221,30 +221,30 @@
│ │ │  -i:
│ │ │   Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it.
│ │ │  -w:
│ │ │   Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-16342-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16342-0/190
│ │ │ +using temporary file /tmp/M2-20462-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20462-0/190
│ │ │  This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list.
│ │ │  Options:
│ │ │  --ideal:
│ │ │   Treat input as an ideal. This will make the program compute the initial ideal of the ideal generated by the input polynomials. The computation is done by computing a Groebner basis with respect to the given vector. The vector must be positive or the input polynomials must be homogeneous in a positive grading. None of these conditions are checked by the program.
│ │ │  
│ │ │  --pair:
│ │ │   Produce a pair of polynomial lists. Used together with --ideal this option will also write a compatible reduced Groebner basis for the input ideal to the output. This is useful for finding the Groebner cone of a non-monomial initial ideal.
│ │ │  
│ │ │  --mark:
│ │ │   If the --pair option is and the --ideal option is not used this option will still make sure that the second output basis is marked consistently with the vector.
│ │ │  --list:
│ │ │   Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output.
│ │ │ -using temporary file /tmp/M2-16342-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16342-0/192
│ │ │ +using temporary file /tmp/M2-20462-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20462-0/192
│ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -259,57 +259,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-16342-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16342-0/194
│ │ │ +using temporary file /tmp/M2-20462-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20462-0/194
│ │ │  This program checks if a set of marked polynomials is a Groebner basis with respect to its marking. First it is checked if the markings are consistent with respect to a positive vector. Then Buchberger's S-criterion is checked. The output is boolean value.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16342-0/196
│ │ │ +using temporary file /tmp/M2-20462-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20462-0/196
│ │ │  Takes an ideal $I$ and computes the Krull dimension of R/I where R is the polynomial ring. This is done by first computing a Groebner basis.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │ -using temporary file /tmp/M2-16342-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16342-0/198
│ │ │ +using temporary file /tmp/M2-20462-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20462-0/198
│ │ │  This program computes the lattice ideal of a lattice. The input is a list of generators for the lattice.
│ │ │  Options:
│ │ │  -t:
│ │ │   Compute the toric ideal of the matrix whose rows are given on the input instead.
│ │ │  --convert:
│ │ │   Does not do any computation, but just converts the vectors to binomials.
│ │ │ -using temporary file /tmp/M2-16342-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16342-0/200
│ │ │ +using temporary file /tmp/M2-20462-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20462-0/200
│ │ │  This program converts a list of polynomials to a list of their leading terms.
│ │ │  Options:
│ │ │  -m:
│ │ │   Do the same thing for a list of polynomial sets. That is, output the set of sets of leading terms.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16342-0/202
│ │ │ +using temporary file /tmp/M2-20462-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20462-0/202
│ │ │  This program marks a set of polynomials with respect to the vector given at the end of the input, meaning that the largest terms are moved to the front. In case of a tie the lexicographic term order with $a>b>c...$ is used to break it.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16342-0/204
│ │ │ +using temporary file /tmp/M2-20462-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20462-0/204
│ │ │  This is a program for computing the normal fan of the Minkowski sum of the Newton polytopes of a list of polynomials.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │  
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --nocones:
│ │ │   Tell the program to not list cones in the output.
│ │ │ -using temporary file /tmp/M2-16342-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16342-0/206
│ │ │ +using temporary file /tmp/M2-20462-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20462-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -324,16 +324,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-16342-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16342-0/208
│ │ │ +using temporary file /tmp/M2-20462-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20462-0/208
│ │ │  This program computes the mixed volume of the Newton polytopes of a list of polynomials. The ring is specified on the input. After this follows the list of polynomials.
│ │ │  Options:
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  --cyclic value:
│ │ │   Use cyclic-n example instead of reading input.
│ │ │  --noon value:
│ │ │ @@ -346,47 +346,47 @@
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │  --resume:
│ │ │   Continue enumeration from saved state...
│ │ │ -using temporary file /tmp/M2-16342-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16342-0/210
│ │ │ +using temporary file /tmp/M2-20462-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20462-0/210
│ │ │  This program computes the union of a list of polynomial sets given as input. The polynomials must all belong to the same ring. The ring is specified on the input. After this follows the list of polynomial sets.
│ │ │  Options:
│ │ │  -s:
│ │ │   Sort output by degree.
│ │ │  
│ │ │  --twosetswithrings:
│ │ │   Switch to mode where input is: ring polynomialset (same)ring polynomialset.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16342-0/212
│ │ │ +using temporary file /tmp/M2-20462-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20462-0/212
│ │ │  This program renders a Groebner fan as an xfig file. To be more precise, the input is the list of all reduced Groebner bases of an ideal. The output is a drawing of the Groebner fan intersected with a triangle. The corners of the triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. If there are more than three variables in the ring these coordinates are extended with zeros. It is possible to shift the 1 entry cyclic with the option --shiftVariables.
│ │ │  Options:
│ │ │  -L:
│ │ │   Make the triangle larger so that the shape of the Groebner region appears.
│ │ │  --shiftVariables value:
│ │ │   Shift the positions of the variables in the drawing. For example with the value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) and top: (0,0,0,1,...). The shifting is done modulo the number of variables in the polynomial ring. The default value is 0.
│ │ │ -using temporary file /tmp/M2-16342-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16342-0/214
│ │ │ +using temporary file /tmp/M2-20462-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20462-0/214
│ │ │  This program renders a staircase diagram of a monomial initial ideal to an xfig file. The input is a Groebner basis of a (not necessarily monomial) polynomial ideal. The initial ideal is given by the leading terms in the Groebner basis. Using the -m option it is possible to render more than one staircase diagram. The program only works for ideals in a polynomial ring with three variables.
│ │ │  Options:
│ │ │  -m:
│ │ │   Read multiple ideals from the input. The ideals are given as a list of lists of polynomials. For each polynomial list in the list a staircase diagram is drawn.
│ │ │  
│ │ │  -d value:
│ │ │   Specifies the number of boxes being shown along each axis. Be sure that this number is large enough to give a correct picture of the standard monomials. The default value is 8.
│ │ │  
│ │ │  -w value:
│ │ │   Width. Specifies the number of staircase diagrams per row in the xfig file. The default value is 5.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16342-0/216
│ │ │ +using temporary file /tmp/M2-20462-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20462-0/216
│ │ │  This program computes the resultant fan as defined in "Computing Tropical Resultants" by Jensen and Yu. The input is a polynomial ring followed by polynomials, whose coefficients are ignored. The output is the fan of coefficients such that the input system has a tropical solution.
│ │ │  Options:
│ │ │  --codimension:
│ │ │   Compute only the codimension of the resultant fan and return.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program DOES NOT checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │ @@ -399,25 +399,25 @@
│ │ │  
│ │ │  --vectorinput:
│ │ │   Read in a list of point configurations instead of a polynomial ring and a list of polynomials.
│ │ │  
│ │ │  --projection:
│ │ │   Use the projection method to compute the resultant fan. This works only if the resultant fan is a hypersurface. If this option is combined with --special, then the output fan lives in the subspace of the non-specialized coordinates.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16342-0/218
│ │ │ +using temporary file /tmp/M2-20462-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20462-0/218
│ │ │  This program computes the saturation of the input ideal with the product of the variables x_1,...,x_n. The ideal does not have to be homogeneous.
│ │ │  Options:
│ │ │  -h:
│ │ │   Tell the program that the input is a homogeneous ideal (with homogeneous generators).
│ │ │  
│ │ │  --noideal:
│ │ │   Do not treat input as an ideal but just factor out common monomial factors of the input polynomials.
│ │ │ -using temporary file /tmp/M2-16342-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16342-0/220
│ │ │ +using temporary file /tmp/M2-20462-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20462-0/220
│ │ │  This program computes the secondary fan of a vector configuration. The configuration is given as an ordered list of vectors. In order to compute the secondary fan of a point configuration an additional coordinate of ones must be added. For example {(1,0),(1,1),(1,2),(1,3)}.
│ │ │  Options:
│ │ │  --unimodular:
│ │ │   Use heuristics to search for unimodular triangulation rather than computing the complete secondary fan
│ │ │  --scale value:
│ │ │   Assuming that the first coordinate of each vector is 1, this option will take the polytope in the 1 plane and scale it. The point configuration will be all lattice points in that scaled polytope. The polytope must have maximal dimension. When this option is used the vector configuration must have full rank. This option may be removed in the future.
│ │ │  --restrictingfan value:
│ │ │ @@ -426,72 +426,72 @@
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the vector configuration. The program checks that the configuration stays fixed when permuting the variables with respect to elements in the group. The output is grouped according to the symmetry.
│ │ │  
│ │ │  --nocones:
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16342-0/222
│ │ │ +using temporary file /tmp/M2-20462-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20462-0/222
│ │ │  This program takes a list of reduced Groebner bases for the same ideal and computes various statistics. The following information is listed: the number of bases in the input, the number of variables, the dimension of the homogeneity space, the maximal total degree of any polynomial in the input and the minimal total degree of any basis in the input, the maximal number of polynomials and terms in a basis in the input.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16342-0/224
│ │ │ +using temporary file /tmp/M2-20462-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20462-0/224
│ │ │  This program changes the variable names of a polynomial ring. The input is a polynomial ring, a polynomial set in the ring and a new polynomial ring with the same coefficient field but different variable names. The output is the polynomial set written with the variable names of the second polynomial ring.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x]
│ │ │  Output:
│ │ │  Q[b,a,c,x]{2*b-3*a,c+x}
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16342-0/226
│ │ │ +using temporary file /tmp/M2-20462-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20462-0/226
│ │ │  This program converts ASCII math to TeX math. The data-type is specified by the options.
│ │ │  Options:
│ │ │  -h:
│ │ │   Add a header to the output. Using this option the output will be LaTeXable right away.
│ │ │  --polynomialset_:
│ │ │   The data to be converted is a list of polynomials.
│ │ │  --polynomialsetlist_:
│ │ │   The data to be converted is a list of lists of polynomials.
│ │ │ -using temporary file /tmp/M2-16342-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16342-0/228
│ │ │ +using temporary file /tmp/M2-20462-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20462-0/228
│ │ │  This program takes a list of reduced Groebner bases and produces the fan of all faces of these. In this way by giving the complete list of reduced Groebner bases, the Groebner fan can be computed as a polyhedral complex. The option --restrict lets the user choose between computing the Groebner fan or the restricted Groebner fan.
│ │ │  Options:
│ │ │  --restrict:
│ │ │   Add an inequality for each coordinate, so that the the cones are restricted to the non-negative orthant.
│ │ │  --generatedcones:
│ │ │   Instead of a list of reduced Groebner bases, read in a list of closed generated cones given in modern format (like _cone). In this case --restrict is ignored.
│ │ │  --symmetry:
│ │ │   Tell the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ring. The output is grouped according to these symmetries. Only one representative for each orbit is needed on the input.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16342-0/230
│ │ │ +using temporary file /tmp/M2-20462-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20462-0/230
│ │ │  This program computes a tropical basis for an ideal defining a tropical curve. Defining a tropical curve means that the Krull dimension of R/I is at most 1 + the dimension of the homogeneity space of I where R is the polynomial ring. The input is a generating set for the ideal. If the input is not homogeneous option -h must be used.
│ │ │  Options:
│ │ │  -h:
│ │ │   Homogenise the input before computing a tropical basis and dehomogenise the output. This is needed if the input generators are not already homogeneous.
│ │ │ -using temporary file /tmp/M2-16342-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16342-0/232
│ │ │ +using temporary file /tmp/M2-20462-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20462-0/232
│ │ │  This program takes a marked reduced Groebner basis for a homogeneous ideal and computes the tropical variety of the ideal as a subfan of the Groebner fan. The program is slow but works for any homogeneous ideal. If you know that your ideal is prime over the complex numbers or you simply know that its tropical variety is pure and connected in codimension one then use gfan_tropicalstartingcone and gfan_tropicaltraverse instead.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16342-0/234
│ │ │ +using temporary file /tmp/M2-20462-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20462-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16342-0/236
│ │ │ +using temporary file /tmp/M2-20462-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20462-0/236
│ │ │  This program takes a polynomial and tropicalizes it. The output is piecewise linear function represented by a fan whose cones are the linear regions. Each ray of the fan gets the value of the tropical function assigned to it. In other words this program computes the normal fan of the Newton polytope of the input polynomial with additional information.Options:
│ │ │  --exponents:
│ │ │   Tell program to read a list of exponent vectors instead.
│ │ │ -using temporary file /tmp/M2-16342-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16342-0/238
│ │ │ +using temporary file /tmp/M2-20462-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20462-0/238
│ │ │  This program computes the tropical hypersurface defined by a principal ideal. The input is the polynomial ring followed by a set containing just a generator of the ideal.Options:
│ │ │ -using temporary file /tmp/M2-16342-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16342-0/240
│ │ │ +using temporary file /tmp/M2-20462-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20462-0/240
│ │ │  This program computes the set theoretical intersection of a set of tropical hypersurfaces (or to be precise, their common refinement as a fan). The input is a list of polynomials with each polynomial defining a hypersurface. Considering tropical hypersurfaces as fans, the intersection can be computed as the common refinement of these. Thus the output is a fan whose support is the intersection of the tropical hypersurfaces.
│ │ │  Options:
│ │ │  --tropicalbasistest:
│ │ │   This option will test that the input polynomials for a tropical basis of the ideal they generate by computing the tropical prevariety of the input polynomials and then refine each cone with the Groebner fan and testing whether each cone in the refinement has an associated monomial free initial ideal. If so, then we have a tropical basis and 1 is written as output. If not, then a zero is written to the output together with a vector in the tropical prevariety but not in the variety. The actual check is done on a homogenization of the input ideal, but this does not affect the result. (This option replaces the -t option from earlier gfan versions.)
│ │ │  
│ │ │  --tplane:
│ │ │   This option intersects the resulting fan with the plane x_0=-1, where x_0 is the first variable. To simplify the implementation the output is actually the common refinement with the non-negative half space. This means that "stuff at infinity" (where x_0=0) is not removed.
│ │ │ @@ -503,16 +503,16 @@
│ │ │   Tells the program not to output the CONES and MAXIMAL_CONES sections, but still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is used.
│ │ │  --restrict:
│ │ │   Restrict the computation to a full-dimensional cone given by a list of marked polynomials. The cone is the closure of all weight vectors choosing these marked terms.
│ │ │  --stable:
│ │ │   Find the stable intersection of the input polynomials using tropical intersection theory. This can be slow. Most other options are ignored.
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16342-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16342-0/242
│ │ │ +using temporary file /tmp/M2-20462-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20462-0/242
│ │ │  This program is part of the Puiseux lifting algorithm implemented in Gfan and Singular. The Singular part of the implementation can be found in:
│ │ │  
│ │ │  Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig:
│ │ │   tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007 
│ │ │  
│ │ │  See also
│ │ │  
│ │ │ @@ -537,48 +537,48 @@
│ │ │  Options:
│ │ │  --noMult:
│ │ │   Disable the multiplicity computation.
│ │ │  -n value:
│ │ │   Number of variables that should have negative weight.
│ │ │  -c:
│ │ │   Only output a list of vectors being the possible choices.
│ │ │ -using temporary file /tmp/M2-16342-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16342-0/244
│ │ │ +using temporary file /tmp/M2-20462-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20462-0/244
│ │ │  This program generates tropical equations for a tropical linear space in the Speyer sense given the tropical Pluecker coordinates as input.
│ │ │  Options:
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │   Specify n.
│ │ │  --trees:
│ │ │   list the boundary trees (assumes d=3)
│ │ │ -using temporary file /tmp/M2-16342-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16342-0/246
│ │ │ +using temporary file /tmp/M2-20462-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20462-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16342-0/248
│ │ │ +using temporary file /tmp/M2-20462-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20462-0/248
│ │ │  This program will compute the tropical rank of matrix given as input. Tropical addition is MAXIMUM.
│ │ │  Options:
│ │ │  --kapranov:
│ │ │   Compute Kapranov rank instead of tropical rank.
│ │ │  --determinant:
│ │ │   Compute the tropical determinant instead.
│ │ │ -using temporary file /tmp/M2-16342-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16342-0/250
│ │ │ +using temporary file /tmp/M2-20462-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20462-0/250
│ │ │  This program computes a starting pair of marked reduced Groebner bases to be used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose tropical variety is a pure d-dimensional polyhedral complex.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tell the program that the input is already a reduced Groebner basis.
│ │ │  -d:
│ │ │   Output dimension information to standard error.
│ │ │  --stable:
│ │ │   Find starting cone in the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │ -using temporary file /tmp/M2-16342-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16342-0/252
│ │ │ +using temporary file /tmp/M2-20462-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20462-0/252
│ │ │  This program computes a polyhedral fan representation of the tropical variety of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let $\omega$ be a relative interior point of $d$-dimensional Groebner cone contained in the tropical variety. The input for this program is a pair of marked reduced Groebner bases with respect to the term order represented by $\omega$, tie-broken in some way. The first one is for the initial ideal $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point for a traversal of the $d$-dimensional Groebner cones contained in the tropical variety. If the ideal is not prime but with the tropical variety still being pure $d$-dimensional the program will only compute a codimension $1$ connected component of the tropical variety.
│ │ │  Options:
│ │ │  --symmetry:
│ │ │   Do computations up to symmetry and group the output accordingly. If this option is used the program will read in a list of generators for a symmetry group after the pair of Groebner bases have been read. Two advantages of using this option is that the output is nicely grouped and that the computation can be done faster.
│ │ │  --symsigns:
│ │ │   Specify for each generator of the symmetry group an element of ${-1,+1}^n$ which by its multiplication on the variables together with the permutation will keep the ideal fixed. The vectors are given as the rows of a matrix.
│ │ │  --nocones:
│ │ │ @@ -586,24 +586,24 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --stable:
│ │ │   Traverse the stable intersection or, equivalently, pretend that the coefficients are genereric.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16342-0/254
│ │ │ +using temporary file /tmp/M2-20462-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20462-0/254
│ │ │  This program computes the tropical Weil divisor of piecewise linear (or tropical rational) function on a tropical k-cycle. See the Gfan manual for more information.
│ │ │  Options:
│ │ │  -i1 value:
│ │ │   Specify the name of the Polymake input file containing the k-cycle.
│ │ │  -i2 value:
│ │ │   Specify the name of the Polymake input file containing the piecewise linear function.
│ │ │ -using temporary file /tmp/M2-16342-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16342-0/256
│ │ │ +using temporary file /tmp/M2-20462-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20462-0/256
│ │ │  This program is an experimental implementation of Groebner bases for ideals in Z[x_1,...,x_n].
│ │ │  Several operations are supported by specifying the appropriate option:
│ │ │   (1) computation of the reduced Groebner basis with respect to a given vector (tiebroken lexicographically),
│ │ │   (2) computation of an initial ideal,
│ │ │   (3) computation of the Groebner fan,
│ │ │   (4) computation of a single Groebner cone.
│ │ │  Since Gfan only knows polynomial rings with coefficients being elements of a field, the ideal is specified by giving a set of polynomials in the polynomial ring Q[x_1,...,x_n]. That is, by using Q instead of Z when specifying the ring. The ideal MUST BE HOMOGENEOUS (in a positive grading) for computation of the Groebner fan. Non-homogeneous ideals are allowed for the other computations if the specified weight vectors are positive.
│ │ │ @@ -623,32 +623,32 @@
│ │ │  --groebnerCone:
│ │ │   Asks the program to compute a single Groebner cone containing the specified vector in its relative interior. The output is stored as a fan. The input order is: Ring ideal vector.
│ │ │  -m:
│ │ │   For the operations taking a vector as input, read in a list of vectors instead, and perform the operation for each vector in the list.
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if the usual --groebnerFan is too slow.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/256
    │ │ │ +using temporary file /tmp/M2-20462-0/256 │ │ │
    │ │ │ 
    i6 : QQ[x,y];
    │ │ │
    │ │ │ 
    i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16342-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20462-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16342-0/258
    │ │ │ +using temporary file /tmp/M2-20462-0/258 │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -43,15 +43,15 @@ │ │ │ │ If you would like to see the input and output files used to communicate with │ │ │ │ gfan you can set the "keepfiles" configuration option to true. If "verbose" is │ │ │ │ set to true, gfanInterface will output the names of the temporary files used. │ │ │ │ i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, │ │ │ │ "verbose" => true}, Reload => true); │ │ │ │ -- warning: reloading gfanInterface; recreate instances of types from this │ │ │ │ package │ │ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16342-0/172 │ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20462-0/172 │ │ │ │ This is a program for computing all reduced Groebner bases of a polynomial │ │ │ │ ideal. It takes the ring and a generating set for the ideal as input. By │ │ │ │ default the enumeration is done by an almost memoryless reverse search. If the │ │ │ │ ideal is symmetric the symmetry option is useful and enumeration will be done │ │ │ │ up to symmetry using a breadth first search. The program needs a starting │ │ │ │ Groebner basis to do its computations. If the -g option is not specified it │ │ │ │ will compute one using Buchberger's algorithm. │ │ │ │ @@ -81,16 +81,16 @@ │ │ │ │ With this option you can specify how many variables to treat as parameters │ │ │ │ instead of variables. This makes it possible to do computations where the │ │ │ │ coefficient field is the field of rational functions in the parameters. │ │ │ │ --interrupt value: │ │ │ │ Interrupt the enumeration after a specified number of facets have been │ │ │ │ computed (works for usual symmetric traversals, but may not work in general for │ │ │ │ non-symmetric traversals or for traversals restricted to fans). │ │ │ │ -using temporary file /tmp/M2-16342-0/172 │ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16342-0/174 │ │ │ │ +using temporary file /tmp/M2-20462-0/172 │ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20462-0/174 │ │ │ │ This program computes a reduced lexicographic Groebner basis of the polynomial │ │ │ │ ideal given as input. The default behavior is to use Buchberger's algorithm. │ │ │ │ The ordering of the variables is $a>b>c...$ (assuming that the ring is Q │ │ │ │ [a,b,c,...]). │ │ │ │ Options: │ │ │ │ -w: │ │ │ │ Compute a Groebner basis with respect to a degree lexicographic order with │ │ │ │ @@ -111,63 +111,63 @@ │ │ │ │ minimal Groebner basis with respect to the reverse lexicographic term order. │ │ │ │ The target term order is always lexicographic. The -W option must be used. │ │ │ │ │ │ │ │ --parameters value: │ │ │ │ With this option you can specify how many variables to treat as parameters │ │ │ │ instead of variables. This makes it possible to do computations where the │ │ │ │ coefficient field is the field of rational functions in the parameters. │ │ │ │ -using temporary file /tmp/M2-16342-0/174 │ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16342-0/176 │ │ │ │ +using temporary file /tmp/M2-20462-0/174 │ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20462-0/176 │ │ │ │ This program takes a marked Groebner basis of an ideal I and a set of │ │ │ │ polynomials on its input and tests if the polynomial set is contained in I by │ │ │ │ applying the division algorithm for each element. The output is 1 for true and │ │ │ │ 0 for false. │ │ │ │ Options: │ │ │ │ --remainder: │ │ │ │ Tell the program to output the remainders of the divisions rather than │ │ │ │ outputting 0 or 1. │ │ │ │ --multiplier: │ │ │ │ Reads in a polynomial that will be multiplied to the polynomial to be divided │ │ │ │ before doing the division. │ │ │ │ -using temporary file /tmp/M2-16342-0/176 │ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16342-0/178 │ │ │ │ +using temporary file /tmp/M2-20462-0/176 │ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20462-0/178 │ │ │ │ This program takes two polyhedral fans and computes their common refinement. │ │ │ │ Options: │ │ │ │ -i1 value: │ │ │ │ Specify the name of the first input file. │ │ │ │ -i2 value: │ │ │ │ Specify the name of the second input file. │ │ │ │ --stable: │ │ │ │ Compute the stable intersection. │ │ │ │ -using temporary file /tmp/M2-16342-0/178 │ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16342-0/180 │ │ │ │ +using temporary file /tmp/M2-20462-0/178 │ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20462-0/180 │ │ │ │ This program takes a polyhedral fan and a vector and computes the link of the │ │ │ │ polyhedral fan around that vertex. The link will have lineality space dimension │ │ │ │ equal to the dimension of the relative open polyhedral cone of the original fan │ │ │ │ containing the vector. │ │ │ │ Options: │ │ │ │ -i value: │ │ │ │ Specify the name of the input file. │ │ │ │ --symmetry: │ │ │ │ Reads in a fan stored with symmetry. The generators of the symmetry group must │ │ │ │ be given on the standard input. │ │ │ │ │ │ │ │ --star: │ │ │ │ Computes the star instead. The star is defined as the smallest polyhedral fan │ │ │ │ containing all cones of the original fan containing the vector. │ │ │ │ -using temporary file /tmp/M2-16342-0/180 │ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16342-0/182 │ │ │ │ +using temporary file /tmp/M2-20462-0/180 │ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20462-0/182 │ │ │ │ This program takes two polyhedral fans and computes their product. │ │ │ │ Options: │ │ │ │ -i1 value: │ │ │ │ Specify the name of the first input file. │ │ │ │ -i2 value: │ │ │ │ Specify the name of the second input file. │ │ │ │ -using temporary file /tmp/M2-16342-0/182 │ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16342-0/184 │ │ │ │ +using temporary file /tmp/M2-20462-0/182 │ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20462-0/184 │ │ │ │ This program computes a Groebner cone. Three different cases are handled. The │ │ │ │ input may be a marked reduced Groebner basis in which case its Groebner cone is │ │ │ │ computed. The input may be just a marked minimal basis in which case the cone │ │ │ │ computed is not a Groebner cone in the usual sense but smaller. (These cones │ │ │ │ are described in [Fukuda, Jensen, Lauritzen, Thomas]). The third possible case │ │ │ │ is that the Groebner cone is possibly lower dimensional and given by a pair of │ │ │ │ Groebner bases as it is useful to do for tropical varieties, see option --pair. │ │ │ │ @@ -184,24 +184,24 @@ │ │ │ │ --asfan: │ │ │ │ Writes the cone as a polyhedral fan with all its faces instead. In this way │ │ │ │ the extreme rays of the cone are also computed. │ │ │ │ --vectorinput: │ │ │ │ Compute a cone given list of inequalities rather than a Groebner cone. The │ │ │ │ input is an integer which specifies the dimension of the ambient space, a list │ │ │ │ of inequalities given as vectors and a list of equations. │ │ │ │ -using temporary file /tmp/M2-16342-0/184 │ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16342-0/186 │ │ │ │ +using temporary file /tmp/M2-20462-0/184 │ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20462-0/186 │ │ │ │ This program computes the homogeneity space of a list of polynomials - as a │ │ │ │ cone. Thus generators for the homogeneity space are found in the section │ │ │ │ LINEALITY_SPACE. If you wish the homogeneity space of an ideal you should first │ │ │ │ compute a set of homogeneous generators and call the program on these. A │ │ │ │ reduced Groebner basis will always suffice for this purpose. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/186 │ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16342-0/188 │ │ │ │ +using temporary file /tmp/M2-20462-0/186 │ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20462-0/188 │ │ │ │ This program homogenises a list of polynomials by introducing an extra │ │ │ │ variable. The name of the variable to be introduced is read from the input │ │ │ │ after the list of polynomials. Without the -w option the homogenisation is done │ │ │ │ with respect to total degree. │ │ │ │ Example: │ │ │ │ Input: │ │ │ │ Q[x,y]{y-1} │ │ │ │ @@ -217,16 +217,16 @@ │ │ │ │ Specify a homogenisation vector. The length of the vector must be the same as │ │ │ │ the number of variables in the ring. The vector is read from the input after │ │ │ │ the list of polynomials. │ │ │ │ │ │ │ │ -H: │ │ │ │ Let the name of the new variable be H rather than reading in a name from the │ │ │ │ input. │ │ │ │ -using temporary file /tmp/M2-16342-0/188 │ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16342-0/190 │ │ │ │ +using temporary file /tmp/M2-20462-0/188 │ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20462-0/190 │ │ │ │ This program converts a list of polynomials to a list of their initial forms │ │ │ │ with respect to the vector given after the list. │ │ │ │ Options: │ │ │ │ --ideal: │ │ │ │ Treat input as an ideal. This will make the program compute the initial ideal │ │ │ │ of the ideal generated by the input polynomials. The computation is done by │ │ │ │ computing a Groebner basis with respect to the given vector. The vector must be │ │ │ │ @@ -242,16 +242,16 @@ │ │ │ │ --mark: │ │ │ │ If the --pair option is and the --ideal option is not used this option will │ │ │ │ still make sure that the second output basis is marked consistently with the │ │ │ │ vector. │ │ │ │ --list: │ │ │ │ Read in a list of vectors instead of a single vector and produce a list of │ │ │ │ polynomial sets as output. │ │ │ │ -using temporary file /tmp/M2-16342-0/190 │ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16342-0/192 │ │ │ │ +using temporary file /tmp/M2-20462-0/190 │ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20462-0/192 │ │ │ │ This is a program for doing interactive walks in the Groebner fan of an ideal. │ │ │ │ The input is a Groebner basis defining the starting Groebner cone of the walk. │ │ │ │ The program will list all flippable facets of the Groebner cone and ask the │ │ │ │ user to choose one. The user types in the index (number) of the facet in the │ │ │ │ list. The program will walk through the selected facet and display the new │ │ │ │ Groebner basis and a list of new facet normals for the user to choose from. │ │ │ │ Since the program reads the user's choices through the the standard input it is │ │ │ │ @@ -281,54 +281,54 @@ │ │ │ │ -W: │ │ │ │ Print weight vector. This will make the program print an interior vector of │ │ │ │ the current Groebner cone and a relative interior point for each flippable │ │ │ │ facet of the current Groebner cone. │ │ │ │ │ │ │ │ --tropical: │ │ │ │ Traverse a tropical variety interactively. │ │ │ │ -using temporary file /tmp/M2-16342-0/192 │ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16342-0/194 │ │ │ │ +using temporary file /tmp/M2-20462-0/192 │ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20462-0/194 │ │ │ │ This program checks if a set of marked polynomials is a Groebner basis with │ │ │ │ respect to its marking. First it is checked if the markings are consistent with │ │ │ │ respect to a positive vector. Then Buchberger's S-criterion is checked. The │ │ │ │ output is boolean value. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/194 │ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16342-0/196 │ │ │ │ +using temporary file /tmp/M2-20462-0/194 │ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20462-0/196 │ │ │ │ Takes an ideal $I$ and computes the Krull dimension of R/I where R is the │ │ │ │ polynomial ring. This is done by first computing a Groebner basis. │ │ │ │ Options: │ │ │ │ -g: │ │ │ │ Tell the program that the input is already a reduced Groebner basis. │ │ │ │ -using temporary file /tmp/M2-16342-0/196 │ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16342-0/198 │ │ │ │ +using temporary file /tmp/M2-20462-0/196 │ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20462-0/198 │ │ │ │ This program computes the lattice ideal of a lattice. The input is a list of │ │ │ │ generators for the lattice. │ │ │ │ Options: │ │ │ │ -t: │ │ │ │ Compute the toric ideal of the matrix whose rows are given on the input │ │ │ │ instead. │ │ │ │ --convert: │ │ │ │ Does not do any computation, but just converts the vectors to binomials. │ │ │ │ -using temporary file /tmp/M2-16342-0/198 │ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16342-0/200 │ │ │ │ +using temporary file /tmp/M2-20462-0/198 │ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20462-0/200 │ │ │ │ This program converts a list of polynomials to a list of their leading terms. │ │ │ │ Options: │ │ │ │ -m: │ │ │ │ Do the same thing for a list of polynomial sets. That is, output the set of │ │ │ │ sets of leading terms. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/200 │ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16342-0/202 │ │ │ │ +using temporary file /tmp/M2-20462-0/200 │ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20462-0/202 │ │ │ │ This program marks a set of polynomials with respect to the vector given at the │ │ │ │ end of the input, meaning that the largest terms are moved to the front. In │ │ │ │ case of a tie the lexicographic term order with $a>b>c...$ is used to break it. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/202 │ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16342-0/204 │ │ │ │ +using temporary file /tmp/M2-20462-0/202 │ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20462-0/204 │ │ │ │ This is a program for computing the normal fan of the Minkowski sum of the │ │ │ │ Newton polytopes of a list of polynomials. │ │ │ │ Options: │ │ │ │ --symmetry: │ │ │ │ Tells the program to read in generators for a group of symmetries (subgroup of │ │ │ │ $S_n$) after having read in the ideal. The program checks that the ideal stays │ │ │ │ fixed when permuting the variables with respect to elements in the group. The │ │ │ │ @@ -338,16 +338,16 @@ │ │ │ │ --disableSymmetryTest: │ │ │ │ When using --symmetry this option will disable the check that the group read │ │ │ │ off from the input actually is a symmetry group with respect to the input │ │ │ │ ideal. │ │ │ │ │ │ │ │ --nocones: │ │ │ │ Tell the program to not list cones in the output. │ │ │ │ -using temporary file /tmp/M2-16342-0/204 │ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16342-0/206 │ │ │ │ +using temporary file /tmp/M2-20462-0/204 │ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20462-0/206 │ │ │ │ This program will generate the r*r minors of a d*n matrix of indeterminates. │ │ │ │ Options: │ │ │ │ -r value: │ │ │ │ Specify r. │ │ │ │ -d value: │ │ │ │ Specify d. │ │ │ │ -n value: │ │ │ │ @@ -365,16 +365,16 @@ │ │ │ │ Produces a list of generators for the group of symmetries keeping the set of │ │ │ │ minors fixed. (Only without --names). │ │ │ │ --parametrize: │ │ │ │ Parametrize the set of d times n matrices of Barvinok rank less than or equal │ │ │ │ to r-1 by a list of tropical polynomials. │ │ │ │ --ultrametric: │ │ │ │ Produce tropical equations cutting out the ultrametrics. │ │ │ │ -using temporary file /tmp/M2-16342-0/206 │ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16342-0/208 │ │ │ │ +using temporary file /tmp/M2-20462-0/206 │ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20462-0/208 │ │ │ │ This program computes the mixed volume of the Newton polytopes of a list of │ │ │ │ polynomials. The ring is specified on the input. After this follows the list of │ │ │ │ polynomials. │ │ │ │ Options: │ │ │ │ --vectorinput: │ │ │ │ Read in a list of point configurations instead of a polynomial ring and a list │ │ │ │ of polynomials. │ │ │ │ @@ -390,28 +390,28 @@ │ │ │ │ Use Gaukwa-n example instead of reading input. │ │ │ │ --eco value: │ │ │ │ Use Eco-n example instead of reading input. │ │ │ │ -j value: │ │ │ │ Number of threads │ │ │ │ --resume: │ │ │ │ Continue enumeration from saved state... │ │ │ │ -using temporary file /tmp/M2-16342-0/208 │ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16342-0/210 │ │ │ │ +using temporary file /tmp/M2-20462-0/208 │ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20462-0/210 │ │ │ │ This program computes the union of a list of polynomial sets given as input. │ │ │ │ The polynomials must all belong to the same ring. The ring is specified on the │ │ │ │ input. After this follows the list of polynomial sets. │ │ │ │ Options: │ │ │ │ -s: │ │ │ │ Sort output by degree. │ │ │ │ │ │ │ │ --twosetswithrings: │ │ │ │ Switch to mode where input is: ring polynomialset (same)ring polynomialset. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/210 │ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16342-0/212 │ │ │ │ +using temporary file /tmp/M2-20462-0/210 │ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20462-0/212 │ │ │ │ This program renders a Groebner fan as an xfig file. To be more precise, the │ │ │ │ input is the list of all reduced Groebner bases of an ideal. The output is a │ │ │ │ drawing of the Groebner fan intersected with a triangle. The corners of the │ │ │ │ triangle are (1,0,0) to the right, (0,1,0) to the left and (0,0,1) at the top. │ │ │ │ If there are more than three variables in the ring these coordinates are │ │ │ │ extended with zeros. It is possible to shift the 1 entry cyclic with the option │ │ │ │ --shiftVariables. │ │ │ │ @@ -419,16 +419,16 @@ │ │ │ │ -L: │ │ │ │ Make the triangle larger so that the shape of the Groebner region appears. │ │ │ │ --shiftVariables value: │ │ │ │ Shift the positions of the variables in the drawing. For example with the │ │ │ │ value equal to 1 the corners will be right: (0,1,0,0,...), left: (0,0,1,0,...) │ │ │ │ and top: (0,0,0,1,...). The shifting is done modulo the number of variables in │ │ │ │ the polynomial ring. The default value is 0. │ │ │ │ -using temporary file /tmp/M2-16342-0/212 │ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16342-0/214 │ │ │ │ +using temporary file /tmp/M2-20462-0/212 │ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20462-0/214 │ │ │ │ This program renders a staircase diagram of a monomial initial ideal to an xfig │ │ │ │ file. The input is a Groebner basis of a (not necessarily monomial) polynomial │ │ │ │ ideal. The initial ideal is given by the leading terms in the Groebner basis. │ │ │ │ Using the -m option it is possible to render more than one staircase diagram. │ │ │ │ The program only works for ideals in a polynomial ring with three variables. │ │ │ │ Options: │ │ │ │ -m: │ │ │ │ @@ -441,16 +441,16 @@ │ │ │ │ number is large enough to give a correct picture of the standard monomials. The │ │ │ │ default value is 8. │ │ │ │ │ │ │ │ -w value: │ │ │ │ Width. Specifies the number of staircase diagrams per row in the xfig file. │ │ │ │ The default value is 5. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/214 │ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16342-0/216 │ │ │ │ +using temporary file /tmp/M2-20462-0/214 │ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20462-0/216 │ │ │ │ This program computes the resultant fan as defined in "Computing Tropical │ │ │ │ Resultants" by Jensen and Yu. The input is a polynomial ring followed by │ │ │ │ polynomials, whose coefficients are ignored. The output is the fan of │ │ │ │ coefficients such that the input system has a tropical solution. │ │ │ │ Options: │ │ │ │ --codimension: │ │ │ │ Compute only the codimension of the resultant fan and return. │ │ │ │ @@ -478,28 +478,28 @@ │ │ │ │ of polynomials. │ │ │ │ │ │ │ │ --projection: │ │ │ │ Use the projection method to compute the resultant fan. This works only if the │ │ │ │ resultant fan is a hypersurface. If this option is combined with --special, │ │ │ │ then the output fan lives in the subspace of the non-specialized coordinates. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/216 │ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16342-0/218 │ │ │ │ +using temporary file /tmp/M2-20462-0/216 │ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20462-0/218 │ │ │ │ This program computes the saturation of the input ideal with the product of the │ │ │ │ variables x_1,...,x_n. The ideal does not have to be homogeneous. │ │ │ │ Options: │ │ │ │ -h: │ │ │ │ Tell the program that the input is a homogeneous ideal (with homogeneous │ │ │ │ generators). │ │ │ │ │ │ │ │ --noideal: │ │ │ │ Do not treat input as an ideal but just factor out common monomial factors of │ │ │ │ the input polynomials. │ │ │ │ -using temporary file /tmp/M2-16342-0/218 │ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16342-0/220 │ │ │ │ +using temporary file /tmp/M2-20462-0/218 │ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20462-0/220 │ │ │ │ This program computes the secondary fan of a vector configuration. The │ │ │ │ configuration is given as an ordered list of vectors. In order to compute the │ │ │ │ secondary fan of a point configuration an additional coordinate of ones must be │ │ │ │ added. For example {(1,0),(1,1),(1,2),(1,3)}. │ │ │ │ Options: │ │ │ │ --unimodular: │ │ │ │ Use heuristics to search for unimodular triangulation rather than computing │ │ │ │ @@ -528,49 +528,49 @@ │ │ │ │ Tells the program not to output the CONES and MAXIMAL_CONES sections, but │ │ │ │ still output CONES_COMPRESSED and MAXIMAL_CONES_COMPRESSED if --symmetry is │ │ │ │ used. │ │ │ │ --interrupt value: │ │ │ │ Interrupt the enumeration after a specified number of facets have been │ │ │ │ computed (works for usual symmetric traversals, but may not work in general for │ │ │ │ non-symmetric traversals or for traversals restricted to fans). │ │ │ │ -using temporary file /tmp/M2-16342-0/220 │ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16342-0/222 │ │ │ │ +using temporary file /tmp/M2-20462-0/220 │ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20462-0/222 │ │ │ │ This program takes a list of reduced Groebner bases for the same ideal and │ │ │ │ computes various statistics. The following information is listed: the number of │ │ │ │ bases in the input, the number of variables, the dimension of the homogeneity │ │ │ │ space, the maximal total degree of any polynomial in the input and the minimal │ │ │ │ total degree of any basis in the input, the maximal number of polynomials and │ │ │ │ terms in a basis in the input. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/222 │ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16342-0/224 │ │ │ │ +using temporary file /tmp/M2-20462-0/222 │ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20462-0/224 │ │ │ │ This program changes the variable names of a polynomial ring. The input is a │ │ │ │ polynomial ring, a polynomial set in the ring and a new polynomial ring with │ │ │ │ the same coefficient field but different variable names. The output is the │ │ │ │ polynomial set written with the variable names of the second polynomial ring. │ │ │ │ Example: │ │ │ │ Input: │ │ │ │ Q[a,b,c,d]{2a-3b,c+d}Q[b,a,c,x] │ │ │ │ Output: │ │ │ │ Q[b,a,c,x]{2*b-3*a,c+x} │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/224 │ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16342-0/226 │ │ │ │ +using temporary file /tmp/M2-20462-0/224 │ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20462-0/226 │ │ │ │ This program converts ASCII math to TeX math. The data-type is specified by the │ │ │ │ options. │ │ │ │ Options: │ │ │ │ -h: │ │ │ │ Add a header to the output. Using this option the output will be LaTeXable │ │ │ │ right away. │ │ │ │ --polynomialset_: │ │ │ │ The data to be converted is a list of polynomials. │ │ │ │ --polynomialsetlist_: │ │ │ │ The data to be converted is a list of lists of polynomials. │ │ │ │ -using temporary file /tmp/M2-16342-0/226 │ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16342-0/228 │ │ │ │ +using temporary file /tmp/M2-20462-0/226 │ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20462-0/228 │ │ │ │ This program takes a list of reduced Groebner bases and produces the fan of all │ │ │ │ faces of these. In this way by giving the complete list of reduced Groebner │ │ │ │ bases, the Groebner fan can be computed as a polyhedral complex. The option -- │ │ │ │ restrict lets the user choose between computing the Groebner fan or the │ │ │ │ restricted Groebner fan. │ │ │ │ Options: │ │ │ │ --restrict: │ │ │ │ @@ -581,54 +581,54 @@ │ │ │ │ generated cones given in modern format (like _cone). In this case --restrict is │ │ │ │ ignored. │ │ │ │ --symmetry: │ │ │ │ Tell the program to read in generators for a group of symmetries (subgroup of │ │ │ │ $S_n$) after having read in the ring. The output is grouped according to these │ │ │ │ symmetries. Only one representative for each orbit is needed on the input. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/228 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16342-0/230 │ │ │ │ +using temporary file /tmp/M2-20462-0/228 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20462-0/230 │ │ │ │ This program computes a tropical basis for an ideal defining a tropical curve. │ │ │ │ Defining a tropical curve means that the Krull dimension of R/I is at most 1 + │ │ │ │ the dimension of the homogeneity space of I where R is the polynomial ring. The │ │ │ │ input is a generating set for the ideal. If the input is not homogeneous option │ │ │ │ -h must be used. │ │ │ │ Options: │ │ │ │ -h: │ │ │ │ Homogenise the input before computing a tropical basis and dehomogenise the │ │ │ │ output. This is needed if the input generators are not already homogeneous. │ │ │ │ -using temporary file /tmp/M2-16342-0/230 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16342-0/232 │ │ │ │ +using temporary file /tmp/M2-20462-0/230 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20462-0/232 │ │ │ │ This program takes a marked reduced Groebner basis for a homogeneous ideal and │ │ │ │ computes the tropical variety of the ideal as a subfan of the Groebner fan. The │ │ │ │ program is slow but works for any homogeneous ideal. If you know that your │ │ │ │ ideal is prime over the complex numbers or you simply know that its tropical │ │ │ │ variety is pure and connected in codimension one then use │ │ │ │ gfan_tropicalstartingcone and gfan_tropicaltraverse instead. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/232 │ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16342-0/234 │ │ │ │ +using temporary file /tmp/M2-20462-0/232 │ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20462-0/234 │ │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/234 │ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16342-0/236 │ │ │ │ +using temporary file /tmp/M2-20462-0/234 │ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20462-0/236 │ │ │ │ This program takes a polynomial and tropicalizes it. The output is piecewise │ │ │ │ linear function represented by a fan whose cones are the linear regions. Each │ │ │ │ ray of the fan gets the value of the tropical function assigned to it. In other │ │ │ │ words this program computes the normal fan of the Newton polytope of the input │ │ │ │ polynomial with additional information.Options: │ │ │ │ --exponents: │ │ │ │ Tell program to read a list of exponent vectors instead. │ │ │ │ -using temporary file /tmp/M2-16342-0/236 │ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16342-0/238 │ │ │ │ +using temporary file /tmp/M2-20462-0/236 │ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20462-0/238 │ │ │ │ This program computes the tropical hypersurface defined by a principal ideal. │ │ │ │ The input is the polynomial ring followed by a set containing just a generator │ │ │ │ of the ideal.Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/238 │ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16342-0/240 │ │ │ │ +using temporary file /tmp/M2-20462-0/238 │ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20462-0/240 │ │ │ │ This program computes the set theoretical intersection of a set of tropical │ │ │ │ hypersurfaces (or to be precise, their common refinement as a fan). The input │ │ │ │ is a list of polynomials with each polynomial defining a hypersurface. │ │ │ │ Considering tropical hypersurfaces as fans, the intersection can be computed as │ │ │ │ the common refinement of these. Thus the output is a fan whose support is the │ │ │ │ intersection of the tropical hypersurfaces. │ │ │ │ Options: │ │ │ │ @@ -665,16 +665,16 @@ │ │ │ │ --stable: │ │ │ │ Find the stable intersection of the input polynomials using tropical │ │ │ │ intersection theory. This can be slow. Most other options are ignored. │ │ │ │ --parameters value: │ │ │ │ With this option you can specify how many variables to treat as parameters │ │ │ │ instead of variables. This makes it possible to do computations where the │ │ │ │ coefficient field is the field of rational functions in the parameters. │ │ │ │ -using temporary file /tmp/M2-16342-0/240 │ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16342-0/242 │ │ │ │ +using temporary file /tmp/M2-20462-0/240 │ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20462-0/242 │ │ │ │ This program is part of the Puiseux lifting algorithm implemented in Gfan and │ │ │ │ Singular. The Singular part of the implementation can be found in: │ │ │ │ │ │ │ │ Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig: │ │ │ │ tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, │ │ │ │ 2007 │ │ │ │ │ │ │ │ @@ -702,54 +702,54 @@ │ │ │ │ Options: │ │ │ │ --noMult: │ │ │ │ Disable the multiplicity computation. │ │ │ │ -n value: │ │ │ │ Number of variables that should have negative weight. │ │ │ │ -c: │ │ │ │ Only output a list of vectors being the possible choices. │ │ │ │ -using temporary file /tmp/M2-16342-0/242 │ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16342-0/244 │ │ │ │ +using temporary file /tmp/M2-20462-0/242 │ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20462-0/244 │ │ │ │ This program generates tropical equations for a tropical linear space in the │ │ │ │ Speyer sense given the tropical Pluecker coordinates as input. │ │ │ │ Options: │ │ │ │ -d value: │ │ │ │ Specify d. │ │ │ │ -n value: │ │ │ │ Specify n. │ │ │ │ --trees: │ │ │ │ list the boundary trees (assumes d=3) │ │ │ │ -using temporary file /tmp/M2-16342-0/244 │ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16342-0/246 │ │ │ │ +using temporary file /tmp/M2-20462-0/244 │ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20462-0/246 │ │ │ │ This program computes the multiplicity of a tropical cone given a marked │ │ │ │ reduced Groebner basis for its initial ideal. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/246 │ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16342-0/248 │ │ │ │ +using temporary file /tmp/M2-20462-0/246 │ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20462-0/248 │ │ │ │ This program will compute the tropical rank of matrix given as input. Tropical │ │ │ │ addition is MAXIMUM. │ │ │ │ Options: │ │ │ │ --kapranov: │ │ │ │ Compute Kapranov rank instead of tropical rank. │ │ │ │ --determinant: │ │ │ │ Compute the tropical determinant instead. │ │ │ │ -using temporary file /tmp/M2-16342-0/248 │ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16342-0/250 │ │ │ │ +using temporary file /tmp/M2-20462-0/248 │ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20462-0/250 │ │ │ │ This program computes a starting pair of marked reduced Groebner bases to be │ │ │ │ used as input for gfan_tropicaltraverse. The input is a homogeneous ideal whose │ │ │ │ tropical variety is a pure d-dimensional polyhedral complex. │ │ │ │ Options: │ │ │ │ -g: │ │ │ │ Tell the program that the input is already a reduced Groebner basis. │ │ │ │ -d: │ │ │ │ Output dimension information to standard error. │ │ │ │ --stable: │ │ │ │ Find starting cone in the stable intersection or, equivalently, pretend that │ │ │ │ the coefficients are genereric. │ │ │ │ -using temporary file /tmp/M2-16342-0/250 │ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16342-0/252 │ │ │ │ +using temporary file /tmp/M2-20462-0/250 │ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20462-0/252 │ │ │ │ This program computes a polyhedral fan representation of the tropical variety │ │ │ │ of a homogeneous prime ideal $I$. Let $d$ be the Krull dimension of $I$ and let │ │ │ │ $\omega$ be a relative interior point of $d$-dimensional Groebner cone │ │ │ │ contained in the tropical variety. The input for this program is a pair of │ │ │ │ marked reduced Groebner bases with respect to the term order represented by │ │ │ │ $\omega$, tie-broken in some way. The first one is for the initial ideal │ │ │ │ $in_\omega(I)$ the second one for $I$ itself. The pair is the starting point │ │ │ │ @@ -779,27 +779,27 @@ │ │ │ │ --stable: │ │ │ │ Traverse the stable intersection or, equivalently, pretend that the │ │ │ │ coefficients are genereric. │ │ │ │ --interrupt value: │ │ │ │ Interrupt the enumeration after a specified number of facets have been │ │ │ │ computed (works for usual symmetric traversals, but may not work in general for │ │ │ │ non-symmetric traversals or for traversals restricted to fans). │ │ │ │ -using temporary file /tmp/M2-16342-0/252 │ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16342-0/254 │ │ │ │ +using temporary file /tmp/M2-20462-0/252 │ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20462-0/254 │ │ │ │ This program computes the tropical Weil divisor of piecewise linear (or │ │ │ │ tropical rational) function on a tropical k-cycle. See the Gfan manual for more │ │ │ │ information. │ │ │ │ Options: │ │ │ │ -i1 value: │ │ │ │ Specify the name of the Polymake input file containing the k-cycle. │ │ │ │ -i2 value: │ │ │ │ Specify the name of the Polymake input file containing the piecewise linear │ │ │ │ function. │ │ │ │ -using temporary file /tmp/M2-16342-0/254 │ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16342-0/256 │ │ │ │ +using temporary file /tmp/M2-20462-0/254 │ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20462-0/256 │ │ │ │ This program is an experimental implementation of Groebner bases for ideals in │ │ │ │ Z[x_1,...,x_n]. │ │ │ │ Several operations are supported by specifying the appropriate option: │ │ │ │ (1) computation of the reduced Groebner basis with respect to a given vector │ │ │ │ (tiebroken lexicographically), │ │ │ │ (2) computation of an initial ideal, │ │ │ │ (3) computation of the Groebner fan, │ │ │ │ @@ -834,23 +834,23 @@ │ │ │ │ For the operations taking a vector as input, read in a list of vectors │ │ │ │ instead, and perform the operation for each vector in the list. │ │ │ │ -g: │ │ │ │ Tells the program that the input is already a Groebner basis (with the initial │ │ │ │ term of each polynomial being the first ones listed). Use this option if the │ │ │ │ usual --groebnerFan is too slow. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/256 │ │ │ │ +using temporary file /tmp/M2-20462-0/256 │ │ │ │ i6 : QQ[x,y]; │ │ │ │ i7 : gfan {x,y}; │ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16342-0/258 │ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20462-0/258 │ │ │ │ Q[x1,x2] │ │ │ │ {{ │ │ │ │ x2, │ │ │ │ x1} │ │ │ │ } │ │ │ │ -using temporary file /tmp/M2-16342-0/258 │ │ │ │ +using temporary file /tmp/M2-20462-0/258 │ │ │ │ Finally, if you want to be able to render Groebner fans and monomial staircases │ │ │ │ to .png files, you should install fig2dev. If it is installed in a non-standard │ │ │ │ location, then you may specify its path using _p_r_o_g_r_a_m_P_a_t_h_s. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/gfanInterface.m2:2631:0. │ │ ├── ./usr/share/info/AInfinity.info.gz │ │ │ ├── AInfinity.info │ │ │ │ @@ -6133,16 +6133,16 @@ │ │ │ │ 00017f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017f70: 2d2d 2d2d 2b0a 7c69 3320 3a20 656c 6170 ----+.|i3 : elap │ │ │ │ 00017f80: 7365 6454 696d 6520 6275 726b 6552 6573 sedTime burkeRes │ │ │ │ 00017f90: 6f6c 7574 696f 6e28 4d2c 2037 2c20 4368 olution(M, 7, Ch │ │ │ │ 00017fa0: 6563 6b20 3d3e 2066 616c 7365 2920 2020 eck => false) │ │ │ │ -00017fb0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2032 |.| -- 2 │ │ │ │ -00017fc0: 2e34 3430 3938 7320 656c 6170 7365 6420 .44098s elapsed │ │ │ │ +00017fb0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2031 |.| -- 1 │ │ │ │ +00017fc0: 2e33 3334 3538 7320 656c 6170 7365 6420 .33458s elapsed │ │ │ │ 00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00018000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6176,15 +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 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00018220: 3420 3a20 656c 6170 7365 6454 696d 6520 4 : elapsedTime │ │ │ │ 00018230: 6275 726b 6552 6573 6f6c 7574 696f 6e28 burkeResolution( │ │ │ │ 00018240: 4d2c 2037 2c20 4368 6563 6b20 3d3e 2074 M, 7, Check => t │ │ │ │ 00018250: 7275 6529 2020 2020 2020 2020 2020 2020 rue) │ │ │ │ -00018260: 7c0a 7c20 2d2d 2032 2e33 3931 3435 7320 |.| -- 2.39145s │ │ │ │ +00018260: 7c0a 7c20 2d2d 2031 2e36 3631 3039 7320 |.| -- 1.66109s │ │ │ │ 00018270: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00018280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000182b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/AdjunctionForSurfaces.info.gz │ │ │ ├── AdjunctionForSurfaces.info │ │ │ │ @@ -741,16 +741,16 @@ │ │ │ │ 00002e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00002e70: 7c69 3130 203a 2065 6c61 7073 6564 5469 |i10 : elapsedTi │ │ │ │ 00002e80: 6d65 2066 493d 7265 7320 4920 2020 2020 me fI=res I │ │ │ │ 00002e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3235 3738 |.| -- .02578 │ │ │ │ -00002ec0: 3338 7320 656c 6170 7365 6420 2020 2020 38s elapsed │ │ │ │ +00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3436 3034 |.| -- .04604 │ │ │ │ +00002ec0: 3535 7320 656c 6170 7365 6420 2020 2020 55s elapsed │ │ │ │ 00002ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00002f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -1596,15 +1596,15 @@ │ │ │ │ 000063b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000063c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000063d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000063e0: 7c69 3135 203a 2065 6c61 7073 6564 5469 |i15 : elapsedTi │ │ │ │ 000063f0: 6d65 2062 6574 7469 2849 273d 7472 696d me betti(I'=trim │ │ │ │ 00006400: 206b 6572 2070 6869 2920 2020 2020 2020 ker phi) │ │ │ │ 00006410: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00006420: 2e36 3039 3138 3573 2065 6c61 7073 6564 .609185s elapsed │ │ │ │ +00006420: 2e35 3839 3331 3373 2065 6c61 7073 6564 .589313s elapsed │ │ │ │ 00006430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006450: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00006460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006480: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00006490: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ @@ -1651,15 +1651,15 @@ │ │ │ │ 00006720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006750: 2d2d 2d2b 0a7c 6931 3720 3a20 656c 6170 ---+.|i17 : elap │ │ │ │ 00006760: 7365 6454 696d 6520 6261 7365 5074 733d sedTime basePts= │ │ │ │ 00006770: 7072 696d 6172 7944 6563 6f6d 706f 7369 primaryDecomposi │ │ │ │ 00006780: 7469 6f6e 2069 6465 616c 2048 3b20 7c0a tion ideal H; |. │ │ │ │ -00006790: 7c20 2d2d 2036 2e39 3831 3434 7320 656c | -- 6.98144s el │ │ │ │ +00006790: 7c20 2d2d 2035 2e39 3836 3434 7320 656c | -- 5.98644s 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: 3230 3035 3236 7320 656c 6170 7365 6420 200526s elapsed │ │ │ │ +0000a360: 3132 3539 3733 7320 656c 6170 7365 6420 125973s elapsed │ │ │ │ 0000a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a390: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000a3d0: 6f31 3420 3d20 6964 6561 6c20 2830 2c20 o14 = ideal (0, │ │ │ │ @@ -2648,15 +2648,15 @@ │ │ │ │ 0000a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a5a0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 656c -----+.|i16 : el │ │ │ │ 0000a5b0: 6170 7365 6454 696d 6520 6265 7474 6928 apsedTime betti( │ │ │ │ 0000a5c0: 4927 3d74 7269 6d20 6b65 7220 7068 6929 I'=trim ker phi) │ │ │ │ 0000a5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a5e0: 7c0a 7c20 2d2d 202e 3035 3635 3536 3673 |.| -- .0565566s │ │ │ │ +0000a5e0: 7c0a 7c20 2d2d 202e 3036 3132 3836 3973 |.| -- .0612869s │ │ │ │ 0000a5f0: 2065 6c61 7073 6564 2020 2020 2020 2020 elapsed │ │ │ │ 0000a600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a610: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a650: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ @@ -2700,15 +2700,15 @@ │ │ │ │ 0000a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000a8e0: 0a7c 6931 3820 3a20 656c 6170 7365 6454 .|i18 : elapsedT │ │ │ │ 0000a8f0: 696d 6520 6261 7365 5074 733d 7072 696d ime basePts=prim │ │ │ │ 0000a900: 6172 7944 6563 6f6d 706f 7369 7469 6f6e aryDecomposition │ │ │ │ 0000a910: 2069 6465 616c 2048 3b20 7c0a 7c20 2d2d ideal H; |.| -- │ │ │ │ -0000a920: 2031 2e39 3630 3431 7320 656c 6170 7365 1.96041s elapse │ │ │ │ +0000a920: 2031 2e37 3333 3037 7320 656c 6170 7365 1.73307s 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 │ │ │ │ @@ -4277,1075 +4277,1073 @@ │ │ │ │ 00010b40: 696f 6e73 206f 6620 7468 6520 7072 6f6a ions of the proj │ │ │ │ 00010b50: 6563 7469 7665 2073 7061 6365 7320 6672 ective spaces fr │ │ │ │ 00010b60: 6f6d 2077 686f 7365 0a70 726f 6475 6374 om whose.product │ │ │ │ 00010b70: 2077 6520 6172 6520 7072 6f6a 6563 7469 we are projecti │ │ │ │ 00010b80: 6e67 2e29 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ng.)..+--------- │ │ │ │ 00010b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00010bc0: 3132 203a 2041 203d 206b 6b5b 612c 625d 12 : A = kk[a,b] │ │ │ │ +00010bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00010bc0: 3220 3a20 4120 3d20 6b6b 5b61 2c62 5d20 2 : A = kk[a,b] │ │ │ │ 00010bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010bf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00010bf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00010c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010c20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00010c30: 3132 203d 2041 2020 2020 2020 2020 2020 12 = A │ │ │ │ +00010c20: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ +00010c30: 3d20 4120 2020 2020 2020 2020 2020 2020 = A │ │ │ │ 00010c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010c60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00010c60: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00010c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010c90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00010ca0: 3132 203a 2050 6f6c 796e 6f6d 6961 6c52 12 : PolynomialR │ │ │ │ -00010cb0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -00010cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010cd0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00010c90: 2020 2020 2020 207c 0a7c 6f31 3220 3a20 |.|o12 : │ │ │ │ +00010ca0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +00010cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010cc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00010cd0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00010ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00010d10: 3133 203a 204d 203d 2072 616e 646f 6d28 13 : M = random( │ │ │ │ -00010d20: 415e 342c 2041 5e7b 343a 2d31 7d29 2020 A^4, A^{4:-1}) │ │ │ │ -00010d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010d40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00010d00: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 4d20 -----+.|i13 : M │ │ │ │ +00010d10: 3d20 7261 6e64 6f6d 2841 5e34 2c20 415e = random(A^4, A^ │ │ │ │ +00010d20: 7b34 3a2d 317d 2920 2020 2020 2020 2020 {4:-1}) │ │ │ │ +00010d30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00010d40: 2020 2020 2020 2020 2020 2020 2020 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 7c0a 7c6f |.|o │ │ │ │ -00010d80: 3133 203d 207c 2032 3461 2d33 3662 2020 13 = | 24a-36b │ │ │ │ -00010d90: 2d38 612d 3232 6220 2033 3461 2b31 3962 -8a-22b 34a+19b │ │ │ │ -00010da0: 2020 2d32 3861 2d34 3762 207c 2020 2020 -28a-47b | │ │ │ │ -00010db0: 2020 2020 7c0a 7c20 2020 2020 207c 202d |.| | - │ │ │ │ -00010dc0: 3330 612d 3239 6220 2d32 3961 2d32 3462 30a-29b -29a-24b │ │ │ │ -00010dd0: 202d 3437 612d 3339 6220 3338 612b 3262 -47a-39b 38a+2b │ │ │ │ -00010de0: 2020 207c 2020 2020 2020 2020 7c0a 7c20 | |.| │ │ │ │ -00010df0: 2020 2020 207c 2031 3961 2b31 3962 2020 | 19a+19b │ │ │ │ -00010e00: 2d33 3861 2d31 3662 202d 3138 612d 3133 -38a-16b -18a-13 │ │ │ │ -00010e10: 6220 3136 612b 3232 6220 207c 2020 2020 b 16a+22b | │ │ │ │ -00010e20: 2020 2020 7c0a 7c20 2020 2020 207c 202d |.| | - │ │ │ │ -00010e30: 3130 612d 3239 6220 3339 612b 3231 6220 10a-29b 39a+21b │ │ │ │ -00010e40: 202d 3433 612d 3135 6220 3435 612d 3334 -43a-15b 45a-34 │ │ │ │ -00010e50: 6220 207c 2020 2020 2020 2020 7c0a 7c20 b | |.| │ │ │ │ +00010d70: 2020 207c 0a7c 6f31 3320 3d20 7c20 3234 |.|o13 = | 24 │ │ │ │ +00010d80: 612d 3336 6220 202d 3861 2d32 3262 2020 a-36b -8a-22b │ │ │ │ +00010d90: 3334 612b 3139 6220 202d 3238 612d 3437 34a+19b -28a-47 │ │ │ │ +00010da0: 6220 7c20 2020 2020 2020 7c0a 7c20 2020 b | |.| │ │ │ │ +00010db0: 2020 207c 202d 3330 612d 3239 6220 2d32 | -30a-29b -2 │ │ │ │ +00010dc0: 3961 2d32 3462 202d 3437 612d 3339 6220 9a-24b -47a-39b │ │ │ │ +00010dd0: 3338 612b 3262 2020 207c 2020 2020 2020 38a+2b | │ │ │ │ +00010de0: 207c 0a7c 2020 2020 2020 7c20 3139 612b |.| | 19a+ │ │ │ │ +00010df0: 3139 6220 202d 3338 612d 3136 6220 2d31 19b -38a-16b -1 │ │ │ │ +00010e00: 3861 2d31 3362 2031 3661 2b32 3262 2020 8a-13b 16a+22b │ │ │ │ +00010e10: 7c20 2020 2020 2020 7c0a 7c20 2020 2020 | |.| │ │ │ │ +00010e20: 207c 202d 3130 612d 3239 6220 3339 612b | -10a-29b 39a+ │ │ │ │ +00010e30: 3231 6220 202d 3433 612d 3135 6220 3435 21b -43a-15b 45 │ │ │ │ +00010e40: 612d 3334 6220 207c 2020 2020 2020 207c a-34b | | │ │ │ │ +00010e50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00010e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010e90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00010ea0: 2020 2020 2034 2020 2020 2020 3420 2020 4 4 │ │ │ │ -00010eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010ec0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00010ed0: 3133 203a 204d 6174 7269 7820 4120 203c 13 : Matrix A < │ │ │ │ -00010ee0: 2d2d 2041 2020 2020 2020 2020 2020 2020 -- A │ │ │ │ -00010ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010f00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00010e80: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00010e90: 2020 2020 2020 2034 2020 2020 2020 3420 4 4 │ │ │ │ +00010ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010eb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00010ec0: 6f31 3320 3a20 4d61 7472 6978 2041 2020 o13 : Matrix A │ │ │ │ +00010ed0: 3c2d 2d20 4120 2020 2020 2020 2020 2020 <-- A │ │ │ │ +00010ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00010f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00010f40: 3134 203a 2074 696d 6520 6265 7474 6920 14 : time betti │ │ │ │ -00010f50: 2846 203d 2070 7572 6552 6573 6f6c 7574 (F = pureResolut │ │ │ │ -00010f60: 696f 6e28 4d2c 7b30 2c32 2c34 7d29 2920 ion(M,{0,2,4})) │ │ │ │ -00010f70: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00010f80: 302e 3435 3030 3137 7320 2863 7075 293b 0.450017s (cpu); │ │ │ │ -00010f90: 2030 2e33 3736 3836 3373 2028 7468 7265 0.376863s (thre │ │ │ │ -00010fa0: 6164 293b 2030 7320 2867 6329 7c0a 7c20 ad); 0s (gc)|.| │ │ │ │ +00010f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00010f30: 3420 3a20 7469 6d65 2062 6574 7469 2028 4 : time betti ( │ │ │ │ +00010f40: 4620 3d20 7075 7265 5265 736f 6c75 7469 F = pureResoluti │ │ │ │ +00010f50: 6f6e 284d 2c7b 302c 322c 347d 2929 2020 on(M,{0,2,4})) │ │ │ │ +00010f60: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ +00010f70: 3539 3234 3638 7320 2863 7075 293b 2030 592468s (cpu); 0 │ │ │ │ +00010f80: 2e34 3136 3831 7320 2874 6872 6561 6429 .41681s (thread) │ │ │ │ +00010f90: 3b20 3073 2028 6763 297c 0a7c 2020 2020 ; 0s (gc)|.| │ │ │ │ +00010fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 │ │ │ │ -00011010: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00011020: 3134 203d 2074 6f74 616c 3a20 3320 3620 14 = total: 3 6 │ │ │ │ -00011030: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -00011040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011050: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00011060: 2030 3a20 3320 2e20 2e20 2020 2020 2020 0: 3 . . │ │ │ │ -00011070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011080: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00011090: 2020 2020 2020 2020 2031 3a20 2e20 3620 1: . 6 │ │ │ │ -000110a0: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -000110b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000110c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000110d0: 2032 3a20 2e20 2e20 3320 2020 2020 2020 2: . . 3 │ │ │ │ -000110e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000110f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00010fd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00010fe0: 3020 3120 3220 2020 2020 2020 2020 2020 0 1 2 │ │ │ │ +00010ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011000: 2020 2020 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ +00011010: 746f 7461 6c3a 2033 2036 2033 2020 2020 total: 3 6 3 │ │ │ │ +00011020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00011040: 7c20 2020 2020 2020 2020 2030 3a20 3320 | 0: 3 │ │ │ │ +00011050: 2e20 2e20 2020 2020 2020 2020 2020 2020 . . │ │ │ │ +00011060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011070: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00011080: 2020 313a 202e 2036 202e 2020 2020 2020 1: . 6 . │ │ │ │ +00011090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000110a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000110b0: 2020 2020 2020 2020 2032 3a20 2e20 2e20 2: . . │ │ │ │ +000110c0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +000110d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000110e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000110f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011130: 2020 2020 7c0a 7c6f 3134 203a 2042 6574 |.|o14 : Bet │ │ │ │ -00011140: 7469 5461 6c6c 7920 2020 2020 2020 2020 tiTally │ │ │ │ -00011150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011160: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00011110: 2020 2020 2020 2020 2020 7c0a 7c6f 3134 |.|o14 │ │ │ │ +00011120: 203a 2042 6574 7469 5461 6c6c 7920 2020 : BettiTally │ │ │ │ +00011130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011150: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000111a0: 2d2d 2d2d 2b0a 0a57 6974 6820 7468 6520 ----+..With the │ │ │ │ -000111b0: 666f 726d 2070 7572 6552 6573 6f6c 7574 form pureResolut │ │ │ │ -000111c0: 696f 6e28 702c 712c 4429 2077 6520 6361 ion(p,q,D) we ca │ │ │ │ -000111d0: 6e20 6469 7265 6374 6c79 2063 7265 6174 n directly creat │ │ │ │ -000111e0: 6520 7468 6520 7369 7475 6174 696f 6e20 e the situation │ │ │ │ -000111f0: 6f66 0a70 7572 6552 6573 6f6c 7574 696f of.pureResolutio │ │ │ │ -00011200: 6e28 4d2c 4429 2077 6865 7265 204d 2069 n(M,D) where M i │ │ │ │ -00011210: 7320 6765 6e65 7269 6320 7072 6f64 7563 s generic produc │ │ │ │ -00011220: 7428 6d5f 692b 3129 2078 2023 442d 312b t(m_i+1) x #D-1+ │ │ │ │ -00011230: 7375 6d28 6d5f 6929 206d 6174 7269 7820 sum(m_i) matrix │ │ │ │ -00011240: 6f66 0a6c 696e 6561 7220 666f 726d 7320 of.linear forms │ │ │ │ -00011250: 6465 6669 6e65 6420 6f76 6572 2061 2072 defined over a r │ │ │ │ -00011260: 696e 6720 7769 7468 2070 726f 6475 6374 ing with product │ │ │ │ -00011270: 286d 5f69 2b31 2920 2a20 2344 2d31 2b73 (m_i+1) * #D-1+s │ │ │ │ -00011280: 756d 286d 5f69 2920 7661 7269 6162 6c65 um(m_i) variable │ │ │ │ -00011290: 730a 6f66 2063 6861 7261 6374 6572 6973 s.of characteris │ │ │ │ -000112a0: 7469 6320 702c 2063 7265 6174 6564 2062 tic p, created b │ │ │ │ -000112b0: 7920 7468 6520 7363 7269 7074 2e20 466f y the script. Fo │ │ │ │ -000112c0: 7220 6120 6769 7665 6e20 6e75 6d62 6572 r a given number │ │ │ │ -000112d0: 206f 6620 7661 7269 6162 6c65 7320 696e of variables in │ │ │ │ -000112e0: 0a41 2074 6869 7320 7275 6e73 206d 7563 .A this runs muc │ │ │ │ -000112f0: 6820 6661 7374 6572 2074 6861 6e20 7461 h faster than ta │ │ │ │ -00011300: 6b69 6e67 2061 2072 616e 646f 6d20 6d61 king a random ma │ │ │ │ -00011310: 7472 6978 204d 2e0a 0a2b 2d2d 2d2d 2d2d trix M...+------ │ │ │ │ +00011180: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6974 6820 --------+..With │ │ │ │ +00011190: 7468 6520 666f 726d 2070 7572 6552 6573 the form pureRes │ │ │ │ +000111a0: 6f6c 7574 696f 6e28 702c 712c 4429 2077 olution(p,q,D) w │ │ │ │ +000111b0: 6520 6361 6e20 6469 7265 6374 6c79 2063 e can directly c │ │ │ │ +000111c0: 7265 6174 6520 7468 6520 7369 7475 6174 reate the situat │ │ │ │ +000111d0: 696f 6e20 6f66 0a70 7572 6552 6573 6f6c ion of.pureResol │ │ │ │ +000111e0: 7574 696f 6e28 4d2c 4429 2077 6865 7265 ution(M,D) where │ │ │ │ +000111f0: 204d 2069 7320 6765 6e65 7269 6320 7072 M is generic pr │ │ │ │ +00011200: 6f64 7563 7428 6d5f 692b 3129 2078 2023 oduct(m_i+1) x # │ │ │ │ +00011210: 442d 312b 7375 6d28 6d5f 6929 206d 6174 D-1+sum(m_i) mat │ │ │ │ +00011220: 7269 7820 6f66 0a6c 696e 6561 7220 666f rix of.linear fo │ │ │ │ +00011230: 726d 7320 6465 6669 6e65 6420 6f76 6572 rms defined over │ │ │ │ +00011240: 2061 2072 696e 6720 7769 7468 2070 726f a ring with pro │ │ │ │ +00011250: 6475 6374 286d 5f69 2b31 2920 2a20 2344 duct(m_i+1) * #D │ │ │ │ +00011260: 2d31 2b73 756d 286d 5f69 2920 7661 7269 -1+sum(m_i) vari │ │ │ │ +00011270: 6162 6c65 730a 6f66 2063 6861 7261 6374 ables.of charact │ │ │ │ +00011280: 6572 6973 7469 6320 702c 2063 7265 6174 eristic p, creat │ │ │ │ +00011290: 6564 2062 7920 7468 6520 7363 7269 7074 ed by the script │ │ │ │ +000112a0: 2e20 466f 7220 6120 6769 7665 6e20 6e75 . For a given nu │ │ │ │ +000112b0: 6d62 6572 206f 6620 7661 7269 6162 6c65 mber of variable │ │ │ │ +000112c0: 7320 696e 0a41 2074 6869 7320 7275 6e73 s in.A this runs │ │ │ │ +000112d0: 206d 7563 6820 6661 7374 6572 2074 6861 much faster tha │ │ │ │ +000112e0: 6e20 7461 6b69 6e67 2061 2072 616e 646f n taking a rando │ │ │ │ +000112f0: 6d20 6d61 7472 6978 204d 2e0a 0a2b 2d2d m matrix M...+-- │ │ │ │ +00011300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00011350: 0a7c 6931 3520 3a20 7469 6d65 2062 6574 .|i15 : time bet │ │ │ │ -00011360: 7469 2028 4620 3d20 7075 7265 5265 736f ti (F = pureReso │ │ │ │ -00011370: 6c75 7469 6f6e 2831 312c 342c 7b30 2c32 lution(11,4,{0,2 │ │ │ │ -00011380: 2c34 7d29 2920 207c 0a7c 202d 2d20 7573 ,4})) |.| -- us │ │ │ │ -00011390: 6564 2030 2e34 3638 3335 3173 2028 6370 ed 0.468351s (cp │ │ │ │ -000113a0: 7529 3b20 302e 3430 3337 3332 7320 2874 u); 0.403732s (t │ │ │ │ -000113b0: 6872 6561 6429 3b20 3073 2028 6763 297c hread); 0s (gc)| │ │ │ │ -000113c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -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 │ │ │ │ -00011420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011430: 0a7c 6f31 3520 3d20 746f 7461 6c3a 2033 .|o15 = total: 3 │ │ │ │ -00011440: 2036 2033 2020 2020 2020 2020 2020 2020 6 3 │ │ │ │ -00011450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00011470: 2020 2020 303a 2033 202e 202e 2020 2020 0: 3 . . │ │ │ │ -00011480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000114a0: 0a7c 2020 2020 2020 2020 2020 313a 202e .| 1: . │ │ │ │ -000114b0: 2036 202e 2020 2020 2020 2020 2020 2020 6 . │ │ │ │ -000114c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000114d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000114e0: 2020 2020 323a 202e 202e 2033 2020 2020 2: . . 3 │ │ │ │ -000114f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011500: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011510: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00011520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011540: 2020 2020 2020 207c 0a7c 6f31 3520 3a20 |.|o15 : │ │ │ │ -00011550: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ -00011560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011570: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00011590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000115a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000115b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 -------+.|i16 : │ │ │ │ -000115c0: 7269 6e67 2046 2020 2020 2020 2020 2020 ring F │ │ │ │ -000115d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000115e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000115f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00011600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011620: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00011630: 5a5a 2020 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ -00011640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011650: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011660: 0a7c 6f31 3620 3d20 2d2d 5b61 202e 2e61 .|o16 = --[a ..a │ │ │ │ -00011670: 2020 5d20 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00011680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011690: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000116a0: 3131 2020 3020 2020 3135 2020 2020 2020 11 0 15 │ │ │ │ -000116b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000116c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000116d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000116e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000116f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011700: 2020 2020 2020 207c 0a7c 6f31 3620 3a20 |.|o16 : │ │ │ │ -00011710: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ -00011720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011740: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00011750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011770: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -00011780: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00011790: 202a 6e6f 7465 2064 6972 6563 7449 6d61 *note directIma │ │ │ │ -000117a0: 6765 436f 6d70 6c65 783a 2064 6972 6563 geComplex: direc │ │ │ │ -000117b0: 7449 6d61 6765 436f 6d70 6c65 782c 202d tImageComplex, - │ │ │ │ -000117c0: 2d20 6469 7265 6374 2069 6d61 6765 2063 - direct image c │ │ │ │ -000117d0: 6f6d 706c 6578 0a20 202a 202a 6e6f 7465 omplex. * *note │ │ │ │ -000117e0: 2075 6e69 7665 7273 616c 4578 7465 6e73 universalExtens │ │ │ │ -000117f0: 696f 6e3a 2075 6e69 7665 7273 616c 4578 ion: universalEx │ │ │ │ -00011800: 7465 6e73 696f 6e2c 202d 2d20 556e 6976 tension, -- Univ │ │ │ │ -00011810: 6572 7361 6c20 6578 7465 6e73 696f 6e20 ersal extension │ │ │ │ -00011820: 6f66 0a20 2020 2076 6563 746f 7220 6275 of. vector bu │ │ │ │ -00011830: 6e64 6c65 7320 6f6e 2050 5e31 0a0a 5761 ndles on P^1..Wa │ │ │ │ -00011840: 7973 2074 6f20 7573 6520 7075 7265 5265 ys to use pureRe │ │ │ │ -00011850: 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d 3d3d solution:.====== │ │ │ │ -00011860: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011870: 3d3d 3d3d 3d0a 0a20 202a 2022 7075 7265 =====.. * "pure │ │ │ │ -00011880: 5265 736f 6c75 7469 6f6e 284d 6174 7269 Resolution(Matri │ │ │ │ -00011890: 782c 4c69 7374 2922 0a20 202a 2022 7075 x,List)". * "pu │ │ │ │ -000118a0: 7265 5265 736f 6c75 7469 6f6e 2852 696e reResolution(Rin │ │ │ │ -000118b0: 672c 4c69 7374 2922 0a20 202a 2022 7075 g,List)". * "pu │ │ │ │ -000118c0: 7265 5265 736f 6c75 7469 6f6e 285a 5a2c reResolution(ZZ, │ │ │ │ -000118d0: 4c69 7374 2922 0a20 202a 2022 7075 7265 List)". * "pure │ │ │ │ -000118e0: 5265 736f 6c75 7469 6f6e 285a 5a2c 5a5a Resolution(ZZ,ZZ │ │ │ │ -000118f0: 2c4c 6973 7429 220a 0a46 6f72 2074 6865 ,List)"..For the │ │ │ │ -00011900: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00011910: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00011920: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00011930: 2070 7572 6552 6573 6f6c 7574 696f 6e3a pureResolution: │ │ │ │ -00011940: 2070 7572 6552 6573 6f6c 7574 696f 6e2c pureResolution, │ │ │ │ -00011950: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ -00011960: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61 od function:.(Ma │ │ │ │ -00011970: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -00011980: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ +00011330: 2d2d 2d2b 0a7c 6931 3520 3a20 7469 6d65 ---+.|i15 : time │ │ │ │ +00011340: 2062 6574 7469 2028 4620 3d20 7075 7265 betti (F = pure │ │ │ │ +00011350: 5265 736f 6c75 7469 6f6e 2831 312c 342c Resolution(11,4, │ │ │ │ +00011360: 7b30 2c32 2c34 7d29 2920 207c 0a7c 202d {0,2,4})) |.| - │ │ │ │ +00011370: 2d20 7573 6564 2030 2e34 3837 3737 3873 - used 0.487778s │ │ │ │ +00011380: 2028 6370 7529 3b20 302e 3339 3839 3138 (cpu); 0.398918 │ │ │ │ +00011390: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +000113a0: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ +000113b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000113c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000113d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000113e0: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +000113f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011410: 2020 207c 0a7c 6f31 3520 3d20 746f 7461 |.|o15 = tota │ │ │ │ +00011420: 6c3a 2033 2036 2033 2020 2020 2020 2020 l: 3 6 3 │ │ │ │ +00011430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011440: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00011450: 2020 2020 2020 2020 303a 2033 202e 202e 0: 3 . . │ │ │ │ +00011460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011480: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00011490: 313a 202e 2036 202e 2020 2020 2020 2020 1: . 6 . │ │ │ │ +000114a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000114b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000114c0: 2020 2020 2020 2020 323a 202e 202e 2033 2: . . 3 │ │ │ │ +000114d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000114e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000114f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00011500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011520: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00011530: 3520 3a20 4265 7474 6954 616c 6c79 2020 5 : BettiTally │ │ │ │ +00011540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011560: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00011570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +000115a0: 3620 3a20 7269 6e67 2046 2020 2020 2020 6 : ring F │ │ │ │ +000115b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000115c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000115d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000115e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000115f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011600: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00011610: 2020 2020 5a5a 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00011620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011640: 2020 207c 0a7c 6f31 3620 3d20 2d2d 5b61 |.|o16 = --[a │ │ │ │ +00011650: 202e 2e61 2020 5d20 2020 2020 2020 2020 ..a ] │ │ │ │ +00011660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011670: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00011680: 2020 2020 3131 2020 3020 2020 3135 2020 11 0 15 │ │ │ │ +00011690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000116a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000116b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000116c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000116d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000116e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000116f0: 3620 3a20 506f 6c79 6e6f 6d69 616c 5269 6 : PolynomialRi │ │ │ │ +00011700: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ +00011710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011720: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00011730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 -----------+..Se │ │ │ │ +00011760: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +00011770: 0a20 202a 202a 6e6f 7465 2064 6972 6563 . * *note direc │ │ │ │ +00011780: 7449 6d61 6765 436f 6d70 6c65 783a 2064 tImageComplex: d │ │ │ │ +00011790: 6972 6563 7449 6d61 6765 436f 6d70 6c65 irectImageComple │ │ │ │ +000117a0: 782c 202d 2d20 6469 7265 6374 2069 6d61 x, -- direct ima │ │ │ │ +000117b0: 6765 2063 6f6d 706c 6578 0a20 202a 202a ge complex. * * │ │ │ │ +000117c0: 6e6f 7465 2075 6e69 7665 7273 616c 4578 note universalEx │ │ │ │ +000117d0: 7465 6e73 696f 6e3a 2075 6e69 7665 7273 tension: univers │ │ │ │ +000117e0: 616c 4578 7465 6e73 696f 6e2c 202d 2d20 alExtension, -- │ │ │ │ +000117f0: 556e 6976 6572 7361 6c20 6578 7465 6e73 Universal extens │ │ │ │ +00011800: 696f 6e20 6f66 0a20 2020 2076 6563 746f ion of. vecto │ │ │ │ +00011810: 7220 6275 6e64 6c65 7320 6f6e 2050 5e31 r bundles on P^1 │ │ │ │ +00011820: 0a0a 5761 7973 2074 6f20 7573 6520 7075 ..Ways to use pu │ │ │ │ +00011830: 7265 5265 736f 6c75 7469 6f6e 3a0a 3d3d reResolution:.== │ │ │ │ +00011840: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00011850: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +00011860: 7075 7265 5265 736f 6c75 7469 6f6e 284d pureResolution(M │ │ │ │ +00011870: 6174 7269 782c 4c69 7374 2922 0a20 202a atrix,List)". * │ │ │ │ +00011880: 2022 7075 7265 5265 736f 6c75 7469 6f6e "pureResolution │ │ │ │ +00011890: 2852 696e 672c 4c69 7374 2922 0a20 202a (Ring,List)". * │ │ │ │ +000118a0: 2022 7075 7265 5265 736f 6c75 7469 6f6e "pureResolution │ │ │ │ +000118b0: 285a 5a2c 4c69 7374 2922 0a20 202a 2022 (ZZ,List)". * " │ │ │ │ +000118c0: 7075 7265 5265 736f 6c75 7469 6f6e 285a pureResolution(Z │ │ │ │ +000118d0: 5a2c 5a5a 2c4c 6973 7429 220a 0a46 6f72 Z,ZZ,List)"..For │ │ │ │ +000118e0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +000118f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00011900: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00011910: 6e6f 7465 2070 7572 6552 6573 6f6c 7574 note pureResolut │ │ │ │ +00011920: 696f 6e3a 2070 7572 6552 6573 6f6c 7574 ion: pureResolut │ │ │ │ +00011930: 696f 6e2c 2069 7320 6120 2a6e 6f74 6520 ion, is a *note │ │ │ │ +00011940: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +00011950: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +00011960: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +00011970: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00011980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000119a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000119b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -000119e0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -000119f0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00011a00: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00011a10: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00011a20: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00011a30: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00011a40: 6765 732f 4247 472e 6d32 3a31 3232 360a ges/BGG.m2:1226. │ │ │ │ -00011a50: 3a30 2e0a 1f0a 4669 6c65 3a20 4247 472e :0....File: BGG. │ │ │ │ -00011a60: 696e 666f 2c20 4e6f 6465 3a20 5265 6775 info, Node: Regu │ │ │ │ -00011a70: 6c61 7269 7479 2c20 4e65 7874 3a20 7379 larity, Next: sy │ │ │ │ -00011a80: 6d45 7874 2c20 5072 6576 3a20 7075 7265 mExt, Prev: pure │ │ │ │ -00011a90: 5265 736f 6c75 7469 6f6e 2c20 5570 3a20 Resolution, Up: │ │ │ │ -00011aa0: 546f 700a 0a52 6567 756c 6172 6974 7920 Top..Regularity │ │ │ │ -00011ab0: 2d2d 204f 7074 696f 6e20 666f 7220 6469 -- Option for di │ │ │ │ -00011ac0: 7265 6374 496d 6167 6543 6f6d 706c 6578 rectImageComplex │ │ │ │ -00011ad0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -00011ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00011af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4361 ************..Ca │ │ │ │ -00011b00: 7665 6174 0a3d 3d3d 3d3d 3d0a 0a43 7572 veat.======..Cur │ │ │ │ -00011b10: 7265 6e74 6c79 206e 6f74 2073 7570 706f rently not suppo │ │ │ │ -00011b20: 7274 6564 0a0a 4675 6e63 7469 6f6e 7320 rted..Functions │ │ │ │ -00011b30: 7769 7468 206f 7074 696f 6e61 6c20 6172 with optional ar │ │ │ │ -00011b40: 6775 6d65 6e74 206e 616d 6564 2052 6567 gument named Reg │ │ │ │ -00011b50: 756c 6172 6974 793a 0a3d 3d3d 3d3d 3d3d ularity:.======= │ │ │ │ -00011b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011b80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00011b90: 2022 6469 7265 6374 496d 6167 6543 6f6d "directImageCom │ │ │ │ -00011ba0: 706c 6578 282e 2e2e 2c52 6567 756c 6172 plex(...,Regular │ │ │ │ -00011bb0: 6974 793d 3e2e 2e2e 2922 202d 2d20 7365 ity=>...)" -- se │ │ │ │ -00011bc0: 6520 2a6e 6f74 6520 6469 7265 6374 496d e *note directIm │ │ │ │ -00011bd0: 6167 6543 6f6d 706c 6578 3a0a 2020 2020 ageComplex:. │ │ │ │ -00011be0: 6469 7265 6374 496d 6167 6543 6f6d 706c directImageCompl │ │ │ │ -00011bf0: 6578 2c20 2d2d 2064 6972 6563 7420 696d ex, -- direct im │ │ │ │ -00011c00: 6167 6520 636f 6d70 6c65 780a 0a46 6f72 age complex..For │ │ │ │ -00011c10: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -00011c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011c30: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -00011c40: 6e6f 7465 2052 6567 756c 6172 6974 793a note Regularity: │ │ │ │ -00011c50: 2028 5265 6775 6c61 7269 7479 2954 6f70 (Regularity)Top │ │ │ │ -00011c60: 2c20 6973 2061 202a 6e6f 7465 2073 796d , is a *note sym │ │ │ │ -00011c70: 626f 6c3a 0a28 4d61 6361 756c 6179 3244 bol:.(Macaulay2D │ │ │ │ -00011c80: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d oc)Symbol,...--- │ │ │ │ +000119c0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +000119d0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000119e0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000119f0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00011a00: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +00011a10: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00011a20: 6163 6b61 6765 732f 4247 472e 6d32 3a31 ackages/BGG.m2:1 │ │ │ │ +00011a30: 3232 360a 3a30 2e0a 1f0a 4669 6c65 3a20 226.:0....File: │ │ │ │ +00011a40: 4247 472e 696e 666f 2c20 4e6f 6465 3a20 BGG.info, Node: │ │ │ │ +00011a50: 5265 6775 6c61 7269 7479 2c20 4e65 7874 Regularity, Next │ │ │ │ +00011a60: 3a20 7379 6d45 7874 2c20 5072 6576 3a20 : symExt, Prev: │ │ │ │ +00011a70: 7075 7265 5265 736f 6c75 7469 6f6e 2c20 pureResolution, │ │ │ │ +00011a80: 5570 3a20 546f 700a 0a52 6567 756c 6172 Up: Top..Regular │ │ │ │ +00011a90: 6974 7920 2d2d 204f 7074 696f 6e20 666f ity -- Option fo │ │ │ │ +00011aa0: 7220 6469 7265 6374 496d 6167 6543 6f6d r directImageCom │ │ │ │ +00011ab0: 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a plex.*********** │ │ │ │ +00011ac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00011ad0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00011ae0: 0a0a 4361 7665 6174 0a3d 3d3d 3d3d 3d0a ..Caveat.======. │ │ │ │ +00011af0: 0a43 7572 7265 6e74 6c79 206e 6f74 2073 .Currently not s │ │ │ │ +00011b00: 7570 706f 7274 6564 0a0a 4675 6e63 7469 upported..Functi │ │ │ │ +00011b10: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ +00011b20: 6c20 6172 6775 6d65 6e74 206e 616d 6564 l argument named │ │ │ │ +00011b30: 2052 6567 756c 6172 6974 793a 0a3d 3d3d Regularity:.=== │ │ │ │ +00011b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00011b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00011b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00011b70: 0a20 202a 2022 6469 7265 6374 496d 6167 . * "directImag │ │ │ │ +00011b80: 6543 6f6d 706c 6578 282e 2e2e 2c52 6567 eComplex(...,Reg │ │ │ │ +00011b90: 756c 6172 6974 793d 3e2e 2e2e 2922 202d ularity=>...)" - │ │ │ │ +00011ba0: 2d20 7365 6520 2a6e 6f74 6520 6469 7265 - see *note dire │ │ │ │ +00011bb0: 6374 496d 6167 6543 6f6d 706c 6578 3a0a ctImageComplex:. │ │ │ │ +00011bc0: 2020 2020 6469 7265 6374 496d 6167 6543 directImageC │ │ │ │ +00011bd0: 6f6d 706c 6578 2c20 2d2d 2064 6972 6563 omplex, -- direc │ │ │ │ +00011be0: 7420 696d 6167 6520 636f 6d70 6c65 780a t image complex. │ │ │ │ +00011bf0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +00011c00: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +00011c10: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +00011c20: 6374 202a 6e6f 7465 2052 6567 756c 6172 ct *note Regular │ │ │ │ +00011c30: 6974 793a 2028 5265 6775 6c61 7269 7479 ity: (Regularity │ │ │ │ +00011c40: 2954 6f70 2c20 6973 2061 202a 6e6f 7465 )Top, is a *note │ │ │ │ +00011c50: 2073 796d 626f 6c3a 0a28 4d61 6361 756c symbol:.(Macaul │ │ │ │ +00011c60: 6179 3244 6f63 2953 796d 626f 6c2c 2e0a ay2Doc)Symbol,.. │ │ │ │ +00011c70: 0a2d 2d2d 2d2d 2d2d 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00011ce0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00011cf0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00011d00: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00011d10: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00011d20: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00011d30: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00011d40: 6765 732f 4247 472e 6d32 3a38 3831 3a0a ges/BGG.m2:881:. │ │ │ │ -00011d50: 302e 0a1f 0a46 696c 653a 2042 4747 2e69 0....File: BGG.i │ │ │ │ -00011d60: 6e66 6f2c 204e 6f64 653a 2073 796d 4578 nfo, Node: symEx │ │ │ │ -00011d70: 742c 204e 6578 743a 2074 6174 6552 6573 t, Next: tateRes │ │ │ │ -00011d80: 6f6c 7574 696f 6e2c 2050 7265 763a 2052 olution, Prev: R │ │ │ │ -00011d90: 6567 756c 6172 6974 792c 2055 703a 2054 egularity, Up: T │ │ │ │ -00011da0: 6f70 0a0a 7379 6d45 7874 202d 2d20 7468 op..symExt -- th │ │ │ │ -00011db0: 6520 6669 7273 7420 6469 6666 6572 656e e first differen │ │ │ │ -00011dc0: 7469 616c 206f 6620 7468 6520 636f 6d70 tial of the comp │ │ │ │ -00011dd0: 6c65 7820 5228 4d29 0a2a 2a2a 2a2a 2a2a lex R(M).******* │ │ │ │ +00011cc0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00011cd0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +00011ce0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +00011cf0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00011d00: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +00011d10: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00011d20: 6163 6b61 6765 732f 4247 472e 6d32 3a38 ackages/BGG.m2:8 │ │ │ │ +00011d30: 3831 3a0a 302e 0a1f 0a46 696c 653a 2042 81:.0....File: B │ │ │ │ +00011d40: 4747 2e69 6e66 6f2c 204e 6f64 653a 2073 GG.info, Node: s │ │ │ │ +00011d50: 796d 4578 742c 204e 6578 743a 2074 6174 ymExt, Next: tat │ │ │ │ +00011d60: 6552 6573 6f6c 7574 696f 6e2c 2050 7265 eResolution, Pre │ │ │ │ +00011d70: 763a 2052 6567 756c 6172 6974 792c 2055 v: Regularity, U │ │ │ │ +00011d80: 703a 2054 6f70 0a0a 7379 6d45 7874 202d p: Top..symExt - │ │ │ │ +00011d90: 2d20 7468 6520 6669 7273 7420 6469 6666 - the first diff │ │ │ │ +00011da0: 6572 656e 7469 616c 206f 6620 7468 6520 erential of the │ │ │ │ +00011db0: 636f 6d70 6c65 7820 5228 4d29 0a2a 2a2a complex R(M).*** │ │ │ │ +00011dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00011dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00011de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00011df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00011e00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -00011e10: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -00011e20: 2020 2073 796d 4578 7428 6d2c 4529 0a20 symExt(m,E). │ │ │ │ -00011e30: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -00011e40: 202a 206d 2c20 6120 2a6e 6f74 6520 6d61 * m, a *note ma │ │ │ │ -00011e50: 7472 6978 3a20 284d 6163 6175 6c61 7932 trix: (Macaulay2 │ │ │ │ -00011e60: 446f 6329 4d61 7472 6978 2c2c 2061 2070 Doc)Matrix,, a p │ │ │ │ -00011e70: 7265 7365 6e74 6174 696f 6e20 6d61 7472 resentation matr │ │ │ │ -00011e80: 6978 2066 6f72 2061 0a20 2020 2020 2020 ix for a. │ │ │ │ -00011e90: 2070 6f73 6974 6976 656c 7920 6772 6164 positively grad │ │ │ │ -00011ea0: 6564 206d 6f64 756c 6520 4d20 6f76 6572 ed module M over │ │ │ │ -00011eb0: 2061 2070 6f6c 796e 6f6d 6961 6c20 7269 a polynomial ri │ │ │ │ -00011ec0: 6e67 0a20 2020 2020 202a 2045 2c20 6120 ng. * E, a │ │ │ │ -00011ed0: 2a6e 6f74 6520 706f 6c79 6e6f 6d69 616c *note polynomial │ │ │ │ -00011ee0: 2072 696e 673a 2028 4d61 6361 756c 6179 ring: (Macaulay │ │ │ │ -00011ef0: 3244 6f63 2950 6f6c 796e 6f6d 6961 6c52 2Doc)PolynomialR │ │ │ │ -00011f00: 696e 672c 2c20 6578 7465 7269 6f72 0a20 ing,, exterior. │ │ │ │ -00011f10: 2020 2020 2020 2061 6c67 6562 7261 0a20 algebra. │ │ │ │ -00011f20: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -00011f30: 2020 2a20 6120 2a6e 6f74 6520 6d61 7472 * a *note matr │ │ │ │ -00011f40: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -00011f50: 6329 4d61 7472 6978 2c2c 2061 206d 6174 c)Matrix,, a mat │ │ │ │ -00011f60: 7269 7820 7265 7072 6573 656e 7469 6e67 rix representing │ │ │ │ -00011f70: 2074 6865 206d 6170 0a20 2020 2020 2020 the map. │ │ │ │ -00011f80: 204d 5f31 202a 2a20 6f6d 6567 615f 4520 M_1 ** omega_E │ │ │ │ -00011f90: 3c2d 2d20 4d5f 3020 2a2a 206f 6d65 6761 <-- M_0 ** omega │ │ │ │ -00011fa0: 5f45 0a0a 4465 7363 7269 7074 696f 6e0a _E..Description. │ │ │ │ -00011fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -00011fc0: 7320 6675 6e63 7469 6f6e 2074 616b 6573 s function takes │ │ │ │ -00011fd0: 2061 7320 696e 7075 7420 6120 6d61 7472 as input a matr │ │ │ │ -00011fe0: 6978 206d 2077 6974 6820 6c69 6e65 6172 ix m with linear │ │ │ │ -00011ff0: 2065 6e74 7269 6573 2c20 7768 6963 6820 entries, which │ │ │ │ -00012000: 7765 2074 6869 6e6b 206f 660a 6173 2061 we think of.as a │ │ │ │ -00012010: 2070 7265 7365 6e74 6174 696f 6e20 6d61 presentation ma │ │ │ │ -00012020: 7472 6978 2066 6f72 2061 2070 6f73 6974 trix for a posit │ │ │ │ -00012030: 6976 656c 7920 6772 6164 6564 2053 2d6d ively graded S-m │ │ │ │ -00012040: 6f64 756c 6520 4d20 6d61 7472 6978 2072 odule M matrix r │ │ │ │ -00012050: 6570 7265 7365 6e74 696e 670a 7468 6520 epresenting.the │ │ │ │ -00012060: 6d61 7020 4d5f 3120 2a2a 206f 6d65 6761 map M_1 ** omega │ │ │ │ -00012070: 5f45 203c 2d2d 204d 5f30 202a 2a20 6f6d _E <-- M_0 ** om │ │ │ │ -00012080: 6567 615f 4520 7768 6963 6820 6973 2074 ega_E which is t │ │ │ │ -00012090: 6865 2066 6972 7374 2064 6966 6665 7265 he first differe │ │ │ │ -000120a0: 6e74 6961 6c20 6f66 0a74 6865 2063 6f6d ntial of.the com │ │ │ │ -000120b0: 706c 6578 2052 284d 292e 0a2b 2d2d 2d2d plex R(M)..+---- │ │ │ │ +00011df0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +00011e00: 2020 2020 2020 2073 796d 4578 7428 6d2c symExt(m, │ │ │ │ +00011e10: 4529 0a20 202a 2049 6e70 7574 733a 0a20 E). * Inputs:. │ │ │ │ +00011e20: 2020 2020 202a 206d 2c20 6120 2a6e 6f74 * m, a *not │ │ │ │ +00011e30: 6520 6d61 7472 6978 3a20 284d 6163 6175 e matrix: (Macau │ │ │ │ +00011e40: 6c61 7932 446f 6329 4d61 7472 6978 2c2c lay2Doc)Matrix,, │ │ │ │ +00011e50: 2061 2070 7265 7365 6e74 6174 696f 6e20 a presentation │ │ │ │ +00011e60: 6d61 7472 6978 2066 6f72 2061 0a20 2020 matrix for a. │ │ │ │ +00011e70: 2020 2020 2070 6f73 6974 6976 656c 7920 positively │ │ │ │ +00011e80: 6772 6164 6564 206d 6f64 756c 6520 4d20 graded module M │ │ │ │ +00011e90: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961 over a polynomia │ │ │ │ +00011ea0: 6c20 7269 6e67 0a20 2020 2020 202a 2045 l ring. * E │ │ │ │ +00011eb0: 2c20 6120 2a6e 6f74 6520 706f 6c79 6e6f , a *note polyno │ │ │ │ +00011ec0: 6d69 616c 2072 696e 673a 2028 4d61 6361 mial ring: (Maca │ │ │ │ +00011ed0: 756c 6179 3244 6f63 2950 6f6c 796e 6f6d ulay2Doc)Polynom │ │ │ │ +00011ee0: 6961 6c52 696e 672c 2c20 6578 7465 7269 ialRing,, exteri │ │ │ │ +00011ef0: 6f72 0a20 2020 2020 2020 2061 6c67 6562 or. algeb │ │ │ │ +00011f00: 7261 0a20 202a 204f 7574 7075 7473 3a0a ra. * Outputs:. │ │ │ │ +00011f10: 2020 2020 2020 2a20 6120 2a6e 6f74 6520 * a *note │ │ │ │ +00011f20: 6d61 7472 6978 3a20 284d 6163 6175 6c61 matrix: (Macaula │ │ │ │ +00011f30: 7932 446f 6329 4d61 7472 6978 2c2c 2061 y2Doc)Matrix,, a │ │ │ │ +00011f40: 206d 6174 7269 7820 7265 7072 6573 656e matrix represen │ │ │ │ +00011f50: 7469 6e67 2074 6865 206d 6170 0a20 2020 ting the map. │ │ │ │ +00011f60: 2020 2020 204d 5f31 202a 2a20 6f6d 6567 M_1 ** omeg │ │ │ │ +00011f70: 615f 4520 3c2d 2d20 4d5f 3020 2a2a 206f a_E <-- M_0 ** o │ │ │ │ +00011f80: 6d65 6761 5f45 0a0a 4465 7363 7269 7074 mega_E..Descript │ │ │ │ +00011f90: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +00011fa0: 0a54 6869 7320 6675 6e63 7469 6f6e 2074 .This function t │ │ │ │ +00011fb0: 616b 6573 2061 7320 696e 7075 7420 6120 akes as input a │ │ │ │ +00011fc0: 6d61 7472 6978 206d 2077 6974 6820 6c69 matrix m with li │ │ │ │ +00011fd0: 6e65 6172 2065 6e74 7269 6573 2c20 7768 near entries, wh │ │ │ │ +00011fe0: 6963 6820 7765 2074 6869 6e6b 206f 660a ich we think of. │ │ │ │ +00011ff0: 6173 2061 2070 7265 7365 6e74 6174 696f as a presentatio │ │ │ │ +00012000: 6e20 6d61 7472 6978 2066 6f72 2061 2070 n matrix for a p │ │ │ │ +00012010: 6f73 6974 6976 656c 7920 6772 6164 6564 ositively graded │ │ │ │ +00012020: 2053 2d6d 6f64 756c 6520 4d20 6d61 7472 S-module M matr │ │ │ │ +00012030: 6978 2072 6570 7265 7365 6e74 696e 670a ix representing. │ │ │ │ +00012040: 7468 6520 6d61 7020 4d5f 3120 2a2a 206f the map M_1 ** o │ │ │ │ +00012050: 6d65 6761 5f45 203c 2d2d 204d 5f30 202a mega_E <-- M_0 * │ │ │ │ +00012060: 2a20 6f6d 6567 615f 4520 7768 6963 6820 * omega_E which │ │ │ │ +00012070: 6973 2074 6865 2066 6972 7374 2064 6966 is the first dif │ │ │ │ +00012080: 6665 7265 6e74 6961 6c20 6f66 0a74 6865 ferential of.the │ │ │ │ +00012090: 2063 6f6d 706c 6578 2052 284d 292e 0a2b complex R(M)..+ │ │ │ │ +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 ---------------- │ │ │ │ -000120d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000120e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000120f0: 0a7c 6931 203a 2053 203d 205a 5a2f 3332 .|i1 : S = ZZ/32 │ │ │ │ -00012100: 3030 335b 785f 302e 2e78 5f32 5d3b 2020 003[x_0..x_2]; │ │ │ │ -00012110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012120: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00012130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00012160: 203a 2045 203d 205a 5a2f 3332 3030 335b : E = ZZ/32003[ │ │ │ │ -00012170: 655f 302e 2e65 5f32 2c20 536b 6577 436f e_0..e_2, SkewCo │ │ │ │ -00012180: 6d6d 7574 6174 6976 653d 3e74 7275 655d mmutative=>true] │ │ │ │ -00012190: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ;|.+------------ │ │ │ │ -000121a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000121b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000121c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 204d -------+.|i3 : M │ │ │ │ -000121d0: 203d 2063 6f6b 6572 206d 6174 7269 7820 = coker matrix │ │ │ │ -000121e0: 7b7b 785f 305e 322c 2078 5f31 5e32 7d7d {{x_0^2, x_1^2}} │ │ │ │ -000121f0: 3b20 2020 2020 2020 2020 2020 207c 0a2b ; |.+ │ │ │ │ +000120d0: 2d2d 2d2b 0a7c 6931 203a 2053 203d 205a ---+.|i1 : S = Z │ │ │ │ +000120e0: 5a2f 3332 3030 335b 785f 302e 2e78 5f32 Z/32003[x_0..x_2 │ │ │ │ +000120f0: 5d3b 2020 2020 2020 2020 2020 2020 2020 ]; │ │ │ │ +00012100: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00012110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00012140: 0a7c 6932 203a 2045 203d 205a 5a2f 3332 .|i2 : E = ZZ/32 │ │ │ │ +00012150: 3030 335b 655f 302e 2e65 5f32 2c20 536b 003[e_0..e_2, Sk │ │ │ │ +00012160: 6577 436f 6d6d 7574 6174 6976 653d 3e74 ewCommutative=>t │ │ │ │ +00012170: 7275 655d 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d rue];|.+-------- │ │ │ │ +00012180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000121a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ +000121b0: 203a 204d 203d 2063 6f6b 6572 206d 6174 : M = coker mat │ │ │ │ +000121c0: 7269 7820 7b7b 785f 305e 322c 2078 5f31 rix {{x_0^2, x_1 │ │ │ │ +000121d0: 5e32 7d7d 3b20 2020 2020 2020 2020 2020 ^2}}; │ │ │ │ +000121e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +000121f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012230: 2d2d 2d2b 0a7c 6934 203a 206d 203d 2070 ---+.|i4 : m = p │ │ │ │ -00012240: 7265 7365 6e74 6174 696f 6e20 7472 756e resentation trun │ │ │ │ -00012250: 6361 7465 2872 6567 756c 6172 6974 7920 cate(regularity │ │ │ │ -00012260: 4d2c 4d29 3b20 2020 207c 0a7c 2020 2020 M,M); |.| │ │ │ │ +00012210: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 206d -------+.|i4 : m │ │ │ │ +00012220: 203d 2070 7265 7365 6e74 6174 696f 6e20 = presentation │ │ │ │ +00012230: 7472 756e 6361 7465 2872 6567 756c 6172 truncate(regular │ │ │ │ +00012240: 6974 7920 4d2c 4d29 3b20 2020 207c 0a7c ity M,M); |.| │ │ │ │ +00012250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012290: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000122a0: 0a7c 2020 2020 2020 2020 2020 2020 2034 .| 4 │ │ │ │ -000122b0: 2020 2020 2020 3820 2020 2020 2020 2020 8 │ │ │ │ -000122c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000122d0: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ -000122e0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ -000122f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012300: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00012280: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012290: 2020 2034 2020 2020 2020 3820 2020 2020 4 8 │ │ │ │ +000122a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000122b0: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ +000122c0: 204d 6174 7269 7820 5320 203c 2d2d 2053 Matrix S <-- S │ │ │ │ +000122d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000122e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000122f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00012300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012340: 2d2b 0a7c 6935 203a 2073 796d 4578 7428 -+.|i5 : symExt( │ │ │ │ -00012350: 6d2c 4529 2020 2020 2020 2020 2020 2020 m,E) │ │ │ │ +00012320: 2d2d 2d2d 2d2b 0a7c 6935 203a 2073 796d -----+.|i5 : sym │ │ │ │ +00012330: 4578 7428 6d2c 4529 2020 2020 2020 2020 Ext(m,E) │ │ │ │ +00012340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012350: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00012360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012370: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00012370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000123b0: 6f35 203d 207b 2d31 7d20 7c20 655f 3220 o5 = {-1} | e_2 │ │ │ │ -000123c0: 3020 2020 3020 2020 3020 2020 7c20 2020 0 0 0 | │ │ │ │ -000123d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123e0: 2020 207c 0a7c 2020 2020 207b 2d31 7d20 |.| {-1} │ │ │ │ -000123f0: 7c20 655f 3120 655f 3220 3020 2020 3020 | e_1 e_2 0 0 │ │ │ │ -00012400: 2020 7c20 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00012410: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00012420: 207b 2d31 7d20 7c20 655f 3020 3020 2020 {-1} | e_0 0 │ │ │ │ -00012430: 655f 3220 3020 2020 7c20 2020 2020 2020 e_2 0 | │ │ │ │ -00012440: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00012450: 0a7c 2020 2020 207b 2d31 7d20 7c20 3020 .| {-1} | 0 │ │ │ │ -00012460: 2020 655f 3020 655f 3120 655f 3220 7c20 e_0 e_1 e_2 | │ │ │ │ +00012390: 207c 0a7c 6f35 203d 207b 2d31 7d20 7c20 |.|o5 = {-1} | │ │ │ │ +000123a0: 655f 3220 3020 2020 3020 2020 3020 2020 e_2 0 0 0 │ │ │ │ +000123b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000123c0: 2020 2020 2020 207c 0a7c 2020 2020 207b |.| { │ │ │ │ +000123d0: 2d31 7d20 7c20 655f 3120 655f 3220 3020 -1} | e_1 e_2 0 │ │ │ │ +000123e0: 2020 3020 2020 7c20 2020 2020 2020 2020 0 | │ │ │ │ +000123f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00012400: 2020 2020 207b 2d31 7d20 7c20 655f 3020 {-1} | e_0 │ │ │ │ +00012410: 3020 2020 655f 3220 3020 2020 7c20 2020 0 e_2 0 | │ │ │ │ +00012420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012430: 2020 207c 0a7c 2020 2020 207b 2d31 7d20 |.| {-1} │ │ │ │ +00012440: 7c20 3020 2020 655f 3020 655f 3120 655f | 0 e_0 e_1 e_ │ │ │ │ +00012450: 3220 7c20 2020 2020 2020 2020 2020 2020 2 | │ │ │ │ +00012460: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00012470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012480: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00012490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000124a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000124b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000124c0: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ -000124d0: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -000124e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000124f0: 207c 0a7c 6f35 203a 204d 6174 7269 7820 |.|o5 : Matrix │ │ │ │ -00012500: 4520 203c 2d2d 2045 2020 2020 2020 2020 E <-- E │ │ │ │ -00012510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012520: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00012480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000124a0: 0a7c 2020 2020 2020 2020 2020 2020 2034 .| 4 │ │ │ │ +000124b0: 2020 2020 2020 3420 2020 2020 2020 2020 4 │ │ │ │ +000124c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000124d0: 2020 2020 207c 0a7c 6f35 203a 204d 6174 |.|o5 : Mat │ │ │ │ +000124e0: 7269 7820 4520 203c 2d2d 2045 2020 2020 rix E <-- E │ │ │ │ +000124f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012500: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00012510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -00012560: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54 Caveat.======..T │ │ │ │ -00012570: 6869 7320 6675 6e63 7469 6f6e 2069 7320 his function is │ │ │ │ -00012580: 6120 7175 6963 6b2d 616e 642d 6469 7274 a quick-and-dirt │ │ │ │ -00012590: 7920 746f 6f6c 2077 6869 6368 2072 6571 y tool which req │ │ │ │ -000125a0: 7569 7265 7320 6c69 7474 6c65 2063 6f6d uires little com │ │ │ │ -000125b0: 7075 7461 7469 6f6e 2e0a 486f 7765 7665 putation..Howeve │ │ │ │ -000125c0: 7220 6966 2069 7420 6973 2063 616c 6c65 r if it is calle │ │ │ │ -000125d0: 6420 6f6e 2074 776f 2073 7563 6365 7373 d on two success │ │ │ │ -000125e0: 6976 6520 7472 756e 6361 7469 6f6e 7320 ive truncations │ │ │ │ -000125f0: 6f66 2061 206d 6f64 756c 652c 2074 6865 of a module, the │ │ │ │ -00012600: 6e20 7468 650a 6d61 7073 2069 7420 7072 n the.maps it pr │ │ │ │ -00012610: 6f64 7563 6573 206d 6179 204e 4f54 2063 oduces may NOT c │ │ │ │ -00012620: 6f6d 706f 7365 2074 6f20 7a65 726f 2062 ompose to zero b │ │ │ │ -00012630: 6563 6175 7365 2074 6865 2063 686f 6963 ecause the choic │ │ │ │ -00012640: 6520 6f66 2062 6173 6573 2069 7320 6e6f e of bases is no │ │ │ │ -00012650: 740a 636f 6e73 6973 7465 6e74 2e0a 0a53 t.consistent...S │ │ │ │ -00012660: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -00012670: 0a0a 2020 2a20 2a6e 6f74 6520 6267 673a .. * *note bgg: │ │ │ │ -00012680: 2062 6767 2c20 2d2d 2074 6865 2069 7468 bgg, -- the ith │ │ │ │ -00012690: 2064 6966 6665 7265 6e74 6961 6c20 6f66 differential of │ │ │ │ -000126a0: 2074 6865 2063 6f6d 706c 6578 2052 284d the complex R(M │ │ │ │ -000126b0: 290a 0a57 6179 7320 746f 2075 7365 2073 )..Ways to use s │ │ │ │ -000126c0: 796d 4578 743a 0a3d 3d3d 3d3d 3d3d 3d3d ymExt:.========= │ │ │ │ -000126d0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -000126e0: 2273 796d 4578 7428 4d61 7472 6978 2c50 "symExt(Matrix,P │ │ │ │ -000126f0: 6f6c 796e 6f6d 6961 6c52 696e 6729 220a olynomialRing)". │ │ │ │ -00012700: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00012710: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00012720: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00012730: 6374 202a 6e6f 7465 2073 796d 4578 743a ct *note symExt: │ │ │ │ -00012740: 2073 796d 4578 742c 2069 7320 6120 2a6e symExt, is a *n │ │ │ │ -00012750: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00012760: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00012770: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00012780: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00012540: 2d2b 0a0a 4361 7665 6174 0a3d 3d3d 3d3d -+..Caveat.===== │ │ │ │ +00012550: 3d0a 0a54 6869 7320 6675 6e63 7469 6f6e =..This function │ │ │ │ +00012560: 2069 7320 6120 7175 6963 6b2d 616e 642d is a quick-and- │ │ │ │ +00012570: 6469 7274 7920 746f 6f6c 2077 6869 6368 dirty tool which │ │ │ │ +00012580: 2072 6571 7569 7265 7320 6c69 7474 6c65 requires little │ │ │ │ +00012590: 2063 6f6d 7075 7461 7469 6f6e 2e0a 486f computation..Ho │ │ │ │ +000125a0: 7765 7665 7220 6966 2069 7420 6973 2063 wever if it is c │ │ │ │ +000125b0: 616c 6c65 6420 6f6e 2074 776f 2073 7563 alled on two suc │ │ │ │ +000125c0: 6365 7373 6976 6520 7472 756e 6361 7469 cessive truncati │ │ │ │ +000125d0: 6f6e 7320 6f66 2061 206d 6f64 756c 652c ons of a module, │ │ │ │ +000125e0: 2074 6865 6e20 7468 650a 6d61 7073 2069 then the.maps i │ │ │ │ +000125f0: 7420 7072 6f64 7563 6573 206d 6179 204e t produces may N │ │ │ │ +00012600: 4f54 2063 6f6d 706f 7365 2074 6f20 7a65 OT compose to ze │ │ │ │ +00012610: 726f 2062 6563 6175 7365 2074 6865 2063 ro because the c │ │ │ │ +00012620: 686f 6963 6520 6f66 2062 6173 6573 2069 hoice of bases i │ │ │ │ +00012630: 7320 6e6f 740a 636f 6e73 6973 7465 6e74 s not.consistent │ │ │ │ +00012640: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ +00012650: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +00012660: 6267 673a 2062 6767 2c20 2d2d 2074 6865 bgg: bgg, -- the │ │ │ │ +00012670: 2069 7468 2064 6966 6665 7265 6e74 6961 ith differentia │ │ │ │ +00012680: 6c20 6f66 2074 6865 2063 6f6d 706c 6578 l of the complex │ │ │ │ +00012690: 2052 284d 290a 0a57 6179 7320 746f 2075 R(M)..Ways to u │ │ │ │ +000126a0: 7365 2073 796d 4578 743a 0a3d 3d3d 3d3d se symExt:.===== │ │ │ │ +000126b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +000126c0: 2020 2a20 2273 796d 4578 7428 4d61 7472 * "symExt(Matr │ │ │ │ +000126d0: 6978 2c50 6f6c 796e 6f6d 6961 6c52 696e ix,PolynomialRin │ │ │ │ +000126e0: 6729 220a 0a46 6f72 2074 6865 2070 726f g)"..For the pro │ │ │ │ +000126f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00012700: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00012710: 6f62 6a65 6374 202a 6e6f 7465 2073 796d object *note sym │ │ │ │ +00012720: 4578 743a 2073 796d 4578 742c 2069 7320 Ext: symExt, is │ │ │ │ +00012730: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +00012740: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +00012750: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00012760: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00012770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000127a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000127b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000127c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000127d0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -000127e0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -000127f0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00012800: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00012810: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00012820: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00012830: 7932 2f70 6163 6b61 6765 732f 4247 472e y2/packages/BGG. │ │ │ │ -00012840: 6d32 3a36 3937 3a0a 302e 0a1f 0a46 696c m2:697:.0....Fil │ │ │ │ -00012850: 653a 2042 4747 2e69 6e66 6f2c 204e 6f64 e: BGG.info, Nod │ │ │ │ -00012860: 653a 2074 6174 6552 6573 6f6c 7574 696f e: tateResolutio │ │ │ │ -00012870: 6e2c 204e 6578 743a 2075 6e69 7665 7273 n, Next: univers │ │ │ │ -00012880: 616c 4578 7465 6e73 696f 6e2c 2050 7265 alExtension, Pre │ │ │ │ -00012890: 763a 2073 796d 4578 742c 2055 703a 2054 v: symExt, Up: T │ │ │ │ -000128a0: 6f70 0a0a 7461 7465 5265 736f 6c75 7469 op..tateResoluti │ │ │ │ -000128b0: 6f6e 202d 2d20 6669 6e69 7465 2070 6965 on -- finite pie │ │ │ │ -000128c0: 6365 206f 6620 7468 6520 5461 7465 2072 ce of the Tate r │ │ │ │ -000128d0: 6573 6f6c 7574 696f 6e0a 2a2a 2a2a 2a2a esolution.****** │ │ │ │ +000127b0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +000127c0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +000127d0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +000127e0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +000127f0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00012800: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00012810: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00012820: 4247 472e 6d32 3a36 3937 3a0a 302e 0a1f BGG.m2:697:.0... │ │ │ │ +00012830: 0a46 696c 653a 2042 4747 2e69 6e66 6f2c .File: BGG.info, │ │ │ │ +00012840: 204e 6f64 653a 2074 6174 6552 6573 6f6c Node: tateResol │ │ │ │ +00012850: 7574 696f 6e2c 204e 6578 743a 2075 6e69 ution, Next: uni │ │ │ │ +00012860: 7665 7273 616c 4578 7465 6e73 696f 6e2c versalExtension, │ │ │ │ +00012870: 2050 7265 763a 2073 796d 4578 742c 2055 Prev: symExt, U │ │ │ │ +00012880: 703a 2054 6f70 0a0a 7461 7465 5265 736f p: Top..tateReso │ │ │ │ +00012890: 6c75 7469 6f6e 202d 2d20 6669 6e69 7465 lution -- finite │ │ │ │ +000128a0: 2070 6965 6365 206f 6620 7468 6520 5461 piece of the Ta │ │ │ │ +000128b0: 7465 2072 6573 6f6c 7574 696f 6e0a 2a2a te resolution.** │ │ │ │ +000128c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000128d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000128e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000128f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00012900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00012910: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -00012920: 2020 2020 2074 6174 6552 6573 6f6c 7574 tateResolut │ │ │ │ -00012930: 696f 6e28 6d2c 452c 6c2c 6829 0a20 202a ion(m,E,l,h). * │ │ │ │ -00012940: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00012950: 206d 2c20 6120 2a6e 6f74 6520 6d61 7472 m, a *note matr │ │ │ │ -00012960: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -00012970: 6329 4d61 7472 6978 2c2c 2061 2070 7265 c)Matrix,, a pre │ │ │ │ -00012980: 7365 6e74 6174 696f 6e20 6d61 7472 6978 sentation matrix │ │ │ │ -00012990: 2066 6f72 2061 0a20 2020 2020 2020 206d for a. m │ │ │ │ -000129a0: 6f64 756c 650a 2020 2020 2020 2a20 452c odule. * E, │ │ │ │ -000129b0: 2061 202a 6e6f 7465 2070 6f6c 796e 6f6d a *note polynom │ │ │ │ -000129c0: 6961 6c20 7269 6e67 3a20 284d 6163 6175 ial ring: (Macau │ │ │ │ -000129d0: 6c61 7932 446f 6329 506f 6c79 6e6f 6d69 lay2Doc)Polynomi │ │ │ │ -000129e0: 616c 5269 6e67 2c2c 2065 7874 6572 696f alRing,, exterio │ │ │ │ -000129f0: 720a 2020 2020 2020 2020 616c 6765 6272 r. algebr │ │ │ │ -00012a00: 610a 2020 2020 2020 2a20 6c2c 2061 6e20 a. * l, an │ │ │ │ -00012a10: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ -00012a20: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -00012a30: 2c20 6c6f 7765 7220 636f 686f 6d6f 6c6f , lower cohomolo │ │ │ │ -00012a40: 6769 6361 6c20 6465 6772 6565 0a20 2020 gical degree. │ │ │ │ -00012a50: 2020 202a 2068 2c20 616e 202a 6e6f 7465 * h, an *note │ │ │ │ -00012a60: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ -00012a70: 6c61 7932 446f 6329 5a5a 2c2c 2075 7070 lay2Doc)ZZ,, upp │ │ │ │ -00012a80: 6572 2062 6f75 6e64 206f 6e20 7468 650a er bound on the. │ │ │ │ -00012a90: 2020 2020 2020 2020 636f 686f 6d6f 6c6f cohomolo │ │ │ │ -00012aa0: 6769 6361 6c20 6465 6772 6565 0a20 202a gical degree. * │ │ │ │ -00012ab0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00012ac0: 2a20 6120 2a6e 6f74 6520 636f 6d70 6c65 * a *note comple │ │ │ │ -00012ad0: 783a 2028 436f 6d70 6c65 7865 7329 436f x: (Complexes)Co │ │ │ │ -00012ae0: 6d70 6c65 782c 2c20 6120 6669 6e69 7465 mplex,, a finite │ │ │ │ -00012af0: 2070 6965 6365 206f 6620 7468 6520 5461 piece of the Ta │ │ │ │ -00012b00: 7465 0a20 2020 2020 2020 2072 6573 6f6c te. resol │ │ │ │ -00012b10: 7574 696f 6e0a 0a44 6573 6372 6970 7469 ution..Descripti │ │ │ │ -00012b20: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00012b30: 5468 6973 2066 756e 6374 696f 6e20 7461 This function ta │ │ │ │ -00012b40: 6b65 7320 6173 2069 6e70 7574 2061 2070 kes as input a p │ │ │ │ -00012b50: 7265 7365 6e74 6174 696f 6e20 6d61 7472 resentation matr │ │ │ │ -00012b60: 6978 206d 206f 6620 6120 6669 6e69 7465 ix m of a finite │ │ │ │ -00012b70: 6c79 2067 656e 6572 6174 6564 0a67 7261 ly generated.gra │ │ │ │ -00012b80: 6465 6420 532d 6d6f 6475 6c65 204d 2061 ded S-module M a │ │ │ │ -00012b90: 6e20 6578 7465 7269 6f72 2061 6c67 6562 n exterior algeb │ │ │ │ -00012ba0: 7261 2045 2061 6e64 2074 776f 2069 6e74 ra E and two int │ │ │ │ -00012bb0: 6567 6572 7320 6c20 616e 6420 682e 2049 egers l and h. I │ │ │ │ -00012bc0: 6620 7220 6973 2074 6865 0a72 6567 756c f r is the.regul │ │ │ │ -00012bd0: 6172 6974 7920 6f66 204d 2c20 7468 656e arity of M, then │ │ │ │ -00012be0: 2074 6869 7320 6675 6e63 7469 6f6e 2063 this function c │ │ │ │ -00012bf0: 6f6d 7075 7465 7320 7468 6520 7069 6563 omputes the piec │ │ │ │ -00012c00: 6520 6f66 2074 6865 2054 6174 6520 7265 e of the Tate re │ │ │ │ -00012c10: 736f 6c75 7469 6f6e 0a66 726f 6d20 636f solution.from co │ │ │ │ -00012c20: 686f 6d6f 6c6f 6769 6361 6c20 6465 6772 homological degr │ │ │ │ -00012c30: 6565 206c 2074 6f20 636f 686f 6d6f 6c6f ee l to cohomolo │ │ │ │ -00012c40: 6769 6361 6c20 6465 6772 6565 206d 6178 gical degree max │ │ │ │ -00012c50: 2872 2b32 2c68 292e 2046 6f72 2069 6e73 (r+2,h). For ins │ │ │ │ -00012c60: 7461 6e63 652c 0a66 6f72 2074 6865 2068 tance,.for the h │ │ │ │ -00012c70: 6f6d 6f67 656e 656f 7573 2063 6f6f 7264 omogeneous coord │ │ │ │ -00012c80: 696e 6174 6520 7269 6e67 206f 6620 6120 inate ring of a │ │ │ │ -00012c90: 706f 696e 7420 696e 2074 6865 2070 726f point in the pro │ │ │ │ -00012ca0: 6a65 6374 6976 6520 706c 616e 653a 0a2b jective plane:.+ │ │ │ │ +000128f0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00012900: 0a20 2020 2020 2020 2074 6174 6552 6573 . tateRes │ │ │ │ +00012910: 6f6c 7574 696f 6e28 6d2c 452c 6c2c 6829 olution(m,E,l,h) │ │ │ │ +00012920: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +00012930: 2020 202a 206d 2c20 6120 2a6e 6f74 6520 * m, a *note │ │ │ │ +00012940: 6d61 7472 6978 3a20 284d 6163 6175 6c61 matrix: (Macaula │ │ │ │ +00012950: 7932 446f 6329 4d61 7472 6978 2c2c 2061 y2Doc)Matrix,, a │ │ │ │ +00012960: 2070 7265 7365 6e74 6174 696f 6e20 6d61 presentation ma │ │ │ │ +00012970: 7472 6978 2066 6f72 2061 0a20 2020 2020 trix for a. │ │ │ │ +00012980: 2020 206d 6f64 756c 650a 2020 2020 2020 module. │ │ │ │ +00012990: 2a20 452c 2061 202a 6e6f 7465 2070 6f6c * E, a *note pol │ │ │ │ +000129a0: 796e 6f6d 6961 6c20 7269 6e67 3a20 284d ynomial ring: (M │ │ │ │ +000129b0: 6163 6175 6c61 7932 446f 6329 506f 6c79 acaulay2Doc)Poly │ │ │ │ +000129c0: 6e6f 6d69 616c 5269 6e67 2c2c 2065 7874 nomialRing,, ext │ │ │ │ +000129d0: 6572 696f 720a 2020 2020 2020 2020 616c erior. al │ │ │ │ +000129e0: 6765 6272 610a 2020 2020 2020 2a20 6c2c gebra. * l, │ │ │ │ +000129f0: 2061 6e20 2a6e 6f74 6520 696e 7465 6765 an *note intege │ │ │ │ +00012a00: 723a 2028 4d61 6361 756c 6179 3244 6f63 r: (Macaulay2Doc │ │ │ │ +00012a10: 295a 5a2c 2c20 6c6f 7765 7220 636f 686f )ZZ,, lower coho │ │ │ │ +00012a20: 6d6f 6c6f 6769 6361 6c20 6465 6772 6565 mological degree │ │ │ │ +00012a30: 0a20 2020 2020 202a 2068 2c20 616e 202a . * h, an * │ │ │ │ +00012a40: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ +00012a50: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ +00012a60: 2075 7070 6572 2062 6f75 6e64 206f 6e20 upper bound on │ │ │ │ +00012a70: 7468 650a 2020 2020 2020 2020 636f 686f the. coho │ │ │ │ +00012a80: 6d6f 6c6f 6769 6361 6c20 6465 6772 6565 mological degree │ │ │ │ +00012a90: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00012aa0: 2020 2020 2a20 6120 2a6e 6f74 6520 636f * a *note co │ │ │ │ +00012ab0: 6d70 6c65 783a 2028 436f 6d70 6c65 7865 mplex: (Complexe │ │ │ │ +00012ac0: 7329 436f 6d70 6c65 782c 2c20 6120 6669 s)Complex,, a fi │ │ │ │ +00012ad0: 6e69 7465 2070 6965 6365 206f 6620 7468 nite piece of th │ │ │ │ +00012ae0: 6520 5461 7465 0a20 2020 2020 2020 2072 e Tate. r │ │ │ │ +00012af0: 6573 6f6c 7574 696f 6e0a 0a44 6573 6372 esolution..Descr │ │ │ │ +00012b00: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00012b10: 3d3d 0a0a 5468 6973 2066 756e 6374 696f ==..This functio │ │ │ │ +00012b20: 6e20 7461 6b65 7320 6173 2069 6e70 7574 n takes as input │ │ │ │ +00012b30: 2061 2070 7265 7365 6e74 6174 696f 6e20 a presentation │ │ │ │ +00012b40: 6d61 7472 6978 206d 206f 6620 6120 6669 matrix m of a fi │ │ │ │ +00012b50: 6e69 7465 6c79 2067 656e 6572 6174 6564 nitely generated │ │ │ │ +00012b60: 0a67 7261 6465 6420 532d 6d6f 6475 6c65 .graded S-module │ │ │ │ +00012b70: 204d 2061 6e20 6578 7465 7269 6f72 2061 M an exterior a │ │ │ │ +00012b80: 6c67 6562 7261 2045 2061 6e64 2074 776f lgebra E and two │ │ │ │ +00012b90: 2069 6e74 6567 6572 7320 6c20 616e 6420 integers l and │ │ │ │ +00012ba0: 682e 2049 6620 7220 6973 2074 6865 0a72 h. If r is the.r │ │ │ │ +00012bb0: 6567 756c 6172 6974 7920 6f66 204d 2c20 egularity of M, │ │ │ │ +00012bc0: 7468 656e 2074 6869 7320 6675 6e63 7469 then this functi │ │ │ │ +00012bd0: 6f6e 2063 6f6d 7075 7465 7320 7468 6520 on computes the │ │ │ │ +00012be0: 7069 6563 6520 6f66 2074 6865 2054 6174 piece of the Tat │ │ │ │ +00012bf0: 6520 7265 736f 6c75 7469 6f6e 0a66 726f e resolution.fro │ │ │ │ +00012c00: 6d20 636f 686f 6d6f 6c6f 6769 6361 6c20 m cohomological │ │ │ │ +00012c10: 6465 6772 6565 206c 2074 6f20 636f 686f degree l to coho │ │ │ │ +00012c20: 6d6f 6c6f 6769 6361 6c20 6465 6772 6565 mological degree │ │ │ │ +00012c30: 206d 6178 2872 2b32 2c68 292e 2046 6f72 max(r+2,h). For │ │ │ │ +00012c40: 2069 6e73 7461 6e63 652c 0a66 6f72 2074 instance,.for t │ │ │ │ +00012c50: 6865 2068 6f6d 6f67 656e 656f 7573 2063 he homogeneous c │ │ │ │ +00012c60: 6f6f 7264 696e 6174 6520 7269 6e67 206f oordinate ring o │ │ │ │ +00012c70: 6620 6120 706f 696e 7420 696e 2074 6865 f a point in the │ │ │ │ +00012c80: 2070 726f 6a65 6374 6976 6520 706c 616e projective plan │ │ │ │ +00012c90: 653a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d e:.+------------ │ │ │ │ +00012ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012ce0: 2d2d 2d2b 0a7c 6931 203a 2053 203d 205a ---+.|i1 : S = Z │ │ │ │ -00012cf0: 5a2f 3332 3030 335b 785f 302e 2e78 5f32 Z/32003[x_0..x_2 │ │ │ │ -00012d00: 5d3b 2020 2020 2020 2020 2020 2020 2020 ]; │ │ │ │ -00012d10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00012cc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 -------+.|i1 : S │ │ │ │ +00012cd0: 203d 205a 5a2f 3332 3030 335b 785f 302e = ZZ/32003[x_0. │ │ │ │ +00012ce0: 2e78 5f32 5d3b 2020 2020 2020 2020 2020 .x_2]; │ │ │ │ +00012cf0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00012d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00012d50: 0a7c 6932 203a 2045 203d 205a 5a2f 3332 .|i2 : E = ZZ/32 │ │ │ │ -00012d60: 3030 335b 655f 302e 2e65 5f32 2c20 536b 003[e_0..e_2, Sk │ │ │ │ -00012d70: 6577 436f 6d6d 7574 6174 6976 653d 3e74 ewCommutative=>t │ │ │ │ -00012d80: 7275 655d 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d rue];|.+-------- │ │ │ │ -00012d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -00012dc0: 203a 206d 203d 206d 6174 7269 787b 7b78 : m = matrix{{x │ │ │ │ -00012dd0: 5f30 2c78 5f31 7d7d 3b20 2020 2020 2020 _0,x_1}}; │ │ │ │ +00012d30: 2d2d 2d2b 0a7c 6932 203a 2045 203d 205a ---+.|i2 : E = Z │ │ │ │ +00012d40: 5a2f 3332 3030 335b 655f 302e 2e65 5f32 Z/32003[e_0..e_2 │ │ │ │ +00012d50: 2c20 536b 6577 436f 6d6d 7574 6174 6976 , SkewCommutativ │ │ │ │ +00012d60: 653d 3e74 7275 655d 3b7c 0a2b 2d2d 2d2d e=>true];|.+---- │ │ │ │ +00012d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00012da0: 0a7c 6933 203a 206d 203d 206d 6174 7269 .|i3 : m = matri │ │ │ │ +00012db0: 787b 7b78 5f30 2c78 5f31 7d7d 3b20 2020 x{{x_0,x_1}}; │ │ │ │ +00012dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012dd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00012de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012df0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00012e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00012e30: 2020 2020 2020 2031 2020 2020 2020 3220 1 2 │ │ │ │ -00012e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00012e60: 6f33 203a 204d 6174 7269 7820 5320 203c o3 : Matrix S < │ │ │ │ -00012e70: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ -00012e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00012ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012ec0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -00012ed0: 2072 6567 756c 6172 6974 7920 636f 6b65 regularity coke │ │ │ │ -00012ee0: 7220 6d20 2020 2020 2020 2020 2020 2020 r m │ │ │ │ -00012ef0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00012f00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00012f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f30: 2020 2020 207c 0a7c 6f34 203d 2030 2020 |.|o4 = 0 │ │ │ │ -00012f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00012df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00012e10: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +00012e20: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00012e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e40: 207c 0a7c 6f33 203a 204d 6174 7269 7820 |.|o3 : Matrix │ │ │ │ +00012e50: 5320 203c 2d2d 2053 2020 2020 2020 2020 S <-- S │ │ │ │ +00012e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00012e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00012eb0: 6934 203a 2072 6567 756c 6172 6974 7920 i4 : regularity │ │ │ │ +00012ec0: 636f 6b65 7220 6d20 2020 2020 2020 2020 coker m │ │ │ │ +00012ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012ee0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012f10: 2020 2020 2020 2020 207c 0a7c 6f34 203d |.|o4 = │ │ │ │ +00012f20: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00012f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012f40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00012f50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00012f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012fa0: 2d2b 0a7c 6935 203a 2054 203d 2074 6174 -+.|i5 : T = tat │ │ │ │ -00012fb0: 6552 6573 6f6c 7574 696f 6e28 6d2c 452c eResolution(m,E, │ │ │ │ -00012fc0: 2d32 2c34 2920 2020 2020 2020 2020 2020 -2,4) │ │ │ │ -00012fd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00012f80: 2d2d 2d2d 2d2b 0a7c 6935 203a 2054 203d -----+.|i5 : T = │ │ │ │ +00012f90: 2074 6174 6552 6573 6f6c 7574 696f 6e28 tateResolution( │ │ │ │ +00012fa0: 6d2c 452c 2d32 2c34 2920 2020 2020 2020 m,E,-2,4) │ │ │ │ +00012fb0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00012fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013000: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00012ff0: 207c 0a7c 2020 2020 2020 3120 2020 2020 |.| 1 │ │ │ │ +00013000: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ 00013010: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ -00013020: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00013030: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ -00013040: 3120 207c 0a7c 6f35 203d 2045 2020 3c2d 1 |.|o5 = E <- │ │ │ │ -00013050: 2d20 4520 203c 2d2d 2045 2020 3c2d 2d20 - E <-- E <-- │ │ │ │ -00013060: 4520 203c 2d2d 2045 2020 3c2d 2d20 4520 E <-- E <-- E │ │ │ │ -00013070: 203c 2d2d 2045 2020 207c 0a7c 2020 2020 <-- E |.| │ │ │ │ +00013020: 2020 2020 3120 207c 0a7c 6f35 203d 2045 1 |.|o5 = E │ │ │ │ +00013030: 2020 3c2d 2d20 4520 203c 2d2d 2045 2020 <-- E <-- E │ │ │ │ +00013040: 3c2d 2d20 4520 203c 2d2d 2045 2020 3c2d <-- E <-- E <- │ │ │ │ +00013050: 2d20 4520 203c 2d2d 2045 2020 207c 0a7c - E <-- E |.| │ │ │ │ +00013060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000130a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000130b0: 0a7c 2020 2020 2030 2020 2020 2020 3120 .| 0 1 │ │ │ │ -000130c0: 2020 2020 2032 2020 2020 2020 3320 2020 2 3 │ │ │ │ -000130d0: 2020 2034 2020 2020 2020 3520 2020 2020 4 5 │ │ │ │ -000130e0: 2036 2020 207c 0a7c 2020 2020 2020 2020 6 |.| │ │ │ │ -000130f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013110: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -00013120: 203a 2043 6f6d 706c 6578 2020 2020 2020 : Complex │ │ │ │ -00013130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013150: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00013160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013180: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2062 -------+.|i6 : b │ │ │ │ -00013190: 6574 7469 2054 2020 2020 2020 2020 2020 etti T │ │ │ │ -000131a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00013090: 2020 207c 0a7c 2020 2020 2030 2020 2020 |.| 0 │ │ │ │ +000130a0: 2020 3120 2020 2020 2032 2020 2020 2020 1 2 │ │ │ │ +000130b0: 3320 2020 2020 2034 2020 2020 2020 3520 3 4 5 │ │ │ │ +000130c0: 2020 2020 2036 2020 207c 0a7c 2020 2020 6 |.| │ │ │ │ +000130d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000130e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000130f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00013100: 0a7c 6f35 203a 2043 6f6d 706c 6578 2020 .|o5 : Complex │ │ │ │ +00013110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013130: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ +00013170: 203a 2062 6574 7469 2054 2020 2020 2020 : betti T │ │ │ │ +00013180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000131a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000131b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000131c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00013200: 2020 3020 3120 3220 3320 3420 3520 3620 0 1 2 3 4 5 6 │ │ │ │ -00013210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013220: 2020 2020 2020 2020 207c 0a7c 6f36 203d |.|o6 = │ │ │ │ -00013230: 2074 6f74 616c 3a20 3120 3120 3120 3120 total: 1 1 1 1 │ │ │ │ -00013240: 3120 3120 3120 2020 2020 2020 2020 2020 1 1 1 │ │ │ │ -00013250: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00013260: 0a7c 2020 2020 2020 2020 2d34 3a20 3120 .| -4: 1 │ │ │ │ -00013270: 3120 3120 3120 3120 3120 3120 2020 2020 1 1 1 1 1 1 │ │ │ │ +000131d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000131e0: 2020 2020 2020 3020 3120 3220 3320 3420 0 1 2 3 4 │ │ │ │ +000131f0: 3520 3620 2020 2020 2020 2020 2020 2020 5 6 │ │ │ │ +00013200: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00013210: 6f36 203d 2074 6f74 616c 3a20 3120 3120 o6 = total: 1 1 │ │ │ │ +00013220: 3120 3120 3120 3120 3120 2020 2020 2020 1 1 1 1 1 │ │ │ │ +00013230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013240: 2020 207c 0a7c 2020 2020 2020 2020 2d34 |.| -4 │ │ │ │ +00013250: 3a20 3120 3120 3120 3120 3120 3120 3120 : 1 1 1 1 1 1 1 │ │ │ │ +00013260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013270: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00013280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013290: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000132a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000132b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000132c0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -000132d0: 203a 2042 6574 7469 5461 6c6c 7920 2020 : BettiTally │ │ │ │ -000132e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000132f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013300: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00013310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013330: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2054 -------+.|i7 : T │ │ │ │ -00013340: 2e64 645f 3120 2020 2020 2020 2020 2020 .dd_1 │ │ │ │ -00013350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00013290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000132a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000132b0: 0a7c 6f36 203a 2042 6574 7469 5461 6c6c .|o6 : BettiTall │ │ │ │ +000132c0: 7920 2020 2020 2020 2020 2020 2020 2020 y │ │ │ │ +000132d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000132e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000132f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2020 2020 2020 2020 2020 │ │ │ │ +00013420: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ +00013430: 204d 6174 7269 7820 4520 203c 2d2d 2045 Matrix E <-- E │ │ │ │ +00013440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013450: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00013460: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00013470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000134a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000134b0: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -000134c0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -000134d0: 2073 796d 4578 743a 2073 796d 4578 742c symExt: symExt, │ │ │ │ -000134e0: 202d 2d20 7468 6520 6669 7273 7420 6469 -- the first di │ │ │ │ -000134f0: 6666 6572 656e 7469 616c 206f 6620 7468 fferential of th │ │ │ │ -00013500: 6520 636f 6d70 6c65 7820 5228 4d29 0a0a e complex R(M).. │ │ │ │ -00013510: 5761 7973 2074 6f20 7573 6520 7461 7465 Ways to use tate │ │ │ │ -00013520: 5265 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d Resolution:.==== │ │ │ │ -00013530: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00013540: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7461 =======.. * "ta │ │ │ │ -00013550: 7465 5265 736f 6c75 7469 6f6e 284d 6174 teResolution(Mat │ │ │ │ -00013560: 7269 782c 506f 6c79 6e6f 6d69 616c 5269 rix,PolynomialRi │ │ │ │ -00013570: 6e67 2c5a 5a2c 5a5a 2922 0a0a 466f 7220 ng,ZZ,ZZ)"..For │ │ │ │ -00013580: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00013590: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000135a0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -000135b0: 6f74 6520 7461 7465 5265 736f 6c75 7469 ote tateResoluti │ │ │ │ -000135c0: 6f6e 3a20 7461 7465 5265 736f 6c75 7469 on: tateResoluti │ │ │ │ -000135d0: 6f6e 2c20 6973 2061 202a 6e6f 7465 206d on, is a *note m │ │ │ │ -000135e0: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ -000135f0: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00013600: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +00013490: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ +000134a0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +000134b0: 6e6f 7465 2073 796d 4578 743a 2073 796d note symExt: sym │ │ │ │ +000134c0: 4578 742c 202d 2d20 7468 6520 6669 7273 Ext, -- the firs │ │ │ │ +000134d0: 7420 6469 6666 6572 656e 7469 616c 206f t differential o │ │ │ │ +000134e0: 6620 7468 6520 636f 6d70 6c65 7820 5228 f the complex R( │ │ │ │ +000134f0: 4d29 0a0a 5761 7973 2074 6f20 7573 6520 M)..Ways to use │ │ │ │ +00013500: 7461 7465 5265 736f 6c75 7469 6f6e 3a0a tateResolution:. │ │ │ │ +00013510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00013520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00013530: 2022 7461 7465 5265 736f 6c75 7469 6f6e "tateResolution │ │ │ │ +00013540: 284d 6174 7269 782c 506f 6c79 6e6f 6d69 (Matrix,Polynomi │ │ │ │ +00013550: 616c 5269 6e67 2c5a 5a2c 5a5a 2922 0a0a alRing,ZZ,ZZ)".. │ │ │ │ +00013560: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +00013570: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +00013580: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +00013590: 7420 2a6e 6f74 6520 7461 7465 5265 736f t *note tateReso │ │ │ │ +000135a0: 6c75 7469 6f6e 3a20 7461 7465 5265 736f lution: tateReso │ │ │ │ +000135b0: 6c75 7469 6f6e 2c20 6973 2061 202a 6e6f lution, is a *no │ │ │ │ +000135c0: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +000135d0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ +000135e0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +000135f0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +00013600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00013660: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00013670: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00013680: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00013690: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -000136a0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -000136b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -000136c0: 636b 6167 6573 2f42 4747 2e6d 323a 3735 ckages/BGG.m2:75 │ │ │ │ -000136d0: 333a 0a30 2e0a 1f0a 4669 6c65 3a20 4247 3:.0....File: BG │ │ │ │ -000136e0: 472e 696e 666f 2c20 4e6f 6465 3a20 756e G.info, Node: un │ │ │ │ -000136f0: 6976 6572 7361 6c45 7874 656e 7369 6f6e iversalExtension │ │ │ │ -00013700: 2c20 5072 6576 3a20 7461 7465 5265 736f , Prev: tateReso │ │ │ │ -00013710: 6c75 7469 6f6e 2c20 5570 3a20 546f 700a lution, Up: Top. │ │ │ │ -00013720: 0a75 6e69 7665 7273 616c 4578 7465 6e73 .universalExtens │ │ │ │ -00013730: 696f 6e20 2d2d 2055 6e69 7665 7273 616c ion -- Universal │ │ │ │ -00013740: 2065 7874 656e 7369 6f6e 206f 6620 7665 extension of ve │ │ │ │ -00013750: 6374 6f72 2062 756e 646c 6573 206f 6e20 ctor bundles on │ │ │ │ -00013760: 505e 310a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a P^1.************ │ │ │ │ +00013640: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +00013650: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +00013660: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +00013670: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +00013680: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +00013690: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +000136a0: 322f 7061 636b 6167 6573 2f42 4747 2e6d 2/packages/BGG.m │ │ │ │ +000136b0: 323a 3735 333a 0a30 2e0a 1f0a 4669 6c65 2:753:.0....File │ │ │ │ +000136c0: 3a20 4247 472e 696e 666f 2c20 4e6f 6465 : BGG.info, Node │ │ │ │ +000136d0: 3a20 756e 6976 6572 7361 6c45 7874 656e : universalExten │ │ │ │ +000136e0: 7369 6f6e 2c20 5072 6576 3a20 7461 7465 sion, Prev: tate │ │ │ │ +000136f0: 5265 736f 6c75 7469 6f6e 2c20 5570 3a20 Resolution, Up: │ │ │ │ +00013700: 546f 700a 0a75 6e69 7665 7273 616c 4578 Top..universalEx │ │ │ │ +00013710: 7465 6e73 696f 6e20 2d2d 2055 6e69 7665 tension -- Unive │ │ │ │ +00013720: 7273 616c 2065 7874 656e 7369 6f6e 206f rsal extension o │ │ │ │ +00013730: 6620 7665 6374 6f72 2062 756e 646c 6573 f vector bundles │ │ │ │ +00013740: 206f 6e20 505e 310a 2a2a 2a2a 2a2a 2a2a on P^1.******** │ │ │ │ +00013750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00013760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00013770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000137a0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -000137b0: 653a 200a 2020 2020 2020 2020 4520 3d20 e: . E = │ │ │ │ -000137c0: 756e 6976 6572 7361 6c45 7874 656e 7369 universalExtensi │ │ │ │ -000137d0: 6f6e 284c 612c 204c 6229 0a20 202a 2049 on(La, Lb). * I │ │ │ │ -000137e0: 6e70 7574 733a 0a20 2020 2020 202a 204c nputs:. * L │ │ │ │ -000137f0: 612c 2061 202a 6e6f 7465 206c 6973 743a a, a *note list: │ │ │ │ -00013800: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -00013810: 6973 742c 2c20 6f66 2069 6e74 6567 6572 ist,, of integer │ │ │ │ -00013820: 730a 2020 2020 2020 2a20 4c62 2c20 6120 s. * Lb, a │ │ │ │ -00013830: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ -00013840: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ -00013850: 206f 6620 696e 7465 6765 7273 0a20 202a of integers. * │ │ │ │ -00013860: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00013870: 2a20 452c 2061 202a 6e6f 7465 206d 6f64 * E, a *note mod │ │ │ │ -00013880: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -00013890: 6f63 294d 6f64 756c 652c 2c20 7265 7072 oc)Module,, repr │ │ │ │ -000138a0: 6573 656e 7469 6e67 2074 6865 2065 7874 esenting the ext │ │ │ │ -000138b0: 656e 7369 6f6e 0a0a 4465 7363 7269 7074 ension..Descript │ │ │ │ -000138c0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -000138d0: 0a45 7665 7279 2076 6563 746f 7220 6275 .Every vector bu │ │ │ │ -000138e0: 6e64 6c65 2045 206f 6e20 245c 5050 5e31 ndle E on $\PP^1 │ │ │ │ -000138f0: 2420 7370 6c69 7473 2061 7320 6120 7375 $ splits as a su │ │ │ │ -00013900: 6d20 6f66 206c 696e 6520 6275 6e64 6c65 m of line bundle │ │ │ │ -00013910: 7320 4f4f 2861 5f69 292e 2049 6620 4c61 s OO(a_i). If La │ │ │ │ -00013920: 0a69 7320 6120 6c69 7374 206f 6620 696e .is a list of in │ │ │ │ -00013930: 7465 6765 7273 2c20 7765 2077 7269 7465 tegers, we write │ │ │ │ -00013940: 2045 284c 6129 2066 6f72 2074 6865 2064 E(La) for the d │ │ │ │ -00013950: 6972 6563 7420 7375 6d20 6f66 2074 6865 irect sum of the │ │ │ │ -00013960: 206c 696e 6520 6275 6e64 6c65 0a4f 4f28 line bundle.OO( │ │ │ │ -00013970: 4c61 5f69 292e 2020 4769 7665 6e20 7477 La_i). Given tw │ │ │ │ -00013980: 6f20 7375 6368 2062 756e 646c 6573 2073 o such bundles s │ │ │ │ -00013990: 7065 6369 6669 6564 2062 7920 7468 6520 pecified by the │ │ │ │ -000139a0: 6c69 7374 7320 4c61 2061 6e64 204c 6220 lists La and Lb │ │ │ │ -000139b0: 7468 6973 2073 6372 6970 740a 636f 6e73 this script.cons │ │ │ │ -000139c0: 7472 7563 7473 2061 206d 6f64 756c 6520 tructs a module │ │ │ │ -000139d0: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ -000139e0: 2075 6e69 7665 7273 616c 2065 7874 656e universal exten │ │ │ │ -000139f0: 7369 6f6e 206f 6620 4528 4c62 2920 6279 sion of E(Lb) by │ │ │ │ -00013a00: 2045 284c 6129 2e20 4974 0a69 7320 6465 E(La). It.is de │ │ │ │ -00013a10: 6669 6e65 6420 6f6e 2074 6865 2070 726f fined on the pro │ │ │ │ -00013a20: 6475 6374 2076 6172 6965 7479 2045 7874 duct variety Ext │ │ │ │ -00013a30: 5e31 2845 284c 6129 2c20 4528 4c62 2929 ^1(E(La), E(Lb)) │ │ │ │ -00013a40: 2078 2024 5c50 505e 3124 2c20 616e 640a x $\PP^1$, and. │ │ │ │ -00013a50: 7265 7072 6573 656e 7465 6420 6865 7265 represented here │ │ │ │ -00013a60: 2062 7920 6120 6772 6164 6564 206d 6f64 by a graded mod │ │ │ │ -00013a70: 756c 6520 6f76 6572 2074 6865 2063 6f6f ule over the coo │ │ │ │ -00013a80: 7264 696e 6174 6520 7269 6e67 2053 203d rdinate ring S = │ │ │ │ -00013a90: 2041 5b79 5f30 2c79 5f31 5d20 6f66 0a74 A[y_0,y_1] of.t │ │ │ │ -00013aa0: 6869 7320 7661 7269 6574 793b 2068 6572 his variety; her │ │ │ │ -00013ab0: 6520 4120 6973 2074 6865 2063 6f6f 7264 e A is the coord │ │ │ │ -00013ac0: 696e 6174 6520 7269 6e67 206f 6620 4578 inate ring of Ex │ │ │ │ -00013ad0: 745e 3128 4528 4c61 292c 2045 284c 6229 t^1(E(La), E(Lb) │ │ │ │ -00013ae0: 292c 2077 6869 6368 2069 7320 610a 706f ), which is a.po │ │ │ │ -00013af0: 6c79 6e6f 6d69 616c 2072 696e 672e 0a0a lynomial ring... │ │ │ │ -00013b00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00013780: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +00013790: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +000137a0: 4520 3d20 756e 6976 6572 7361 6c45 7874 E = universalExt │ │ │ │ +000137b0: 656e 7369 6f6e 284c 612c 204c 6229 0a20 ension(La, Lb). │ │ │ │ +000137c0: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +000137d0: 202a 204c 612c 2061 202a 6e6f 7465 206c * La, a *note l │ │ │ │ +000137e0: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +000137f0: 6f63 294c 6973 742c 2c20 6f66 2069 6e74 oc)List,, of int │ │ │ │ +00013800: 6567 6572 730a 2020 2020 2020 2a20 4c62 egers. * Lb │ │ │ │ +00013810: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ +00013820: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ +00013830: 7374 2c2c 206f 6620 696e 7465 6765 7273 st,, of integers │ │ │ │ +00013840: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00013850: 2020 2020 2a20 452c 2061 202a 6e6f 7465 * E, a *note │ │ │ │ +00013860: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +00013870: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +00013880: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ +00013890: 2065 7874 656e 7369 6f6e 0a0a 4465 7363 extension..Desc │ │ │ │ +000138a0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +000138b0: 3d3d 3d0a 0a45 7665 7279 2076 6563 746f ===..Every vecto │ │ │ │ +000138c0: 7220 6275 6e64 6c65 2045 206f 6e20 245c r bundle E on $\ │ │ │ │ +000138d0: 5050 5e31 2420 7370 6c69 7473 2061 7320 PP^1$ splits as │ │ │ │ +000138e0: 6120 7375 6d20 6f66 206c 696e 6520 6275 a sum of line bu │ │ │ │ +000138f0: 6e64 6c65 7320 4f4f 2861 5f69 292e 2049 ndles OO(a_i). I │ │ │ │ +00013900: 6620 4c61 0a69 7320 6120 6c69 7374 206f f La.is a list o │ │ │ │ +00013910: 6620 696e 7465 6765 7273 2c20 7765 2077 f integers, we w │ │ │ │ +00013920: 7269 7465 2045 284c 6129 2066 6f72 2074 rite E(La) for t │ │ │ │ +00013930: 6865 2064 6972 6563 7420 7375 6d20 6f66 he direct sum of │ │ │ │ +00013940: 2074 6865 206c 696e 6520 6275 6e64 6c65 the line bundle │ │ │ │ +00013950: 0a4f 4f28 4c61 5f69 292e 2020 4769 7665 .OO(La_i). Give │ │ │ │ +00013960: 6e20 7477 6f20 7375 6368 2062 756e 646c n two such bundl │ │ │ │ +00013970: 6573 2073 7065 6369 6669 6564 2062 7920 es specified by │ │ │ │ +00013980: 7468 6520 6c69 7374 7320 4c61 2061 6e64 the lists La and │ │ │ │ +00013990: 204c 6220 7468 6973 2073 6372 6970 740a Lb this script. │ │ │ │ +000139a0: 636f 6e73 7472 7563 7473 2061 206d 6f64 constructs a mod │ │ │ │ +000139b0: 756c 6520 7265 7072 6573 656e 7469 6e67 ule representing │ │ │ │ +000139c0: 2074 6865 2075 6e69 7665 7273 616c 2065 the universal e │ │ │ │ +000139d0: 7874 656e 7369 6f6e 206f 6620 4528 4c62 xtension of E(Lb │ │ │ │ +000139e0: 2920 6279 2045 284c 6129 2e20 4974 0a69 ) by E(La). It.i │ │ │ │ +000139f0: 7320 6465 6669 6e65 6420 6f6e 2074 6865 s defined on the │ │ │ │ +00013a00: 2070 726f 6475 6374 2076 6172 6965 7479 product variety │ │ │ │ +00013a10: 2045 7874 5e31 2845 284c 6129 2c20 4528 Ext^1(E(La), E( │ │ │ │ +00013a20: 4c62 2929 2078 2024 5c50 505e 3124 2c20 Lb)) x $\PP^1$, │ │ │ │ +00013a30: 616e 640a 7265 7072 6573 656e 7465 6420 and.represented │ │ │ │ +00013a40: 6865 7265 2062 7920 6120 6772 6164 6564 here by a graded │ │ │ │ +00013a50: 206d 6f64 756c 6520 6f76 6572 2074 6865 module over the │ │ │ │ +00013a60: 2063 6f6f 7264 696e 6174 6520 7269 6e67 coordinate ring │ │ │ │ +00013a70: 2053 203d 2041 5b79 5f30 2c79 5f31 5d20 S = A[y_0,y_1] │ │ │ │ +00013a80: 6f66 0a74 6869 7320 7661 7269 6574 793b of.this variety; │ │ │ │ +00013a90: 2068 6572 6520 4120 6973 2074 6865 2063 here A is the c │ │ │ │ +00013aa0: 6f6f 7264 696e 6174 6520 7269 6e67 206f oordinate ring o │ │ │ │ +00013ab0: 6620 4578 745e 3128 4528 4c61 292c 2045 f Ext^1(E(La), E │ │ │ │ +00013ac0: 284c 6229 292c 2077 6869 6368 2069 7320 (Lb)), which is │ │ │ │ +00013ad0: 610a 706f 6c79 6e6f 6d69 616c 2072 696e a.polynomial rin │ │ │ │ +00013ae0: 672e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d g...+----------- │ │ │ │ +00013af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00013b50: 7c69 3120 3a20 4d20 3d20 756e 6976 6572 |i1 : M = univer │ │ │ │ -00013b60: 7361 6c45 7874 656e 7369 6f6e 287b 2d32 salExtension({-2 │ │ │ │ -00013b70: 7d2c 207b 327d 2920 2020 2020 2020 2020 }, {2}) │ │ │ │ -00013b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013b90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013ba0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013b30: 2d2d 2b0a 7c69 3120 3a20 4d20 3d20 756e --+.|i1 : M = un │ │ │ │ +00013b40: 6976 6572 7361 6c45 7874 656e 7369 6f6e iversalExtension │ │ │ │ +00013b50: 287b 2d32 7d2c 207b 327d 2920 2020 2020 ({-2}, {2}) │ │ │ │ +00013b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013b80: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00013b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013be0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013bf0: 7c6f 3120 3d20 636f 6b65 726e 656c 207b |o1 = cokernel { │ │ │ │ -00013c00: 322c 2030 7d20 7c20 785f 3020 785f 3120 2, 0} | x_0 x_1 │ │ │ │ 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|.|o1 : ---[x │ │ │ │ +00013e10: 2e2e 7820 5d5b 7920 2e2e 7920 5d2d 6d6f ..x ][y ..y ]-mo │ │ │ │ +00013e20: 6475 6c65 2c20 7175 6f74 6965 6e74 206f dule, quotient o │ │ │ │ +00013e30: 6620 282d 2d2d 5b78 202e 2e78 205d 5b79 f (---[x ..x ][y │ │ │ │ +00013e40: 202e 2e79 205d 2920 2020 2020 2020 2020 ..y ]) │ │ │ │ +00013e50: 2020 7c0a 7c20 2020 2020 3130 3120 2030 |.| 101 0 │ │ │ │ +00013e60: 2020 2032 2020 2030 2020 2031 2020 2020 2 0 1 │ │ │ │ +00013e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013e80: 2020 2031 3031 2020 3020 2020 3220 2020 101 0 2 │ │ │ │ +00013e90: 3020 2020 3120 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00013ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00013eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00013f10: 7c69 3220 3a20 4d20 3d20 756e 6976 6572 |i2 : M = univer │ │ │ │ -00013f20: 7361 6c45 7874 656e 7369 6f6e 287b 2d32 salExtension({-2 │ │ │ │ -00013f30: 2c2d 337d 2c20 7b32 2c33 7d29 2020 2020 ,-3}, {2,3}) │ │ │ │ -00013f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013f50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013f60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013ef0: 2d2d 2b0a 7c69 3220 3a20 4d20 3d20 756e --+.|i2 : M = un │ │ │ │ +00013f00: 6976 6572 7361 6c45 7874 656e 7369 6f6e iversalExtension │ │ │ │ +00013f10: 287b 2d32 2c2d 337d 2c20 7b32 2c33 7d29 ({-2,-3}, {2,3}) │ │ │ │ +00013f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013f40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 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│ │ │ │ -00014040: 2020 785f 3135 2020 2020 2020 2020 7c0a x_15 |. │ │ │ │ -00014050: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014060: 322c 2031 7d20 7c20 795f 3020 2020 2030 2, 1} | y_0 0 │ │ │ │ -00014070: 2020 2020 2020 3020 2020 2020 2030 2020 0 0 │ │ │ │ -00014080: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -00014090: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -000140a0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -000140b0: 322c 2031 7d20 7c20 795f 3120 2020 2079 2, 1} | y_1 y │ │ │ │ -000140c0: 5f30 2020 2020 3020 2020 2020 2030 2020 _0 0 0 │ │ │ │ -000140d0: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -000140e0: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -000140f0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014100: 322c 2031 7d20 7c20 3020 2020 2020 2079 2, 1} | 0 y │ │ │ │ -00014110: 5f31 2020 2020 795f 3020 2020 2030 2020 _1 y_0 0 │ │ │ │ -00014120: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ 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2020 3020 2020 2020 2030 0 0 0 │ │ │ │ +000140c0: 2020 2020 2020 3020 2020 2020 2020 2020 0 │ │ │ │ +000140d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000140e0: 2020 207b 322c 2031 7d20 7c20 3020 2020 {2, 1} | 0 │ │ │ │ +000140f0: 2020 2079 5f31 2020 2020 795f 3020 2020 y_1 y_0 │ │ │ │ +00014100: 2030 2020 2020 2020 3020 2020 2020 2030 0 0 0 │ │ │ │ +00014110: 2020 2020 2020 3020 2020 2020 2020 2020 0 │ │ │ │ +00014120: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00014130: 2020 207b 322c 2031 7d20 7c20 3020 2020 {2, 1} | 0 │ │ │ │ +00014140: 2020 2030 2020 2020 2020 795f 3120 2020 0 y_1 │ │ │ │ +00014150: 2079 5f30 2020 2020 3020 2020 2020 2030 y_0 0 0 │ │ │ │ +00014160: 2020 2020 2020 3020 2020 2020 2020 2020 0 │ │ │ │ +00014170: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00014180: 2020 207b 322c 2031 7d20 7c20 3020 2020 {2, 1} | 0 │ │ │ │ +00014190: 2020 2030 2020 2020 2020 3020 2020 2020 0 0 │ │ │ │ +000141a0: 2079 5f31 2020 2020 3020 2020 2020 2030 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+00014ad0: 616c 4578 7465 6e73 696f 6e3a 0a3d 3d3d alExtension:.=== │ │ │ │ +00014ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00014af0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +00014b00: 2a20 2275 6e69 7665 7273 616c 4578 7465 * "universalExte │ │ │ │ +00014b10: 6e73 696f 6e28 4c69 7374 2c4c 6973 7429 nsion(List,List) │ │ │ │ +00014b20: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +00014b30: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00014b40: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00014b50: 6a65 6374 202a 6e6f 7465 2075 6e69 7665 ject *note unive │ │ │ │ +00014b60: 7273 616c 4578 7465 6e73 696f 6e3a 2075 rsalExtension: u │ │ │ │ +00014b70: 6e69 7665 7273 616c 4578 7465 6e73 696f niversalExtensio │ │ │ │ +00014b80: 6e2c 2069 7320 6120 2a6e 6f74 6520 6d65 n, is a *note me │ │ │ │ +00014b90: 7468 6f64 0a66 756e 6374 696f 6e3a 2028 thod.function: ( │ │ │ │ +00014ba0: 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6174 .73043.Node: tat │ │ │ │ -00014e20: 6552 6573 6f6c 7574 696f 6e7f 3735 3835 eResolution.7585 │ │ │ │ -00014e30: 310a 4e6f 6465 3a20 756e 6976 6572 7361 1.Node: universa │ │ │ │ -00014e40: 6c45 7874 656e 7369 6f6e 7f37 3935 3734 lExtension.79574 │ │ │ │ -00014e50: 0a1f 0a45 6e64 2054 6167 2054 6162 6c65 ...End Tag Table │ │ │ │ -00014e60: 0a . │ │ │ │ +00014c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00014c10: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00014c20: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00014c30: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +00014c40: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +00014c50: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +00014c60: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +00014c70: 6b61 6765 732f 4247 472e 6d32 3a31 3131 kages/BGG.m2:111 │ │ │ │ +00014c80: 330a 3a30 2e0a 1f0a 5461 6720 5461 626c 3.:0....Tag Tabl │ │ │ │ +00014c90: 653a 0a4e 6f64 653a 2054 6f70 7f32 3339 e:.Node: Top.239 │ │ │ │ +00014ca0: 0a4e 6f64 653a 2062 6569 6c69 6e73 6f6e .Node: beilinson │ │ │ │ +00014cb0: 7f35 3938 300a 4e6f 6465 3a20 6267 677f .5980.Node: bgg. │ │ │ │ +00014cc0: 3138 3236 320a 4e6f 6465 3a20 636f 686f 18262.Node: coho │ │ │ │ +00014cd0: 6d6f 6c6f 6779 5461 626c 657f 3230 3736 mologyTable.2076 │ │ │ │ +00014ce0: 380a 4e6f 6465 3a20 6469 7265 6374 496d 8.Node: directIm │ │ │ │ +00014cf0: 6167 6543 6f6d 706c 6578 7f33 3030 3037 ageComplex.30007 │ │ │ │ +00014d00: 0a4e 6f64 653a 2064 6972 6563 7449 6d61 .Node: directIma │ │ │ │ +00014d10: 6765 436f 6d70 6c65 785f 6c70 436f 6d70 geComplex_lpComp │ │ │ │ +00014d20: 6c65 785f 7270 7f33 3134 3033 0a4e 6f64 lex_rp.31403.Nod │ │ │ │ +00014d30: 653a 2064 6972 6563 7449 6d61 6765 436f e: directImageCo │ │ │ │ +00014d40: 6d70 6c65 785f 6c70 4d61 7472 6978 5f72 mplex_lpMatrix_r │ │ │ │ +00014d50: 707f 3337 3839 310a 4e6f 6465 3a20 6469 p.37891.Node: di │ │ │ │ +00014d60: 7265 6374 496d 6167 6543 6f6d 706c 6578 rectImageComplex │ │ │ │ +00014d70: 5f6c 704d 6f64 756c 655f 7270 7f34 3433 _lpModule_rp.443 │ │ │ │ +00014d80: 3539 0a4e 6f64 653a 2045 7874 6572 696f 59.Node: Exterio │ │ │ │ +00014d90: 727f 3537 3537 350a 4e6f 6465 3a20 7072 r.57575.Node: pr │ │ │ │ +00014da0: 6f6a 6563 7469 7665 5072 6f64 7563 747f ojectiveProduct. │ │ │ │ +00014db0: 3538 3231 310a 4e6f 6465 3a20 7075 7265 58211.Node: pure │ │ │ │ +00014dc0: 5265 736f 6c75 7469 6f6e 7f35 3937 3435 Resolution.59745 │ │ │ │ +00014dd0: 0a4e 6f64 653a 2052 6567 756c 6172 6974 .Node: Regularit │ │ │ │ +00014de0: 797f 3732 3234 380a 4e6f 6465 3a20 7379 y.72248.Node: sy │ │ │ │ +00014df0: 6d45 7874 7f37 3330 3135 0a4e 6f64 653a mExt.73015.Node: │ │ │ │ +00014e00: 2074 6174 6552 6573 6f6c 7574 696f 6e7f tateResolution. │ │ │ │ +00014e10: 3735 3832 330a 4e6f 6465 3a20 756e 6976 75823.Node: univ │ │ │ │ +00014e20: 6572 7361 6c45 7874 656e 7369 6f6e 7f37 ersalExtension.7 │ │ │ │ +00014e30: 3935 3436 0a1f 0a45 6e64 2054 6167 2054 9546...End Tag T │ │ │ │ +00014e40: 6162 6c65 0a able. │ │ ├── ./usr/share/info/Benchmark.info.gz │ │ │ ├── Benchmark.info │ │ │ │ @@ -200,46 +200,46 @@ │ │ │ │ 00000c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00000c80: 3120 3a20 7275 6e42 656e 6368 6d61 726b 1 : runBenchmark │ │ │ │ 00000c90: 7320 2272 6573 3339 2220 2020 2020 2020 s "res39" │ │ │ │ 00000ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00000cd0: 2d20 6265 6769 6e6e 696e 6720 636f 6d70 - beginning comp │ │ │ │ -00000ce0: 7574 6174 696f 6e20 5361 7420 4f63 7420 utation Sat Oct │ │ │ │ -00000cf0: 3234 2030 333a 3230 3a30 3620 5554 4320 24 03:20:06 UTC │ │ │ │ +00000ce0: 7574 6174 696f 6e20 5468 7520 4f63 7420 utation Thu Oct │ │ │ │ +00000cf0: 3239 2031 323a 3131 3a31 3220 5554 4320 29 12:11:12 UTC │ │ │ │ 00000d00: 3230 3236 2020 2020 2020 2020 2020 2020 2026 │ │ │ │ 00000d10: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00000d20: 2d20 4c69 6e75 7820 7362 7569 6c64 2036 - Linux sbuild 6 │ │ │ │ -00000d30: 2e31 322e 3131 332b 6465 6231 332d 616d .12.113+deb13-am │ │ │ │ -00000d40: 6436 3420 2331 2053 4d50 2050 5245 454d d64 #1 SMP PREEM │ │ │ │ -00000d50: 5054 5f44 594e 414d 4943 2044 6562 6961 PT_DYNAMIC Debia │ │ │ │ -00000d60: 6e20 362e 3132 2e31 3133 2d31 7c0a 7c2d n 6.12.113-1|.|- │ │ │ │ -00000d70: 2d20 414d 4420 4550 5943 2037 3730 3250 - AMD EPYC 7702P │ │ │ │ -00000d80: 2036 342d 436f 7265 2050 726f 6365 7373 64-Core Process │ │ │ │ -00000d90: 6f72 2020 4175 7468 656e 7469 6341 4d44 or AuthenticAMD │ │ │ │ -00000da0: 2020 6370 7520 4d48 7a20 3139 3936 2e32 cpu MHz 1996.2 │ │ │ │ -00000db0: 3530 2020 2020 2020 2020 2020 7c0a 7c2d 50 |.|- │ │ │ │ +00000d30: 2e31 322e 3130 302b 6465 6231 332d 636c .12.100+deb13-cl │ │ │ │ +00000d40: 6f75 642d 616d 6436 3420 2331 2053 4d50 oud-amd64 #1 SMP │ │ │ │ +00000d50: 2050 5245 454d 5054 5f44 594e 414d 4943 PREEMPT_DYNAMIC │ │ │ │ +00000d60: 2044 6562 6961 6e20 2020 2020 7c0a 7c2d Debian |.|- │ │ │ │ +00000d70: 2d20 496e 7465 6c20 5865 6f6e 2050 726f - Intel Xeon Pro │ │ │ │ +00000d80: 6365 7373 6f72 2028 536b 796c 616b 652c cessor (Skylake, │ │ │ │ +00000d90: 2049 4252 5329 2020 4765 6e75 696e 6549 IBRS) GenuineI │ │ │ │ +00000da0: 6e74 656c 2020 6370 7520 4d48 7a20 3231 ntel cpu MHz 21 │ │ │ │ +00000db0: 3030 2e30 3238 2020 2020 2020 7c0a 7c2d 00.028 |.|- │ │ │ │ 00000dc0: 2d20 4d61 6361 756c 6179 3220 312e 3236 - Macaulay2 1.26 │ │ │ │ 00000dd0: 2e30 362c 2063 6f6d 7069 6c65 6420 7769 .06, compiled wi │ │ │ │ 00000de0: 7468 2067 6363 2031 362e 322e 3020 2020 th gcc 16.2.0 │ │ │ │ 00000df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000e00: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00000e10: 2d20 7265 7333 393a 2072 6573 206f 6620 - res39: res of │ │ │ │ 00000e20: 6120 6765 6e65 7269 6320 3320 6279 2039 a generic 3 by 9 │ │ │ │ 00000e30: 206d 6174 7269 7820 6f76 6572 205a 5a2f matrix over ZZ/ │ │ │ │ -00000e40: 3130 313a 202e 3330 3331 3520 7365 636f 101: .30315 seco │ │ │ │ -00000e50: 6e64 7320 2020 2020 2020 2020 7c0a 7c2d nds |.|- │ │ │ │ +00000e40: 3130 313a 202e 3134 3734 3236 2073 6563 101: .147426 sec │ │ │ │ +00000e50: 6f6e 6473 2020 2020 2020 2020 7c0a 7c2d onds |.|- │ │ │ │ 00000e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c28 ------------|.|( │ │ │ │ -00000eb0: 3230 3236 2d31 302d 3135 2920 7838 365f 2026-10-15) x86_ │ │ │ │ -00000ec0: 3634 2047 4e55 2f4c 696e 7578 2020 2020 64 GNU/Linux │ │ │ │ -00000ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c36 ------------|.|6 │ │ │ │ +00000eb0: 2e31 322e 3130 302d 3120 2832 3032 362d .12.100-1 (2026- │ │ │ │ +00000ec0: 3037 2d33 3029 2078 3836 5f36 3420 474e 07-30) x86_64 GN │ │ │ │ +00000ed0: 552f 4c69 6e75 7820 2020 2020 2020 2020 U/Linux │ │ │ │ 00000ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000ef0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00000f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 ------------+..F │ │ ├── ./usr/share/info/Bertini.info.gz │ │ │ ├── Bertini.info │ │ │ │ @@ -2253,16 +2253,16 @@ │ │ │ │ 00008cc0: 616c 206e 756d 6265 720a 2020 2020 2020 al number. │ │ │ │ 00008cd0: 2020 6f72 2072 616e 646f 6d20 636f 6d70 or random comp │ │ │ │ 00008ce0: 6c65 7820 6e75 6d62 6572 0a20 2020 2020 lex number. │ │ │ │ 00008cf0: 202a 202a 6e6f 7465 2054 6f70 4469 7265 * *note TopDire │ │ │ │ 00008d00: 6374 6f72 793a 2054 6f70 4469 7265 6374 ctory: TopDirect │ │ │ │ 00008d10: 6f72 792c 203d 3e20 2e2e 2e2c 2064 6566 ory, => ..., def │ │ │ │ 00008d20: 6175 6c74 2076 616c 7565 0a20 2020 2020 ault value. │ │ │ │ -00008d30: 2020 2022 2f74 6d70 2f4d 322d 3233 3636 "/tmp/M2-2366 │ │ │ │ -00008d40: 392d 302f 3022 2c20 4f70 7469 6f6e 2074 9-0/0", Option t │ │ │ │ +00008d30: 2020 2022 2f74 6d70 2f4d 322d 3330 3130 "/tmp/M2-3010 │ │ │ │ +00008d40: 322d 302f 3022 2c20 4f70 7469 6f6e 2074 2-0/0", 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 3336 3639 2d30 2f30 222c p/M2-23669-0/0", │ │ │ │ +00013710: 702f 4d32 2d33 3031 3032 2d30 2f30 222c p/M2-30102-0/0", │ │ │ │ 00013720: 204f 7074 696f 6e20 746f 2063 6861 6e67 Option to chang │ │ │ │ 00013730: 6520 6469 7265 6374 6f72 7920 666f 7220 e directory for │ │ │ │ 00013740: 6669 6c65 2073 746f 7261 6765 2e0a 2020 file storage.. │ │ │ │ 00013750: 2020 2020 2a20 2a6e 6f74 6520 5665 7262 * *note Verb │ │ │ │ 00013760: 6f73 653a 2062 6572 7469 6e69 5472 6163 ose: bertiniTrac │ │ │ │ 00013770: 6b48 6f6d 6f74 6f70 795f 6c70 5f70 645f kHomotopy_lp_pd_ │ │ │ │ 00013780: 7064 5f70 645f 636d 5665 7262 6f73 653d pd_pd_cmVerbose= │ │ │ │ @@ -5472,16 +5472,16 @@ │ │ │ │ 000155f0: 6561 6c20 6e75 6d62 6572 0a20 2020 2020 eal number. │ │ │ │ 00015600: 2020 206f 7220 7261 6e64 6f6d 2063 6f6d or random com │ │ │ │ 00015610: 706c 6578 206e 756d 6265 720a 2020 2020 plex number. │ │ │ │ 00015620: 2020 2a20 2a6e 6f74 6520 546f 7044 6972 * *note TopDir │ │ │ │ 00015630: 6563 746f 7279 3a20 546f 7044 6972 6563 ectory: TopDirec │ │ │ │ 00015640: 746f 7279 2c20 3d3e 202e 2e2e 2c20 6465 tory, => ..., de │ │ │ │ 00015650: 6661 756c 7420 7661 6c75 650a 2020 2020 fault value. │ │ │ │ -00015660: 2020 2020 222f 746d 702f 4d32 2d32 3336 "/tmp/M2-236 │ │ │ │ -00015670: 3639 2d30 2f30 222c 204f 7074 696f 6e20 69-0/0", Option │ │ │ │ +00015660: 2020 2020 222f 746d 702f 4d32 2d33 3031 "/tmp/M2-301 │ │ │ │ +00015670: 3032 2d30 2f30 222c 204f 7074 696f 6e20 02-0/0", Option │ │ │ │ 00015680: 746f 2063 6861 6e67 6520 6469 7265 6374 to change direct │ │ │ │ 00015690: 6f72 7920 666f 7220 6669 6c65 2073 746f ory for file sto │ │ │ │ 000156a0: 7261 6765 2e0a 2020 2020 2020 2a20 5573 rage.. * Us │ │ │ │ 000156b0: 6552 6567 656e 6572 6174 696f 6e20 286d eRegeneration (m │ │ │ │ 000156c0: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ 000156d0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ 000156e0: 6661 756c 7420 7661 6c75 6520 2d31 2c20 fault value -1, │ │ ├── ./usr/share/info/BettiCharacters.info.gz │ │ │ ├── BettiCharacters.info │ │ │ │ @@ -16843,15 +16843,15 @@ │ │ │ │ 00041ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00041cc0: 3920 3a20 656c 6170 7365 6454 696d 6520 9 : elapsedTime │ │ │ │ 00041cd0: 6320 3d20 6368 6172 6163 7465 7220 4120 c = character A │ │ │ │ 00041ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00041d10: 2d2d 202e 3434 3931 3832 7320 656c 6170 -- .449182s elap │ │ │ │ +00041d10: 2d2d 202e 3330 3431 3933 7320 656c 6170 -- .304193s elap │ │ │ │ 00041d20: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 00041d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00041d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -18058,16 +18058,16 @@ │ │ │ │ 00046890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000468a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000468b0: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 656c ------+.|i7 : el │ │ │ │ 000468c0: 6170 7365 6454 696d 6520 633d 6368 6172 apsedTime c=char │ │ │ │ 000468d0: 6163 7465 7220 4120 2020 2020 2020 2020 acter A │ │ │ │ 000468e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000468f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046900: 2020 2020 2020 7c0a 7c20 2d2d 2031 2e30 |.| -- 1.0 │ │ │ │ -00046910: 3632 3635 7320 656c 6170 7365 6420 2020 6265s elapsed │ │ │ │ +00046900: 2020 2020 2020 7c0a 7c20 2d2d 202e 3433 |.| -- .43 │ │ │ │ +00046910: 3037 3837 7320 656c 6170 7365 6420 2020 0787s elapsed │ │ │ │ 00046920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046950: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00046960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -19501,15 +19501,15 @@ │ │ │ │ 0004c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c2f0: 2b0a 7c69 3230 203a 2065 6c61 7073 6564 +.|i20 : elapsed │ │ │ │ 0004c300: 5469 6d65 2061 3120 3d20 6368 6172 6163 Time a1 = charac │ │ │ │ 0004c310: 7465 7220 4131 2020 2020 2020 2020 2020 ter A1 │ │ │ │ 0004c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c330: 2020 7c0a 7c20 2d2d 2031 2e30 3637 3132 |.| -- 1.06712 │ │ │ │ +0004c330: 2020 7c0a 7c20 2d2d 202e 3537 3333 3334 |.| -- .573334 │ │ │ │ 0004c340: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ 0004c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c370: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0004c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -19555,15 +19555,15 @@ │ │ │ │ 0004c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 ----------+.|i21 │ │ │ │ 0004c650: 203a 2065 6c61 7073 6564 5469 6d65 2061 : elapsedTime a │ │ │ │ 0004c660: 3220 3d20 6368 6172 6163 7465 7220 4132 2 = character A2 │ │ │ │ 0004c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c680: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004c690: 2d2d 2033 362e 3335 3339 7320 656c 6170 -- 36.3539s elap │ │ │ │ +0004c690: 2d2d 2032 322e 3237 3433 7320 656c 6170 -- 22.2743s elap │ │ │ │ 0004c6a0: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0004c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0004c6d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0004c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -20066,15 +20066,15 @@ │ │ │ │ 0004e610: 4f6e 4772 6164 6564 4d6f 6475 6c65 2020 OnGradedModule │ │ │ │ 0004e620: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0004e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004e650: 2d2b 0a7c 6933 3220 3a20 656c 6170 7365 -+.|i32 : elapse │ │ │ │ 0004e660: 6454 696d 6520 6220 3d20 6368 6172 6163 dTime b = charac │ │ │ │ 0004e670: 7465 7228 422c 3231 297c 0a7c 202d 2d20 ter(B,21)|.| -- │ │ │ │ -0004e680: 3136 2e34 3237 3773 2065 6c61 7073 6564 16.4277s elapsed │ │ │ │ +0004e680: 392e 3532 3738 3473 2065 6c61 7073 6564 9.52784s elapsed │ │ │ │ 0004e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004e6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0004e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004e6c0: 2020 2020 2020 2020 207c 0a7c 6f33 3220 |.|o32 │ │ │ │ 0004e6d0: 3d20 4368 6172 6163 7465 7220 6f76 6572 = Character over │ │ │ │ 0004e6e0: 206b 6b20 2020 2020 2020 2020 2020 2020 kk │ │ │ │ 0004e6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ ├── ./usr/share/info/Bruns.info.gz │ │ │ ├── Bruns.info │ │ │ │ @@ -1095,17 +1095,17 @@ │ │ │ │ 00004460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00004480: 6932 3320 3a20 7469 6d65 206a 3d62 7275 i23 : time j=bru │ │ │ │ 00004490: 6e73 2046 2e64 645f 333b 2020 2020 2020 ns F.dd_3; │ │ │ │ 000044a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000044b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000044c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000044d0: 202d 2d20 7573 6564 2030 2e34 3032 3236 -- used 0.40226 │ │ │ │ -000044e0: 3873 2028 6370 7529 3b20 302e 3234 3632 8s (cpu); 0.2462 │ │ │ │ -000044f0: 3439 7320 2874 6872 6561 6429 3b20 3073 49s (thread); 0s │ │ │ │ +000044d0: 202d 2d20 7573 6564 2030 2e33 3634 3534 -- used 0.36454 │ │ │ │ +000044e0: 3373 2028 6370 7529 3b20 302e 3233 3332 3s (cpu); 0.2332 │ │ │ │ +000044f0: 3231 7320 2874 6872 6561 6429 3b20 3073 21s (thread); 0s │ │ │ │ 00004500: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00004510: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00004520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004560: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ ├── ./usr/share/info/ChainComplexExtras.info.gz │ │ │ ├── ChainComplexExtras.info │ │ │ │ @@ -6049,17 +6049,17 @@ │ │ │ │ 00017a00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00017a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a30: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2074 ------+.|i13 : t │ │ │ │ 00017a40: 696d 6520 6d20 3d20 6d69 6e69 6d69 7a65 ime m = minimize │ │ │ │ 00017a50: 2028 455b 315d 293b 2020 2020 2020 2020 (E[1]); │ │ │ │ 00017a60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00017a70: 7c20 2d2d 2075 7365 6420 302e 3239 3635 | -- used 0.2965 │ │ │ │ -00017a80: 3733 7320 2863 7075 293b 2030 2e32 3137 73s (cpu); 0.217 │ │ │ │ -00017a90: 3036 3473 2028 7468 7265 6164 293b 2030 064s (thread); 0 │ │ │ │ +00017a70: 7c20 2d2d 2075 7365 6420 302e 3330 3336 | -- used 0.3036 │ │ │ │ +00017a80: 3836 7320 2863 7075 293b 2030 2e32 3234 86s (cpu); 0.224 │ │ │ │ +00017a90: 3130 3573 2028 7468 7265 6164 293b 2030 105s (thread); 0 │ │ │ │ 00017aa0: 7320 2867 6329 7c0a 2b2d 2d2d 2d2d 2d2d s (gc)|.+------- │ │ │ │ 00017ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00017ae0: 7c69 3134 203a 2069 7351 7561 7369 4973 |i14 : isQuasiIs │ │ │ │ 00017af0: 6f6d 6f72 7068 6973 6d20 6d20 2020 2020 omorphism m │ │ │ │ 00017b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7810,32 +7810,32 @@ │ │ │ │ 0001e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e820: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 -------+.|i8 : t │ │ │ │ 0001e830: 696d 6520 6d20 3d20 7265 736f 6c75 7469 ime m = resoluti │ │ │ │ 0001e840: 6f6e 4f66 4368 6169 6e43 6f6d 706c 6578 onOfChainComplex │ │ │ │ 0001e850: 2043 3b20 2020 2020 2020 2020 2020 2020 C; │ │ │ │ 0001e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e870: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0001e880: 6564 2030 2e30 3939 3530 3833 7320 2863 ed 0.0995083s (c │ │ │ │ -0001e890: 7075 293b 2030 2e30 3938 3736 3537 7320 pu); 0.0987657s │ │ │ │ +0001e880: 6564 2030 2e30 3931 3839 3536 7320 2863 ed 0.0918956s (c │ │ │ │ +0001e890: 7075 293b 2030 2e30 3933 3331 3938 7320 pu); 0.0933198s │ │ │ │ 0001e8a0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0001e8b0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0001e8c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0001e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e910: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 -------+.|i9 : t │ │ │ │ 0001e920: 696d 6520 6e20 3d20 6361 7274 616e 4569 ime n = cartanEi │ │ │ │ 0001e930: 6c65 6e62 6572 6752 6573 6f6c 7574 696f lenbergResolutio │ │ │ │ 0001e940: 6e20 433b 2020 2020 2020 2020 2020 2020 n C; │ │ │ │ 0001e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e960: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0001e970: 6564 2030 2e31 3133 3236 3173 2028 6370 ed 0.113261s (cp │ │ │ │ -0001e980: 7529 3b20 302e 3131 3439 7320 2874 6872 u); 0.1149s (thr │ │ │ │ -0001e990: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +0001e970: 6564 2030 2e31 3132 3830 3873 2028 6370 ed 0.112808s (cp │ │ │ │ +0001e980: 7529 3b20 302e 3131 3536 3334 7320 2874 u); 0.115634s (t │ │ │ │ +0001e990: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0001e9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0001e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea00: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ ├── ./usr/share/info/CharacteristicClasses.info.gz │ │ │ ├── CharacteristicClasses.info │ │ │ │ @@ -1215,16 +1215,16 @@ │ │ │ │ 00004be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004bf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ 00004c00: 2074 696d 6520 4353 4d20 5520 2020 2020 time CSM U │ │ │ │ 00004c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004c40: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00004c50: 7573 6564 2030 2e32 3732 3834 3173 2028 used 0.272841s ( │ │ │ │ -00004c60: 6370 7529 3b20 302e 3137 3635 3436 7320 cpu); 0.176546s │ │ │ │ +00004c50: 7573 6564 2030 2e32 3537 3037 3973 2028 used 0.257079s ( │ │ │ │ +00004c60: 6370 7529 3b20 302e 3137 3339 3332 7320 cpu); 0.173932s │ │ │ │ 00004c70: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00004c80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00004c90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1300,16 +1300,16 @@ │ │ │ │ 00005130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005140: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ 00005150: 2074 696d 6520 4353 4d28 552c 4368 6563 time CSM(U,Chec │ │ │ │ 00005160: 6b53 6d6f 6f74 683d 3e66 616c 7365 2920 kSmooth=>false) │ │ │ │ 00005170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005190: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000051a0: 7573 6564 2030 2e34 3134 3636 3273 2028 used 0.414662s ( │ │ │ │ -000051b0: 6370 7529 3b20 302e 3332 3231 3836 7320 cpu); 0.322186s │ │ │ │ +000051a0: 7573 6564 2030 2e33 3731 3632 3973 2028 used 0.371629s ( │ │ │ │ +000051b0: 6370 7529 3b20 302e 3238 3837 3531 7320 cpu); 0.288751s │ │ │ │ 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 3535 3336 3233 -- used 0.553623 │ │ │ │ -00010fc0: 7320 2863 7075 293b 2030 2e33 3833 3036 s (cpu); 0.38306 │ │ │ │ -00010fd0: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ +00010fb0: 2d2d 2075 7365 6420 302e 3836 3039 3433 -- used 0.860943 │ │ │ │ +00010fc0: 7320 2863 7075 293b 2030 2e33 3538 3133 s (cpu); 0.35813 │ │ │ │ +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 │ │ │ │ @@ -4400,16 +4400,16 @@ │ │ │ │ 000112f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011310: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520 ---+.|i6 : time │ │ │ │ 00011320: 4353 4d28 492c 436f 6d70 4d65 7468 6f64 CSM(I,CompMethod │ │ │ │ 00011330: 3d3e 506e 5265 7369 6475 616c 2920 2020 =>PnResidual) │ │ │ │ 00011340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011350: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00011360: 6420 322e 3331 3331 3373 2028 6370 7529 d 2.31313s (cpu) │ │ │ │ -00011370: 3b20 312e 3835 3435 3173 2028 7468 7265 ; 1.85451s (thre │ │ │ │ +00011360: 6420 312e 3935 3635 3873 2028 6370 7529 d 1.95658s (cpu) │ │ │ │ +00011370: 3b20 312e 3732 3732 3473 2028 7468 7265 ; 1.72724s (thre │ │ │ │ 00011380: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00011390: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000113a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000113d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000113e0: 2020 2020 2020 3520 2020 2020 2034 2020 5 4 │ │ │ │ @@ -4488,16 +4488,16 @@ │ │ │ │ 00011870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011890: 2d2d 2b0a 7c69 3130 203a 2074 696d 6520 --+.|i10 : time │ │ │ │ 000118a0: 4353 4d28 4b2c 436f 6d70 4d65 7468 6f64 CSM(K,CompMethod │ │ │ │ 000118b0: 3d3e 5072 6f6a 6563 7469 7665 4465 6772 =>ProjectiveDegr │ │ │ │ 000118c0: 6565 2920 2020 2020 2020 2020 2020 2020 ee) │ │ │ │ 000118d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000118e0: 2030 2e33 3032 3131 3273 2028 6370 7529 0.302112s (cpu) │ │ │ │ -000118f0: 3b20 302e 3230 3531 3735 7320 2874 6872 ; 0.205175s (thr │ │ │ │ +000118e0: 2030 2e32 3539 3837 3873 2028 6370 7529 0.259878s (cpu) │ │ │ │ +000118f0: 3b20 302e 3138 3135 3236 7320 2874 6872 ; 0.181526s (thr │ │ │ │ 00011900: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00011910: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00011920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011950: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00011960: 2020 2020 2020 3320 2020 2020 3220 2020 3 2 │ │ │ │ @@ -4546,18 +4546,18 @@ │ │ │ │ 00011c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00011c40: 3131 203a 2074 696d 6520 4353 4d28 4b2c 11 : time CSM(K, │ │ │ │ 00011c50: 436f 6d70 4d65 7468 6f64 3d3e 506e 5265 CompMethod=>PnRe │ │ │ │ 00011c60: 7369 6475 616c 2920 2020 2020 2020 2020 sidual) │ │ │ │ 00011c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011c80: 0a7c 202d 2d20 7573 6564 2030 2e30 3837 .| -- used 0.087 │ │ │ │ -00011c90: 3934 3638 7320 2863 7075 293b 2030 2e30 9468s (cpu); 0.0 │ │ │ │ -00011ca0: 3837 3935 3573 2028 7468 7265 6164 293b 87955s (thread); │ │ │ │ -00011cb0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +00011c80: 0a7c 202d 2d20 7573 6564 2030 2e30 3836 .| -- used 0.086 │ │ │ │ +00011c90: 3033 3632 7320 2863 7075 293b 2030 2e30 0362s (cpu); 0.0 │ │ │ │ +00011ca0: 3836 3034 3132 7320 2874 6872 6561 6429 860412s (thread) │ │ │ │ +00011cb0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00011cc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00011d10: 3320 2020 2020 3220 2020 2020 2020 2020 3 2 │ │ │ │ 00011d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5446,6790 +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 3339 3939 3339 7320 used 0.399939s │ │ │ │ -00015b00: 2863 7075 293b 2030 2e33 3130 3437 3373 (cpu); 0.310473s │ │ │ │ -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 2e30 3532 3532 -- used 1.05252 │ │ │ │ +00015ae0: 7320 2863 7075 293b 2030 2e33 3439 3836 s (cpu); 0.34986 │ │ │ │ +00015af0: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ +00015b00: 2867 6329 7c0a 7c20 2020 2020 2020 2020 (gc)|.| │ │ │ │ +00015b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015b30: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00015b40: 2020 2020 2020 3220 3220 2020 2020 3220 2 2 2 │ │ │ │ +00015b50: 2020 2020 2020 2020 3220 2020 2032 2020 2 2 │ │ │ │ +00015b60: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00015b70: 2020 7c0a 7c6f 3135 203d 2037 6820 6820 |.|o15 = 7h h │ │ │ │ +00015b80: 202b 2035 6820 6820 202b 2034 6820 6820 + 5h h + 4h h │ │ │ │ +00015b90: 202b 2068 2020 2b20 3368 2068 2020 2b20 + h + 3h h + │ │ │ │ +00015ba0: 6820 2020 2020 2020 207c 0a7c 2020 2020 h |.| │ │ │ │ +00015bb0: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ +00015bc0: 2020 2020 3120 3220 2020 2031 2020 2020 1 2 1 │ │ │ │ +00015bd0: 2031 2032 2020 2020 3220 2020 2020 2020 1 2 2 │ │ │ │ +00015be0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015c10: 2020 2020 2020 207c 0a7c 6f31 3520 3a20 |.|o15 : │ │ │ │ +00015c20: 4120 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ +00015c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015c40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00015c50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00015c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015c80: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 6373 -----+.|i16 : cs │ │ │ │ +00015c90: 6d4b 4861 7368 3d20 4353 4d28 412c 4b2c mKHash= CSM(A,K, │ │ │ │ +00015ca0: 4f75 7470 7574 3d3e 4861 7368 466f 726d Output=>HashForm │ │ │ │ +00015cb0: 2920 2020 2020 2020 2020 2020 7c0a 7c20 ) |.| │ │ │ │ +00015cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015cf0: 2020 207c 0a7c 6f31 3620 3d20 4d75 7461 |.|o16 = Muta │ │ │ │ +00015d00: 626c 6548 6173 6854 6162 6c65 7b2e 2e2e bleHashTable{... │ │ │ │ +00015d10: 342e 2e2e 7d20 2020 2020 2020 2020 2020 4...} │ │ │ │ +00015d20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00015d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00015d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015d80: 2020 2020 2020 2020 2020 7c0a 7c6f 3136 |.|o16 │ │ │ │ -00015d90: 203a 204d 7574 6162 6c65 4861 7368 5461 : MutableHashTa │ │ │ │ -00015da0: 626c 6520 2020 2020 2020 2020 2020 2020 ble │ │ │ │ -00015db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015dc0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00015dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 ----------+.|i17 │ │ │ │ -00015e00: 203a 2063 736d 4b3d 3d63 736d 4b48 6173 : csmK==csmKHas │ │ │ │ -00015e10: 6823 2243 534d 2220 2020 2020 2020 2020 h#"CSM" │ │ │ │ +00015d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015d60: 207c 0a7c 6f31 3620 3a20 4d75 7461 626c |.|o16 : Mutabl │ │ │ │ +00015d70: 6548 6173 6854 6162 6c65 2020 2020 2020 eHashTable │ │ │ │ +00015d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015d90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00015da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00015dd0: 0a7c 6931 3720 3a20 6373 6d4b 3d3d 6373 .|i17 : csmK==cs │ │ │ │ +00015de0: 6d4b 4861 7368 2322 4353 4d22 2020 2020 mKHash#"CSM" │ │ │ │ +00015df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015e00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00015e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015e30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00015e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015e30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00015e40: 6f31 3720 3d20 7472 7565 2020 2020 2020 o17 = true │ │ │ │ 00015e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015e60: 2020 2020 2020 2020 2020 7c0a 7c6f 3137 |.|o17 │ │ │ │ -00015e70: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ -00015e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00015eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 ----------+.|i18 │ │ │ │ -00015ee0: 203a 2043 534d 2841 2c69 6465 616c 284b : CSM(A,ideal(K │ │ │ │ -00015ef0: 5f30 2929 3d3d 6373 6d4b 4861 7368 237b _0))==csmKHash#{ │ │ │ │ -00015f00: 307d 2020 2020 2020 2020 2020 2020 2020 0} │ │ │ │ -00015f10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00015f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015e70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00015e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00015eb0: 3820 3a20 4353 4d28 412c 6964 6561 6c28 8 : CSM(A,ideal( │ │ │ │ +00015ec0: 4b5f 3029 293d 3d63 736d 4b48 6173 6823 K_0))==csmKHash# │ │ │ │ +00015ed0: 7b30 7d20 2020 2020 2020 2020 2020 2020 {0} │ │ │ │ +00015ee0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015f10: 2020 2020 2020 2020 207c 0a7c 6f31 3820 |.|o18 │ │ │ │ +00015f20: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ 00015f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015f40: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ -00015f50: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ -00015f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015f80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00015f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 7570 ----------+..Sup │ │ │ │ -00015fc0: 706f 7365 2077 6520 6861 7665 2061 6c72 pose we have alr │ │ │ │ -00015fd0: 6561 6479 2063 6f6d 7075 7465 6420 736f eady computed so │ │ │ │ -00015fe0: 6d65 206f 6620 4353 4d20 636c 6173 7365 me of CSM classe │ │ │ │ -00015ff0: 7320 6f66 2068 7970 6572 7375 7266 6163 s of hypersurfac │ │ │ │ -00016000: 6573 2069 6e76 6f6c 7665 640a 696e 2074 es involved.in t │ │ │ │ -00016010: 6865 2069 6e63 6c75 7369 6f6e 2d65 7863 he inclusion-exc │ │ │ │ -00016020: 6c75 7369 6f6e 2070 726f 6365 6475 7265 lusion procedure │ │ │ │ -00016030: 2c20 7468 656e 2077 6520 6d61 7920 696e , then we may in │ │ │ │ -00016040: 7075 7420 7468 6573 6520 746f 2062 6520 put these to be │ │ │ │ -00016050: 7573 6564 2062 7920 7468 650a 4353 4d20 used by the.CSM │ │ │ │ -00016060: 6675 6e63 7469 6f6e 2e20 496e 2074 6865 function. In the │ │ │ │ -00016070: 2065 7861 6d70 6c65 2062 656c 6f77 2077 example below w │ │ │ │ -00016080: 6520 696e 7075 7420 7468 6520 4353 4d20 e input the CSM │ │ │ │ -00016090: 636c 6173 7320 6f66 2056 284b 5f30 2920 class of V(K_0) │ │ │ │ -000160a0: 2874 6861 7420 6973 206f 660a 7468 6520 (that is of.the │ │ │ │ -000160b0: 6879 7065 7273 7572 6661 6365 2064 6566 hypersurface def │ │ │ │ -000160c0: 696e 6564 2062 7920 7468 6520 6669 7273 ined by the firs │ │ │ │ -000160d0: 7420 706f 6c79 6e6f 6d69 616c 2067 656e t polynomial gen │ │ │ │ -000160e0: 6572 6174 696e 6720 4b29 2061 6e64 2074 erating K) and t │ │ │ │ -000160f0: 6865 2043 534d 0a63 6c61 7373 206f 6620 he CSM.class of │ │ │ │ -00016100: 7468 6520 6879 7065 7273 7572 6661 6365 the hypersurface │ │ │ │ -00016110: 2064 6566 696e 6564 2062 7920 7468 6520 defined by the │ │ │ │ -00016120: 7072 6f64 7563 7420 6f66 2074 6865 2067 product of the g │ │ │ │ -00016130: 656e 6572 6174 6f72 7320 6f66 204b 2e0a enerators of K.. │ │ │ │ -00016140: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00016150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00016180: 6931 3920 3a20 6d3d 6e65 7720 4d75 7461 i19 : m=new Muta │ │ │ │ -00016190: 626c 6548 6173 6854 6162 6c65 3b20 2020 bleHashTable; │ │ │ │ -000161a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000161b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000161d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000161e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000161f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 ---------+.|i20 │ │ │ │ -00016200: 3a20 6d23 7b30 7d3d 6373 6d4b 4861 7368 : m#{0}=csmKHash │ │ │ │ -00016210: 237b 307d 2020 2020 2020 2020 2020 2020 #{0} │ │ │ │ +00015f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015f50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00015f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015f80: 2d2d 2d2d 2d2d 2d2b 0a0a 5375 7070 6f73 -------+..Suppos │ │ │ │ +00015f90: 6520 7765 2068 6176 6520 616c 7265 6164 e we have alread │ │ │ │ +00015fa0: 7920 636f 6d70 7574 6564 2073 6f6d 6520 y computed some │ │ │ │ +00015fb0: 6f66 2043 534d 2063 6c61 7373 6573 206f of CSM classes o │ │ │ │ +00015fc0: 6620 6879 7065 7273 7572 6661 6365 7320 f hypersurfaces │ │ │ │ +00015fd0: 696e 766f 6c76 6564 0a69 6e20 7468 6520 involved.in the │ │ │ │ +00015fe0: 696e 636c 7573 696f 6e2d 6578 636c 7573 inclusion-exclus │ │ │ │ +00015ff0: 696f 6e20 7072 6f63 6564 7572 652c 2074 ion procedure, t │ │ │ │ +00016000: 6865 6e20 7765 206d 6179 2069 6e70 7574 hen we may input │ │ │ │ +00016010: 2074 6865 7365 2074 6f20 6265 2075 7365 these to be use │ │ │ │ +00016020: 6420 6279 2074 6865 0a43 534d 2066 756e d by the.CSM fun │ │ │ │ +00016030: 6374 696f 6e2e 2049 6e20 7468 6520 6578 ction. In the ex │ │ │ │ +00016040: 616d 706c 6520 6265 6c6f 7720 7765 2069 ample below we i │ │ │ │ +00016050: 6e70 7574 2074 6865 2043 534d 2063 6c61 nput the CSM cla │ │ │ │ +00016060: 7373 206f 6620 5628 4b5f 3029 2028 7468 ss of V(K_0) (th │ │ │ │ +00016070: 6174 2069 7320 6f66 0a74 6865 2068 7970 at is of.the hyp │ │ │ │ +00016080: 6572 7375 7266 6163 6520 6465 6669 6e65 ersurface define │ │ │ │ +00016090: 6420 6279 2074 6865 2066 6972 7374 2070 d by the first p │ │ │ │ +000160a0: 6f6c 796e 6f6d 6961 6c20 6765 6e65 7261 olynomial genera │ │ │ │ +000160b0: 7469 6e67 204b 2920 616e 6420 7468 6520 ting K) and the │ │ │ │ +000160c0: 4353 4d0a 636c 6173 7320 6f66 2074 6865 CSM.class of the │ │ │ │ +000160d0: 2068 7970 6572 7375 7266 6163 6520 6465 hypersurface de │ │ │ │ +000160e0: 6669 6e65 6420 6279 2074 6865 2070 726f fined by the pro │ │ │ │ +000160f0: 6475 6374 206f 6620 7468 6520 6765 6e65 duct of the gene │ │ │ │ +00016100: 7261 746f 7273 206f 6620 4b2e 0a0a 2b2d rators of K...+- │ │ │ │ +00016110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016140: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139 ----------+.|i19 │ │ │ │ +00016150: 203a 206d 3d6e 6577 204d 7574 6162 6c65 : m=new Mutable │ │ │ │ +00016160: 4861 7368 5461 626c 653b 2020 2020 2020 HashTable; │ │ │ │ +00016170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016180: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00016190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000161a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000161b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000161c0: 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a 206d ------+.|i20 : m │ │ │ │ +000161d0: 237b 307d 3d63 736d 4b48 6173 6823 7b30 #{0}=csmKHash#{0 │ │ │ │ +000161e0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +000161f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016200: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016230: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00016240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016270: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00016280: 3220 3220 2020 2020 3220 2020 2020 2020 2 2 2 │ │ │ │ -00016290: 2020 3220 2020 2020 3220 2020 2020 2020 2 2 │ │ │ │ -000162a0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -000162b0: 2020 207c 0a7c 6f32 3020 3d20 3868 2068 |.|o20 = 8h h │ │ │ │ -000162c0: 2020 2b20 3768 2068 2020 2b20 3668 2068 + 7h h + 6h h │ │ │ │ -000162d0: 2020 2b20 3268 2020 2b20 3568 2068 2020 + 2h + 5h h │ │ │ │ -000162e0: 2b20 3268 2020 2b20 3268 2020 2b20 6820 + 2h + 2h + h │ │ │ │ -000162f0: 207c 0a7c 2020 2020 2020 2020 3120 3220 |.| 1 2 │ │ │ │ -00016300: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ -00016310: 2020 2020 3120 2020 2020 3120 3220 2020 1 1 2 │ │ │ │ -00016320: 2020 3220 2020 2020 3120 2020 2032 207c 2 1 2 | │ │ │ │ -00016330: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00016340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016240: 2020 7c0a 7c20 2020 2020 2020 2032 2032 |.| 2 2 │ │ │ │ +00016250: 2020 2020 2032 2020 2020 2020 2020 2032 2 2 │ │ │ │ +00016260: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00016270: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00016280: 7c0a 7c6f 3230 203d 2038 6820 6820 202b |.|o20 = 8h h + │ │ │ │ +00016290: 2037 6820 6820 202b 2036 6820 6820 202b 7h h + 6h h + │ │ │ │ +000162a0: 2032 6820 202b 2035 6820 6820 202b 2032 2h + 5h h + 2 │ │ │ │ +000162b0: 6820 202b 2032 6820 202b 2068 2020 7c0a h + 2h + h |. │ │ │ │ +000162c0: 7c20 2020 2020 2020 2031 2032 2020 2020 | 1 2 │ │ │ │ +000162d0: 2031 2032 2020 2020 2031 2032 2020 2020 1 2 1 2 │ │ │ │ +000162e0: 2031 2020 2020 2031 2032 2020 2020 2032 1 1 2 2 │ │ │ │ +000162f0: 2020 2020 2031 2020 2020 3220 7c0a 7c20 1 2 |.| │ │ │ │ +00016300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016330: 2020 2020 2020 2020 2020 7c0a 7c6f 3230 |.|o20 │ │ │ │ +00016340: 203a 2041 2020 2020 2020 2020 2020 2020 : A │ │ │ │ 00016350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00016370: 6f32 3020 3a20 4120 2020 2020 2020 2020 o20 : A │ │ │ │ -00016380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000163a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000163b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000163c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000163d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000163e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 ---------+.|i21 │ │ │ │ -000163f0: 3a20 6d23 7b30 2c31 7d3d 6373 6d4b 4861 : m#{0,1}=csmKHa │ │ │ │ -00016400: 7368 237b 302c 317d 2020 2020 2020 2020 sh#{0,1} │ │ │ │ +00016360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00016380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000163a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000163b0: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 206d ------+.|i21 : m │ │ │ │ +000163c0: 237b 302c 317d 3d63 736d 4b48 6173 6823 #{0,1}=csmKHash# │ │ │ │ +000163d0: 7b30 2c31 7d20 2020 2020 2020 2020 2020 {0,1} │ │ │ │ +000163e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000163f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016420: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00016430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016460: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00016470: 3220 3220 2020 2020 3220 2020 2020 2020 2 2 2 │ │ │ │ -00016480: 2020 3220 2020 2020 3220 2020 2020 2020 2 2 │ │ │ │ -00016490: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -000164a0: 2020 207c 0a7c 6f32 3120 3d20 3968 2068 |.|o21 = 9h h │ │ │ │ -000164b0: 2020 2b20 3968 2068 2020 2b20 3968 2068 + 9h h + 9h h │ │ │ │ -000164c0: 2020 2b20 3368 2020 2b20 3768 2068 2020 + 3h + 7h h │ │ │ │ -000164d0: 2b20 3368 2020 2b20 3368 2020 2b20 3268 + 3h + 3h + 2h │ │ │ │ -000164e0: 207c 0a7c 2020 2020 2020 2020 3120 3220 |.| 1 2 │ │ │ │ -000164f0: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ -00016500: 2020 2020 3120 2020 2020 3120 3220 2020 1 1 2 │ │ │ │ -00016510: 2020 3220 2020 2020 3120 2020 2020 327c 2 1 2| │ │ │ │ -00016520: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00016530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016430: 2020 7c0a 7c20 2020 2020 2020 2032 2032 |.| 2 2 │ │ │ │ +00016440: 2020 2020 2032 2020 2020 2020 2020 2032 2 2 │ │ │ │ +00016450: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00016460: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00016470: 7c0a 7c6f 3231 203d 2039 6820 6820 202b |.|o21 = 9h h + │ │ │ │ +00016480: 2039 6820 6820 202b 2039 6820 6820 202b 9h h + 9h h + │ │ │ │ +00016490: 2033 6820 202b 2037 6820 6820 202b 2033 3h + 7h h + 3 │ │ │ │ +000164a0: 6820 202b 2033 6820 202b 2032 6820 7c0a h + 3h + 2h |. │ │ │ │ +000164b0: 7c20 2020 2020 2020 2031 2032 2020 2020 | 1 2 │ │ │ │ +000164c0: 2031 2032 2020 2020 2031 2032 2020 2020 1 2 1 2 │ │ │ │ +000164d0: 2031 2020 2020 2031 2032 2020 2020 2032 1 1 2 2 │ │ │ │ +000164e0: 2020 2020 2031 2020 2020 2032 7c0a 7c20 1 2|.| │ │ │ │ +000164f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016520: 2020 2020 2020 2020 2020 7c0a 7c6f 3231 |.|o21 │ │ │ │ +00016530: 203a 2041 2020 2020 2020 2020 2020 2020 : A │ │ │ │ 00016540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016550: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00016560: 6f32 3120 3a20 4120 2020 2020 2020 2020 o21 : A │ │ │ │ -00016570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016590: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000165a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000165b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000165c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000165d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220 ---------+.|i22 │ │ │ │ -000165e0: 3a20 7469 6d65 2043 534d 2841 2c4b 2c6d : time CSM(A,K,m │ │ │ │ -000165f0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00016600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016610: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00016620: 6564 2030 2e31 3832 3638 3673 2028 6370 ed 0.182686s (cp │ │ │ │ -00016630: 7529 3b20 302e 3130 3736 3831 7320 2874 u); 0.107681s (t │ │ │ │ -00016640: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ -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 3139 3436 3231 7320 2863 7075 293b 0.194621s (cpu); │ │ │ │ +00016600: 2030 2e30 3838 3131 3338 7320 2874 6872 0.0881138s (thr │ │ │ │ +00016610: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00016620: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00016630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016660: 7c0a 7c20 2020 2020 2020 2032 2032 2020 |.| 2 2 │ │ │ │ +00016670: 2020 2032 2020 2020 2020 2020 2032 2020 2 2 │ │ │ │ +00016680: 2020 3220 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ +00016690: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000166a0: 7c6f 3232 203d 2037 6820 6820 202b 2035 |o22 = 7h h + 5 │ │ │ │ +000166b0: 6820 6820 202b 2034 6820 6820 202b 2068 h h + 4h h + h │ │ │ │ +000166c0: 2020 2b20 3368 2068 2020 2b20 6820 2020 + 3h h + h │ │ │ │ +000166d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000166e0: 2020 2020 2020 2031 2032 2020 2020 2031 1 2 1 │ │ │ │ +000166f0: 2032 2020 2020 2031 2032 2020 2020 3120 2 1 2 1 │ │ │ │ +00016700: 2020 2020 3120 3220 2020 2032 2020 2020 1 2 2 │ │ │ │ +00016710: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00016720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016750: 2020 2020 2020 2020 7c0a 7c6f 3232 203a |.|o22 : │ │ │ │ +00016760: 2041 2020 2020 2020 2020 2020 2020 2020 A │ │ │ │ 00016770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016780: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00016790: 3220 3a20 4120 2020 2020 2020 2020 2020 2 : A │ │ │ │ -000167a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000167b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000167c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000167d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000167e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000167f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016800: 2d2d 2d2d 2d2d 2d2b 0a0a 496e 2074 6865 -------+..In the │ │ │ │ -00016810: 2063 6173 6520 7768 6572 6520 7468 6520 case where the │ │ │ │ -00016820: 616d 6269 656e 7420 7370 6163 6520 6973 ambient space is │ │ │ │ -00016830: 2061 2074 6f72 6963 2076 6172 6965 7479 a toric variety │ │ │ │ -00016840: 2077 6869 6368 2069 7320 6e6f 7420 6120 which is not a │ │ │ │ -00016850: 7072 6f64 7563 740a 6f66 2070 726f 6a65 product.of proje │ │ │ │ -00016860: 6374 6976 6520 7370 6163 6573 2077 6520 ctive spaces we │ │ │ │ -00016870: 6d75 7374 206c 6f61 6420 7468 6520 4e6f must load the No │ │ │ │ -00016880: 726d 616c 546f 7269 6356 6172 6965 7469 rmalToricVarieti │ │ │ │ -00016890: 6573 2070 6163 6b61 6765 2061 6e64 206d es package and m │ │ │ │ -000168a0: 7573 740a 616c 736f 2069 6e70 7574 2074 ust.also input t │ │ │ │ -000168b0: 6865 2074 6f72 6963 2076 6172 6965 7479 he toric variety │ │ │ │ -000168c0: 2e20 4966 2074 6865 2074 6f72 6963 2076 . If the toric v │ │ │ │ -000168d0: 6172 6965 7479 2069 7320 6120 7072 6f64 ariety is a prod │ │ │ │ -000168e0: 7563 7420 6f66 2070 726f 6a65 6374 6976 uct of projectiv │ │ │ │ -000168f0: 650a 7370 6163 6520 6974 2069 7320 7265 e.space it is re │ │ │ │ -00016900: 636f 6d6d 656e 6420 746f 2075 7365 2074 commend to use t │ │ │ │ -00016910: 6865 2066 6f72 6d20 6162 6f76 6520 7261 he form above ra │ │ │ │ -00016920: 7468 6572 2074 6861 6e20 696e 7075 7474 ther than inputt │ │ │ │ -00016930: 696e 6720 7468 6520 746f 7269 630a 7661 ing the toric.va │ │ │ │ -00016940: 7269 6574 7920 666f 7220 6566 6669 6369 riety for effici │ │ │ │ -00016950: 656e 6379 2072 6561 736f 6e73 2e0a 0a2b ency reasons...+ │ │ │ │ +00016780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016790: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +000167a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000167b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000167c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000167d0: 2d2d 2d2d 2b0a 0a49 6e20 7468 6520 6361 ----+..In the ca │ │ │ │ +000167e0: 7365 2077 6865 7265 2074 6865 2061 6d62 se where the amb │ │ │ │ +000167f0: 6965 6e74 2073 7061 6365 2069 7320 6120 ient space is a │ │ │ │ +00016800: 746f 7269 6320 7661 7269 6574 7920 7768 toric variety wh │ │ │ │ +00016810: 6963 6820 6973 206e 6f74 2061 2070 726f ich is not a pro │ │ │ │ +00016820: 6475 6374 0a6f 6620 7072 6f6a 6563 7469 duct.of projecti │ │ │ │ +00016830: 7665 2073 7061 6365 7320 7765 206d 7573 ve spaces we mus │ │ │ │ +00016840: 7420 6c6f 6164 2074 6865 204e 6f72 6d61 t load the Norma │ │ │ │ +00016850: 6c54 6f72 6963 5661 7269 6574 6965 7320 lToricVarieties │ │ │ │ +00016860: 7061 636b 6167 6520 616e 6420 6d75 7374 package and must │ │ │ │ +00016870: 0a61 6c73 6f20 696e 7075 7420 7468 6520 .also input the │ │ │ │ +00016880: 746f 7269 6320 7661 7269 6574 792e 2049 toric variety. I │ │ │ │ +00016890: 6620 7468 6520 746f 7269 6320 7661 7269 f the toric vari │ │ │ │ +000168a0: 6574 7920 6973 2061 2070 726f 6475 6374 ety is a product │ │ │ │ +000168b0: 206f 6620 7072 6f6a 6563 7469 7665 0a73 of projective.s │ │ │ │ +000168c0: 7061 6365 2069 7420 6973 2072 6563 6f6d pace it is recom │ │ │ │ +000168d0: 6d65 6e64 2074 6f20 7573 6520 7468 6520 mend to use the │ │ │ │ +000168e0: 666f 726d 2061 626f 7665 2072 6174 6865 form above rathe │ │ │ │ +000168f0: 7220 7468 616e 2069 6e70 7574 7469 6e67 r than inputting │ │ │ │ +00016900: 2074 6865 2074 6f72 6963 0a76 6172 6965 the toric.varie │ │ │ │ +00016910: 7479 2066 6f72 2065 6666 6963 6965 6e63 ty for efficienc │ │ │ │ +00016920: 7920 7265 6173 6f6e 732e 0a0a 2b2d 2d2d y reasons...+--- │ │ │ │ +00016930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000169a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a --------+.|i23 : │ │ │ │ -000169b0: 206e 6565 6473 5061 636b 6167 6520 224e needsPackage "N │ │ │ │ -000169c0: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ -000169d0: 6965 7322 2020 2020 2020 2020 2020 2020 ies" │ │ │ │ +00016970: 2d2d 2d2d 2d2b 0a7c 6932 3320 3a20 6e65 -----+.|i23 : ne │ │ │ │ +00016980: 6564 7350 6163 6b61 6765 2022 4e6f 726d edsPackage "Norm │ │ │ │ +00016990: 616c 546f 7269 6356 6172 6965 7469 6573 alToricVarieties │ │ │ │ +000169a0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ +000169b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000169c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000169d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000169e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000169f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00016a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016a30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00016a40: 7c6f 3233 203d 204e 6f72 6d61 6c54 6f72 |o23 = NormalTor │ │ │ │ -00016a50: 6963 5661 7269 6574 6965 7320 2020 2020 icVarieties │ │ │ │ +000169f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016a00: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +00016a10: 3320 3d20 4e6f 726d 616c 546f 7269 6356 3 = NormalToricV │ │ │ │ +00016a20: 6172 6965 7469 6573 2020 2020 2020 2020 arieties │ │ │ │ +00016a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016a50: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00016a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016a80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00016a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016aa0: 207c 0a7c 6f32 3320 3a20 5061 636b 6167 |.|o23 : Packag │ │ │ │ +00016ab0: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ 00016ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ad0: 2020 2020 7c0a 7c6f 3233 203a 2050 6163 |.|o23 : Pac │ │ │ │ -00016ae0: 6b61 6765 2020 2020 2020 2020 2020 2020 kage │ │ │ │ -00016af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016b10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016b20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00016b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234 ----------+.|i24 │ │ │ │ -00016b70: 203a 2052 686f 203d 207b 7b31 2c30 2c30 : Rho = {{1,0,0 │ │ │ │ -00016b80: 7d2c 7b30 2c31 2c30 7d2c 7b30 2c30 2c31 },{0,1,0},{0,0,1 │ │ │ │ -00016b90: 7d2c 7b2d 312c 2d31 2c30 7d2c 7b30 2c30 },{-1,-1,0},{0,0 │ │ │ │ -00016ba0: 2c2d 317d 7d20 2020 2020 2020 2020 2020 ,-1}} │ │ │ │ -00016bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00016bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016c00: 7c0a 7c6f 3234 203d 207b 7b31 2c20 302c |.|o24 = {{1, 0, │ │ │ │ -00016c10: 2030 7d2c 207b 302c 2031 2c20 307d 2c20 0}, {0, 1, 0}, │ │ │ │ -00016c20: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20 {0, 0, 1}, {-1, │ │ │ │ -00016c30: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d -1, 0}, {0, 0, - │ │ │ │ -00016c40: 317d 7d20 2020 2020 2020 207c 0a7c 2020 1}} |.| │ │ │ │ +00016ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016ae0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00016af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016b30: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20 -------+.|i24 : │ │ │ │ +00016b40: 5268 6f20 3d20 7b7b 312c 302c 307d 2c7b Rho = {{1,0,0},{ │ │ │ │ +00016b50: 302c 312c 307d 2c7b 302c 302c 317d 2c7b 0,1,0},{0,0,1},{ │ │ │ │ +00016b60: 2d31 2c2d 312c 307d 2c7b 302c 302c 2d31 -1,-1,0},{0,0,-1 │ │ │ │ +00016b70: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ +00016b80: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00016b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00016bd0: 6f32 3420 3d20 7b7b 312c 2030 2c20 307d o24 = {{1, 0, 0} │ │ │ │ +00016be0: 2c20 7b30 2c20 312c 2030 7d2c 207b 302c , {0, 1, 0}, {0, │ │ │ │ +00016bf0: 2030 2c20 317d 2c20 7b2d 312c 202d 312c 0, 1}, {-1, -1, │ │ │ │ +00016c00: 2030 7d2c 207b 302c 2030 2c20 2d31 7d7d 0}, {0, 0, -1}} │ │ │ │ +00016c10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00016c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016c60: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374 |.|o24 : List │ │ │ │ 00016c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016c90: 2020 2020 2020 7c0a 7c6f 3234 203a 204c |.|o24 : L │ │ │ │ -00016ca0: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ -00016cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ce0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00016cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00016d30: 3235 203a 2053 6967 6d61 203d 207b 7b30 25 : Sigma = {{0 │ │ │ │ -00016d40: 2c31 2c32 7d2c 7b31 2c32 2c33 7d2c 7b30 ,1,2},{1,2,3},{0 │ │ │ │ -00016d50: 2c32 2c33 7d2c 7b30 2c31 2c34 7d2c 7b31 ,2,3},{0,1,4},{1 │ │ │ │ -00016d60: 2c33 2c34 7d2c 7b30 2c33 2c34 7d7d 2020 ,3,4},{0,3,4}} │ │ │ │ -00016d70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00016d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016dc0: 2020 7c0a 7c6f 3235 203d 207b 7b30 2c20 |.|o25 = {{0, │ │ │ │ -00016dd0: 312c 2032 7d2c 207b 312c 2032 2c20 337d 1, 2}, {1, 2, 3} │ │ │ │ -00016de0: 2c20 7b30 2c20 322c 2033 7d2c 207b 302c , {0, 2, 3}, {0, │ │ │ │ -00016df0: 2031 2c20 347d 2c20 7b31 2c20 332c 2034 1, 4}, {1, 3, 4 │ │ │ │ -00016e00: 7d2c 207b 302c 2033 2c20 347d 7d7c 0a7c }, {0, 3, 4}}|.| │ │ │ │ +00016c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016ca0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00016cb0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00016cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016cf0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3520 ---------+.|i25 │ │ │ │ +00016d00: 3a20 5369 676d 6120 3d20 7b7b 302c 312c : Sigma = {{0,1, │ │ │ │ +00016d10: 327d 2c7b 312c 322c 337d 2c7b 302c 322c 2},{1,2,3},{0,2, │ │ │ │ +00016d20: 337d 2c7b 302c 312c 347d 2c7b 312c 332c 3},{0,1,4},{1,3, │ │ │ │ +00016d30: 347d 2c7b 302c 332c 347d 7d20 2020 2020 4},{0,3,4}} │ │ │ │ +00016d40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016d80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00016d90: 0a7c 6f32 3520 3d20 7b7b 302c 2031 2c20 .|o25 = {{0, 1, │ │ │ │ +00016da0: 327d 2c20 7b31 2c20 322c 2033 7d2c 207b 2}, {1, 2, 3}, { │ │ │ │ +00016db0: 302c 2032 2c20 337d 2c20 7b30 2c20 312c 0, 2, 3}, {0, 1, │ │ │ │ +00016dc0: 2034 7d2c 207b 312c 2033 2c20 347d 2c20 4}, {1, 3, 4}, │ │ │ │ +00016dd0: 7b30 2c20 332c 2034 7d7d 7c0a 7c20 2020 {0, 3, 4}}|.| │ │ │ │ +00016de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016e20: 2020 2020 207c 0a7c 6f32 3520 3a20 4c69 |.|o25 : Li │ │ │ │ +00016e30: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ 00016e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e50: 2020 2020 2020 2020 7c0a 7c6f 3235 203a |.|o25 : │ │ │ │ -00016e60: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ -00016e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ea0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00016eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00016ef0: 7c69 3236 203a 2058 203d 206e 6f72 6d61 |i26 : X = norma │ │ │ │ -00016f00: 6c54 6f72 6963 5661 7269 6574 7928 5268 lToricVariety(Rh │ │ │ │ -00016f10: 6f2c 5369 676d 612c 436f 6566 6669 6369 o,Sigma,Coeffici │ │ │ │ -00016f20: 656e 7452 696e 6720 3d3e 5a5a 2f33 3237 entRing =>ZZ/327 │ │ │ │ -00016f30: 3439 2920 2020 2020 207c 0a7c 2020 2020 49) |.| │ │ │ │ +00016e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016e70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00016e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00016eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +00016ec0: 3620 3a20 5820 3d20 6e6f 726d 616c 546f 6 : X = normalTo │ │ │ │ +00016ed0: 7269 6356 6172 6965 7479 2852 686f 2c53 ricVariety(Rho,S │ │ │ │ +00016ee0: 6967 6d61 2c43 6f65 6666 6963 6965 6e74 igma,Coefficient │ │ │ │ +00016ef0: 5269 6e67 203d 3e5a 5a2f 3332 3734 3929 Ring =>ZZ/32749) │ │ │ │ +00016f00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00016f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016f50: 207c 0a7c 6f32 3620 3d20 5820 2020 2020 |.|o26 = X │ │ │ │ 00016f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016f80: 2020 2020 7c0a 7c6f 3236 203d 2058 2020 |.|o26 = X │ │ │ │ -00016f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016fc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017010: 2020 2020 2020 2020 2020 7c0a 7c6f 3236 |.|o26 │ │ │ │ -00017020: 203a 204e 6f72 6d61 6c54 6f72 6963 5661 : NormalToricVa │ │ │ │ -00017030: 7269 6574 7920 2020 2020 2020 2020 2020 riety │ │ │ │ -00017040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017060: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00017070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000170a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000170b0: 2b0a 7c69 3237 203a 2063 736d 583d 4353 +.|i27 : csmX=CS │ │ │ │ -000170c0: 4d20 5820 2020 2020 2020 2020 2020 2020 M X │ │ │ │ +00016fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00016fe0: 2020 2020 2020 207c 0a7c 6f32 3620 3a20 |.|o26 : │ │ │ │ +00016ff0: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ +00017000: 7479 2020 2020 2020 2020 2020 2020 2020 ty │ │ │ │ +00017010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017030: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00017040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00017070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00017080: 6932 3720 3a20 6373 6d58 3d43 534d 2058 i27 : csmX=CSM X │ │ │ │ +00017090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000170a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000170b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000170c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000170d0: 2020 2020 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+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 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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 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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 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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 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-000189d0: 0a43 6861 6e67 696e 6720 7468 6520 6f70 .Changing the op │ │ │ │ -000189e0: 7469 6f6e 202a 6e6f 7465 2043 6f6d 704d tion *note CompM │ │ │ │ -000189f0: 6574 686f 643a 2043 6f6d 704d 6574 686f ethod: CompMetho │ │ │ │ -00018a00: 642c 2074 6f20 6265 7274 696e 6920 7769 d, to bertini wi │ │ │ │ -00018a10: 6c6c 2064 6f20 7468 6520 6d61 696e 0a63 ll do the main.c │ │ │ │ -00018a20: 6f6d 7075 7461 7469 6f6e 7320 6e75 6d65 omputations nume │ │ │ │ -00018a30: 7269 6361 6c6c 792c 2070 726f 7669 6465 rically, provide │ │ │ │ -00018a40: 6420 4265 7274 696e 6920 6973 202a 6e6f d Bertini is *no │ │ │ │ -00018a50: 7465 2069 6e73 7461 6c6c 6564 2061 6e64 te installed and │ │ │ │ -00018a60: 2063 6f6e 6669 6775 7265 643a 0a63 6f6e configured:.con │ │ │ │ -00018a70: 6669 6775 7269 6e67 2042 6572 7469 6e69 figuring Bertini │ │ │ │ -00018a80: 2c2e 204e 6f74 6520 7468 6174 2074 6865 ,. Note that the │ │ │ │ -00018a90: 2062 6572 7469 6e69 2061 6e64 2050 6e52 bertini and PnR │ │ │ │ -00018aa0: 6573 6964 7561 6c20 6f70 7469 6f6e 7320 esidual options │ │ │ │ -00018ab0: 6d61 7920 6f6e 6c79 2062 650a 7573 6564 may only be.used │ │ │ │ -00018ac0: 2066 6f72 2073 7562 7363 6865 6d65 7320 for subschemes │ │ │ │ -00018ad0: 6f66 205c 5050 5e6e 2e0a 0a4f 6273 6572 of \PP^n...Obser │ │ │ │ -00018ae0: 7665 2074 6861 7420 7468 6520 616c 676f ve that the algo │ │ │ │ -00018af0: 7269 7468 6d20 6973 2061 2070 726f 6261 rithm is a proba │ │ │ │ -00018b00: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ -00018b10: 686d 2061 6e64 206d 6179 2067 6976 6520 hm and may give │ │ │ │ -00018b20: 6120 7772 6f6e 670a 616e 7377 6572 2077 a wrong.answer w │ │ │ │ -00018b30: 6974 6820 6120 736d 616c 6c20 6275 7420 ith a small but │ │ │ │ -00018b40: 6e6f 6e7a 6572 6f20 7072 6f62 6162 696c nonzero probabil │ │ │ │ -00018b50: 6974 792e 2052 6561 6420 6d6f 7265 2075 ity. Read more u │ │ │ │ -00018b60: 6e64 6572 202a 6e6f 7465 0a70 726f 6261 nder *note.proba │ │ │ │ -00018b70: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ -00018b80: 686d 3a20 7072 6f62 6162 696c 6973 7469 hm: probabilisti │ │ │ │ -00018b90: 6320 616c 676f 7269 7468 6d2c 2e0a 0a0a c algorithm,.... │ │ │ │ -00018ba0: 0a57 6179 7320 746f 2075 7365 2043 534d .Ways to use CSM │ │ │ │ -00018bb0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00018bc0: 3d3d 0a0a 2020 2a20 2243 534d 2849 6465 ==.. * "CSM(Ide │ │ │ │ -00018bd0: 616c 2922 0a20 202a 2022 4353 4d28 4964 al)". * "CSM(Id │ │ │ │ -00018be0: 6561 6c2c 5379 6d62 6f6c 2922 0a20 202a eal,Symbol)". * │ │ │ │ -00018bf0: 2022 4353 4d28 5175 6f74 6965 6e74 5269 "CSM(QuotientRi │ │ │ │ -00018c00: 6e67 2c49 6465 616c 2922 0a20 202a 2022 ng,Ideal)". * " │ │ │ │ -00018c10: 4353 4d28 5175 6f74 6965 6e74 5269 6e67 CSM(QuotientRing │ │ │ │ -00018c20: 2c49 6465 616c 2c4d 7574 6162 6c65 4861 ,Ideal,MutableHa │ │ │ │ -00018c30: 7368 5461 626c 6529 220a 0a46 6f72 2074 shTable)"..For t │ │ │ │ -00018c40: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00018c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00018c60: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00018c70: 7465 2043 534d 3a20 4353 4d2c 2069 7320 te CSM: CSM, is │ │ │ │ -00018c80: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ -00018c90: 756e 6374 696f 6e20 7769 7468 206f 7074 unction with opt │ │ │ │ -00018ca0: 696f 6e73 3a0a 284d 6163 6175 6c61 7932 ions:.(Macaulay2 │ │ │ │ -00018cb0: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -00018cc0: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a onWithOptions,.. │ │ │ │ -00018cd0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ -00018ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018d20: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00018d30: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00018d40: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -00018d50: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -00018d60: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ -00018d70: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -00018d80: 6163 6b61 6765 732f 0a43 6861 7261 6374 ackages/.Charact │ │ │ │ -00018d90: 6572 6973 7469 6343 6c61 7373 6573 2e6d eristicClasses.m │ │ │ │ -00018da0: 323a 3232 3231 3a30 2e0a 1f0a 4669 6c65 2:2221:0....File │ │ │ │ -00018db0: 3a20 4368 6172 6163 7465 7269 7374 6963 : Characteristic │ │ │ │ -00018dc0: 436c 6173 7365 732e 696e 666f 2c20 4e6f Classes.info, No │ │ │ │ -00018dd0: 6465 3a20 4575 6c65 722c 204e 6578 743a de: Euler, Next: │ │ │ │ -00018de0: 2045 756c 6572 4166 6669 6e65 2c20 5072 EulerAffine, Pr │ │ │ │ -00018df0: 6576 3a20 4353 4d2c 2055 703a 2054 6f70 ev: CSM, Up: Top │ │ │ │ -00018e00: 0a0a 4575 6c65 7220 2d2d 2054 6865 2045 ..Euler -- The E │ │ │ │ -00018e10: 756c 6572 2043 6861 7261 6374 6572 6973 uler Characteris │ │ │ │ -00018e20: 7469 630a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a tic.************ │ │ │ │ -00018e30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00018e40: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -00018e50: 3a20 0a20 2020 2020 2020 2045 756c 6572 : . Euler │ │ │ │ -00018e60: 2049 0a20 2020 2020 2020 2045 756c 6572 I. Euler │ │ │ │ -00018e70: 2858 2c4a 290a 2020 2020 2020 2020 4575 (X,J). Eu │ │ │ │ -00018e80: 6c65 7220 6373 6d0a 2020 2a20 496e 7075 ler csm. * Inpu │ │ │ │ -00018e90: 7473 3a0a 2020 2020 2020 2a20 492c 2061 ts:. * I, a │ │ │ │ -00018ea0: 6e20 2a6e 6f74 6520 6964 6561 6c3a 2028 n *note ideal: ( │ │ │ │ -00018eb0: 4d61 6361 756c 6179 3244 6f63 2949 6465 Macaulay2Doc)Ide │ │ │ │ -00018ec0: 616c 2c2c 2061 206d 756c 7469 2d68 6f6d al,, a multi-hom │ │ │ │ -00018ed0: 6f67 656e 656f 7573 2069 6465 616c 2069 ogeneous ideal i │ │ │ │ -00018ee0: 6e20 610a 2020 2020 2020 2020 6772 6164 n a. grad │ │ │ │ -00018ef0: 6564 2070 6f6c 796e 6f6d 6961 6c20 7269 ed polynomial ri │ │ │ │ -00018f00: 6e67 206f 7665 7220 6120 6669 656c 6420 ng over a field │ │ │ │ -00018f10: 6465 6669 6e69 6e67 2061 2063 6c6f 7365 defining a close │ │ │ │ -00018f20: 6420 7375 6273 6368 656d 6520 5620 6f66 d subscheme V of │ │ │ │ -00018f30: 0a20 2020 2020 2020 205c 5050 5e7b 6e5f . \PP^{n_ │ │ │ │ -00018f40: 317d 782e 2e2e 785c 5050 5e7b 6e5f 6d7d 1}x...x\PP^{n_m} │ │ │ │ -00018f50: 0a20 2020 2020 202a 204a 2c20 616e 202a . * J, an * │ │ │ │ -00018f60: 6e6f 7465 2069 6465 616c 3a20 284d 6163 note ideal: (Mac │ │ │ │ -00018f70: 6175 6c61 7932 446f 6329 4964 6561 6c2c aulay2Doc)Ideal, │ │ │ │ -00018f80: 2c20 616e 2069 6465 616c 2069 6e20 7468 , an ideal in th │ │ │ │ -00018f90: 6520 6772 6164 6564 0a20 2020 2020 2020 e graded. │ │ │ │ -00018fa0: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ -00018fb0: 2077 6869 6368 2069 7320 636f 6f72 6469 which is coordi │ │ │ │ -00018fc0: 6e61 7465 2072 696e 6720 6f66 2074 6865 nate ring of the │ │ │ │ -00018fd0: 204e 6f72 6d61 6c20 546f 7269 6320 5661 Normal Toric Va │ │ │ │ -00018fe0: 7269 6574 7920 580a 2020 2020 2020 2a20 riety X. * │ │ │ │ -00018ff0: 582c 2061 202a 6e6f 7465 206e 6f72 6d61 X, a *note norma │ │ │ │ -00019000: 6c20 746f 7269 6320 7661 7269 6574 793a l toric variety: │ │ │ │ -00019010: 0a20 2020 2020 2020 2028 4e6f 726d 616c . (Normal │ │ │ │ -00019020: 546f 7269 6356 6172 6965 7469 6573 294e ToricVarieties)N │ │ │ │ -00019030: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ -00019040: 792c 2c20 6120 6e6f 726d 616c 2074 6f72 y,, a normal tor │ │ │ │ -00019050: 6963 2076 6172 6965 7479 2077 6869 6368 ic variety which │ │ │ │ -00019060: 0a20 2020 2020 2020 2069 7320 7468 6520 . is the │ │ │ │ -00019070: 616d 6269 656e 7420 7370 6163 6520 7468 ambient space th │ │ │ │ -00019080: 6174 2077 6520 6172 6520 776f 726b 696e at we are workin │ │ │ │ -00019090: 6720 696e 0a20 2020 2020 202a 2063 736d g in. * csm │ │ │ │ -000190a0: 2c20 6120 2a6e 6f74 6520 7269 6e67 2065 , a *note ring e │ │ │ │ -000190b0: 6c65 6d65 6e74 3a20 284d 6163 6175 6c61 lement: (Macaula │ │ │ │ -000190c0: 7932 446f 6329 5269 6e67 456c 656d 656e y2Doc)RingElemen │ │ │ │ -000190d0: 742c 2c20 7468 6520 4353 4d20 636c 6173 t,, the CSM clas │ │ │ │ -000190e0: 7320 6f66 0a20 2020 2020 2020 2073 6f6d s of. som │ │ │ │ -000190f0: 6520 7661 7269 6574 7920 560a 2020 2a20 e variety V. * │ │ │ │ -00019100: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069 *note Optional i │ │ │ │ -00019110: 6e70 7574 733a 2028 4d61 6361 756c 6179 nputs: (Macaulay │ │ │ │ -00019120: 3244 6f63 2975 7369 6e67 2066 756e 6374 2Doc)using funct │ │ │ │ -00019130: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ -00019140: 616c 2069 6e70 7574 732c 3a0a 2020 2020 al inputs,:. │ │ │ │ -00019150: 2020 2a20 436f 6d70 4d65 7468 6f64 2028 * CompMethod ( │ │ │ │ -00019160: 6d69 7373 696e 6720 646f 6375 6d65 6e74 missing document │ │ │ │ -00019170: 6174 696f 6e29 203d 3e20 2e2e 2e2c 2064 ation) => ..., d │ │ │ │ -00019180: 6566 6175 6c74 2076 616c 7565 0a20 2020 efault value. │ │ │ │ -00019190: 2020 2020 2050 726f 6a65 6374 6976 6544 ProjectiveD │ │ │ │ -000191a0: 6567 7265 652c 2050 726f 6a65 6374 6976 egree, Projectiv │ │ │ │ -000191b0: 6544 6567 7265 652c 2061 7070 6c69 6361 eDegree, applica │ │ │ │ -000191c0: 626c 6520 666f 7220 616c 6c20 6361 7365 ble for all case │ │ │ │ -000191d0: 7320 7768 6572 6520 7468 650a 2020 2020 s where the. │ │ │ │ -000191e0: 2020 2020 6d65 7468 6f64 7320 696e 2074 methods in t │ │ │ │ -000191f0: 6865 2070 6163 6b61 6765 206d 6179 2062 he package may b │ │ │ │ -00019200: 6520 7573 6564 0a20 2020 2020 202a 2043 e used. * C │ │ │ │ -00019210: 6f6d 704d 6574 686f 6420 286d 6973 7369 ompMethod (missi │ │ │ │ -00019220: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ -00019230: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ -00019240: 7420 7661 6c75 650a 2020 2020 2020 2020 t value. │ │ │ │ -00019250: 5072 6f6a 6563 7469 7665 4465 6772 6565 ProjectiveDegree │ │ │ │ -00019260: 2c20 506e 5265 7369 6475 616c 2c20 7468 , PnResidual, th │ │ │ │ -00019270: 6973 2061 6c67 6f72 6974 686d 206d 6179 is algorithm may │ │ │ │ -00019280: 2062 6520 7573 6564 2066 6f72 2073 7562 be used for sub │ │ │ │ -00019290: 7363 6865 6d65 730a 2020 2020 2020 2020 schemes. │ │ │ │ -000192a0: 6f66 205c 5050 5e6e 206f 6e6c 790a 2020 of \PP^n only. │ │ │ │ -000192b0: 2020 2020 2a20 4d65 7468 6f64 2028 6d69 * Method (mi │ │ │ │ -000192c0: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -000192d0: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ -000192e0: 6175 6c74 2076 616c 7565 0a20 2020 2020 ault value. │ │ │ │ -000192f0: 2020 2049 6e63 6c75 7369 6f6e 4578 636c InclusionExcl │ │ │ │ -00019300: 7573 696f 6e2c 2049 6e63 6c75 7369 6f6e usion, Inclusion │ │ │ │ -00019310: 4578 636c 7573 696f 6e2c 2061 7070 6c69 Exclusion, appli │ │ │ │ -00019320: 6361 626c 6520 666f 7220 616c 6c20 696e cable for all in │ │ │ │ -00019330: 7075 7473 0a20 2020 2020 202a 204d 6574 puts. * Met │ │ │ │ -00019340: 686f 6420 286d 6973 7369 6e67 2064 6f63 hod (missing doc │ │ │ │ -00019350: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ -00019360: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ -00019370: 650a 2020 2020 2020 2020 496e 636c 7573 e. Inclus │ │ │ │ -00019380: 696f 6e45 7863 6c75 7369 6f6e 2c20 4469 ionExclusion, Di │ │ │ │ -00019390: 7265 6374 436f 6d70 6c65 7465 496e 742c rectCompleteInt, │ │ │ │ -000193a0: 2074 6869 7320 6d65 7468 6f64 206d 6179 this method may │ │ │ │ -000193b0: 2070 726f 7669 6465 2061 0a20 2020 2020 provide a. │ │ │ │ -000193c0: 2020 2070 6572 666f 726d 616e 6365 2069 performance i │ │ │ │ -000193d0: 6d70 726f 7665 6d65 6e74 2077 6865 6e20 mprovement when │ │ │ │ -000193e0: 7468 6520 696e 7075 7420 6973 2061 2063 the input is a c │ │ │ │ -000193f0: 6f6d 706c 6574 6520 696e 7465 7273 6563 omplete intersec │ │ │ │ -00019400: 7469 6f6e 2c20 6966 0a20 2020 2020 2020 tion, if. │ │ │ │ -00019410: 2074 6865 2069 6e70 7574 2069 7320 6e6f the input is no │ │ │ │ -00019420: 7420 6120 636f 6d70 6c65 7465 2069 6e74 t a complete int │ │ │ │ -00019430: 6572 7365 6374 696f 6e20 696e 636c 7573 ersection inclus │ │ │ │ -00019440: 696f 6e2f 6578 636c 7573 696f 6e20 6974 ion/exclusion it │ │ │ │ -00019450: 2077 696c 6c0a 2020 2020 2020 2020 7265 will. re │ │ │ │ -00019460: 7475 726e 2061 6e20 6572 726f 720a 2020 turn an error. │ │ │ │ -00019470: 2020 2020 2a20 496e 7075 7449 7353 6d6f * InputIsSmo │ │ │ │ -00019480: 6f74 6820 286d 6973 7369 6e67 2064 6f63 oth (missing doc │ │ │ │ -00019490: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ -000194a0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ -000194b0: 6520 6661 6c73 652c 2074 6869 730a 2020 e false, this. │ │ │ │ -000194c0: 2020 2020 2020 6f70 7469 6f6e 2068 6173 option has │ │ │ │ -000194d0: 2076 616c 7565 7320 7472 7565 2f66 616c values true/fal │ │ │ │ -000194e0: 7365 2061 6e64 2074 656c 6c73 2074 6865 se and tells the │ │ │ │ -000194f0: 206d 6574 686f 6420 7768 6574 6865 7220 method whether │ │ │ │ -00019500: 746f 2061 7373 756d 6520 7468 650a 2020 to assume the. │ │ │ │ -00019510: 2020 2020 2020 696e 7075 7420 6964 6561 input idea │ │ │ │ -00019520: 6c20 6465 6669 6e65 7320 6120 736d 6f6f l defines a smoo │ │ │ │ -00019530: 7468 2073 6368 656d 652c 2061 6e64 2068 th scheme, and h │ │ │ │ -00019540: 656e 6365 2074 6f20 6361 6c6c 2074 6865 ence to call the │ │ │ │ -00019550: 206d 6574 686f 6420 4368 6572 6e0a 2020 method Chern. │ │ │ │ -00019560: 2020 2020 2020 696e 7374 6561 6420 666f instead fo │ │ │ │ -00019570: 7220 7265 6475 6365 6420 7275 6e20 7469 r reduced run ti │ │ │ │ -00019580: 6d65 2c20 616c 7465 726e 6174 6976 656c me, alternativel │ │ │ │ -00019590: 7920 7468 6520 4368 6572 6e20 6675 6e63 y the Chern func │ │ │ │ -000195a0: 7469 6f6e 2063 616e 2062 650a 2020 2020 tion can be. │ │ │ │ -000195b0: 2020 2020 7573 6564 2064 6972 6563 746c used directl │ │ │ │ -000195c0: 790a 2020 2020 2020 2a20 4f75 7470 7574 y. * Output │ │ │ │ -000195d0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -000195e0: 2076 616c 7565 2043 686f 7752 696e 6745 value ChowRingE │ │ │ │ -000195f0: 6c65 6d65 6e74 2c20 7468 6520 7479 7065 lement, the type │ │ │ │ -00019600: 206f 6620 6f75 7470 7574 2074 6f0a 2020 of output to. │ │ │ │ -00019610: 2020 2020 2020 7265 7475 726e 2074 6865 return the │ │ │ │ -00019620: 2064 6566 6175 6c74 206f 7574 7075 7420 default output │ │ │ │ -00019630: 6973 2061 6e20 696e 7465 6765 720a 2020 is an integer. │ │ │ │ -00019640: 2020 2020 2a20 4f75 7470 7574 203d 3e20 * Output => │ │ │ │ -00019650: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -00019660: 7565 2043 686f 7752 696e 6745 6c65 6d65 ue ChowRingEleme │ │ │ │ -00019670: 6e74 2c20 4861 7368 466f 726d 2c20 7468 nt, HashForm, th │ │ │ │ -00019680: 6520 7479 7065 206f 660a 2020 2020 2020 e type of. │ │ │ │ -00019690: 2020 6f75 7470 7574 2074 6f20 7265 7475 output to retu │ │ │ │ -000196a0: 726e 2c20 4861 7368 466f 726d 2072 6574 rn, HashForm ret │ │ │ │ -000196b0: 7572 6e73 2061 204d 7574 6162 6c65 4861 urns a MutableHa │ │ │ │ -000196c0: 7368 5461 626c 6520 636f 6e74 6169 6e69 shTable containi │ │ │ │ -000196d0: 6e67 2074 6865 0a20 2020 2020 2020 206b ng the. k │ │ │ │ -000196e0: 6579 2022 4353 4d22 2028 7468 6520 4353 ey "CSM" (the CS │ │ │ │ -000196f0: 4d20 636c 6173 7329 2c20 616e 6420 6b65 M class), and ke │ │ │ │ -00019700: 7973 206f 6620 7468 6520 666f 726d 0a20 ys of the form. │ │ │ │ -00019710: 2020 2020 2020 205c 7b30 5c7d 2c5c 7b31 \{0\},\{1 │ │ │ │ -00019720: 5c7d 2c5c 7b32 5c7d 2c2e 2e2e 2c5c 7b30 \},\{2\},...,\{0 │ │ │ │ -00019730: 2c31 5c7d 2c5c 7b30 2c32 5c7d 202e 2e2e ,1\},\{0,2\} ... │ │ │ │ -00019740: 2e5c 7b30 2c31 2c32 5c7d 2e2e 2e20 616e .\{0,1,2\}... an │ │ │ │ -00019750: 6420 736f 206f 6e20 7768 6963 680a 2020 d so on which. │ │ │ │ -00019760: 2020 2020 2020 636f 7272 6573 706f 6e64 correspond │ │ │ │ -00019770: 2074 6f20 7468 6520 696e 6469 6365 7320 to the indices │ │ │ │ -00019780: 6f66 2074 6865 2070 6f73 7369 626c 6520 of the possible │ │ │ │ -00019790: 7375 6273 6574 7320 6f66 2074 6865 2067 subsets of the g │ │ │ │ -000197a0: 656e 6572 6174 6f72 7320 6f66 0a20 2020 enerators of. │ │ │ │ -000197b0: 2020 2020 2074 6865 2069 6e70 7574 2069 the input i │ │ │ │ -000197c0: 6465 616c 2c20 666f 7220 6561 6368 2073 deal, for each s │ │ │ │ -000197d0: 6574 206f 6620 696e 6469 6365 7320 7468 et of indices th │ │ │ │ -000197e0: 6520 4353 4d20 636c 6173 7320 6f66 2074 e CSM class of t │ │ │ │ -000197f0: 6865 0a20 2020 2020 2020 2068 7970 6572 he. hyper │ │ │ │ -00019800: 7375 7266 6163 6520 6769 7665 6e20 6279 surface given by │ │ │ │ -00019810: 2074 6865 2070 726f 6475 6374 206f 6620 the product of │ │ │ │ -00019820: 616c 6c20 706f 6c79 6e6f 6d69 616c 7320 all polynomials │ │ │ │ -00019830: 696e 2074 6865 0a20 2020 2020 2020 2063 in the. c │ │ │ │ -00019840: 6f72 7265 7370 6f6e 6469 6e67 2073 6574 orresponding set │ │ │ │ -00019850: 206f 6620 6765 6e65 7261 746f 7273 2069 of generators i │ │ │ │ -00019860: 7320 7374 6f72 6564 2c20 7468 6572 6520 s stored, there │ │ │ │ -00019870: 6973 206e 6f20 6578 7472 6120 636f 7374 is no extra cost │ │ │ │ -00019880: 2074 6f0a 2020 2020 2020 2020 7573 696e to. usin │ │ │ │ -00019890: 6720 7468 6973 206f 7074 696f 6e0a 2020 g this option. │ │ │ │ -000198a0: 2020 2020 2a20 496e 6473 4f66 536d 6f6f * IndsOfSmoo │ │ │ │ -000198b0: 7468 2028 6d69 7373 696e 6720 646f 6375 th (missing docu │ │ │ │ -000198c0: 6d65 6e74 6174 696f 6e29 203d 3e20 2e2e mentation) => .. │ │ │ │ -000198d0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -000198e0: 207b 7d2c 2074 6869 730a 2020 2020 2020 {}, this. │ │ │ │ -000198f0: 2020 6f70 7469 6f6e 206d 6179 2073 7065 option may spe │ │ │ │ -00019900: 6564 2075 7020 7468 6520 7275 6e20 7469 ed up the run ti │ │ │ │ -00019910: 6d65 2077 6865 6e20 7573 696e 6720 7468 me when using th │ │ │ │ -00019920: 6520 4469 7265 6374 436f 6d70 6c65 7465 e DirectComplete │ │ │ │ -00019930: 496e 740a 2020 2020 2020 2020 4d65 7468 Int. Meth │ │ │ │ -00019940: 6f64 2069 6620 7468 6520 7573 6572 206b od if the user k │ │ │ │ -00019950: 6e6f 7773 2061 6464 6974 696f 6e61 6c20 nows additional │ │ │ │ -00019960: 696e 666f 726d 6174 696f 6e20 6162 6f75 information abou │ │ │ │ -00019970: 7420 7468 6520 696e 7075 7420 6964 6561 t the input idea │ │ │ │ -00019980: 6c2c 0a20 2020 2020 2020 2073 6565 202a l,. see * │ │ │ │ -00019990: 6e6f 7465 2049 6e64 734f 6653 6d6f 6f74 note IndsOfSmoot │ │ │ │ -000199a0: 683a 2049 6e64 734f 6653 6d6f 6f74 682c h: IndsOfSmooth, │ │ │ │ -000199b0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -000199c0: 2020 2020 2a20 6120 2a6e 6f74 6520 7269 * a *note ri │ │ │ │ -000199d0: 6e67 2065 6c65 6d65 6e74 3a20 284d 6163 ng element: (Mac │ │ │ │ -000199e0: 6175 6c61 7932 446f 6329 5269 6e67 456c aulay2Doc)RingEl │ │ │ │ -000199f0: 656d 656e 742c 2c20 7468 6520 4575 6c65 ement,, the Eule │ │ │ │ -00019a00: 720a 2020 2020 2020 2020 6368 6172 6163 r. charac │ │ │ │ -00019a10: 7465 7269 7374 6963 0a0a 4465 7363 7269 teristic..Descri │ │ │ │ -00019a20: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00019a30: 3d0a 0a46 6f72 2061 2073 7562 7363 6865 =..For a subsche │ │ │ │ -00019a40: 6d65 2056 206f 6620 616e 2061 7070 6c69 me V of an appli │ │ │ │ -00019a50: 6361 626c 6520 746f 7269 6320 7661 7269 cable toric vari │ │ │ │ -00019a60: 6574 7920 582c 2074 6869 7320 636f 6d6d ety X, this comm │ │ │ │ -00019a70: 616e 6420 636f 6d70 7574 6573 2074 6865 and computes the │ │ │ │ -00019a80: 0a45 756c 6572 2063 6861 7261 6374 6572 .Euler character │ │ │ │ -00019a90: 6973 7469 630a 0a2b 2d2d 2d2d 2d2d 2d2d istic..+-------- │ │ │ │ +00018920: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00018930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00018940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00018950: 0a41 6c6c 2074 6865 2065 7861 6d70 6c65 .All the example │ │ │ │ +00018960: 7320 7765 7265 2064 6f6e 6520 7573 696e s were done usin │ │ │ │ +00018970: 6720 7379 6d62 6f6c 6963 2063 6f6d 7075 g symbolic compu │ │ │ │ +00018980: 7461 7469 6f6e 7320 7769 7468 2047 725c tations with Gr\ │ │ │ │ +00018990: 226f 626e 6572 2062 6173 6573 2e0a 4368 "obner bases..Ch │ │ │ │ +000189a0: 616e 6769 6e67 2074 6865 206f 7074 696f anging the optio │ │ │ │ +000189b0: 6e20 2a6e 6f74 6520 436f 6d70 4d65 7468 n *note CompMeth │ │ │ │ +000189c0: 6f64 3a20 436f 6d70 4d65 7468 6f64 2c20 od: CompMethod, │ │ │ │ +000189d0: 746f 2062 6572 7469 6e69 2077 696c 6c20 to bertini will │ │ │ │ +000189e0: 646f 2074 6865 206d 6169 6e0a 636f 6d70 do the main.comp │ │ │ │ +000189f0: 7574 6174 696f 6e73 206e 756d 6572 6963 utations numeric │ │ │ │ +00018a00: 616c 6c79 2c20 7072 6f76 6964 6564 2042 ally, provided B │ │ │ │ +00018a10: 6572 7469 6e69 2069 7320 2a6e 6f74 6520 ertini is *note │ │ │ │ +00018a20: 696e 7374 616c 6c65 6420 616e 6420 636f installed and co │ │ │ │ +00018a30: 6e66 6967 7572 6564 3a0a 636f 6e66 6967 nfigured:.config │ │ │ │ +00018a40: 7572 696e 6720 4265 7274 696e 692c 2e20 uring Bertini,. │ │ │ │ +00018a50: 4e6f 7465 2074 6861 7420 7468 6520 6265 Note that the be │ │ │ │ +00018a60: 7274 696e 6920 616e 6420 506e 5265 7369 rtini and PnResi │ │ │ │ +00018a70: 6475 616c 206f 7074 696f 6e73 206d 6179 dual options may │ │ │ │ +00018a80: 206f 6e6c 7920 6265 0a75 7365 6420 666f only be.used fo │ │ │ │ +00018a90: 7220 7375 6273 6368 656d 6573 206f 6620 r subschemes of │ │ │ │ +00018aa0: 5c50 505e 6e2e 0a0a 4f62 7365 7276 6520 \PP^n...Observe │ │ │ │ +00018ab0: 7468 6174 2074 6865 2061 6c67 6f72 6974 that the algorit │ │ │ │ +00018ac0: 686d 2069 7320 6120 7072 6f62 6162 696c hm is a probabil │ │ │ │ +00018ad0: 6973 7469 6320 616c 676f 7269 7468 6d20 istic algorithm │ │ │ │ +00018ae0: 616e 6420 6d61 7920 6769 7665 2061 2077 and may give a w │ │ │ │ +00018af0: 726f 6e67 0a61 6e73 7765 7220 7769 7468 rong.answer with │ │ │ │ +00018b00: 2061 2073 6d61 6c6c 2062 7574 206e 6f6e a small but non │ │ │ │ +00018b10: 7a65 726f 2070 726f 6261 6269 6c69 7479 zero probability │ │ │ │ +00018b20: 2e20 5265 6164 206d 6f72 6520 756e 6465 . 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6f72 2061 6c6c 2069 6e70 7574 le for all input │ │ │ │ +00019300: 730a 2020 2020 2020 2a20 4d65 7468 6f64 s. * Method │ │ │ │ +00019310: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +00019320: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ +00019330: 2064 6566 6175 6c74 2076 616c 7565 0a20 default value. │ │ │ │ +00019340: 2020 2020 2020 2049 6e63 6c75 7369 6f6e Inclusion │ │ │ │ +00019350: 4578 636c 7573 696f 6e2c 2044 6972 6563 Exclusion, Direc │ │ │ │ +00019360: 7443 6f6d 706c 6574 6549 6e74 2c20 7468 tCompleteInt, th │ │ │ │ +00019370: 6973 206d 6574 686f 6420 6d61 7920 7072 is method may pr │ │ │ │ +00019380: 6f76 6964 6520 610a 2020 2020 2020 2020 ovide a. │ │ │ │ +00019390: 7065 7266 6f72 6d61 6e63 6520 696d 7072 performance impr │ │ │ │ +000193a0: 6f76 656d 656e 7420 7768 656e 2074 6865 ovement when the │ │ │ │ +000193b0: 2069 6e70 7574 2069 7320 6120 636f 6d70 input is a comp │ │ │ │ +000193c0: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ +000193d0: 6e2c 2069 660a 2020 2020 2020 2020 7468 n, if. th │ │ │ │ +000193e0: 6520 696e 7075 7420 6973 206e 6f74 2061 e input is not a │ │ │ │ +000193f0: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ +00019400: 6563 7469 6f6e 2069 6e63 6c75 7369 6f6e ection inclusion │ │ │ │ +00019410: 2f65 7863 6c75 7369 6f6e 2069 7420 7769 /exclusion it wi │ │ │ │ +00019420: 6c6c 0a20 2020 2020 2020 2072 6574 7572 ll. retur │ │ │ │ +00019430: 6e20 616e 2065 7272 6f72 0a20 2020 2020 n an error. │ │ │ │ +00019440: 202a 2049 6e70 7574 4973 536d 6f6f 7468 * InputIsSmooth │ │ │ │ +00019450: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +00019460: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ +00019470: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ +00019480: 616c 7365 2c20 7468 6973 0a20 2020 2020 alse, this. │ │ │ │ +00019490: 2020 206f 7074 696f 6e20 6861 7320 7661 option has va │ │ │ │ +000194a0: 6c75 6573 2074 7275 652f 6661 6c73 6520 lues true/false │ │ │ │ +000194b0: 616e 6420 7465 6c6c 7320 7468 6520 6d65 and tells the me │ │ │ │ +000194c0: 7468 6f64 2077 6865 7468 6572 2074 6f20 thod whether to │ │ │ │ +000194d0: 6173 7375 6d65 2074 6865 0a20 2020 2020 assume the. │ │ │ │ +000194e0: 2020 2069 6e70 7574 2069 6465 616c 2064 input ideal d │ │ │ │ +000194f0: 6566 696e 6573 2061 2073 6d6f 6f74 6820 efines a smooth │ │ │ │ +00019500: 7363 6865 6d65 2c20 616e 6420 6865 6e63 scheme, and henc │ │ │ │ +00019510: 6520 746f 2063 616c 6c20 7468 6520 6d65 e to call the me │ │ │ │ +00019520: 7468 6f64 2043 6865 726e 0a20 2020 2020 thod Chern. │ │ │ │ +00019530: 2020 2069 6e73 7465 6164 2066 6f72 2072 instead for r │ │ │ │ +00019540: 6564 7563 6564 2072 756e 2074 696d 652c educed run time, │ │ │ │ +00019550: 2061 6c74 6572 6e61 7469 7665 6c79 2074 alternatively t │ │ │ │ +00019560: 6865 2043 6865 726e 2066 756e 6374 696f he Chern functio │ │ │ │ +00019570: 6e20 6361 6e20 6265 0a20 2020 2020 2020 n can be. │ │ │ │ +00019580: 2075 7365 6420 6469 7265 6374 6c79 0a20 used directly. │ │ │ │ +00019590: 2020 2020 202a 204f 7574 7075 7420 3d3e * Output => │ │ │ │ +000195a0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ +000195b0: 6c75 6520 4368 6f77 5269 6e67 456c 656d lue ChowRingElem │ │ │ │ +000195c0: 656e 742c 2074 6865 2074 7970 6520 6f66 ent, the type of │ │ │ │ +000195d0: 206f 7574 7075 7420 746f 0a20 2020 2020 output to. │ │ │ │ +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 │ │ │ │ +00019720: 6f20 6f6e 2077 6869 6368 0a20 2020 2020 o on which. │ │ │ │ +00019730: 2020 2063 6f72 7265 7370 6f6e 6420 746f correspond to │ │ │ │ +00019740: 2074 6865 2069 6e64 6963 6573 206f 6620 the indices of │ │ │ │ +00019750: 7468 6520 706f 7373 6962 6c65 2073 7562 the possible sub │ │ │ │ +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 │ │ │ │ +000199c0: 6e74 2c2c 2074 6865 2045 756c 6572 0a20 nt,, the Euler. │ │ │ │ +000199d0: 2020 2020 2020 2063 6861 7261 6374 6572 character │ │ │ │ +000199e0: 6973 7469 630a 0a44 6573 6372 6970 7469 istic..Descripti │ │ │ │ +000199f0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +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= │ │ │ │ -00019af0: 5a5a 2f33 3237 3439 3b20 2020 2020 2020 ZZ/32749; │ │ │ │ -00019b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00019ab0: 2d2d 2b0a 7c69 3120 3a20 6b6b 3d5a 5a2f --+.|i1 : kk=ZZ/ │ │ │ │ +00019ac0: 3332 3734 393b 2020 2020 2020 2020 2020 32749; │ │ │ │ +00019ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019b00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00019b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019b80: 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 3d6b -----+.|i2 : R=k │ │ │ │ -00019b90: 6b5b 785f 302e 2e78 5f34 5d20 2020 2020 k[x_0..x_4] │ │ │ │ -00019ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019b50: 2d2d 2b0a 7c69 3220 3a20 523d 6b6b 5b78 --+.|i2 : R=kk[x │ │ │ │ +00019b60: 5f30 2e2e 785f 345d 2020 2020 2020 2020 _0..x_4] │ │ │ │ +00019b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019ba0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00019bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019bd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00019bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019bf0: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020 |.|o2 = R │ │ │ │ 00019c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c20: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ +00019c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019c40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00019c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00019c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019c90: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f |.|o2 : Polyno │ │ │ │ +00019ca0: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ 00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019cc0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c |.|o2 : Pol │ │ │ │ -00019cd0: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ -00019ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019d10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00019cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019ce0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00019cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019d60: 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 3d69 -----+.|i3 : I=i │ │ │ │ -00019d70: 6465 616c 2872 616e 646f 6d28 312c 5229 deal(random(1,R) │ │ │ │ -00019d80: 2c72 616e 646f 6d28 322c 5229 2920 2020 ,random(2,R)) │ │ │ │ +00019d30: 2d2d 2b0a 7c69 3320 3a20 493d 6964 6561 --+.|i3 : I=idea │ │ │ │ +00019d40: 6c28 7261 6e64 6f6d 2831 2c52 292c 7261 l(random(1,R),ra │ │ │ │ +00019d50: 6e64 6f6d 2832 2c52 2929 2020 2020 2020 ndom(2,R)) │ │ │ │ +00019d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019d80: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00019d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019db0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00019db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019dd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00019de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019e00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00019e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019e40: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00019e50: 2020 2020 207c 0a7c 6f33 203d 2069 6465 |.|o3 = ide │ │ │ │ -00019e60: 616c 2028 3130 3778 2020 2b20 3433 3736 al (107x + 4376 │ │ │ │ -00019e70: 7820 202d 2036 3331 3678 2020 2b20 3331 x - 6316x + 31 │ │ │ │ -00019e80: 3837 7820 202b 2033 3738 3378 202c 202d 87x + 3783x , - │ │ │ │ -00019e90: 2036 3035 3378 2020 2b20 3835 3730 7820 6053x + 8570x │ │ │ │ -00019ea0: 7820 202b 207c 0a7c 2020 2020 2020 2020 x + |.| │ │ │ │ -00019eb0: 2020 2020 2020 2020 3020 2020 2020 2020 0 │ │ │ │ -00019ec0: 2031 2020 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -00019ed0: 2020 2033 2020 2020 2020 2020 3420 2020 3 4 │ │ │ │ -00019ee0: 2020 2020 2020 3020 2020 2020 2020 2030 0 0 │ │ │ │ -00019ef0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d 1 |.| --- │ │ │ │ +00019e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019e10: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00019e20: 2020 7c0a 7c6f 3320 3d20 6964 6561 6c20 |.|o3 = ideal │ │ │ │ +00019e30: 2831 3037 7820 202b 2034 3337 3678 2020 (107x + 4376x │ │ │ │ +00019e40: 2d20 3633 3136 7820 202b 2033 3138 3778 - 6316x + 3187x │ │ │ │ +00019e50: 2020 2b20 3337 3833 7820 2c20 2d20 3630 + 3783x , - 60 │ │ │ │ +00019e60: 3533 7820 202b 2038 3537 3078 2078 2020 53x + 8570x x │ │ │ │ +00019e70: 2b20 7c0a 7c20 2020 2020 2020 2020 2020 + |.| │ │ │ │ +00019e80: 2020 2020 2030 2020 2020 2020 2020 3120 0 1 │ │ │ │ +00019e90: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00019ea0: 3320 2020 2020 2020 2034 2020 2020 2020 3 4 │ │ │ │ +00019eb0: 2020 2030 2020 2020 2020 2020 3020 3120 0 0 1 │ │ │ │ +00019ec0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +00019ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00019ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019f40: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -00019f50: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00019f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f70: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00019f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f90: 2020 2020 207c 0a7c 2020 2020 2031 3033 |.| 103 │ │ │ │ -00019fa0: 3539 7820 202d 2031 3630 3930 7820 7820 59x - 16090x x │ │ │ │ -00019fb0: 202d 2038 3231 3078 2078 2020 2b20 3530 - 8210x x + 50 │ │ │ │ -00019fc0: 3731 7820 202b 2038 3434 3478 2078 2020 71x + 8444x x │ │ │ │ -00019fd0: 2d20 3839 3937 7820 7820 202d 2036 3934 - 8997x x - 694 │ │ │ │ -00019fe0: 3978 2078 207c 0a7c 2020 2020 2020 2020 9x x |.| │ │ │ │ -00019ff0: 2020 2031 2020 2020 2020 2020 2030 2032 1 0 2 │ │ │ │ -0001a000: 2020 2020 2020 2020 3120 3220 2020 2020 1 2 │ │ │ │ -0001a010: 2020 2032 2020 2020 2020 2020 3020 3320 2 0 3 │ │ │ │ -0001a020: 2020 2020 2020 2031 2033 2020 2020 2020 1 3 │ │ │ │ -0001a030: 2020 3220 337c 0a7c 2020 2020 202d 2d2d 2 3|.| --- │ │ │ │ +00019f10: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ +00019f20: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00019f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019f40: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00019f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019f60: 2020 7c0a 7c20 2020 2020 3130 3335 3978 |.| 10359x │ │ │ │ +00019f70: 2020 2d20 3136 3039 3078 2078 2020 2d20 - 16090x x - │ │ │ │ +00019f80: 3832 3130 7820 7820 202b 2035 3037 3178 8210x x + 5071x │ │ │ │ +00019f90: 2020 2b20 3834 3434 7820 7820 202d 2038 + 8444x x - 8 │ │ │ │ +00019fa0: 3939 3778 2078 2020 2d20 3639 3439 7820 997x x - 6949x │ │ │ │ +00019fb0: 7820 7c0a 7c20 2020 2020 2020 2020 2020 x |.| │ │ │ │ +00019fc0: 3120 2020 2020 2020 2020 3020 3220 2020 1 0 2 │ │ │ │ +00019fd0: 2020 2020 2031 2032 2020 2020 2020 2020 1 2 │ │ │ │ +00019fe0: 3220 2020 2020 2020 2030 2033 2020 2020 2 0 3 │ │ │ │ +00019ff0: 2020 2020 3120 3320 2020 2020 2020 2032 1 3 2 │ │ │ │ +0001a000: 2033 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 3|.| ------ │ │ │ │ +0001a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a080: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0001a090: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0001a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a0c0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -0001a0d0: 2020 2020 207c 0a7c 2020 2020 202d 2031 |.| - 1 │ │ │ │ -0001a0e0: 3432 3534 7820 202d 2031 3132 3236 7820 4254x - 11226x │ │ │ │ -0001a0f0: 7820 202b 2032 3635 3378 2078 2020 2b20 x + 2653x x + │ │ │ │ -0001a100: 3132 3336 3578 2078 2020 2d20 3130 3232 12365x x - 1022 │ │ │ │ -0001a110: 3678 2078 2020 2d20 3132 3639 3678 2029 6x x - 12696x ) │ │ │ │ -0001a120: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001a130: 2020 2020 2033 2020 2020 2020 2020 2030 3 0 │ │ │ │ -0001a140: 2034 2020 2020 2020 2020 3120 3420 2020 4 1 4 │ │ │ │ -0001a150: 2020 2020 2020 3220 3420 2020 2020 2020 2 4 │ │ │ │ -0001a160: 2020 3320 3420 2020 2020 2020 2020 3420 3 4 4 │ │ │ │ -0001a170: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a050: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ +0001a060: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a090: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0001a0a0: 2020 7c0a 7c20 2020 2020 2d20 3134 3235 |.| - 1425 │ │ │ │ +0001a0b0: 3478 2020 2d20 3131 3232 3678 2078 2020 4x - 11226x x │ │ │ │ +0001a0c0: 2b20 3236 3533 7820 7820 202b 2031 3233 + 2653x x + 123 │ │ │ │ +0001a0d0: 3635 7820 7820 202d 2031 3032 3236 7820 65x x - 10226x │ │ │ │ +0001a0e0: 7820 202d 2031 3236 3936 7820 2920 2020 x - 12696x ) │ │ │ │ +0001a0f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001a100: 2020 3320 2020 2020 2020 2020 3020 3420 3 0 4 │ │ │ │ +0001a110: 2020 2020 2020 2031 2034 2020 2020 2020 1 4 │ │ │ │ +0001a120: 2020 2032 2034 2020 2020 2020 2020 2033 2 4 3 │ │ │ │ +0001a130: 2034 2020 2020 2020 2020 2034 2020 2020 4 4 │ │ │ │ +0001a140: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a190: 2020 7c0a 7c6f 3320 3a20 4964 6561 6c20 |.|o3 : Ideal │ │ │ │ +0001a1a0: 6f66 2052 2020 2020 2020 2020 2020 2020 of R │ │ │ │ 0001a1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a1c0: 2020 2020 207c 0a7c 6f33 203a 2049 6465 |.|o3 : Ide │ │ │ │ -0001a1d0: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ -0001a1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a210: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001a1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a1e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a260: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 696d -----+.|i4 : tim │ │ │ │ -0001a270: 6520 4575 6c65 7228 492c 496e 7075 7449 e Euler(I,InputI │ │ │ │ -0001a280: 7353 6d6f 6f74 683d 3e74 7275 6529 2020 sSmooth=>true) │ │ │ │ -0001a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a2b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001a2c0: 2030 2e30 3337 3639 3636 7320 2863 7075 0.0376966s (cpu │ │ │ │ -0001a2d0: 293b 2030 2e30 3336 3939 3531 7320 2874 ); 0.0369951s (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: 3038 3536 3638 3673 2028 6370 7529 3b20 0856686s (cpu); │ │ │ │ +0001a2a0: 302e 3035 3236 3473 2028 7468 7265 6164 0.05264s (thread │ │ │ │ +0001a2b0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 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 3537 3335 3473 2028 6370 7529 0.257354s (cpu) │ │ │ │ -0001a460: 3b20 302e 3136 3434 3435 7320 2874 6872 ; 0.164445s (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: 3235 3535 3836 7320 2863 7075 293b 2030 255586s (cpu); 0 │ │ │ │ +0001a430: 2e31 3535 3431 3573 2028 7468 7265 6164 .155415s (thread │ │ │ │ +0001a440: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +0001a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a460: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001a470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a490: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a4b0: 2020 7c0a 7c6f 3520 3d20 3420 2020 2020 |.|o5 = 4 │ │ │ │ 0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a4e0: 2020 2020 207c 0a7c 6f35 203d 2034 2020 |.|o5 = 4 │ │ │ │ +0001a4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a530: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001a500: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a580: 2d2d 2d2d 2d2b 0a7c 6936 203a 2045 756c -----+.|i6 : Eul │ │ │ │ -0001a590: 6572 4948 6173 683d 4575 6c65 7228 492c erIHash=Euler(I, │ │ │ │ -0001a5a0: 4f75 7470 7574 3d3e 4861 7368 466f 726d Output=>HashForm │ │ │ │ -0001a5b0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -0001a5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a5d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001a550: 2d2d 2b0a 7c69 3620 3a20 4575 6c65 7249 --+.|i6 : EulerI │ │ │ │ +0001a560: 4861 7368 3d45 756c 6572 2849 2c4f 7574 Hash=Euler(I,Out │ │ │ │ +0001a570: 7075 743d 3e48 6173 6846 6f72 6d29 3b20 put=>HashForm); │ │ │ │ +0001a580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a5a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a620: 2d2d 2d2d 2d2b 0a7c 6937 203a 2041 3d72 -----+.|i7 : A=r │ │ │ │ -0001a630: 696e 6720 4575 6c65 7249 4861 7368 2322 ing EulerIHash#" │ │ │ │ -0001a640: 4353 4d22 2020 2020 2020 2020 2020 2020 CSM" │ │ │ │ +0001a5f0: 2d2d 2b0a 7c69 3720 3a20 413d 7269 6e67 --+.|i7 : A=ring │ │ │ │ +0001a600: 2045 756c 6572 4948 6173 6823 2243 534d EulerIHash#"CSM │ │ │ │ +0001a610: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ +0001a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a640: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a670: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a690: 2020 7c0a 7c6f 3720 3d20 4120 2020 2020 |.|o7 = A │ │ │ │ 0001a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a6c0: 2020 2020 207c 0a7c 6f37 203d 2041 2020 |.|o7 = A │ │ │ │ +0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a6e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a710: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a730: 2020 7c0a 7c6f 3720 3a20 5175 6f74 6965 |.|o7 : Quotie │ │ │ │ +0001a740: 6e74 5269 6e67 2020 2020 2020 2020 2020 ntRing │ │ │ │ 0001a750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a760: 2020 2020 207c 0a7c 6f37 203a 2051 756f |.|o7 : Quo │ │ │ │ -0001a770: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ -0001a780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a7b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001a760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a780: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a800: 2d2d 2d2d 2d2b 0a7c 6938 203a 2045 756c -----+.|i8 : Eul │ │ │ │ -0001a810: 6572 4948 6173 6823 7b30 2c31 7d3d 3d43 erIHash#{0,1}==C │ │ │ │ -0001a820: 534d 2841 2c69 6465 616c 2849 5f30 2a49 SM(A,ideal(I_0*I │ │ │ │ -0001a830: 5f31 2929 2020 2020 2020 2020 2020 2020 _1)) │ │ │ │ +0001a7d0: 2d2d 2b0a 7c69 3820 3a20 4575 6c65 7249 --+.|i8 : EulerI │ │ │ │ +0001a7e0: 4861 7368 237b 302c 317d 3d3d 4353 4d28 Hash#{0,1}==CSM( │ │ │ │ +0001a7f0: 412c 6964 6561 6c28 495f 302a 495f 3129 A,ideal(I_0*I_1) │ │ │ │ +0001a800: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0001a810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a820: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001a830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a850: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a870: 2020 7c0a 7c6f 3820 3d20 7472 7565 2020 |.|o8 = true │ │ │ │ 0001a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8a0: 2020 2020 207c 0a7c 6f38 203d 2074 7275 |.|o8 = tru │ │ │ │ -0001a8b0: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a8c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a940: 2d2d 2d2d 2d2b 0a7c 6939 203a 204a 3d49 -----+.|i9 : J=I │ │ │ │ -0001a950: 2b69 6465 616c 2878 5f30 2a78 5f32 2d78 +ideal(x_0*x_2-x │ │ │ │ -0001a960: 5f33 2a78 5f30 2920 2020 2020 2020 2020 _3*x_0) │ │ │ │ +0001a910: 2d2d 2b0a 7c69 3920 3a20 4a3d 492b 6964 --+.|i9 : J=I+id │ │ │ │ +0001a920: 6561 6c28 785f 302a 785f 322d 785f 332a eal(x_0*x_2-x_3* │ │ │ │ +0001a930: 785f 3029 2020 2020 2020 2020 2020 2020 x_0) │ │ │ │ +0001a940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a960: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a990: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a9b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001a9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a9e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa20: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0001aa30: 2020 2020 207c 0a7c 6f39 203d 2069 6465 |.|o9 = ide │ │ │ │ -0001aa40: 616c 2028 3130 3778 2020 2b20 3433 3736 al (107x + 4376 │ │ │ │ -0001aa50: 7820 202d 2036 3331 3678 2020 2b20 3331 x - 6316x + 31 │ │ │ │ -0001aa60: 3837 7820 202b 2033 3738 3378 202c 202d 87x + 3783x , - │ │ │ │ -0001aa70: 2036 3035 3378 2020 2b20 3835 3730 7820 6053x + 8570x │ │ │ │ -0001aa80: 7820 202b 207c 0a7c 2020 2020 2020 2020 x + |.| │ │ │ │ -0001aa90: 2020 2020 2020 2020 3020 2020 2020 2020 0 │ │ │ │ -0001aaa0: 2031 2020 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -0001aab0: 2020 2033 2020 2020 2020 2020 3420 2020 3 4 │ │ │ │ -0001aac0: 2020 2020 2020 3020 2020 2020 2020 2030 0 0 │ │ │ │ -0001aad0: 2031 2020 207c 0a7c 2020 2020 202d 2d2d 1 |.| --- │ │ │ │ +0001a9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a9f0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001aa00: 2020 7c0a 7c6f 3920 3d20 6964 6561 6c20 |.|o9 = ideal │ │ │ │ +0001aa10: 2831 3037 7820 202b 2034 3337 3678 2020 (107x + 4376x │ │ │ │ +0001aa20: 2d20 3633 3136 7820 202b 2033 3138 3778 - 6316x + 3187x │ │ │ │ +0001aa30: 2020 2b20 3337 3833 7820 2c20 2d20 3630 + 3783x , - 60 │ │ │ │ +0001aa40: 3533 7820 202b 2038 3537 3078 2078 2020 53x + 8570x x │ │ │ │ +0001aa50: 2b20 7c0a 7c20 2020 2020 2020 2020 2020 + |.| │ │ │ │ +0001aa60: 2020 2020 2030 2020 2020 2020 2020 3120 0 1 │ │ │ │ +0001aa70: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0001aa80: 3320 2020 2020 2020 2034 2020 2020 2020 3 4 │ │ │ │ +0001aa90: 2020 2030 2020 2020 2020 2020 3020 3120 0 0 1 │ │ │ │ +0001aaa0: 2020 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +0001aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001aae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001aaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0001ab30: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ab40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ab50: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ab60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ab70: 2020 2020 207c 0a7c 2020 2020 2031 3033 |.| 103 │ │ │ │ -0001ab80: 3539 7820 202d 2031 3630 3930 7820 7820 59x - 16090x x │ │ │ │ -0001ab90: 202d 2038 3231 3078 2078 2020 2b20 3530 - 8210x x + 50 │ │ │ │ -0001aba0: 3731 7820 202b 2038 3434 3478 2078 2020 71x + 8444x x │ │ │ │ -0001abb0: 2d20 3839 3937 7820 7820 202d 2036 3934 - 8997x x - 694 │ │ │ │ -0001abc0: 3978 2078 207c 0a7c 2020 2020 2020 2020 9x x |.| │ │ │ │ -0001abd0: 2020 2031 2020 2020 2020 2020 2030 2032 1 0 2 │ │ │ │ -0001abe0: 2020 2020 2020 2020 3120 3220 2020 2020 1 2 │ │ │ │ -0001abf0: 2020 2032 2020 2020 2020 2020 3020 3320 2 0 3 │ │ │ │ -0001ac00: 2020 2020 2020 2031 2033 2020 2020 2020 1 3 │ │ │ │ -0001ac10: 2020 3220 337c 0a7c 2020 2020 202d 2d2d 2 3|.| --- │ │ │ │ +0001aaf0: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ +0001ab00: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001ab10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ab20: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001ab30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ab40: 2020 7c0a 7c20 2020 2020 3130 3335 3978 |.| 10359x │ │ │ │ +0001ab50: 2020 2d20 3136 3039 3078 2078 2020 2d20 - 16090x x - │ │ │ │ +0001ab60: 3832 3130 7820 7820 202b 2035 3037 3178 8210x x + 5071x │ │ │ │ +0001ab70: 2020 2b20 3834 3434 7820 7820 202d 2038 + 8444x x - 8 │ │ │ │ +0001ab80: 3939 3778 2078 2020 2d20 3639 3439 7820 997x x - 6949x │ │ │ │ +0001ab90: 7820 7c0a 7c20 2020 2020 2020 2020 2020 x |.| │ │ │ │ +0001aba0: 3120 2020 2020 2020 2020 3020 3220 2020 1 0 2 │ │ │ │ +0001abb0: 2020 2020 2031 2032 2020 2020 2020 2020 1 2 │ │ │ │ +0001abc0: 3220 2020 2020 2020 2030 2033 2020 2020 2 0 3 │ │ │ │ +0001abd0: 2020 2020 3120 3320 2020 2020 2020 2032 1 3 2 │ │ │ │ +0001abe0: 2033 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 3|.| ------ │ │ │ │ +0001abf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ac00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ac10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ac20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ac30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ac40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ac50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ac60: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0001ac70: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aca0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -0001acb0: 2020 2020 207c 0a7c 2020 2020 202d 2031 |.| - 1 │ │ │ │ -0001acc0: 3432 3534 7820 202d 2031 3132 3236 7820 4254x - 11226x │ │ │ │ -0001acd0: 7820 202b 2032 3635 3378 2078 2020 2b20 x + 2653x x + │ │ │ │ -0001ace0: 3132 3336 3578 2078 2020 2d20 3130 3232 12365x x - 1022 │ │ │ │ -0001acf0: 3678 2078 2020 2d20 3132 3639 3678 202c 6x x - 12696x , │ │ │ │ -0001ad00: 2078 2078 207c 0a7c 2020 2020 2020 2020 x x |.| │ │ │ │ -0001ad10: 2020 2020 2033 2020 2020 2020 2020 2030 3 0 │ │ │ │ -0001ad20: 2034 2020 2020 2020 2020 3120 3420 2020 4 1 4 │ │ │ │ -0001ad30: 2020 2020 2020 3220 3420 2020 2020 2020 2 4 │ │ │ │ -0001ad40: 2020 3320 3420 2020 2020 2020 2020 3420 3 4 4 │ │ │ │ -0001ad50: 2020 3020 327c 0a7c 2020 2020 202d 2d2d 0 2|.| --- │ │ │ │ +0001ac30: 2d2d 7c0a 7c20 2020 2020 2020 2020 2020 --|.| │ │ │ │ +0001ac40: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ac60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ac70: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0001ac80: 2020 7c0a 7c20 2020 2020 2d20 3134 3235 |.| - 1425 │ │ │ │ +0001ac90: 3478 2020 2d20 3131 3232 3678 2078 2020 4x - 11226x x │ │ │ │ +0001aca0: 2b20 3236 3533 7820 7820 202b 2031 3233 + 2653x x + 123 │ │ │ │ +0001acb0: 3635 7820 7820 202d 2031 3032 3236 7820 65x x - 10226x │ │ │ │ +0001acc0: 7820 202d 2031 3236 3936 7820 2c20 7820 x - 12696x , x │ │ │ │ +0001acd0: 7820 7c0a 7c20 2020 2020 2020 2020 2020 x |.| │ │ │ │ +0001ace0: 2020 3320 2020 2020 2020 2020 3020 3420 3 0 4 │ │ │ │ +0001acf0: 2020 2020 2020 2031 2034 2020 2020 2020 1 4 │ │ │ │ +0001ad00: 2020 2032 2034 2020 2020 2020 2020 2033 2 4 3 │ │ │ │ +0001ad10: 2034 2020 2020 2020 2020 2034 2020 2030 4 4 0 │ │ │ │ +0001ad20: 2032 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2|.| ------ │ │ │ │ +0001ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ad80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ada0: 2d2d 2d2d 2d7c 0a7c 2020 2020 202d 2078 -----|.| - x │ │ │ │ -0001adb0: 2078 2029 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ -0001adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ad70: 2d2d 7c0a 7c20 2020 2020 2d20 7820 7820 --|.| - x x │ │ │ │ +0001ad80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0001ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001adc0: 2020 7c0a 7c20 2020 2020 2020 2030 2033 |.| 0 3 │ │ │ │ 0001add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001adf0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001ae00: 3020 3320 2020 2020 2020 2020 2020 2020 0 3 │ │ │ │ -0001ae10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001adf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ae10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ae50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ae60: 2020 7c0a 7c6f 3920 3a20 4964 6561 6c20 |.|o9 : Ideal │ │ │ │ +0001ae70: 6f66 2052 2020 2020 2020 2020 2020 2020 of R │ │ │ │ 0001ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae90: 2020 2020 207c 0a7c 6f39 203a 2049 6465 |.|o9 : Ide │ │ │ │ -0001aea0: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ -0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aee0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001ae90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001aea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001aeb0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001af10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001af20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001af30: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2074 6861 -----+..Note tha │ │ │ │ -0001af40: 7420 7468 6520 6964 6561 6c20 4a20 6162 t the ideal J ab │ │ │ │ -0001af50: 6f76 6520 6973 2061 2063 6f6d 706c 6574 ove is a complet │ │ │ │ -0001af60: 6520 696e 7465 7273 6563 7469 6f6e 2c20 e intersection, │ │ │ │ -0001af70: 7468 7573 2077 6520 6d61 7920 6368 616e thus we may chan │ │ │ │ -0001af80: 6765 2074 6865 0a6d 6574 686f 6420 6f70 ge the.method op │ │ │ │ -0001af90: 7469 6f6e 2077 6869 6368 206d 6179 2073 tion which may s │ │ │ │ -0001afa0: 7065 6564 2063 6f6d 7075 7461 7469 6f6e peed computation │ │ │ │ -0001afb0: 2069 6e20 736f 6d65 2063 6173 6573 2e20 in some cases. │ │ │ │ -0001afc0: 5765 206d 6179 2061 6c73 6f20 6e6f 7465 We may also note │ │ │ │ -0001afd0: 2074 6861 740a 7468 6520 6964 6561 6c20 that.the ideal │ │ │ │ -0001afe0: 6765 6e65 7261 7465 6420 6279 2074 6865 generated by the │ │ │ │ -0001aff0: 2066 6972 7374 2032 2067 656e 6572 6174 first 2 generat │ │ │ │ -0001b000: 6f72 7320 6f66 2049 2064 6566 696e 6573 ors of I defines │ │ │ │ -0001b010: 2061 2073 6d6f 6f74 6820 7363 6865 6d65 a smooth scheme │ │ │ │ -0001b020: 2061 6e64 0a69 6e70 7574 2074 6869 7320 and.input this │ │ │ │ -0001b030: 696e 666f 726d 6174 696f 6e20 696e 746f information into │ │ │ │ -0001b040: 2074 6865 206d 6574 686f 642e 2054 6869 the method. Thi │ │ │ │ -0001b050: 7320 6d61 7920 616c 736f 2069 6d70 726f s may also impro │ │ │ │ -0001b060: 7665 2063 6f6d 7075 7461 7469 6f6e 0a73 ve computation.s │ │ │ │ -0001b070: 7065 6564 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d peed...+-------- │ │ │ │ -0001b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b0b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 ---------+.|i10 │ │ │ │ -0001b0c0: 3a20 7469 6d65 2045 756c 6572 284a 2c4d : time Euler(J,M │ │ │ │ -0001b0d0: 6574 686f 643d 3e44 6972 6563 7443 6f6d ethod=>DirectCom │ │ │ │ -0001b0e0: 706c 6574 6549 6e74 2920 2020 2020 2020 pleteInt) │ │ │ │ -0001b0f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001b100: 202d 2d20 7573 6564 2030 2e30 3739 3233 -- used 0.07923 │ │ │ │ -0001b110: 3233 7320 2863 7075 293b 2030 2e30 3735 23s (cpu); 0.075 │ │ │ │ -0001b120: 3436 3438 7320 2874 6872 6561 6429 3b20 4648s (thread); │ │ │ │ -0001b130: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ -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 3633 3232 7320 used 0.146322s │ │ │ │ +0001b0e0: 2863 7075 293b 2030 2e30 3739 3335 3935 (cpu); 0.0793595 │ │ │ │ +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: 3935 3536 3273 2028 6370 7529 3b20 302e 95562s (cpu); 0. │ │ │ │ -0001b270: 3130 3337 3035 7320 2874 6872 6561 6429 103705s (thread) │ │ │ │ -0001b280: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ -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 3230 3134 | -- used 0.2014 │ │ │ │ +0001b230: 3232 7320 2863 7075 293b 2030 2e30 3834 22s (cpu); 0.084 │ │ │ │ +0001b240: 3935 3635 7320 2874 6872 6561 6429 3b20 9565s (thread); │ │ │ │ +0001b250: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +0001b260: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b2a0: 2020 2020 2020 7c0a 7c6f 3131 203d 2032 |.|o11 = 2 │ │ │ │ 0001b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b2d0: 2020 2020 2020 2020 207c 0a7c 6f31 3120 |.|o11 │ │ │ │ -0001b2e0: 3d20 3220 2020 2020 2020 2020 2020 2020 = 2 │ │ │ │ -0001b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b310: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0001b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b360: 2d2b 0a0a 4e6f 7720 636f 6e73 6964 6572 -+..Now consider │ │ │ │ -0001b370: 2061 6e20 6578 616d 706c 6520 696e 205c an example in \ │ │ │ │ -0001b380: 5050 5e32 205c 7469 6d65 7320 5c50 505e PP^2 \times \PP^ │ │ │ │ -0001b390: 322e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2...+----------- │ │ │ │ -0001b3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b3c0: 2d2d 2d2b 0a7c 6931 3220 3a20 523d 4d75 ---+.|i12 : R=Mu │ │ │ │ -0001b3d0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ -0001b3e0: 287b 322c 327d 2920 2020 2020 2020 2020 ({2,2}) │ │ │ │ -0001b3f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b2e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0001b2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001b330: 0a4e 6f77 2063 6f6e 7369 6465 7220 616e .Now consider an │ │ │ │ +0001b340: 2065 7861 6d70 6c65 2069 6e20 5c50 505e example in \PP^ │ │ │ │ +0001b350: 3220 5c74 696d 6573 205c 5050 5e32 2e0a 2 \times \PP^2.. │ │ │ │ +0001b360: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b390: 2b0a 7c69 3132 203a 2052 3d4d 756c 7469 +.|i12 : R=Multi │ │ │ │ +0001b3a0: 5072 6f6a 436f 6f72 6452 696e 6728 7b32 ProjCoordRing({2 │ │ │ │ +0001b3b0: 2c32 7d29 2020 2020 2020 2020 2020 2020 ,2}) │ │ │ │ +0001b3c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b3f0: 2020 7c0a 7c6f 3132 203d 2052 2020 2020 |.|o12 = R │ │ │ │ 0001b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b420: 2020 2020 207c 0a7c 6f31 3220 3d20 5220 |.|o12 = R │ │ │ │ +0001b420: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b450: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b450: 2020 2020 7c0a 7c6f 3132 203a 2050 6f6c |.|o12 : Pol │ │ │ │ +0001b460: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ 0001b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b480: 2020 2020 2020 207c 0a7c 6f31 3220 3a20 |.|o12 : │ │ │ │ -0001b490: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ -0001b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b4b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001b4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b4e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ -0001b4f0: 3a20 723d 6765 6e73 2052 2020 2020 2020 : r=gens R │ │ │ │ +0001b480: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001b490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b4b0: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2072 ------+.|i13 : r │ │ │ │ +0001b4c0: 3d67 656e 7320 5220 2020 2020 2020 2020 =gens R │ │ │ │ +0001b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b4e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b510: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b540: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0001b550: 3320 3d20 7b78 202c 2078 202c 2078 202c 3 = {x , x , x , │ │ │ │ -0001b560: 2078 202c 2078 202c 2078 207d 2020 2020 x , x , x } │ │ │ │ -0001b570: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001b580: 2020 2020 2020 2030 2020 2031 2020 2032 0 1 2 │ │ │ │ -0001b590: 2020 2033 2020 2034 2020 2035 2020 2020 3 4 5 │ │ │ │ -0001b5a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001b5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b510: 2020 2020 2020 2020 7c0a 7c6f 3133 203d |.|o13 = │ │ │ │ +0001b520: 207b 7820 2c20 7820 2c20 7820 2c20 7820 {x , x , x , x │ │ │ │ +0001b530: 2c20 7820 2c20 7820 7d20 2020 2020 2020 , x , x } │ │ │ │ +0001b540: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001b550: 2020 2020 3020 2020 3120 2020 3220 2020 0 1 2 │ │ │ │ +0001b560: 3320 2020 3420 2020 3520 2020 2020 2020 3 4 5 │ │ │ │ +0001b570: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b5a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0001b5b0: 3320 3a20 4c69 7374 2020 2020 2020 2020 3 : List │ │ │ │ 0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b5d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b5e0: 7c6f 3133 203a 204c 6973 7420 2020 2020 |o13 : List │ │ │ │ -0001b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b600: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001b610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0001b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b640: 2b0a 7c69 3134 203a 204b 3d69 6465 616c +.|i14 : K=ideal │ │ │ │ -0001b650: 2872 5f30 5e32 2a72 5f33 2d72 5f34 2a72 (r_0^2*r_3-r_4*r │ │ │ │ -0001b660: 5f31 2a72 5f32 2c72 5f32 5e32 2a72 5f35 _1*r_2,r_2^2*r_5 │ │ │ │ -0001b670: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ -0001b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b5d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001b610: 6931 3420 3a20 4b3d 6964 6561 6c28 725f i14 : K=ideal(r_ │ │ │ │ +0001b620: 305e 322a 725f 332d 725f 342a 725f 312a 0^2*r_3-r_4*r_1* │ │ │ │ +0001b630: 725f 322c 725f 325e 322a 725f 3529 7c0a r_2,r_2^2*r_5)|. │ │ │ │ +0001b640: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001b650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b660: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b670: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001b680: 3220 2020 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ 0001b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b6a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b6b0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001b6c0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001b6d0: 2020 207c 0a7c 6f31 3420 3d20 6964 6561 |.|o14 = idea │ │ │ │ -0001b6e0: 6c20 2878 2078 2020 2d20 7820 7820 7820 l (x x - x x x │ │ │ │ -0001b6f0: 2c20 7820 7820 2920 2020 2020 2020 2020 , x x ) │ │ │ │ -0001b700: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001b710: 2020 2020 2030 2033 2020 2020 3120 3220 0 3 1 2 │ │ │ │ -0001b720: 3420 2020 3220 3520 2020 2020 2020 2020 4 2 5 │ │ │ │ -0001b730: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001b740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b6a0: 7c0a 7c6f 3134 203d 2069 6465 616c 2028 |.|o14 = ideal ( │ │ │ │ +0001b6b0: 7820 7820 202d 2078 2078 2078 202c 2078 x x - x x x , x │ │ │ │ +0001b6c0: 2078 2029 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ +0001b6d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b6e0: 2020 3020 3320 2020 2031 2032 2034 2020 0 3 1 2 4 │ │ │ │ +0001b6f0: 2032 2035 2020 2020 2020 2020 2020 2020 2 5 │ │ │ │ +0001b700: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b730: 2020 207c 0a7c 6f31 3420 3a20 4964 6561 |.|o14 : Idea │ │ │ │ +0001b740: 6c20 6f66 2052 2020 2020 2020 2020 2020 l of R │ │ │ │ 0001b750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b760: 2020 2020 2020 7c0a 7c6f 3134 203a 2049 |.|o14 : I │ │ │ │ -0001b770: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ -0001b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b790: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0001b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a --------+.|i15 : │ │ │ │ -0001b7d0: 2045 756c 6572 4b3d 4575 6c65 7228 4b29 EulerK=Euler(K) │ │ │ │ +0001b760: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b790: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 4575 -----+.|i15 : Eu │ │ │ │ +0001b7a0: 6c65 724b 3d45 756c 6572 284b 2920 2020 lerK=Euler(K) │ │ │ │ +0001b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b7c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b7f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001b800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b7f0: 2020 2020 2020 207c 0a7c 6f31 3520 3d20 |.|o15 = │ │ │ │ +0001b800: 3720 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ 0001b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b820: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -0001b830: 203d 2037 2020 2020 2020 2020 2020 2020 = 7 │ │ │ │ -0001b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b850: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0001b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001b890: 3136 203a 2063 736d 4b3d 2043 534d 284b 16 : csmK= CSM(K │ │ │ │ -0001b8a0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0001b8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001b8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b8e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b8f0: 7c20 2020 2020 2020 2032 2032 2020 2020 | 2 2 │ │ │ │ -0001b900: 2032 2020 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ -0001b910: 3220 2020 2020 2020 2020 2020 2032 207c 2 2 | │ │ │ │ -0001b920: 0a7c 6f31 3620 3d20 3768 2068 2020 2b20 .|o16 = 7h h + │ │ │ │ -0001b930: 3568 2068 2020 2b20 3468 2068 2020 2b20 5h h + 4h h + │ │ │ │ -0001b940: 6820 202b 2033 6820 6820 202b 2068 2020 h + 3h h + h │ │ │ │ -0001b950: 7c0a 7c20 2020 2020 2020 2031 2032 2020 |.| 1 2 │ │ │ │ -0001b960: 2020 2031 2032 2020 2020 2031 2032 2020 1 2 1 2 │ │ │ │ -0001b970: 2020 3120 2020 2020 3120 3220 2020 2032 1 1 2 2 │ │ │ │ -0001b980: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b820: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b850: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 ---------+.|i16 │ │ │ │ +0001b860: 3a20 6373 6d4b 3d20 4353 4d28 4b29 2020 : csmK= CSM(K) │ │ │ │ +0001b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b880: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b8b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001b8c0: 2020 2020 2020 3220 3220 2020 2020 3220 2 2 2 │ │ │ │ +0001b8d0: 2020 2020 2020 2020 3220 2020 2032 2020 2 2 │ │ │ │ +0001b8e0: 2020 2020 2020 2020 2020 3220 7c0a 7c6f 2 |.|o │ │ │ │ +0001b8f0: 3136 203d 2037 6820 6820 202b 2035 6820 16 = 7h h + 5h │ │ │ │ +0001b900: 6820 202b 2034 6820 6820 202b 2068 2020 h + 4h h + h │ │ │ │ +0001b910: 2b20 3368 2068 2020 2b20 6820 207c 0a7c + 3h h + h |.| │ │ │ │ +0001b920: 2020 2020 2020 2020 3120 3220 2020 2020 1 2 │ │ │ │ +0001b930: 3120 3220 2020 2020 3120 3220 2020 2031 1 2 1 2 1 │ │ │ │ +0001b940: 2020 2020 2031 2032 2020 2020 3220 7c0a 1 2 2 |. │ │ │ │ +0001b950: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001b960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b980: 0a7c 2020 2020 2020 5a5a 5b68 202e 2e68 .| ZZ[h ..h │ │ │ │ +0001b990: 205d 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9b0: 2020 7c0a 7c20 2020 2020 205a 5a5b 6820 |.| ZZ[h │ │ │ │ -0001b9c0: 2e2e 6820 5d20 2020 2020 2020 2020 2020 ..h ] │ │ │ │ +0001b9b0: 7c0a 7c20 2020 2020 2020 2020 2031 2020 |.| 1 │ │ │ │ +0001b9c0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b9f0: 3120 2020 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ +0001b9e0: 207c 0a7c 6f31 3620 3a20 2d2d 2d2d 2d2d |.|o16 : ------ │ │ │ │ +0001b9f0: 2d2d 2d2d 2020 2020 2020 2020 2020 2020 ---- │ │ │ │ 0001ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba10: 2020 2020 7c0a 7c6f 3136 203a 202d 2d2d |.|o16 : --- │ │ │ │ -0001ba20: 2d2d 2d2d 2d2d 2d20 2020 2020 2020 2020 ------- │ │ │ │ +0001ba10: 2020 7c0a 7c20 2020 2020 2020 2020 3320 |.| 3 │ │ │ │ +0001ba20: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0001ba30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001ba50: 2033 2020 2033 2020 2020 2020 2020 2020 3 3 │ │ │ │ +0001ba40: 2020 207c 0a7c 2020 2020 2020 2028 6820 |.| (h │ │ │ │ +0001ba50: 2c20 6820 2920 2020 2020 2020 2020 2020 , h ) │ │ │ │ 0001ba60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba70: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001ba80: 2868 202c 2068 2029 2020 2020 2020 2020 (h , h ) │ │ │ │ +0001ba70: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001ba80: 3120 2020 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ 0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001baa0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0001bab0: 2020 2031 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ -0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bad0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001bae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001baf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bb00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3720 ---------+.|i17 │ │ │ │ -0001bb10: 3a20 4575 6c65 724b 3d3d 4575 6c65 7228 : EulerK==Euler( │ │ │ │ -0001bb20: 6373 6d4b 2920 2020 2020 2020 2020 2020 csmK) │ │ │ │ -0001bb30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001baa0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bad0: 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a 2045 ------+.|i17 : E │ │ │ │ +0001bae0: 756c 6572 4b3d 3d45 756c 6572 2863 736d ulerK==Euler(csm │ │ │ │ +0001baf0: 4b29 2020 2020 2020 2020 2020 2020 2020 K) │ │ │ │ +0001bb00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb30: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ +0001bb40: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ 0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb60: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0001bb70: 3720 3d20 7472 7565 2020 2020 2020 2020 7 = true │ │ │ │ -0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0001bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0001bbd0: 496e 2074 6865 2063 6173 6520 7768 6572 In the case wher │ │ │ │ -0001bbe0: 6520 7468 6520 616d 6269 656e 7420 7370 e the ambient sp │ │ │ │ -0001bbf0: 6163 6520 6973 2061 2074 6f72 6963 2076 ace is a toric v │ │ │ │ -0001bc00: 6172 6965 7479 2077 6869 6368 2069 7320 ariety which is │ │ │ │ -0001bc10: 6e6f 7420 6120 7072 6f64 7563 740a 6f66 not a product.of │ │ │ │ -0001bc20: 2070 726f 6a65 6374 6976 6520 7370 6163 projective spac │ │ │ │ -0001bc30: 6573 2077 6520 6d75 7374 206c 6f61 6420 es we must load │ │ │ │ -0001bc40: 7468 6520 4e6f 726d 616c 546f 7269 6356 the NormalToricV │ │ │ │ -0001bc50: 6172 6965 7469 6573 2070 6163 6b61 6765 arieties package │ │ │ │ -0001bc60: 2061 6e64 206d 7573 740a 616c 736f 2069 and must.also i │ │ │ │ -0001bc70: 6e70 7574 2074 6865 2074 6f72 6963 2076 nput the toric v │ │ │ │ -0001bc80: 6172 6965 7479 2e20 4966 2074 6865 2074 ariety. If the t │ │ │ │ -0001bc90: 6f72 6963 2076 6172 6965 7479 2069 7320 oric variety is │ │ │ │ -0001bca0: 6120 7072 6f64 7563 7420 6f66 2070 726f a product of pro │ │ │ │ -0001bcb0: 6a65 6374 6976 650a 7370 6163 6520 6974 jective.space it │ │ │ │ -0001bcc0: 2069 7320 7265 636f 6d6d 656e 6465 6420 is recommended │ │ │ │ -0001bcd0: 746f 2075 7365 2074 6865 2066 6f72 6d20 to use the form │ │ │ │ -0001bce0: 6162 6f76 6520 7261 7468 6572 2074 6861 above rather tha │ │ │ │ -0001bcf0: 6e20 696e 7075 7474 696e 6720 7468 6520 n inputting the │ │ │ │ -0001bd00: 746f 7269 630a 7661 7269 6574 7920 666f toric.variety fo │ │ │ │ -0001bd10: 7220 6566 6669 6369 656e 6379 2072 6561 r efficiency rea │ │ │ │ -0001bd20: 736f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d sons...+-------- │ │ │ │ -0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd70: 2b0a 7c69 3138 203a 206e 6565 6473 5061 +.|i18 : needsPa │ │ │ │ -0001bd80: 636b 6167 6520 224e 6f72 6d61 6c54 6f72 ckage "NormalTor │ │ │ │ -0001bd90: 6963 5661 7269 6574 6965 7322 2020 2020 icVarieties" │ │ │ │ +0001bb60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 ----------+..In │ │ │ │ +0001bba0: 7468 6520 6361 7365 2077 6865 7265 2074 the case where t │ │ │ │ +0001bbb0: 6865 2061 6d62 6965 6e74 2073 7061 6365 he ambient space │ │ │ │ +0001bbc0: 2069 7320 6120 746f 7269 6320 7661 7269 is a toric vari │ │ │ │ +0001bbd0: 6574 7920 7768 6963 6820 6973 206e 6f74 ety which is not │ │ │ │ +0001bbe0: 2061 2070 726f 6475 6374 0a6f 6620 7072 a product.of pr │ │ │ │ +0001bbf0: 6f6a 6563 7469 7665 2073 7061 6365 7320 ojective spaces │ │ │ │ +0001bc00: 7765 206d 7573 7420 6c6f 6164 2074 6865 we must load the │ │ │ │ +0001bc10: 204e 6f72 6d61 6c54 6f72 6963 5661 7269 NormalToricVari │ │ │ │ +0001bc20: 6574 6965 7320 7061 636b 6167 6520 616e eties package an │ │ │ │ +0001bc30: 6420 6d75 7374 0a61 6c73 6f20 696e 7075 d must.also inpu │ │ │ │ +0001bc40: 7420 7468 6520 746f 7269 6320 7661 7269 t the toric vari │ │ │ │ +0001bc50: 6574 792e 2049 6620 7468 6520 746f 7269 ety. If the tori │ │ │ │ +0001bc60: 6320 7661 7269 6574 7920 6973 2061 2070 c variety is a p │ │ │ │ +0001bc70: 726f 6475 6374 206f 6620 7072 6f6a 6563 roduct of projec │ │ │ │ +0001bc80: 7469 7665 0a73 7061 6365 2069 7420 6973 tive.space it is │ │ │ │ +0001bc90: 2072 6563 6f6d 6d65 6e64 6564 2074 6f20 recommended to │ │ │ │ +0001bca0: 7573 6520 7468 6520 666f 726d 2061 626f use the form abo │ │ │ │ +0001bcb0: 7665 2072 6174 6865 7220 7468 616e 2069 ve rather than i │ │ │ │ +0001bcc0: 6e70 7574 7469 6e67 2074 6865 2074 6f72 nputting the tor │ │ │ │ +0001bcd0: 6963 0a76 6172 6965 7479 2066 6f72 2065 ic.variety for e │ │ │ │ +0001bce0: 6666 6963 6965 6e63 7920 7265 6173 6f6e fficiency reason │ │ │ │ +0001bcf0: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s...+----------- │ │ │ │ +0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001bd40: 6931 3820 3a20 6e65 6564 7350 6163 6b61 i18 : needsPacka │ │ │ │ +0001bd50: 6765 2022 4e6f 726d 616c 546f 7269 6356 ge "NormalToricV │ │ │ │ +0001bd60: 6172 6965 7469 6573 2220 2020 2020 2020 arieties" │ │ │ │ +0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bd80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001bd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bdb0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bdd0: 2020 207c 0a7c 6f31 3820 3d20 4e6f 726d |.|o18 = Norm │ │ │ │ +0001bde0: 616c 546f 7269 6356 6172 6965 7469 6573 alToricVarieties │ │ │ │ 0001bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be00: 2020 2020 2020 7c0a 7c6f 3138 203d 204e |.|o18 = N │ │ │ │ -0001be10: 6f72 6d61 6c54 6f72 6963 5661 7269 6574 ormalToricVariet │ │ │ │ -0001be20: 6965 7320 2020 2020 2020 2020 2020 2020 ies │ │ │ │ +0001be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001be20: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be50: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be60: 2020 2020 2020 2020 207c 0a7c 6f31 3820 |.|o18 │ │ │ │ +0001be70: 3a20 5061 636b 6167 6520 2020 2020 2020 : Package │ │ │ │ 0001be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001bea0: 3138 203a 2050 6163 6b61 6765 2020 2020 18 : Package │ │ │ │ -0001beb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bee0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bf00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bf10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bf20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bf30: 2d2d 2b0a 7c69 3139 203a 2052 686f 203d --+.|i19 : Rho = │ │ │ │ -0001bf40: 207b 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 {{1,0,0},{0,1,0 │ │ │ │ -0001bf50: 7d2c 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 },{0,0,1},{-1,-1 │ │ │ │ -0001bf60: 2c30 7d2c 7b30 2c30 2c2d 317d 7d20 2020 ,0},{0,0,-1}} │ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001beb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001bf00: 0a7c 6931 3920 3a20 5268 6f20 3d20 7b7b .|i19 : Rho = {{ │ │ │ │ +0001bf10: 312c 302c 307d 2c7b 302c 312c 307d 2c7b 1,0,0},{0,1,0},{ │ │ │ │ +0001bf20: 302c 302c 317d 2c7b 2d31 2c2d 312c 307d 0,0,1},{-1,-1,0} │ │ │ │ +0001bf30: 2c7b 302c 302c 2d31 7d7d 2020 2020 2020 ,{0,0,-1}} │ │ │ │ +0001bf40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bfc0: 2020 2020 2020 2020 7c0a 7c6f 3139 203d |.|o19 = │ │ │ │ -0001bfd0: 207b 7b31 2c20 302c 2030 7d2c 207b 302c {{1, 0, 0}, {0, │ │ │ │ -0001bfe0: 2031 2c20 307d 2c20 7b30 2c20 302c 2031 1, 0}, {0, 0, 1 │ │ │ │ -0001bff0: 7d2c 207b 2d31 2c20 2d31 2c20 307d 2c20 }, {-1, -1, 0}, │ │ │ │ -0001c000: 7b30 2c20 302c 202d 317d 7d20 2020 2020 {0, 0, -1}} │ │ │ │ -0001c010: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bf90: 2020 2020 207c 0a7c 6f31 3920 3d20 7b7b |.|o19 = {{ │ │ │ │ +0001bfa0: 312c 2030 2c20 307d 2c20 7b30 2c20 312c 1, 0, 0}, {0, 1, │ │ │ │ +0001bfb0: 2030 7d2c 207b 302c 2030 2c20 317d 2c20 0}, {0, 0, 1}, │ │ │ │ +0001bfc0: 7b2d 312c 202d 312c 2030 7d2c 207b 302c {-1, -1, 0}, {0, │ │ │ │ +0001bfd0: 2030 2c20 2d31 7d7d 2020 2020 2020 2020 0, -1}} │ │ │ │ +0001bfe0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001bff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c020: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0001c030: 3920 3a20 4c69 7374 2020 2020 2020 2020 9 : List │ │ │ │ 0001c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001c060: 7c6f 3139 203a 204c 6973 7420 2020 2020 |o19 : List │ │ │ │ -0001c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c0a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001c050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c070: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c0f0: 2d2d 2d2d 2b0a 7c69 3230 203a 2053 6967 ----+.|i20 : Sig │ │ │ │ -0001c100: 6d61 203d 207b 7b30 2c31 2c32 7d2c 7b31 ma = {{0,1,2},{1 │ │ │ │ -0001c110: 2c32 2c33 7d2c 7b30 2c32 2c33 7d2c 7b30 ,2,3},{0,2,3},{0 │ │ │ │ -0001c120: 2c31 2c34 7d2c 7b31 2c33 2c34 7d2c 7b30 ,1,4},{1,3,4},{0 │ │ │ │ -0001c130: 2c33 2c34 7d7d 2020 2020 2020 2020 207c ,3,4}} | │ │ │ │ -0001c140: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c180: 2020 2020 2020 2020 2020 7c0a 7c6f 3230 |.|o20 │ │ │ │ -0001c190: 203d 207b 7b30 2c20 312c 2032 7d2c 207b = {{0, 1, 2}, { │ │ │ │ -0001c1a0: 312c 2032 2c20 337d 2c20 7b30 2c20 322c 1, 2, 3}, {0, 2, │ │ │ │ -0001c1b0: 2033 7d2c 207b 302c 2031 2c20 347d 2c20 3}, {0, 1, 4}, │ │ │ │ -0001c1c0: 7b31 2c20 332c 2034 7d2c 207b 302c 2033 {1, 3, 4}, {0, 3 │ │ │ │ -0001c1d0: 2c20 347d 7d7c 0a7c 2020 2020 2020 2020 , 4}}|.| │ │ │ │ -0001c1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c0c0: 2d2b 0a7c 6932 3020 3a20 5369 676d 6120 -+.|i20 : Sigma │ │ │ │ +0001c0d0: 3d20 7b7b 302c 312c 327d 2c7b 312c 322c = {{0,1,2},{1,2, │ │ │ │ +0001c0e0: 337d 2c7b 302c 322c 337d 2c7b 302c 312c 3},{0,2,3},{0,1, │ │ │ │ +0001c0f0: 347d 2c7b 312c 332c 347d 2c7b 302c 332c 4},{1,3,4},{0,3, │ │ │ │ +0001c100: 347d 7d20 2020 2020 2020 2020 7c0a 7c20 4}} |.| │ │ │ │ +0001c110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c150: 2020 2020 2020 207c 0a7c 6f32 3020 3d20 |.|o20 = │ │ │ │ +0001c160: 7b7b 302c 2031 2c20 327d 2c20 7b31 2c20 {{0, 1, 2}, {1, │ │ │ │ +0001c170: 322c 2033 7d2c 207b 302c 2032 2c20 337d 2, 3}, {0, 2, 3} │ │ │ │ +0001c180: 2c20 7b30 2c20 312c 2034 7d2c 207b 312c , {0, 1, 4}, {1, │ │ │ │ +0001c190: 2033 2c20 347d 2c20 7b30 2c20 332c 2034 3, 4}, {0, 3, 4 │ │ │ │ +0001c1a0: 7d7d 7c0a 7c20 2020 2020 2020 2020 2020 }}|.| │ │ │ │ +0001c1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c1e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c1f0: 6f32 3020 3a20 4c69 7374 2020 2020 2020 o20 : List │ │ │ │ 0001c200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c220: 7c0a 7c6f 3230 203a 204c 6973 7420 2020 |.|o20 : List │ │ │ │ -0001c230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c260: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001c220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c230: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c2b0: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 2058 ------+.|i21 : X │ │ │ │ -0001c2c0: 203d 206e 6f72 6d61 6c54 6f72 6963 5661 = normalToricVa │ │ │ │ -0001c2d0: 7269 6574 7928 5268 6f2c 5369 676d 612c riety(Rho,Sigma, │ │ │ │ -0001c2e0: 436f 6566 6669 6369 656e 7452 696e 6720 CoefficientRing │ │ │ │ -0001c2f0: 3d3e 5a5a 2f33 3237 3439 2920 2020 2020 =>ZZ/32749) │ │ │ │ -0001c300: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c280: 2d2d 2d2b 0a7c 6932 3120 3a20 5820 3d20 ---+.|i21 : X = │ │ │ │ +0001c290: 6e6f 726d 616c 546f 7269 6356 6172 6965 normalToricVarie │ │ │ │ +0001c2a0: 7479 2852 686f 2c53 6967 6d61 2c43 6f65 ty(Rho,Sigma,Coe │ │ │ │ +0001c2b0: 6666 6963 6965 6e74 5269 6e67 203d 3e5a fficientRing =>Z │ │ │ │ +0001c2c0: 5a2f 3332 3734 3929 2020 2020 2020 7c0a Z/32749) |. │ │ │ │ +0001c2d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001c2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c310: 2020 2020 2020 2020 207c 0a7c 6f32 3120 |.|o21 │ │ │ │ +0001c320: 3d20 5820 2020 2020 2020 2020 2020 2020 = X │ │ │ │ 0001c330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c340: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001c350: 3231 203d 2058 2020 2020 2020 2020 2020 21 = X │ │ │ │ -0001c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c360: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0001c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c390: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0001c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c3a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c3b0: 0a7c 6f32 3120 3a20 4e6f 726d 616c 546f .|o21 : NormalTo │ │ │ │ +0001c3c0: 7269 6356 6172 6965 7479 2020 2020 2020 ricVariety │ │ │ │ 0001c3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3e0: 2020 7c0a 7c6f 3231 203a 204e 6f72 6d61 |.|o21 : Norma │ │ │ │ -0001c3f0: 6c54 6f72 6963 5661 7269 6574 7920 2020 lToricVariety │ │ │ │ -0001c400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c420: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001c3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c3f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0001c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a --------+.|i22 : │ │ │ │ -0001c480: 2043 6865 636b 546f 7269 6356 6172 6965 CheckToricVarie │ │ │ │ -0001c490: 7479 5661 6c69 6428 5829 2020 2020 2020 tyValid(X) │ │ │ │ +0001c440: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 4368 -----+.|i22 : Ch │ │ │ │ +0001c450: 6563 6b54 6f72 6963 5661 7269 6574 7956 eckToricVarietyV │ │ │ │ +0001c460: 616c 6964 2858 2920 2020 2020 2020 2020 alid(X) │ │ │ │ +0001c470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c490: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c4d0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0001c4e0: 3220 3d20 7472 7565 2020 2020 2020 2020 2 = true │ │ │ │ 0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c500: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001c510: 7c6f 3232 203d 2074 7275 6520 2020 2020 |o22 = true │ │ │ │ -0001c520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c550: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001c500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c520: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c5a0: 2d2d 2d2d 2b0a 7c69 3233 203a 2052 3d72 ----+.|i23 : R=r │ │ │ │ -0001c5b0: 696e 6728 5829 2020 2020 2020 2020 2020 ing(X) │ │ │ │ +0001c570: 2d2b 0a7c 6932 3320 3a20 523d 7269 6e67 -+.|i23 : R=ring │ │ │ │ +0001c580: 2858 2920 2020 2020 2020 2020 2020 2020 (X) │ │ │ │ +0001c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001c5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c5e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001c5f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001c600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c600: 2020 2020 2020 207c 0a7c 6f32 3320 3d20 |.|o23 = │ │ │ │ +0001c610: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0001c620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c630: 2020 2020 2020 2020 2020 7c0a 7c6f 3233 |.|o23 │ │ │ │ -0001c640: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ -0001c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c650: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c680: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001c690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c690: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c6a0: 6f32 3320 3a20 506f 6c79 6e6f 6d69 616c o23 : Polynomial │ │ │ │ +0001c6b0: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 0001c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c6d0: 7c0a 7c6f 3233 203a 2050 6f6c 796e 6f6d |.|o23 : Polynom │ │ │ │ -0001c6e0: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ -0001c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c710: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c6e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001c6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c760: 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a 2049 ------+.|i24 : I │ │ │ │ -0001c770: 3d69 6465 616c 2852 5f30 5e34 2a52 5f31 =ideal(R_0^4*R_1 │ │ │ │ -0001c780: 2c52 5f30 2a52 5f33 2a52 5f34 2a52 5f32 ,R_0*R_3*R_4*R_2 │ │ │ │ -0001c790: 2d52 5f32 5e32 2a52 5f30 5e32 2920 2020 -R_2^2*R_0^2) │ │ │ │ +0001c730: 2d2d 2d2b 0a7c 6932 3420 3a20 493d 6964 ---+.|i24 : I=id │ │ │ │ +0001c740: 6561 6c28 525f 305e 342a 525f 312c 525f eal(R_0^4*R_1,R_ │ │ │ │ +0001c750: 302a 525f 332a 525f 342a 525f 322d 525f 0*R_3*R_4*R_2-R_ │ │ │ │ +0001c760: 325e 322a 525f 305e 3229 2020 2020 2020 2^2*R_0^2) │ │ │ │ +0001c770: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001c780: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001c800: 2020 2020 2020 2020 2020 2020 2034 2020 4 │ │ │ │ -0001c810: 2020 2020 2032 2032 2020 2020 2020 2020 2 2 │ │ │ │ -0001c820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c840: 2020 2020 2020 207c 0a7c 6f32 3420 3d20 |.|o24 = │ │ │ │ -0001c850: 6964 6561 6c20 2878 2078 202c 202d 2078 ideal (x x , - x │ │ │ │ -0001c860: 2078 2020 2b20 7820 7820 7820 7820 2920 x + x x x x ) │ │ │ │ -0001c870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c890: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c8a0: 2020 2030 2031 2020 2020 2030 2032 2020 0 1 0 2 │ │ │ │ -0001c8b0: 2020 3020 3220 3320 3420 2020 2020 2020 0 2 3 4 │ │ │ │ +0001c7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c7c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001c7d0: 2020 2020 2020 2020 2020 3420 2020 2020 4 │ │ │ │ +0001c7e0: 2020 3220 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ +0001c7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c810: 2020 2020 7c0a 7c6f 3234 203d 2069 6465 |.|o24 = ide │ │ │ │ +0001c820: 616c 2028 7820 7820 2c20 2d20 7820 7820 al (x x , - x x │ │ │ │ +0001c830: 202b 2078 2078 2078 2078 2029 2020 2020 + x x x x ) │ │ │ │ +0001c840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c850: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c860: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001c870: 3020 3120 2020 2020 3020 3220 2020 2030 0 1 0 2 0 │ │ │ │ +0001c880: 2032 2033 2034 2020 2020 2020 2020 2020 2 3 4 │ │ │ │ +0001c890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c8a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c8d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c8f0: 2020 2020 207c 0a7c 6f32 3420 3a20 4964 |.|o24 : Id │ │ │ │ +0001c900: 6561 6c20 6f66 2052 2020 2020 2020 2020 eal of R │ │ │ │ 0001c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c920: 2020 2020 2020 2020 7c0a 7c6f 3234 203a |.|o24 : │ │ │ │ -0001c930: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ │ │ -0001c940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c970: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001c9c0: 7c69 3235 203a 2063 736d 493d 4353 4d28 |i25 : csmI=CSM( │ │ │ │ -0001c9d0: 582c 4929 2020 2020 2020 2020 2020 2020 X,I) │ │ │ │ +0001c920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c940: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0001c990: 3520 3a20 6373 6d49 3d43 534d 2858 2c49 5 : csmI=CSM(X,I │ │ │ │ +0001c9a0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0001c9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c9d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001c9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001ca00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ca10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ca20: 207c 0a7c 2020 2020 2020 2020 3220 2020 |.| 2 │ │ │ │ +0001ca30: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 0001ca40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca50: 2020 2020 7c0a 7c20 2020 2020 2020 2032 |.| 2 │ │ │ │ -0001ca60: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -0001ca70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001caa0: 0a7c 6f32 3520 3d20 3578 2078 2020 2b20 .|o25 = 5x x + │ │ │ │ -0001cab0: 3378 2020 2b20 3478 2078 2020 2b20 7820 3x + 4x x + x │ │ │ │ -0001cac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001caf0: 2020 2020 2033 2034 2020 2020 2033 2020 3 4 3 │ │ │ │ -0001cb00: 2020 2033 2034 2020 2020 3320 2020 2020 3 4 3 │ │ │ │ +0001ca50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ca60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0001ca70: 3235 203d 2035 7820 7820 202b 2033 7820 25 = 5x x + 3x │ │ │ │ +0001ca80: 202b 2034 7820 7820 202b 2078 2020 2020 + 4x x + x │ │ │ │ +0001ca90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001caa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cab0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001cac0: 2020 3320 3420 2020 2020 3320 2020 2020 3 4 3 │ │ │ │ +0001cad0: 3320 3420 2020 2033 2020 2020 2020 2020 3 4 3 │ │ │ │ +0001cae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cb00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001cb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cb30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001cb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cb40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0001cb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cb60: 2020 2020 2020 5a5a 5b78 202e 2e78 205d ZZ[x ..x ] │ │ │ │ 0001cb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cb80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001cb90: 2020 2020 2020 2020 205a 5a5b 7820 2e2e ZZ[x .. │ │ │ │ -0001cba0: 7820 5d20 2020 2020 2020 2020 2020 2020 x ] │ │ │ │ -0001cbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cbc0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001cb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001cba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cbb0: 2020 2020 2030 2020 2034 2020 2020 2020 0 4 │ │ │ │ +0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cbe0: 2020 2020 2020 2020 3020 2020 3420 2020 0 4 │ │ │ │ -0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc10: 2020 2020 2020 7c0a 7c6f 3235 203a 202d |.|o25 : - │ │ │ │ -0001cc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cc40: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020 -------- │ │ │ │ -0001cc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc60: 207c 0a7c 2020 2020 2020 2878 2078 202c |.| (x x , │ │ │ │ -0001cc70: 2078 2078 2078 202c 2078 2020 2d20 7820 x x x , x - x │ │ │ │ -0001cc80: 2c20 7820 202d 2078 202c 2078 2020 2d20 , x - x , x - │ │ │ │ -0001cc90: 7820 2920 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ -0001cca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001ccb0: 2020 2020 2020 2032 2034 2020 2030 2031 2 4 0 1 │ │ │ │ -0001ccc0: 2033 2020 2030 2020 2020 3320 2020 3120 3 0 3 1 │ │ │ │ -0001ccd0: 2020 2033 2020 2032 2020 2020 3420 2020 3 2 4 │ │ │ │ -0001cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ccf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cd40: 2d2d 2b0a 7c69 3236 203a 2045 756c 6572 --+.|i26 : Euler │ │ │ │ -0001cd50: 493d 4575 6c65 7228 582c 4929 2020 2020 I=Euler(X,I) │ │ │ │ +0001cbe0: 2020 207c 0a7c 6f32 3520 3a20 2d2d 2d2d |.|o25 : ---- │ │ │ │ +0001cbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cc10: 2d2d 2d2d 2d20 2020 2020 2020 2020 2020 ----- │ │ │ │ +0001cc20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001cc30: 7c20 2020 2020 2028 7820 7820 2c20 7820 | (x x , x │ │ │ │ +0001cc40: 7820 7820 2c20 7820 202d 2078 202c 2078 x x , x - x , x │ │ │ │ +0001cc50: 2020 2d20 7820 2c20 7820 202d 2078 2029 - x , x - x ) │ │ │ │ +0001cc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cc70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001cc80: 2020 2020 3220 3420 2020 3020 3120 3320 2 4 0 1 3 │ │ │ │ +0001cc90: 2020 3020 2020 2033 2020 2031 2020 2020 0 3 1 │ │ │ │ +0001cca0: 3320 2020 3220 2020 2034 2020 2020 2020 3 2 4 │ │ │ │ +0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ccc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001ccd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ccf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001cd10: 0a7c 6932 3620 3a20 4575 6c65 7249 3d45 .|i26 : EulerI=E │ │ │ │ +0001cd20: 756c 6572 2858 2c49 2920 2020 2020 2020 uler(X,I) │ │ │ │ +0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cd50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0001cd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cda0: 2020 2020 207c 0a7c 6f32 3620 3d20 3520 |.|o26 = 5 │ │ │ │ 0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cdd0: 2020 2020 2020 2020 7c0a 7c6f 3236 203d |.|o26 = │ │ │ │ -0001cde0: 2035 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ -0001cdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001ce70: 7c69 3237 203a 2045 756c 6572 2863 736d |i27 : Euler(csm │ │ │ │ -0001ce80: 4929 3d3d 4575 6c65 7249 2020 2020 2020 I)==EulerI │ │ │ │ +0001cdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cdf0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ce30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0001ce40: 3720 3a20 4575 6c65 7228 6373 6d49 293d 7 : Euler(csmI)= │ │ │ │ +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 │ │ │ │ -0001cfc0: 6d62 6f6c 6963 2063 6f6d 7075 7461 7469 mbolic computati │ │ │ │ -0001cfd0: 6f6e 7320 7769 7468 2047 725c 226f 626e ons with Gr\"obn │ │ │ │ -0001cfe0: 6572 2062 6173 6573 2e0a 4368 616e 6769 er bases..Changi │ │ │ │ -0001cff0: 6e67 2074 6865 206f 7074 696f 6e20 2a6e ng the option *n │ │ │ │ -0001d000: 6f74 6520 436f 6d70 4d65 7468 6f64 3a20 ote CompMethod: │ │ │ │ -0001d010: 436f 6d70 4d65 7468 6f64 2c20 746f 2062 CompMethod, to b │ │ │ │ -0001d020: 6572 7469 6e69 2077 696c 6c20 646f 2074 ertini will do t │ │ │ │ -0001d030: 6865 206d 6169 6e0a 636f 6d70 7574 6174 he main.computat │ │ │ │ -0001d040: 696f 6e73 206e 756d 6572 6963 616c 6c79 ions numerically │ │ │ │ -0001d050: 2c20 7072 6f76 6964 6564 2042 6572 7469 , provided Berti │ │ │ │ -0001d060: 6e69 2069 7320 2a6e 6f74 6520 696e 7374 ni is *note inst │ │ │ │ -0001d070: 616c 6c65 6420 616e 6420 636f 6e66 6967 alled and config │ │ │ │ -0001d080: 7572 6564 3a0a 636f 6e66 6967 7572 696e ured:.configurin │ │ │ │ -0001d090: 6720 4265 7274 696e 692c 2e20 4e6f 7465 g Bertini,. Note │ │ │ │ -0001d0a0: 2074 6861 7420 7468 6520 6265 7274 696e that the bertin │ │ │ │ -0001d0b0: 6920 616e 6420 506e 5265 7369 6475 616c i and PnResidual │ │ │ │ -0001d0c0: 206f 7074 696f 6e73 206d 6179 206f 6e6c options may onl │ │ │ │ -0001d0d0: 7920 6265 0a75 7365 6420 666f 7220 7375 y be.used for su │ │ │ │ -0001d0e0: 6273 6368 656d 6573 206f 6620 5c50 505e bschemes of \PP^ │ │ │ │ -0001d0f0: 6e2e 0a0a 4f62 7365 7276 6520 7468 6174 n...Observe that │ │ │ │ -0001d100: 2074 6865 2061 6c67 6f72 6974 686d 2069 the algorithm i │ │ │ │ -0001d110: 7320 6120 7072 6f62 6162 696c 6973 7469 s a probabilisti │ │ │ │ -0001d120: 6320 616c 676f 7269 7468 6d20 616e 6420 c algorithm and │ │ │ │ -0001d130: 6d61 7920 6769 7665 2061 2077 726f 6e67 may give a wrong │ │ │ │ -0001d140: 0a61 6e73 7765 7220 7769 7468 2061 2073 .answer with a s │ │ │ │ -0001d150: 6d61 6c6c 2062 7574 206e 6f6e 7a65 726f mall but nonzero │ │ │ │ -0001d160: 2070 726f 6261 6269 6c69 7479 2e20 5265 probability. Re │ │ │ │ -0001d170: 6164 206d 6f72 6520 756e 6465 7220 2a6e ad more under *n │ │ │ │ -0001d180: 6f74 650a 7072 6f62 6162 696c 6973 7469 ote.probabilisti │ │ │ │ -0001d190: 6320 616c 676f 7269 7468 6d3a 2070 726f c algorithm: pro │ │ │ │ -0001d1a0: 6261 6269 6c69 7374 6963 2061 6c67 6f72 babilistic algor │ │ │ │ -0001d1b0: 6974 686d 2c2e 0a0a 5761 7973 2074 6f20 ithm,...Ways to │ │ │ │ -0001d1c0: 7573 6520 4575 6c65 723a 0a3d 3d3d 3d3d use Euler:.===== │ │ │ │ -0001d1d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0001d1e0: 202a 2022 4575 6c65 7228 4964 6561 6c29 * "Euler(Ideal) │ │ │ │ -0001d1f0: 220a 2020 2a20 2245 756c 6572 2852 696e ". * "Euler(Rin │ │ │ │ -0001d200: 6745 6c65 6d65 6e74 2922 0a0a 466f 7220 gElement)"..For │ │ │ │ -0001d210: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0001d220: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d230: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0001d240: 6f74 6520 4575 6c65 723a 2045 756c 6572 ote Euler: Euler │ │ │ │ -0001d250: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0001d260: 686f 6420 6675 6e63 7469 6f6e 2077 6974 hod function wit │ │ │ │ -0001d270: 6820 6f70 7469 6f6e 733a 0a28 4d61 6361 h options:.(Maca │ │ │ │ -0001d280: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -0001d290: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ -0001d2a0: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ +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 │ │ │ │ +0001d050: 643a 0a63 6f6e 6669 6775 7269 6e67 2042 d:.configuring B │ │ │ │ +0001d060: 6572 7469 6e69 2c2e 204e 6f74 6520 7468 ertini,. Note th │ │ │ │ +0001d070: 6174 2074 6865 2062 6572 7469 6e69 2061 at the bertini a │ │ │ │ +0001d080: 6e64 2050 6e52 6573 6964 7561 6c20 6f70 nd PnResidual op │ │ │ │ +0001d090: 7469 6f6e 7320 6d61 7920 6f6e 6c79 2062 tions may only b │ │ │ │ +0001d0a0: 650a 7573 6564 2066 6f72 2073 7562 7363 e.used for subsc │ │ │ │ +0001d0b0: 6865 6d65 7320 6f66 205c 5050 5e6e 2e0a hemes of \PP^n.. │ │ │ │ +0001d0c0: 0a4f 6273 6572 7665 2074 6861 7420 7468 .Observe that th │ │ │ │ +0001d0d0: 6520 616c 676f 7269 7468 6d20 6973 2061 e algorithm is a │ │ │ │ +0001d0e0: 2070 726f 6261 6269 6c69 7374 6963 2061 probabilistic a │ │ │ │ +0001d0f0: 6c67 6f72 6974 686d 2061 6e64 206d 6179 lgorithm and may │ │ │ │ +0001d100: 2067 6976 6520 6120 7772 6f6e 670a 616e give a wrong.an │ │ │ │ +0001d110: 7377 6572 2077 6974 6820 6120 736d 616c swer with a smal │ │ │ │ +0001d120: 6c20 6275 7420 6e6f 6e7a 6572 6f20 7072 l but nonzero pr │ │ │ │ +0001d130: 6f62 6162 696c 6974 792e 2052 6561 6420 obability. Read │ │ │ │ +0001d140: 6d6f 7265 2075 6e64 6572 202a 6e6f 7465 more under *note │ │ │ │ +0001d150: 0a70 726f 6261 6269 6c69 7374 6963 2061 .probabilistic a │ │ │ │ +0001d160: 6c67 6f72 6974 686d 3a20 7072 6f62 6162 lgorithm: probab │ │ │ │ +0001d170: 696c 6973 7469 6320 616c 676f 7269 7468 ilistic algorith │ │ │ │ +0001d180: 6d2c 2e0a 0a57 6179 7320 746f 2075 7365 m,...Ways to use │ │ │ │ +0001d190: 2045 756c 6572 3a0a 3d3d 3d3d 3d3d 3d3d Euler:.======== │ │ │ │ +0001d1a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0001d1b0: 2245 756c 6572 2849 6465 616c 2922 0a20 "Euler(Ideal)". │ │ │ │ +0001d1c0: 202a 2022 4575 6c65 7228 5269 6e67 456c * "Euler(RingEl │ │ │ │ +0001d1d0: 656d 656e 7429 220a 0a46 6f72 2074 6865 ement)"..For the │ │ │ │ +0001d1e0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +0001d1f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0001d200: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +0001d210: 2045 756c 6572 3a20 4575 6c65 722c 2069 Euler: Euler, i │ │ │ │ +0001d220: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +0001d230: 2066 756e 6374 696f 6e20 7769 7468 206f function with o │ │ │ │ +0001d240: 7074 696f 6e73 3a0a 284d 6163 6175 6c61 ptions:.(Macaula │ │ │ │ +0001d250: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +0001d260: 7469 6f6e 5769 7468 4f70 7469 6f6e 732c tionWithOptions, │ │ │ │ +0001d270: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0001d280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d2f0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -0001d300: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0001d310: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -0001d320: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -0001d330: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -0001d340: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -0001d350: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ -0001d360: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ -0001d370: 7365 732e 6d32 3a32 3331 323a 302e 0a1f ses.m2:2312:0... │ │ │ │ -0001d380: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ -0001d390: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ -0001d3a0: 6f2c 204e 6f64 653a 2045 756c 6572 4166 o, Node: EulerAf │ │ │ │ -0001d3b0: 6669 6e65 2c20 4e65 7874 3a20 496e 6473 fine, Next: Inds │ │ │ │ -0001d3c0: 4f66 536d 6f6f 7468 2c20 5072 6576 3a20 OfSmooth, Prev: │ │ │ │ -0001d3d0: 4575 6c65 722c 2055 703a 2054 6f70 0a0a Euler, Up: Top.. │ │ │ │ -0001d3e0: 4575 6c65 7241 6666 696e 6520 2d2d 2054 EulerAffine -- T │ │ │ │ -0001d3f0: 6865 2045 756c 6572 2043 6861 7261 6374 he Euler Charact │ │ │ │ -0001d400: 6572 6973 7469 6320 6f66 2061 6e20 6166 eristic of an af │ │ │ │ -0001d410: 6669 6e65 2076 6172 6965 7479 2e0a 2a2a fine variety..** │ │ │ │ -0001d420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0001d460: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0001d470: 2045 756c 6572 4166 6669 6e65 2049 0a20 EulerAffine I. │ │ │ │ -0001d480: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -0001d490: 202a 2049 2c20 616e 202a 6e6f 7465 2069 * I, an *note i │ │ │ │ -0001d4a0: 6465 616c 3a20 284d 6163 6175 6c61 7932 deal: (Macaulay2 │ │ │ │ -0001d4b0: 446f 6329 4964 6561 6c2c 2c20 616e 2069 Doc)Ideal,, an i │ │ │ │ -0001d4c0: 6465 616c 2069 6e20 6120 706f 6c79 6e6f deal in a polyno │ │ │ │ -0001d4d0: 6d69 616c 2072 696e 670a 2020 2020 2020 mial ring. │ │ │ │ -0001d4e0: 2020 6f76 6572 2061 2066 6965 6c64 2064 over a field d │ │ │ │ -0001d4f0: 6566 696e 696e 6720 616e 2061 6666 696e efining an affin │ │ │ │ -0001d500: 6520 7661 7269 6574 792e 0a20 202a 204f e variety.. * O │ │ │ │ -0001d510: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -0001d520: 6120 2a6e 6f74 6520 7269 6e67 2065 6c65 a *note ring ele │ │ │ │ -0001d530: 6d65 6e74 3a20 284d 6163 6175 6c61 7932 ment: (Macaulay2 │ │ │ │ -0001d540: 446f 6329 5269 6e67 456c 656d 656e 742c Doc)RingElement, │ │ │ │ -0001d550: 2c20 7468 6520 4575 6c65 720a 2020 2020 , the Euler. │ │ │ │ -0001d560: 2020 2020 6368 6172 6163 7465 7269 7374 characterist │ │ │ │ -0001d570: 6963 0a0a 4465 7363 7269 7074 696f 6e0a ic..Description. │ │ │ │ -0001d580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -0001d590: 7320 636f 6d6d 616e 6420 636f 6d70 7574 s command comput │ │ │ │ -0001d5a0: 6573 2074 6865 2045 756c 6572 2063 6861 es the Euler cha │ │ │ │ -0001d5b0: 7261 6374 6572 6973 7469 6320 6f66 2061 racteristic of a │ │ │ │ -0001d5c0: 2063 6f6d 706c 6578 2061 6666 696e 6520 complex affine │ │ │ │ -0001d5d0: 7661 7269 6574 792e 0a0a 2b2d 2d2d 2d2d variety...+----- │ │ │ │ -0001d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d610: 2d2b 0a7c 6931 203a 206b 6b3d 5a5a 2f33 -+.|i1 : kk=ZZ/3 │ │ │ │ -0001d620: 3237 3439 3b20 2020 2020 2020 2020 2020 2749; │ │ │ │ -0001d630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d640: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0001d650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d680: 2d2d 2d2b 0a7c 6932 203a 2052 3d6b 6b5b ---+.|i2 : R=kk[ │ │ │ │ -0001d690: 785f 312e 2e78 5f33 5d20 2020 2020 2020 x_1..x_3] │ │ │ │ +0001d2c0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0001d2d0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0001d2e0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0001d2f0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0001d300: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +0001d310: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0001d320: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ +0001d330: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ +0001d340: 2e6d 323a 3233 3132 3a30 2e0a 1f0a 4669 .m2:2312:0....Fi │ │ │ │ +0001d350: 6c65 3a20 4368 6172 6163 7465 7269 7374 le: Characterist │ │ │ │ +0001d360: 6963 436c 6173 7365 732e 696e 666f 2c20 icClasses.info, │ │ │ │ +0001d370: 4e6f 6465 3a20 4575 6c65 7241 6666 696e Node: EulerAffin │ │ │ │ +0001d380: 652c 204e 6578 743a 2049 6e64 734f 6653 e, Next: IndsOfS │ │ │ │ +0001d390: 6d6f 6f74 682c 2050 7265 763a 2045 756c mooth, Prev: Eul │ │ │ │ +0001d3a0: 6572 2c20 5570 3a20 546f 700a 0a45 756c er, Up: Top..Eul │ │ │ │ +0001d3b0: 6572 4166 6669 6e65 202d 2d20 5468 6520 erAffine -- The │ │ │ │ +0001d3c0: 4575 6c65 7220 4368 6172 6163 7465 7269 Euler Characteri │ │ │ │ +0001d3d0: 7374 6963 206f 6620 616e 2061 6666 696e stic of an affin │ │ │ │ +0001d3e0: 6520 7661 7269 6574 792e 0a2a 2a2a 2a2a e variety..***** │ │ │ │ +0001d3f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001d400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001d410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001d420: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +0001d430: 6167 653a 200a 2020 2020 2020 2020 4575 age: . Eu │ │ │ │ +0001d440: 6c65 7241 6666 696e 6520 490a 2020 2a20 lerAffine I. * │ │ │ │ +0001d450: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ +0001d460: 492c 2061 6e20 2a6e 6f74 6520 6964 6561 I, an *note idea │ │ │ │ +0001d470: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ +0001d480: 2949 6465 616c 2c2c 2061 6e20 6964 6561 )Ideal,, an idea │ │ │ │ +0001d490: 6c20 696e 2061 2070 6f6c 796e 6f6d 6961 l in a polynomia │ │ │ │ +0001d4a0: 6c20 7269 6e67 0a20 2020 2020 2020 206f l ring. o │ │ │ │ +0001d4b0: 7665 7220 6120 6669 656c 6420 6465 6669 ver a field defi │ │ │ │ +0001d4c0: 6e69 6e67 2061 6e20 6166 6669 6e65 2076 ning an affine v │ │ │ │ +0001d4d0: 6172 6965 7479 2e0a 2020 2a20 4f75 7470 ariety.. * Outp │ │ │ │ +0001d4e0: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ +0001d4f0: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ +0001d500: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +0001d510: 2952 696e 6745 6c65 6d65 6e74 2c2c 2074 )RingElement,, t │ │ │ │ +0001d520: 6865 2045 756c 6572 0a20 2020 2020 2020 he Euler. │ │ │ │ +0001d530: 2063 6861 7261 6374 6572 6973 7469 630a characteristic. │ │ │ │ +0001d540: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ +0001d550: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2063 ========..This c │ │ │ │ +0001d560: 6f6d 6d61 6e64 2063 6f6d 7075 7465 7320 ommand computes │ │ │ │ +0001d570: 7468 6520 4575 6c65 7220 6368 6172 6163 the Euler charac │ │ │ │ +0001d580: 7465 7269 7374 6963 206f 6620 6120 636f teristic of a co │ │ │ │ +0001d590: 6d70 6c65 7820 6166 6669 6e65 2076 6172 mplex affine var │ │ │ │ +0001d5a0: 6965 7479 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d iety...+-------- │ │ │ │ +0001d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001d5e0: 0a7c 6931 203a 206b 6b3d 5a5a 2f33 3237 .|i1 : kk=ZZ/327 │ │ │ │ +0001d5f0: 3439 3b20 2020 2020 2020 2020 2020 2020 49; │ │ │ │ +0001d600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d610: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d650: 2d2d 2d2b 0a7c 6932 203a 2052 3d6b 6b5b ---+.|i2 : R=kk[ │ │ │ │ +0001d660: 785f 312e 2e78 5f33 5d20 2020 2020 2020 x_1..x_3] │ │ │ │ +0001d670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d680: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001d690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001d6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d6c0: 2020 2020 2020 207c 0a7c 6f32 203d 2052 |.|o2 = R │ │ │ │ 0001d6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d6f0: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ -0001d700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d700: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d720: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d730: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0001d740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d760: 2020 2020 2020 207c 0a7c 6f32 203a 2050 |.|o2 : P │ │ │ │ -0001d770: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ -0001d780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d7a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0001d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d7d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -0001d7e0: 2049 3d69 6465 616c 2878 5f31 5e32 2b78 I=ideal(x_1^2+x │ │ │ │ -0001d7f0: 5f32 5e32 2b78 5f33 5e32 2d31 2920 2020 _2^2+x_3^2-1) │ │ │ │ +0001d720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d730: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0001d740: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ +0001d750: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +0001d760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d770: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001d780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001d7b0: 0a7c 6933 203a 2049 3d69 6465 616c 2878 .|i3 : I=ideal(x │ │ │ │ +0001d7c0: 5f31 5e32 2b78 5f32 5e32 2b78 5f33 5e32 _1^2+x_2^2+x_3^2 │ │ │ │ +0001d7d0: 2d31 2920 2020 2020 2020 2020 2020 2020 -1) │ │ │ │ +0001d7e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001d7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d810: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001d820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d840: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0001d850: 2020 2020 2020 2020 2020 3220 2020 2032 2 2 │ │ │ │ -0001d860: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0001d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d880: 2020 2020 7c0a 7c6f 3320 3d20 6964 6561 |.|o3 = idea │ │ │ │ -0001d890: 6c28 7820 202b 2078 2020 2b20 7820 202d l(x + x + x - │ │ │ │ -0001d8a0: 2031 2920 2020 2020 2020 2020 2020 2020 1) │ │ │ │ -0001d8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001d8c0: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -0001d8d0: 2032 2020 2020 3320 2020 2020 2020 2020 2 3 │ │ │ │ +0001d810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d820: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001d830: 2020 3220 2020 2032 2020 2020 3220 2020 2 2 2 │ │ │ │ +0001d840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d850: 2020 2020 2020 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: 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 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001d930: 0a7c 6f33 203a 2049 6465 616c 206f 6620 .|o3 : Ideal of │ │ │ │ -0001d940: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -0001d950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d960: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d9a0: 2d2b 0a7c 6934 203a 2074 696d 6520 4575 -+.|i4 : time Eu │ │ │ │ -0001d9b0: 6c65 7241 6666 696e 6520 4920 2020 2020 lerAffine I │ │ │ │ -0001d9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d9d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0001d9e0: 2075 7365 6420 302e 3034 3731 3038 7320 used 0.047108s │ │ │ │ -0001d9f0: 2863 7075 293b 2030 2e30 3436 3733 3439 (cpu); 0.0467349 │ │ │ │ -0001da00: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -0001da10: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ -0001da20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001da50: 3420 3d20 3220 2020 2020 2020 2020 2020 4 = 2 │ │ │ │ -0001da60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001d8f0: 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 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 3639 3636 3934 7320 used 0.0696694s │ │ │ │ +0001d9d0: 2863 7075 293b 2030 2e30 3534 3135 3434 (cpu); 0.0541544 │ │ │ │ +0001d9e0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +0001d9f0: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ +0001da00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001da30: 6f34 203d 2032 2020 2020 2020 2020 2020 o4 = 2 │ │ │ │ +0001da40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001dac0: 0a4f 6273 6572 7665 2074 6861 7420 7468 .Observe that th │ │ │ │ -0001dad0: 6520 616c 676f 7269 7468 6d20 6973 2061 e algorithm is a │ │ │ │ -0001dae0: 2070 726f 6261 6269 6c69 7374 6963 2061 probabilistic a │ │ │ │ -0001daf0: 6c67 6f72 6974 686d 2061 6e64 206d 6179 lgorithm and may │ │ │ │ -0001db00: 2067 6976 6520 6120 7772 6f6e 670a 616e give a wrong.an │ │ │ │ -0001db10: 7377 6572 2077 6974 6820 6120 736d 616c swer with a smal │ │ │ │ -0001db20: 6c20 6275 7420 6e6f 6e7a 6572 6f20 7072 l but nonzero pr │ │ │ │ -0001db30: 6f62 6162 696c 6974 792e 2052 6561 6420 obability. Read │ │ │ │ -0001db40: 6d6f 7265 2075 6e64 6572 202a 6e6f 7465 more under *note │ │ │ │ -0001db50: 0a70 726f 6261 6269 6c69 7374 6963 2061 .probabilistic a │ │ │ │ -0001db60: 6c67 6f72 6974 686d 3a20 7072 6f62 6162 lgorithm: probab │ │ │ │ -0001db70: 696c 6973 7469 6320 616c 676f 7269 7468 ilistic algorith │ │ │ │ -0001db80: 6d2c 2e0a 0a57 6179 7320 746f 2075 7365 m,...Ways to use │ │ │ │ -0001db90: 2045 756c 6572 4166 6669 6e65 3a0a 3d3d EulerAffine:.== │ │ │ │ -0001dba0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001dbb0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 756c ======.. * "Eul │ │ │ │ -0001dbc0: 6572 4166 6669 6e65 2849 6465 616c 2922 erAffine(Ideal)" │ │ │ │ -0001dbd0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0001dbe0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0001dbf0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0001dc00: 6563 7420 2a6e 6f74 6520 4575 6c65 7241 ect *note EulerA │ │ │ │ -0001dc10: 6666 696e 653a 2045 756c 6572 4166 6669 ffine: EulerAffi │ │ │ │ -0001dc20: 6e65 2c20 6973 2061 202a 6e6f 7465 206d ne, is a *note m │ │ │ │ -0001dc30: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ -0001dc40: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -0001dc50: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +0001daa0: 2d2b 0a0a 4f62 7365 7276 6520 7468 6174 -+..Observe that │ │ │ │ +0001dab0: 2074 6865 2061 6c67 6f72 6974 686d 2069 the algorithm i │ │ │ │ +0001dac0: 7320 6120 7072 6f62 6162 696c 6973 7469 s a probabilisti │ │ │ │ +0001dad0: 6320 616c 676f 7269 7468 6d20 616e 6420 c algorithm and │ │ │ │ +0001dae0: 6d61 7920 6769 7665 2061 2077 726f 6e67 may give a wrong │ │ │ │ +0001daf0: 0a61 6e73 7765 7220 7769 7468 2061 2073 .answer with a s │ │ │ │ +0001db00: 6d61 6c6c 2062 7574 206e 6f6e 7a65 726f mall but nonzero │ │ │ │ +0001db10: 2070 726f 6261 6269 6c69 7479 2e20 5265 probability. Re │ │ │ │ +0001db20: 6164 206d 6f72 6520 756e 6465 7220 2a6e ad more under *n │ │ │ │ +0001db30: 6f74 650a 7072 6f62 6162 696c 6973 7469 ote.probabilisti │ │ │ │ +0001db40: 6320 616c 676f 7269 7468 6d3a 2070 726f c algorithm: pro │ │ │ │ +0001db50: 6261 6269 6c69 7374 6963 2061 6c67 6f72 babilistic algor │ │ │ │ +0001db60: 6974 686d 2c2e 0a0a 5761 7973 2074 6f20 ithm,...Ways to │ │ │ │ +0001db70: 7573 6520 4575 6c65 7241 6666 696e 653a use EulerAffine: │ │ │ │ +0001db80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +0001db90: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +0001dba0: 4575 6c65 7241 6666 696e 6528 4964 6561 EulerAffine(Idea │ │ │ │ +0001dbb0: 6c29 220a 0a46 6f72 2074 6865 2070 726f l)"..For the pro │ │ │ │ +0001dbc0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +0001dbd0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0001dbe0: 6f62 6a65 6374 202a 6e6f 7465 2045 756c object *note Eul │ │ │ │ +0001dbf0: 6572 4166 6669 6e65 3a20 4575 6c65 7241 erAffine: EulerA │ │ │ │ +0001dc00: 6666 696e 652c 2069 7320 6120 2a6e 6f74 ffine, is a *not │ │ │ │ +0001dc10: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +0001dc20: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +0001dc30: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +0001dc40: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0001dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001dc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0001dcb0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0001dcc0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0001dcd0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0001dce0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -0001dcf0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -0001dd00: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -0001dd10: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ -0001dd20: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ -0001dd30: 3a32 3534 313a 302e 0a1f 0a46 696c 653a :2541:0....File: │ │ │ │ -0001dd40: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ -0001dd50: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ -0001dd60: 653a 2049 6e64 734f 6653 6d6f 6f74 682c e: IndsOfSmooth, │ │ │ │ -0001dd70: 204e 6578 743a 2049 6e70 7574 4973 536d Next: InputIsSm │ │ │ │ -0001dd80: 6f6f 7468 2c20 5072 6576 3a20 4575 6c65 ooth, Prev: Eule │ │ │ │ -0001dd90: 7241 6666 696e 652c 2055 703a 2054 6f70 rAffine, Up: Top │ │ │ │ -0001dda0: 0a0a 496e 6473 4f66 536d 6f6f 7468 0a2a ..IndsOfSmooth.* │ │ │ │ -0001ddb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 ***********..Des │ │ │ │ -0001ddc0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0001ddd0: 3d3d 3d3d 0a0a 5468 6520 6f70 7469 6f6e ====..The option │ │ │ │ -0001dde0: 2049 6e64 734f 6653 6d6f 6f74 6820 6973 IndsOfSmooth is │ │ │ │ -0001ddf0: 206f 6e6c 7920 7573 6564 2062 7920 7468 only used by th │ │ │ │ -0001de00: 6520 636f 6d6d 616e 6473 202a 6e6f 7465 e commands *note │ │ │ │ -0001de10: 2043 534d 3a20 4353 4d2c 2c20 616e 6420 CSM: CSM,, and │ │ │ │ -0001de20: 2a6e 6f74 650a 4575 6c65 723a 2045 756c *note.Euler: Eul │ │ │ │ -0001de30: 6572 2c20 696e 2063 6f6d 6269 6e61 7469 er, in combinati │ │ │ │ -0001de40: 6f6e 2077 6974 6820 7468 6520 6f70 7469 on with the opti │ │ │ │ -0001de50: 6f6e 204d 6574 686f 643d 3e44 6972 6563 on Method=>Direc │ │ │ │ -0001de60: 7443 6f6d 706c 6574 496e 742e 2057 6865 tCompletInt. Whe │ │ │ │ -0001de70: 6e0a 7573 6564 2074 6869 7320 6f70 7469 n.used this opti │ │ │ │ -0001de80: 6f6e 206d 6179 2061 6c6c 6f77 2074 6865 on may allow the │ │ │ │ -0001de90: 2075 7365 7220 746f 2073 7065 6564 2075 user to speed u │ │ │ │ -0001dea0: 7020 7468 6520 636f 6d70 7574 6174 696f p the computatio │ │ │ │ -0001deb0: 6e20 6279 2074 656c 6c69 6e67 0a67 6976 n by telling.giv │ │ │ │ -0001dec0: 696e 6720 7468 6520 6d65 7468 6f64 2061 ing the method a │ │ │ │ -0001ded0: 206c 6973 7420 6f66 2069 6e64 6963 6573 list of indices │ │ │ │ -0001dee0: 2066 6f72 2074 6865 2067 656e 6572 6174 for the generat │ │ │ │ -0001def0: 6f72 7320 6f66 2074 6865 2069 6e70 7574 ors of the input │ │ │ │ -0001df00: 2069 6465 616c 2074 6861 742c 0a77 6865 ideal that,.whe │ │ │ │ -0001df10: 6e20 7461 6b65 6e20 746f 6765 7468 6572 n taken together │ │ │ │ -0001df20: 2c20 6465 6669 6e65 2061 2073 6d6f 6f74 , define a smoot │ │ │ │ -0001df30: 6820 7375 6273 6368 656d 6520 6f66 2074 h subscheme of t │ │ │ │ -0001df40: 6865 2061 6d62 6965 6e74 2073 7061 6365 he ambient space │ │ │ │ -0001df50: 2e20 5468 6973 0a6f 7074 696f 6e20 7769 . This.option wi │ │ │ │ -0001df60: 6c6c 2062 6520 6967 6e6f 7265 6420 6f74 ll be ignored ot │ │ │ │ -0001df70: 6865 7277 6973 652e 0a0a 2b2d 2d2d 2d2d herwise...+----- │ │ │ │ +0001dc90: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0001dca0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0001dcb0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0001dcc0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0001dcd0: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +0001dce0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0001dcf0: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ +0001dd00: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ +0001dd10: 2e6d 323a 3235 3431 3a30 2e0a 1f0a 4669 .m2:2541:0....Fi │ │ │ │ +0001dd20: 6c65 3a20 4368 6172 6163 7465 7269 7374 le: Characterist │ │ │ │ +0001dd30: 6963 436c 6173 7365 732e 696e 666f 2c20 icClasses.info, │ │ │ │ +0001dd40: 4e6f 6465 3a20 496e 6473 4f66 536d 6f6f Node: IndsOfSmoo │ │ │ │ +0001dd50: 7468 2c20 4e65 7874 3a20 496e 7075 7449 th, Next: InputI │ │ │ │ +0001dd60: 7353 6d6f 6f74 682c 2050 7265 763a 2045 sSmooth, Prev: E │ │ │ │ +0001dd70: 756c 6572 4166 6669 6e65 2c20 5570 3a20 ulerAffine, Up: │ │ │ │ +0001dd80: 546f 700a 0a49 6e64 734f 6653 6d6f 6f74 Top..IndsOfSmoot │ │ │ │ +0001dd90: 680a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a h.************.. │ │ │ │ +0001dda0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +0001ddb0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 7074 =======..The opt │ │ │ │ +0001ddc0: 696f 6e20 496e 6473 4f66 536d 6f6f 7468 ion IndsOfSmooth │ │ │ │ +0001ddd0: 2069 7320 6f6e 6c79 2075 7365 6420 6279 is only used by │ │ │ │ +0001dde0: 2074 6865 2063 6f6d 6d61 6e64 7320 2a6e the commands *n │ │ │ │ +0001ddf0: 6f74 6520 4353 4d3a 2043 534d 2c2c 2061 ote CSM: CSM,, a │ │ │ │ +0001de00: 6e64 202a 6e6f 7465 0a45 756c 6572 3a20 nd *note.Euler: │ │ │ │ +0001de10: 4575 6c65 722c 2069 6e20 636f 6d62 696e Euler, in combin │ │ │ │ +0001de20: 6174 696f 6e20 7769 7468 2074 6865 206f ation with the o │ │ │ │ +0001de30: 7074 696f 6e20 4d65 7468 6f64 3d3e 4469 ption Method=>Di │ │ │ │ +0001de40: 7265 6374 436f 6d70 6c65 7449 6e74 2e20 rectCompletInt. │ │ │ │ +0001de50: 5768 656e 0a75 7365 6420 7468 6973 206f When.used this o │ │ │ │ +0001de60: 7074 696f 6e20 6d61 7920 616c 6c6f 7720 ption may allow │ │ │ │ +0001de70: 7468 6520 7573 6572 2074 6f20 7370 6565 the user to spee │ │ │ │ +0001de80: 6420 7570 2074 6865 2063 6f6d 7075 7461 d up the computa │ │ │ │ +0001de90: 7469 6f6e 2062 7920 7465 6c6c 696e 670a tion by telling. │ │ │ │ +0001dea0: 6769 7669 6e67 2074 6865 206d 6574 686f giving the metho │ │ │ │ +0001deb0: 6420 6120 6c69 7374 206f 6620 696e 6469 d a list of indi │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001dfc0: 0a7c 6931 203a 2052 203d 204d 756c 7469 .|i1 : R = Multi │ │ │ │ -0001dfd0: 5072 6f6a 436f 6f72 6452 696e 6728 7b32 ProjCoordRing({2 │ │ │ │ -0001dfe0: 2c32 7d29 2020 2020 2020 2020 2020 2020 ,2}) │ │ │ │ +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 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e040: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001e050: 6f31 203d 2052 2020 2020 2020 2020 2020 o1 = R │ │ │ │ +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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e070: 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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001e090: 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 2020 │ │ │ │ -0001e0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e0d0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0001e0e0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -0001e0f0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ -0001e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e120: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +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: 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 2d2d 2d2d ---------------- │ │ │ │ -0001e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e160: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -0001e170: 2049 3d69 6465 616c 2852 5f30 2a52 5f31 I=ideal(R_0*R_1 │ │ │ │ -0001e180: 2a52 5f33 2d52 5f30 5e32 2a52 5f33 2c72 *R_3-R_0^2*R_3,r │ │ │ │ -0001e190: 616e 646f 6d28 7b30 2c31 7d2c 5229 2c72 andom({0,1},R),r │ │ │ │ -0001e1a0: 616e 646f 6d28 7b31 2c32 7d2c 5229 293b andom({1,2},R)); │ │ │ │ -0001e1b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e1f0: 2020 2020 2020 207c 0a7c 6f32 203a 2049 |.|o2 : I │ │ │ │ -0001e200: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ +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 2020 2020 │ │ │ │ 0001e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e230: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e240: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e280: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 696d -----+.|i3 : tim │ │ │ │ -0001e290: 6520 4353 4d28 492c 4d65 7468 6f64 3d3e e CSM(I,Method=> │ │ │ │ -0001e2a0: 4469 7265 6374 436f 6d70 6c65 7449 6e74 DirectCompletInt │ │ │ │ -0001e2b0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0001e2c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001e2d0: 2d2d 2075 7365 6420 312e 3732 3139 3673 -- used 1.72196s │ │ │ │ -0001e2e0: 2028 6370 7529 3b20 312e 3136 3739 3273 (cpu); 1.16792s │ │ │ │ -0001e2f0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -0001e300: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ -0001e310: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 2033 2e38 3936 .| -- used 3.896 │ │ │ │ +0001e2c0: 3736 7320 2863 7075 293b 2031 2e31 3438 76s (cpu); 1.148 │ │ │ │ +0001e2d0: 3039 7320 2874 6872 6561 6429 3b20 3073 09s (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 2020 2020 │ │ │ │ -0001e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e350: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001e360: 2020 2020 3220 3220 2020 2020 3220 2020 2 2 2 │ │ │ │ -0001e370: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0001e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3a0: 207c 0a7c 6f33 203d 2032 6820 6820 202b |.|o3 = 2h h + │ │ │ │ -0001e3b0: 2032 6820 6820 202b 2035 6820 6820 2020 2h h + 5h h │ │ │ │ -0001e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001e3f0: 2020 3120 3220 2020 2020 3120 3220 2020 1 2 1 2 │ │ │ │ -0001e400: 2020 3120 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ -0001e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001e430: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +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: 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 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e470: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ -0001e480: 5b68 202e 2e68 205d 2020 2020 2020 2020 [h ..h ] │ │ │ │ +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 2020 2020 │ │ │ │ 0001e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001e4c0: 2020 2020 2020 2020 2031 2020 2032 2020 1 2 │ │ │ │ +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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e500: 2020 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d |.|o3 : ---- │ │ │ │ -0001e510: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020 ------ │ │ │ │ -0001e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e540: 2020 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2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001e5e0: 2020 2020 3120 2020 3220 2020 2020 2020 1 2 │ │ │ │ +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 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 2020 │ │ │ │ 0001e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e620: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +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 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e660: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 -------+.|i4 : t │ │ │ │ -0001e670: 696d 6520 4353 4d28 492c 4d65 7468 6f64 ime CSM(I,Method │ │ │ │ -0001e680: 3d3e 4469 7265 6374 436f 6d70 6c65 7449 =>DirectCompletI │ │ │ │ -0001e690: 6e74 2c49 6e64 734f 6653 6d6f 6f74 683d nt,IndsOfSmooth= │ │ │ │ -0001e6a0: 3e7b 312c 327d 2920 2020 2020 2020 7c0a >{1,2}) |. │ │ │ │ -0001e6b0: 7c20 2d2d 2075 7365 6420 312e 3430 3632 | -- used 1.4062 │ │ │ │ -0001e6c0: 3273 2028 6370 7529 3b20 312e 3035 3636 2s (cpu); 1.0566 │ │ │ │ -0001e6d0: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ -0001e6e0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ -0001e6f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +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 2033 2e38 |.| -- used 3.8 │ │ │ │ +0001e6a0: 3335 3639 7320 2863 7075 293b 2031 2e30 3569s (cpu); 1.0 │ │ │ │ +0001e6b0: 3932 3832 7320 2874 6872 6561 6429 3b20 9282s (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 2020 │ │ │ │ -0001e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e730: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001e740: 2020 2020 2020 3220 3220 2020 2020 3220 2 2 2 │ │ │ │ -0001e750: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -0001e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e780: 2020 207c 0a7c 6f34 203d 2032 6820 6820 |.|o4 = 2h h │ │ │ │ -0001e790: 202b 2032 6820 6820 202b 2035 6820 6820 + 2h h + 5h h │ │ │ │ -0001e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001e7d0: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ -0001e7e0: 2020 2020 3120 3220 2020 2020 2020 2020 1 2 │ │ │ │ -0001e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 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: 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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001e810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e850: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001e860: 5a5a 5b68 202e 2e68 205d 2020 2020 2020 ZZ[h ..h ] │ │ │ │ +0001e830: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001e840: 2020 205a 5a5b 6820 2e2e 6820 5d20 2020 ZZ[h ..h ] │ │ │ │ +0001e850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e890: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001e8a0: 0a7c 2020 2020 2020 2020 2031 2020 2032 .| 1 2 │ │ │ │ +0001e880: 2020 7c0a 7c20 2020 2020 2020 2020 3120 |.| 1 │ │ │ │ +0001e890: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001e8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e8e0: 2020 2020 2020 7c0a 7c6f 3420 3a20 2d2d |.|o4 : -- │ │ │ │ -0001e8f0: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020 -------- │ │ │ │ +0001e8c0: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ +0001e8d0: 202d 2d2d 2d2d 2d2d 2d2d 2d20 2020 2020 ---------- │ │ │ │ +0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e920: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001e930: 2020 2020 2020 2020 3320 2020 3320 2020 3 3 │ │ │ │ +0001e910: 7c0a 7c20 2020 2020 2020 2033 2020 2033 |.| 3 3 │ │ │ │ +0001e920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e970: 2020 2020 7c0a 7c20 2020 2020 2028 6820 |.| (h │ │ │ │ -0001e980: 2c20 6820 2920 2020 2020 2020 2020 2020 , h ) │ │ │ │ -0001e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0001e9c0: 2020 2020 2020 3120 2020 3220 2020 2020 1 2 │ │ │ │ +0001e950: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001e960: 2868 202c 2068 2029 2020 2020 2020 2020 (h , h ) │ │ │ │ +0001e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e990: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001e9a0: 7c20 2020 2020 2020 2031 2020 2032 2020 | 1 2 │ │ │ │ +0001e9b0: 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206e 616d nal argument nam │ │ │ │ -0001ea70: 6564 2049 6e64 734f 6653 6d6f 6f74 683a ed IndsOfSmooth: │ │ │ │ -0001ea80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0001ea90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001eaa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001eab0: 3d3d 3d3d 3d0a 0a20 202a 2022 4353 4d28 =====.. * "CSM( │ │ │ │ -0001eac0: 2e2e 2e2c 496e 6473 4f66 536d 6f6f 7468 ...,IndsOfSmooth │ │ │ │ -0001ead0: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ -0001eae0: 6e6f 7465 2043 534d 3a20 4353 4d2c 202d note CSM: CSM, - │ │ │ │ -0001eaf0: 2d20 5468 650a 2020 2020 4368 6572 6e2d - The. Chern- │ │ │ │ -0001eb00: 5363 6877 6172 747a 2d4d 6163 5068 6572 Schwartz-MacPher │ │ │ │ -0001eb10: 736f 6e20 636c 6173 730a 2020 2a20 4575 son class. * Eu │ │ │ │ -0001eb20: 6c65 7228 2e2e 2e2c 496e 6473 4f66 536d ler(...,IndsOfSm │ │ │ │ -0001eb30: 6f6f 7468 3d3e 2e2e 2e29 2028 6d69 7373 ooth=>...) (miss │ │ │ │ -0001eb40: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -0001eb50: 6e29 0a0a 466f 7220 7468 6520 7072 6f67 n)..For the prog │ │ │ │ -0001eb60: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -0001eb70: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -0001eb80: 626a 6563 7420 2a6e 6f74 6520 496e 6473 bject *note Inds │ │ │ │ -0001eb90: 4f66 536d 6f6f 7468 3a20 496e 6473 4f66 OfSmooth: IndsOf │ │ │ │ -0001eba0: 536d 6f6f 7468 2c20 6973 2061 202a 6e6f Smooth, is a *no │ │ │ │ -0001ebb0: 7465 2073 796d 626f 6c3a 0a28 4d61 6361 te symbol:.(Maca │ │ │ │ -0001ebc0: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ -0001ebd0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 ------------+..F │ │ │ │ +0001ea30: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ +0001ea40: 7469 6f6e 616c 2061 7267 756d 656e 7420 tional argument │ │ │ │ +0001ea50: 6e61 6d65 6420 496e 6473 4f66 536d 6f6f named IndsOfSmoo │ │ │ │ +0001ea60: 7468 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d th:.============ │ │ │ │ +0001ea70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001ea80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001ea90: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2243 ========.. * "C │ │ │ │ +0001eaa0: 534d 282e 2e2e 2c49 6e64 734f 6653 6d6f SM(...,IndsOfSmo │ │ │ │ +0001eab0: 6f74 683d 3e2e 2e2e 2922 202d 2d20 7365 oth=>...)" -- se │ │ │ │ +0001eac0: 6520 2a6e 6f74 6520 4353 4d3a 2043 534d e *note CSM: CSM │ │ │ │ +0001ead0: 2c20 2d2d 2054 6865 0a20 2020 2043 6865 , -- The. Che │ │ │ │ +0001eae0: 726e 2d53 6368 7761 7274 7a2d 4d61 6350 rn-Schwartz-MacP │ │ │ │ +0001eaf0: 6865 7273 6f6e 2063 6c61 7373 0a20 202a herson class. * │ │ │ │ +0001eb00: 2045 756c 6572 282e 2e2e 2c49 6e64 734f Euler(...,IndsO │ │ │ │ +0001eb10: 6653 6d6f 6f74 683d 3e2e 2e2e 2920 286d fSmooth=>...) (m │ │ │ │ +0001eb20: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ +0001eb30: 7469 6f6e 290a 0a46 6f72 2074 6865 2070 tion)..For the p │ │ │ │ +0001eb40: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +0001eb50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +0001eb60: 6520 6f62 6a65 6374 202a 6e6f 7465 2049 e object *note I │ │ │ │ +0001eb70: 6e64 734f 6653 6d6f 6f74 683a 2049 6e64 ndsOfSmooth: Ind │ │ │ │ +0001eb80: 734f 6653 6d6f 6f74 682c 2069 7320 6120 sOfSmooth, is a │ │ │ │ +0001eb90: 2a6e 6f74 6520 7379 6d62 6f6c 3a0a 284d *note symbol:.(M │ │ │ │ +0001eba0: 6163 6175 6c61 7932 446f 6329 5379 6d62 acaulay2Doc)Symb │ │ │ │ +0001ebb0: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ol,...---------- │ │ │ │ +0001ebc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ebd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ebe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec20: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0001ec30: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0001ec40: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0001ec50: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0001ec60: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -0001ec70: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0001ec80: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ -0001ec90: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ -0001eca0: 2e6d 323a 3234 3832 3a30 2e0a 1f0a 4669 .m2:2482:0....Fi │ │ │ │ -0001ecb0: 6c65 3a20 4368 6172 6163 7465 7269 7374 le: Characterist │ │ │ │ -0001ecc0: 6963 436c 6173 7365 732e 696e 666f 2c20 icClasses.info, │ │ │ │ -0001ecd0: 4e6f 6465 3a20 496e 7075 7449 7353 6d6f Node: InputIsSmo │ │ │ │ -0001ece0: 6f74 682c 204e 6578 743a 2069 734d 756c oth, Next: isMul │ │ │ │ -0001ecf0: 7469 486f 6d6f 6765 6e65 6f75 732c 2050 tiHomogeneous, P │ │ │ │ -0001ed00: 7265 763a 2049 6e64 734f 6653 6d6f 6f74 rev: IndsOfSmoot │ │ │ │ -0001ed10: 682c 2055 703a 2054 6f70 0a0a 496e 7075 h, Up: Top..Inpu │ │ │ │ -0001ed20: 7449 7353 6d6f 6f74 680a 2a2a 2a2a 2a2a tIsSmooth.****** │ │ │ │ -0001ed30: 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 6970 *******..Descrip │ │ │ │ -0001ed40: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -0001ed50: 0a0a 5468 6520 6f70 7469 6f6e 2049 6e70 ..The option Inp │ │ │ │ -0001ed60: 7574 4973 536d 6f6f 7468 2069 7320 6f6e utIsSmooth is on │ │ │ │ -0001ed70: 6c79 2075 7365 6420 6279 2074 6865 2063 ly used by the c │ │ │ │ -0001ed80: 6f6d 6d61 6e64 7320 2a6e 6f74 6520 4353 ommands *note CS │ │ │ │ -0001ed90: 4d3a 2043 534d 2c2c 2061 6e64 0a2a 6e6f M: CSM,, and.*no │ │ │ │ -0001eda0: 7465 2045 756c 6572 3a20 4575 6c65 722c te Euler: Euler, │ │ │ │ -0001edb0: 2e20 4966 2074 6865 2069 6e70 7574 2069 . If the input i │ │ │ │ -0001edc0: 6465 616c 2069 7320 6b6e 6f77 6e20 746f deal is known to │ │ │ │ -0001edd0: 2064 6566 696e 6520 6120 736d 6f6f 7468 define a smooth │ │ │ │ -0001ede0: 2073 7562 7363 6865 6d65 0a73 6574 7469 subscheme.setti │ │ │ │ -0001edf0: 6e67 2074 6869 7320 6f70 7469 6f6e 2074 ng this option t │ │ │ │ -0001ee00: 6f20 7472 7565 2077 696c 6c20 7370 6565 o true will spee │ │ │ │ -0001ee10: 6420 7570 2063 6f6d 7075 7461 7469 6f6e d up computation │ │ │ │ -0001ee20: 7320 2869 7420 6973 2073 6574 2074 6f20 s (it is set to │ │ │ │ -0001ee30: 6661 6c73 6520 6279 0a64 6566 6175 6c74 false by.default │ │ │ │ -0001ee40: 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d )...+----------- │ │ │ │ -0001ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ee60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0001ee80: 203a 2052 203d 205a 5a2f 3332 3734 395b : R = ZZ/32749[ │ │ │ │ -0001ee90: 785f 302e 2e78 5f34 5d3b 2020 2020 2020 x_0..x_4]; │ │ │ │ -0001eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eeb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001ec00: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0001ec10: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0001ec20: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0001ec30: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0001ec40: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +0001ec50: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +0001ec60: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ +0001ec70: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ +0001ec80: 7365 732e 6d32 3a32 3438 323a 302e 0a1f ses.m2:2482:0... │ │ │ │ +0001ec90: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ +0001eca0: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ +0001ecb0: 6f2c 204e 6f64 653a 2049 6e70 7574 4973 o, Node: InputIs │ │ │ │ +0001ecc0: 536d 6f6f 7468 2c20 4e65 7874 3a20 6973 Smooth, Next: is │ │ │ │ +0001ecd0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ +0001ece0: 2c20 5072 6576 3a20 496e 6473 4f66 536d , Prev: IndsOfSm │ │ │ │ +0001ecf0: 6f6f 7468 2c20 5570 3a20 546f 700a 0a49 ooth, Up: Top..I │ │ │ │ +0001ed00: 6e70 7574 4973 536d 6f6f 7468 0a2a 2a2a nputIsSmooth.*** │ │ │ │ +0001ed10: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 **********..Desc │ │ │ │ +0001ed20: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +0001ed30: 3d3d 3d0a 0a54 6865 206f 7074 696f 6e20 ===..The option │ │ │ │ +0001ed40: 496e 7075 7449 7353 6d6f 6f74 6820 6973 InputIsSmooth is │ │ │ │ +0001ed50: 206f 6e6c 7920 7573 6564 2062 7920 7468 only used by th │ │ │ │ +0001ed60: 6520 636f 6d6d 616e 6473 202a 6e6f 7465 e commands *note │ │ │ │ +0001ed70: 2043 534d 3a20 4353 4d2c 2c20 616e 640a CSM: CSM,, and. │ │ │ │ +0001ed80: 2a6e 6f74 6520 4575 6c65 723a 2045 756c *note Euler: Eul │ │ │ │ +0001ed90: 6572 2c2e 2049 6620 7468 6520 696e 7075 er,. If the inpu │ │ │ │ +0001eda0: 7420 6964 6561 6c20 6973 206b 6e6f 776e t ideal is known │ │ │ │ +0001edb0: 2074 6f20 6465 6669 6e65 2061 2073 6d6f to define a smo │ │ │ │ +0001edc0: 6f74 6820 7375 6273 6368 656d 650a 7365 oth subscheme.se │ │ │ │ +0001edd0: 7474 696e 6720 7468 6973 206f 7074 696f tting this optio │ │ │ │ +0001ede0: 6e20 746f 2074 7275 6520 7769 6c6c 2073 n to true will s │ │ │ │ +0001edf0: 7065 6564 2075 7020 636f 6d70 7574 6174 peed up computat │ │ │ │ +0001ee00: 696f 6e73 2028 6974 2069 7320 7365 7420 ions (it is set │ │ │ │ +0001ee10: 746f 2066 616c 7365 2062 790a 6465 6661 to false by.defa │ │ │ │ +0001ee20: 756c 7429 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d ult)...+-------- │ │ │ │ +0001ee30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ee40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001ee60: 7c69 3120 3a20 5220 3d20 5a5a 2f33 3237 |i1 : R = ZZ/327 │ │ │ │ +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 ---------------- │ │ │ │ 0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0001eef0: 6932 203a 2049 3d69 6465 616c 2872 616e i2 : I=ideal(ran │ │ │ │ -0001ef00: 646f 6d28 322c 5229 2c72 616e 646f 6d28 dom(2,R),random( │ │ │ │ -0001ef10: 322c 5229 2c72 616e 646f 6d28 312c 5229 2,R),random(1,R) │ │ │ │ -0001ef20: 293b 2020 2020 7c0a 7c20 2020 2020 2020 ); |.| │ │ │ │ +0001eed0: 2b0a 7c69 3220 3a20 493d 6964 6561 6c28 +.|i2 : I=ideal( │ │ │ │ +0001eee0: 7261 6e64 6f6d 2832 2c52 292c 7261 6e64 random(2,R),rand │ │ │ │ +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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001ef60: 0a7c 6f32 203a 2049 6465 616c 206f 6620 .|o2 : Ideal of │ │ │ │ -0001ef70: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -0001ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +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 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001efc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001efd0: 2d2b 0a7c 6933 203a 2074 696d 6520 4353 -+.|i3 : time CS │ │ │ │ -0001efe0: 4d20 4920 2020 2020 2020 2020 2020 2020 M I │ │ │ │ -0001eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f000: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0001f010: 2075 7365 6420 302e 3631 3837 3534 7320 used 0.618754s │ │ │ │ -0001f020: 2863 7075 293b 2030 2e34 3636 3337 3673 (cpu); 0.466376s │ │ │ │ -0001f030: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -0001f040: 6329 207c 0a7c 2020 2020 2020 2020 2020 c) |.| │ │ │ │ -0001f050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f070: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001f080: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ -0001f090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f0b0: 2020 2020 207c 0a7c 6f33 203d 2034 6820 |.|o3 = 4h │ │ │ │ +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 2030 2e38 3238 3430 -- used 0.82840 │ │ │ │ +0001f000: 3873 2028 6370 7529 3b20 302e 3432 3130 8s (cpu); 0.4210 │ │ │ │ +0001f010: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ +0001f020: 2867 6329 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 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 2020 2020 2020 2020 2020 │ │ │ │ -0001f0e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f0f0: 7c20 2020 2020 2020 3120 2020 2020 2020 | 1 │ │ │ │ -0001f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 7c0a 7c20 2020 |.| │ │ │ │ 0001f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f120: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f160: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020 |.| ZZ[h ] │ │ │ │ -0001f170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f190: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001f1a0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ -0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1d0: 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d 2d2d |.|o3 : ------ │ │ │ │ -0001f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f200: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0001f210: 2020 2020 2020 3520 2020 2020 2020 2020 5 │ │ │ │ -0001f220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f240: 2020 2020 7c0a 7c20 2020 2020 2020 6820 |.| h │ │ │ │ +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 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 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 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 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 2020 2020 │ │ │ │ -0001f270: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001f280: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ -0001f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +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 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(miss │ │ │ │ +0001fab0: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ +0001fac0: 6e29 0a0a 466f 7220 7468 6520 7072 6f67 n)..For the prog │ │ │ │ +0001fad0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +0001fae0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +0001faf0: 626a 6563 7420 2a6e 6f74 6520 496e 7075 bject *note Inpu │ │ │ │ +0001fb00: 7449 7353 6d6f 6f74 683a 2049 6e70 7574 tIsSmooth: Input │ │ │ │ +0001fb10: 4973 536d 6f6f 7468 2c20 6973 2061 202a IsSmooth, is a * │ │ │ │ +0001fb20: 6e6f 7465 2073 796d 626f 6c3a 0a28 4d61 note symbol:.(Ma │ │ │ │ +0001fb30: 6361 756c 6179 3244 6f63 2953 796d 626f caulay2Doc)Symbo │ │ │ │ +0001fb40: 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d l,...----------- │ │ │ │ +0001fb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fbb0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0001fbc0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0001fbd0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0001fbe0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0001fbf0: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ -0001fc00: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0001fc10: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ -0001fc20: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ -0001fc30: 6d32 3a32 3530 303a 302e 0a1f 0a46 696c m2:2500:0....Fil │ │ │ │ -0001fc40: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ -0001fc50: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ -0001fc60: 6f64 653a 2069 734d 756c 7469 486f 6d6f ode: isMultiHomo │ │ │ │ -0001fc70: 6765 6e65 6f75 732c 204e 6578 743a 204d geneous, Next: M │ │ │ │ -0001fc80: 6574 686f 642c 2050 7265 763a 2049 6e70 ethod, Prev: Inp │ │ │ │ -0001fc90: 7574 4973 536d 6f6f 7468 2c20 5570 3a20 utIsSmooth, Up: │ │ │ │ -0001fca0: 546f 700a 0a69 734d 756c 7469 486f 6d6f Top..isMultiHomo │ │ │ │ -0001fcb0: 6765 6e65 6f75 7320 2d2d 2043 6865 636b geneous -- Check │ │ │ │ -0001fcc0: 7320 6966 2061 6e20 6964 6561 6c20 6973 s if an ideal is │ │ │ │ -0001fcd0: 2068 6f6d 6f67 656e 656f 7573 2077 6974 homogeneous wit │ │ │ │ -0001fce0: 6820 7265 7370 6563 7420 746f 2074 6865 h respect to the │ │ │ │ -0001fcf0: 2067 7261 6469 6e67 206f 6e20 6974 7320 grading on its │ │ │ │ -0001fd00: 7269 6e67 2028 692e 652e 206d 756c 7469 ring (i.e. multi │ │ │ │ -0001fd10: 2d68 6f6d 6f67 656e 656f 7573 2069 6e20 -homogeneous in │ │ │ │ -0001fd20: 7468 6520 6d75 6c74 692d 6772 6164 6564 the multi-graded │ │ │ │ -0001fd30: 2063 6173 6529 0a2a 2a2a 2a2a 2a2a 2a2a case).********* │ │ │ │ +0001fb90: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +0001fba0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +0001fbb0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +0001fbc0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +0001fbd0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +0001fbe0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +0001fbf0: 7932 2f70 6163 6b61 6765 732f 0a43 6861 y2/packages/.Cha │ │ │ │ +0001fc00: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ +0001fc10: 6573 2e6d 323a 3235 3030 3a30 2e0a 1f0a es.m2:2500:0.... │ │ │ │ +0001fc20: 4669 6c65 3a20 4368 6172 6163 7465 7269 File: Characteri │ │ │ │ +0001fc30: 7374 6963 436c 6173 7365 732e 696e 666f sticClasses.info │ │ │ │ +0001fc40: 2c20 4e6f 6465 3a20 6973 4d75 6c74 6948 , Node: isMultiH │ │ │ │ +0001fc50: 6f6d 6f67 656e 656f 7573 2c20 4e65 7874 omogeneous, Next │ │ │ │ +0001fc60: 3a20 4d65 7468 6f64 2c20 5072 6576 3a20 : Method, Prev: │ │ │ │ +0001fc70: 496e 7075 7449 7353 6d6f 6f74 682c 2055 InputIsSmooth, U │ │ │ │ +0001fc80: 703a 2054 6f70 0a0a 6973 4d75 6c74 6948 p: Top..isMultiH │ │ │ │ +0001fc90: 6f6d 6f67 656e 656f 7573 202d 2d20 4368 omogeneous -- Ch │ │ │ │ +0001fca0: 6563 6b73 2069 6620 616e 2069 6465 616c ecks if an ideal │ │ │ │ +0001fcb0: 2069 7320 686f 6d6f 6765 6e65 6f75 7320 is homogeneous │ │ │ │ +0001fcc0: 7769 7468 2072 6573 7065 6374 2074 6f20 with respect to │ │ │ │ +0001fcd0: 7468 6520 6772 6164 696e 6720 6f6e 2069 the grading on i │ │ │ │ +0001fce0: 7473 2072 696e 6720 2869 2e65 2e20 6d75 ts ring (i.e. mu │ │ │ │ +0001fcf0: 6c74 692d 686f 6d6f 6765 6e65 6f75 7320 lti-homogeneous │ │ │ │ +0001fd00: 696e 2074 6865 206d 756c 7469 2d67 7261 in the multi-gra │ │ │ │ +0001fd10: 6465 6420 6361 7365 290a 2a2a 2a2a 2a2a ded case).****** │ │ │ │ +0001fd20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fd30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fdb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fdc0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ -0001fdd0: 6167 653a 200a 2020 2020 2020 2020 6973 age: . is │ │ │ │ +0001fda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0001fdb0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0001fdc0: 2069 734d 756c 7469 486f 6d6f 6765 6e65 isMultiHomogene │ │ │ │ +0001fdd0: 6f75 7320 490a 2020 2020 2020 2020 6973 ous I. is │ │ │ │ 0001fde0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ -0001fdf0: 2049 0a20 2020 2020 2020 2069 734d 756c I. isMul │ │ │ │ -0001fe00: 7469 486f 6d6f 6765 6e65 6f75 7320 660a tiHomogeneous f. │ │ │ │ -0001fe10: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -0001fe20: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520 * I, an *note │ │ │ │ -0001fe30: 6964 6561 6c3a 2028 4d61 6361 756c 6179 ideal: (Macaulay │ │ │ │ -0001fe40: 3244 6f63 2949 6465 616c 2c2c 2061 6e20 2Doc)Ideal,, an │ │ │ │ -0001fe50: 6964 6561 6c20 696e 2061 2067 7261 6465 ideal in a grade │ │ │ │ -0001fe60: 6420 6f72 0a20 2020 2020 2020 206d 756c d or. mul │ │ │ │ -0001fe70: 7469 2d67 7261 6465 6420 7269 6e67 0a20 ti-graded ring. │ │ │ │ -0001fe80: 2020 2020 202a 2066 2c20 6120 2a6e 6f74 * f, a *not │ │ │ │ -0001fe90: 6520 7269 6e67 2065 6c65 6d65 6e74 3a20 e ring element: │ │ │ │ -0001fea0: 284d 6163 6175 6c61 7932 446f 6329 5269 (Macaulay2Doc)Ri │ │ │ │ -0001feb0: 6e67 456c 656d 656e 742c 2c20 6120 656c ngElement,, a el │ │ │ │ -0001fec0: 656d 656e 7420 696e 2061 0a20 2020 2020 ement in a. │ │ │ │ -0001fed0: 2020 2067 7261 6465 6420 6f72 206d 756c graded or mul │ │ │ │ -0001fee0: 7469 2d67 7261 6465 6420 7269 6e67 0a20 ti-graded ring. │ │ │ │ -0001fef0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -0001ff00: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c * a *note Bool │ │ │ │ -0001ff10: 6561 6e20 7661 6c75 653a 2028 4d61 6361 ean value: (Maca │ │ │ │ -0001ff20: 756c 6179 3244 6f63 2942 6f6f 6c65 616e ulay2Doc)Boolean │ │ │ │ -0001ff30: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e ,, ..Description │ │ │ │ -0001ff40: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5465 .===========..Te │ │ │ │ -0001ff50: 7374 7320 6966 2074 6865 2069 6e70 7574 sts if the input │ │ │ │ -0001ff60: 2049 6465 616c 206f 7220 5269 6e67 456c Ideal or RingEl │ │ │ │ -0001ff70: 656d 656e 7420 6973 2048 6f6d 6f67 656e ement is Homogen │ │ │ │ -0001ff80: 656f 7573 2077 6974 6820 7265 7370 6563 eous with respec │ │ │ │ -0001ff90: 7420 746f 2074 6865 0a67 7261 6469 6e67 t to the.grading │ │ │ │ -0001ffa0: 206f 6e20 7468 6520 7269 6e67 2e20 486f on the ring. Ho │ │ │ │ -0001ffb0: 6d6f 6765 6e65 6f75 7320 696e 7075 7420 mogeneous input │ │ │ │ -0001ffc0: 6973 2072 6571 7569 7265 6420 666f 7220 is required for │ │ │ │ -0001ffd0: 616c 6c20 6d65 7468 6f64 7320 746f 2063 all methods to c │ │ │ │ -0001ffe0: 6f6d 7075 7465 0a63 6861 7261 6374 6572 ompute.character │ │ │ │ -0001fff0: 6973 7469 6320 636c 6173 7365 732e 0a0a istic classes... │ │ │ │ -00020000: 5468 6973 206d 6574 686f 6420 776f 726b This method work │ │ │ │ -00020010: 7320 666f 7220 6964 6561 6c73 2069 6e20 s for ideals in │ │ │ │ -00020020: 7468 6520 6772 6164 6564 2063 6f6f 7264 the graded coord │ │ │ │ -00020030: 696e 6174 6520 7269 6e67 7320 6f66 2074 inate rings of t │ │ │ │ -00020040: 6f72 6963 2076 6172 6965 7469 6573 2c0a oric varieties,. │ │ │ │ -00020050: 616e 6420 6865 6e63 6520 666f 7220 7072 and hence for pr │ │ │ │ -00020060: 6f64 7563 7473 206f 6620 7072 6f6a 6563 oducts of projec │ │ │ │ -00020070: 7469 7665 2073 7061 6365 732e 2054 6865 tive spaces. The │ │ │ │ -00020080: 7365 2063 616e 2062 6520 6372 6561 7465 se can be create │ │ │ │ -00020090: 6420 6469 7265 6374 6c79 2c20 6f72 0a75 d directly, or.u │ │ │ │ -000200a0: 7369 6e67 206d 6574 686f 6473 2074 6865 sing methods the │ │ │ │ -000200b0: 202a 6e6f 7465 204d 756c 7469 5072 6f6a *note MultiProj │ │ │ │ -000200c0: 436f 6f72 6452 696e 673a 204d 756c 7469 CoordRing: Multi │ │ │ │ -000200d0: 5072 6f6a 436f 6f72 6452 696e 672c 206d ProjCoordRing, m │ │ │ │ -000200e0: 6574 686f 6420 6f66 2074 6869 730a 7061 ethod of this.pa │ │ │ │ -000200f0: 636b 6167 652c 206f 7220 7769 7468 206d ckage, or with m │ │ │ │ -00020100: 6574 686f 6473 2066 726f 6d20 7468 6520 ethods from the │ │ │ │ -00020110: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ -00020120: 7469 6573 2050 6163 6b61 6765 2e0a 0a2b ties Package...+ │ │ │ │ +0001fdf0: 2066 0a20 202a 2049 6e70 7574 733a 0a20 f. * Inputs:. │ │ │ │ +0001fe00: 2020 2020 202a 2049 2c20 616e 202a 6e6f * I, an *no │ │ │ │ +0001fe10: 7465 2069 6465 616c 3a20 284d 6163 6175 te ideal: (Macau │ │ │ │ +0001fe20: 6c61 7932 446f 6329 4964 6561 6c2c 2c20 lay2Doc)Ideal,, │ │ │ │ +0001fe30: 616e 2069 6465 616c 2069 6e20 6120 6772 an ideal in a gr │ │ │ │ +0001fe40: 6164 6564 206f 720a 2020 2020 2020 2020 aded or. │ │ │ │ +0001fe50: 6d75 6c74 692d 6772 6164 6564 2072 696e multi-graded rin │ │ │ │ +0001fe60: 670a 2020 2020 2020 2a20 662c 2061 202a g. * f, a * │ │ │ │ +0001fe70: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ +0001fe80: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +0001fe90: 2952 696e 6745 6c65 6d65 6e74 2c2c 2061 )RingElement,, a │ │ │ │ +0001fea0: 2065 6c65 6d65 6e74 2069 6e20 610a 2020 element in a. │ │ │ │ +0001feb0: 2020 2020 2020 6772 6164 6564 206f 7220 graded or │ │ │ │ +0001fec0: 6d75 6c74 692d 6772 6164 6564 2072 696e multi-graded rin │ │ │ │ +0001fed0: 670a 2020 2a20 4f75 7470 7574 733a 0a20 g. * Outputs:. │ │ │ │ +0001fee0: 2020 2020 202a 2061 202a 6e6f 7465 2042 * a *note B │ │ │ │ +0001fef0: 6f6f 6c65 616e 2076 616c 7565 3a20 284d oolean value: (M │ │ │ │ +0001ff00: 6163 6175 6c61 7932 446f 6329 426f 6f6c acaulay2Doc)Bool │ │ │ │ +0001ff10: 6561 6e2c 2c20 0a0a 4465 7363 7269 7074 ean,, ..Descript │ │ │ │ +0001ff20: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0001ff30: 0a54 6573 7473 2069 6620 7468 6520 696e .Tests if the in │ │ │ │ +0001ff40: 7075 7420 4964 6561 6c20 6f72 2052 696e put Ideal or Rin │ │ │ │ +0001ff50: 6745 6c65 6d65 6e74 2069 7320 486f 6d6f gElement is Homo │ │ │ │ +0001ff60: 6765 6e65 6f75 7320 7769 7468 2072 6573 geneous with res │ │ │ │ +0001ff70: 7065 6374 2074 6f20 7468 650a 6772 6164 pect to the.grad │ │ │ │ +0001ff80: 696e 6720 6f6e 2074 6865 2072 696e 672e ing on the ring. │ │ │ │ +0001ff90: 2048 6f6d 6f67 656e 656f 7573 2069 6e70 Homogeneous inp │ │ │ │ +0001ffa0: 7574 2069 7320 7265 7175 6972 6564 2066 ut is required f │ │ │ │ +0001ffb0: 6f72 2061 6c6c 206d 6574 686f 6473 2074 or all methods t │ │ │ │ +0001ffc0: 6f20 636f 6d70 7574 650a 6368 6172 6163 o compute.charac │ │ │ │ +0001ffd0: 7465 7269 7374 6963 2063 6c61 7373 6573 teristic classes │ │ │ │ +0001ffe0: 2e0a 0a54 6869 7320 6d65 7468 6f64 2077 ...This method w │ │ │ │ +0001fff0: 6f72 6b73 2066 6f72 2069 6465 616c 7320 orks for ideals │ │ │ │ +00020000: 696e 2074 6865 2067 7261 6465 6420 636f in the graded co │ │ │ │ +00020010: 6f72 6469 6e61 7465 2072 696e 6773 206f ordinate rings o │ │ │ │ +00020020: 6620 746f 7269 6320 7661 7269 6574 6965 f toric varietie │ │ │ │ +00020030: 732c 0a61 6e64 2068 656e 6365 2066 6f72 s,.and hence for │ │ │ │ +00020040: 2070 726f 6475 6374 7320 6f66 2070 726f products of pro │ │ │ │ +00020050: 6a65 6374 6976 6520 7370 6163 6573 2e20 jective spaces. │ │ │ │ +00020060: 5468 6573 6520 6361 6e20 6265 2063 7265 These can be cre │ │ │ │ +00020070: 6174 6564 2064 6972 6563 746c 792c 206f ated directly, o │ │ │ │ +00020080: 720a 7573 696e 6720 6d65 7468 6f64 7320 r.using methods │ │ │ │ +00020090: 7468 6520 2a6e 6f74 6520 4d75 6c74 6950 the *note MultiP │ │ │ │ +000200a0: 726f 6a43 6f6f 7264 5269 6e67 3a20 4d75 rojCoordRing: Mu │ │ │ │ +000200b0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ +000200c0: 2c20 6d65 7468 6f64 206f 6620 7468 6973 , method of this │ │ │ │ +000200d0: 0a70 6163 6b61 6765 2c20 6f72 2077 6974 .package, or wit │ │ │ │ +000200e0: 6820 6d65 7468 6f64 7320 6672 6f6d 2074 h methods from t │ │ │ │ +000200f0: 6865 204e 6f72 6d61 6c54 6f72 6963 5661 he NormalToricVa │ │ │ │ +00020100: 7269 6574 6965 7320 5061 636b 6167 652e rieties Package. │ │ │ │ +00020110: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +00020120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00020170: 0a7c 6931 203a 2052 3d4d 756c 7469 5072 .|i1 : R=MultiPr │ │ │ │ -00020180: 6f6a 436f 6f72 6452 696e 6728 7b31 2c32 ojCoordRing({1,2 │ │ │ │ -00020190: 2c31 7d29 2020 2020 2020 2020 2020 2020 ,1}) │ │ │ │ +00020150: 2d2d 2b0a 7c69 3120 3a20 523d 4d75 6c74 --+.|i1 : R=Mult │ │ │ │ +00020160: 6950 726f 6a43 6f6f 7264 5269 6e67 287b iProjCoordRing({ │ │ │ │ +00020170: 312c 322c 317d 2920 2020 2020 2020 2020 1,2,1}) │ │ │ │ +00020180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020190: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000201a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000201b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000201c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000201d0: 2020 2020 2020 7c0a 7c6f 3120 3d20 5220 |.|o1 = R │ │ │ │ 000201e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201f0: 2020 207c 0a7c 6f31 203d 2052 2020 2020 |.|o1 = R │ │ │ │ +000201f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020210: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00020220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020230: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00020230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020270: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ -00020280: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ -00020290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000202a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000202b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00020250: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +00020260: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +00020270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020290: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000202a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000202b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00020300: 203a 2078 3d67 656e 7328 5229 2020 2020 : x=gens(R) │ │ │ │ +000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000202e0: 7c69 3220 3a20 783d 6765 6e73 2852 2920 |i2 : x=gens(R) │ │ │ │ +000202f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020330: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00020320: 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020370: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020380: 0a7c 6f32 203d 207b 7820 2c20 7820 2c20 .|o2 = {x , x , │ │ │ │ -00020390: 7820 2c20 7820 2c20 7820 2c20 7820 2c20 x , x , x , x , │ │ │ │ -000203a0: 7820 7d20 2020 2020 2020 2020 2020 2020 x } │ │ │ │ -000203b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000203c0: 207c 0a7c 2020 2020 2020 2030 2020 2031 |.| 0 1 │ │ │ │ -000203d0: 2020 2032 2020 2033 2020 2034 2020 2035 2 3 4 5 │ │ │ │ -000203e0: 2020 2036 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +00020360: 2020 7c0a 7c6f 3220 3d20 7b78 202c 2078 |.|o2 = {x , x │ │ │ │ +00020370: 202c 2078 202c 2078 202c 2078 202c 2078 , x , x , x , x │ │ │ │ +00020380: 202c 2078 207d 2020 2020 2020 2020 2020 , x } │ │ │ │ +00020390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ -00020400: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020400: 2020 2020 2020 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 │ │ │ │ -00020440: 2020 2020 207c 0a7c 6f32 203a 204c 6973 |.|o2 : Lis │ │ │ │ -00020450: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00020460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020480: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00020420: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ +00020430: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +00020440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020450: 2020 2020 2020 2020 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 2d2d 2d2d ---------------- │ │ │ │ -000204b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000204c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -000204d0: 2049 3d69 6465 616c 2878 5f30 5e32 2a78 I=ideal(x_0^2*x │ │ │ │ -000204e0: 5f33 2d78 5f31 2a78 5f30 2a78 5f34 2c78 _3-x_1*x_0*x_4,x │ │ │ │ -000204f0: 5f36 5e33 2920 2020 2020 2020 2020 2020 _6^3) │ │ │ │ -00020500: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 │ │ │ │ -00020530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020550: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -00020560: 2020 2020 2020 2020 2020 2020 3320 2020 3 │ │ │ │ -00020570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020580: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020590: 0a7c 6f33 203d 2069 6465 616c 2028 7820 .|o3 = ideal (x │ │ │ │ -000205a0: 7820 202d 2078 2078 2078 202c 2078 2029 x - x x x , x ) │ │ │ │ -000205b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000205c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000205d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000205e0: 2030 2033 2020 2020 3020 3120 3420 2020 0 3 0 1 4 │ │ │ │ -000205f0: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +00020530: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 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 207c 0a7c 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 2020 2020 2020 │ │ │ │ -00020640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020650: 2020 2020 207c 0a7c 6f33 203a 2049 6465 |.|o3 : Ide │ │ │ │ -00020660: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ -00020670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020690: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00020630: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ +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 7c0a 2b2d 2d2d |.+--- │ │ │ │ +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 2d2d 2d2d ---------------- │ │ │ │ -000206c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -000206e0: 2069 734d 756c 7469 486f 6d6f 6765 6e65 isMultiHomogene │ │ │ │ -000206f0: 6f75 7320 4920 2020 2020 2020 2020 2020 ous I │ │ │ │ -00020700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020710: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 │ │ │ │ -00020740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020750: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020760: 6f34 203d 2074 7275 6520 2020 2020 2020 o4 = true │ │ │ │ +00020740: 7c0a 7c6f 3420 3d20 7472 7565 2020 2020 |.|o4 = true │ │ │ │ +00020750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020790: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000207a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00020780: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00020790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000207a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000207b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207e0: 2d2b 0a7c 6935 203a 2069 734d 756c 7469 -+.|i5 : isMulti │ │ │ │ -000207f0: 486f 6d6f 6765 6e65 6f75 7320 6964 6561 Homogeneous idea │ │ │ │ -00020800: 6c28 785f 302a 785f 332d 785f 312a 785f l(x_0*x_3-x_1*x_ │ │ │ │ -00020810: 302a 785f 342c 785f 365e 3329 2020 2020 0*x_4,x_6^3) │ │ │ │ -00020820: 2020 207c 0a7c 496e 7075 7420 7465 726d |.|Input term │ │ │ │ -00020830: 2062 656c 6f77 2069 7320 6e6f 7420 686f below is not ho │ │ │ │ -00020840: 6d6f 6765 6e65 6f75 7320 7769 7468 2072 mogeneous with r │ │ │ │ -00020850: 6573 7065 6374 2074 6f20 7468 6520 6772 espect to the gr │ │ │ │ -00020860: 6164 696e 677c 0a7c 2d20 7820 7820 7820 ading|.|- x x x │ │ │ │ -00020870: 202b 2078 2078 2020 2020 2020 2020 2020 + x x │ │ │ │ -00020880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000208a0: 2020 2020 2020 207c 0a7c 2020 2030 2031 |.| 0 1 │ │ │ │ -000208b0: 2034 2020 2020 3020 3320 2020 2020 2020 4 0 3 │ │ │ │ -000208c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000207c0: 2d2d 2d2d 2b0a 7c69 3520 3a20 6973 4d75 ----+.|i5 : isMu │ │ │ │ +000207d0: 6c74 6948 6f6d 6f67 656e 656f 7573 2069 ltiHomogeneous i │ │ │ │ +000207e0: 6465 616c 2878 5f30 2a78 5f33 2d78 5f31 deal(x_0*x_3-x_1 │ │ │ │ +000207f0: 2a78 5f30 2a78 5f34 2c78 5f36 5e33 2920 *x_0*x_4,x_6^3) │ │ │ │ +00020800: 2020 2020 2020 7c0a 7c49 6e70 7574 2074 |.|Input t │ │ │ │ +00020810: 6572 6d20 6265 6c6f 7720 6973 206e 6f74 erm below is not │ │ │ │ +00020820: 2068 6f6d 6f67 656e 656f 7573 2077 6974 homogeneous wit │ │ │ │ +00020830: 6820 7265 7370 6563 7420 746f 2074 6865 h respect to the │ │ │ │ +00020840: 2067 7261 6469 6e67 7c0a 7c2d 2078 2078 grading|.|- x x │ │ │ │ +00020850: 2078 2020 2b20 7820 7820 2020 2020 2020 x + x x │ │ │ │ +00020860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020880: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00020890: 3020 3120 3420 2020 2030 2033 2020 2020 0 1 4 0 3 │ │ │ │ +000208a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000208b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000208c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000208d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000208e0: 2020 2020 2020 2020 207c 0a7c 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 2020 │ │ │ │ -00020910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020920: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -00020930: 203d 2066 616c 7365 2020 2020 2020 2020 = false │ │ │ │ +00020900: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020910: 7c6f 3520 3d20 6661 6c73 6520 2020 2020 |o5 = false │ │ │ │ +00020920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020960: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +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 ---------------- │ │ │ │ -00020990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000209b0: 0a0a 4e6f 7465 2074 6861 7420 666f 7220 ..Note that for │ │ │ │ -000209c0: 616e 2069 6465 616c 2074 6f20 6265 206d an ideal to be m │ │ │ │ -000209d0: 756c 7469 2d68 6f6d 6f67 656e 656f 7573 ulti-homogeneous │ │ │ │ -000209e0: 2074 6865 2064 6567 7265 6520 7665 6374 the degree vect │ │ │ │ -000209f0: 6f72 206f 6620 616c 6c0a 6d6f 6e6f 6d69 or of all.monomi │ │ │ │ -00020a00: 616c 7320 696e 2061 2067 6976 656e 2067 als in a given g │ │ │ │ -00020a10: 656e 6572 6174 6f72 206d 7573 7420 6265 enerator must be │ │ │ │ -00020a20: 2074 6865 2073 616d 652e 0a0a 5761 7973 the same...Ways │ │ │ │ -00020a30: 2074 6f20 7573 6520 6973 4d75 6c74 6948 to use isMultiH │ │ │ │ -00020a40: 6f6d 6f67 656e 656f 7573 3a0a 3d3d 3d3d omogeneous:.==== │ │ │ │ -00020a50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00020a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00020a70: 2022 6973 4d75 6c74 6948 6f6d 6f67 656e "isMultiHomogen │ │ │ │ -00020a80: 656f 7573 2849 6465 616c 2922 0a20 202a eous(Ideal)". * │ │ │ │ -00020a90: 2022 6973 4d75 6c74 6948 6f6d 6f67 656e "isMultiHomogen │ │ │ │ -00020aa0: 656f 7573 2852 696e 6745 6c65 6d65 6e74 eous(RingElement │ │ │ │ -00020ab0: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ -00020ac0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -00020ad0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -00020ae0: 626a 6563 7420 2a6e 6f74 6520 6973 4d75 bject *note isMu │ │ │ │ -00020af0: 6c74 6948 6f6d 6f67 656e 656f 7573 3a20 ltiHomogeneous: │ │ │ │ -00020b00: 6973 4d75 6c74 6948 6f6d 6f67 656e 656f isMultiHomogeneo │ │ │ │ -00020b10: 7573 2c20 6973 2061 202a 6e6f 7465 206d us, is a *note m │ │ │ │ -00020b20: 6574 686f 640a 6675 6e63 7469 6f6e 3a20 ethod.function: │ │ │ │ -00020b30: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00020b40: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +00020990: 2d2d 2b0a 0a4e 6f74 6520 7468 6174 2066 --+..Note that f │ │ │ │ +000209a0: 6f72 2061 6e20 6964 6561 6c20 746f 2062 or an ideal to b │ │ │ │ +000209b0: 6520 6d75 6c74 692d 686f 6d6f 6765 6e65 e multi-homogene │ │ │ │ +000209c0: 6f75 7320 7468 6520 6465 6772 6565 2076 ous the degree v │ │ │ │ +000209d0: 6563 746f 7220 6f66 2061 6c6c 0a6d 6f6e ector of all.mon │ │ │ │ +000209e0: 6f6d 6961 6c73 2069 6e20 6120 6769 7665 omials in a give │ │ │ │ +000209f0: 6e20 6765 6e65 7261 746f 7220 6d75 7374 n generator must │ │ │ │ +00020a00: 2062 6520 7468 6520 7361 6d65 2e0a 0a57 be the same...W │ │ │ │ +00020a10: 6179 7320 746f 2075 7365 2069 734d 756c ays to use isMul │ │ │ │ +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 │ │ │ │ +00020a60: 6765 6e65 6f75 7328 4964 6561 6c29 220a geneous(Ideal)". │ │ │ │ +00020a70: 2020 2a20 2269 734d 756c 7469 486f 6d6f * "isMultiHomo │ │ │ │ +00020a80: 6765 6e65 6f75 7328 5269 6e67 456c 656d geneous(RingElem │ │ │ │ +00020a90: 656e 7429 220a 0a46 6f72 2074 6865 2070 ent)"..For the p │ │ │ │ +00020aa0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00020ab0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00020ac0: 6520 6f62 6a65 6374 202a 6e6f 7465 2069 e object *note i │ │ │ │ +00020ad0: 734d 756c 7469 486f 6d6f 6765 6e65 6f75 sMultiHomogeneou │ │ │ │ +00020ae0: 733a 2069 734d 756c 7469 486f 6d6f 6765 s: isMultiHomoge │ │ │ │ +00020af0: 6e65 6f75 732c 2069 7320 6120 2a6e 6f74 neous, is a *not │ │ │ │ +00020b00: 6520 6d65 7468 6f64 0a66 756e 6374 696f e method.functio │ │ │ │ +00020b10: 6e3a 2028 4d61 6361 756c 6179 3244 6f63 n: (Macaulay2Doc │ │ │ │ +00020b20: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +00020b30: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +00020b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00020ba0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00020bb0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00020bc0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00020bd0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -00020be0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -00020bf0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -00020c00: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ -00020c10: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ -00020c20: 3a32 3031 323a 302e 0a1f 0a46 696c 653a :2012:0....File: │ │ │ │ -00020c30: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ -00020c40: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ -00020c50: 653a 204d 6574 686f 642c 204e 6578 743a e: Method, Next: │ │ │ │ -00020c60: 204d 756c 7469 5072 6f6a 436f 6f72 6452 MultiProjCoordR │ │ │ │ -00020c70: 696e 672c 2050 7265 763a 2069 734d 756c ing, Prev: isMul │ │ │ │ -00020c80: 7469 486f 6d6f 6765 6e65 6f75 732c 2055 tiHomogeneous, U │ │ │ │ -00020c90: 703a 2054 6f70 0a0a 4d65 7468 6f64 0a2a p: Top..Method.* │ │ │ │ -00020ca0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ -00020cb0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00020cc0: 5468 6520 6f70 7469 6f6e 204d 6574 686f The option Metho │ │ │ │ -00020cd0: 6420 6973 206f 6e6c 7920 7573 6564 2062 d is only used b │ │ │ │ -00020ce0: 7920 7468 6520 636f 6d6d 616e 6473 202a y the commands * │ │ │ │ -00020cf0: 6e6f 7465 2043 534d 3a20 4353 4d2c 2061 note CSM: CSM, a │ │ │ │ -00020d00: 6e64 202a 6e6f 7465 2045 756c 6572 3a0a nd *note Euler:. │ │ │ │ -00020d10: 4575 6c65 722c 2061 6e64 206f 6e6c 7920 Euler, and only │ │ │ │ -00020d20: 696e 2063 6f6d 6269 6e61 7469 6f6e 2077 in combination w │ │ │ │ -00020d30: 6974 6820 2a6e 6f74 6520 436f 6d70 4d65 ith *note CompMe │ │ │ │ -00020d40: 7468 6f64 3a0a 436f 6d70 4d65 7468 6f64 thod:.CompMethod │ │ │ │ -00020d50: 2c3d 3e50 726f 6a65 6374 6976 6544 6567 ,=>ProjectiveDeg │ │ │ │ -00020d60: 7265 652e 2054 6865 204d 6574 686f 6420 ree. The Method │ │ │ │ -00020d70: 496e 636c 7573 696f 6e45 7863 6c75 7369 InclusionExclusi │ │ │ │ -00020d80: 6f6e 2077 696c 6c20 616c 7761 7973 2062 on will always b │ │ │ │ -00020d90: 650a 7573 6564 2077 6974 6820 2a6e 6f74 e.used with *not │ │ │ │ -00020da0: 6520 436f 6d70 4d65 7468 6f64 3a20 436f e CompMethod: Co │ │ │ │ -00020db0: 6d70 4d65 7468 6f64 2c20 506e 5265 7369 mpMethod, PnResi │ │ │ │ -00020dc0: 6475 616c 206f 7220 6265 7274 696e 692e dual or bertini. │ │ │ │ -00020dd0: 2057 6865 6e20 7468 6520 696e 7075 740a When the input. │ │ │ │ -00020de0: 6964 6561 6c20 6973 2061 2063 6f6d 706c ideal is a compl │ │ │ │ -00020df0: 6574 6520 696e 7465 7273 6563 7469 6f6e ete intersection │ │ │ │ -00020e00: 206f 6e65 206d 6179 2c20 706f 7465 6e74 one may, potent │ │ │ │ -00020e10: 6961 6c6c 792c 2073 7065 6564 2075 7020 ially, speed up │ │ │ │ -00020e20: 7468 6520 636f 6d70 7574 6174 696f 6e0a the computation. │ │ │ │ -00020e30: 6279 2073 6574 7469 6e67 204d 6574 686f by setting Metho │ │ │ │ -00020e40: 643d 3e20 4469 7265 6374 436f 6d70 6c65 d=> DirectComple │ │ │ │ -00020e50: 7465 496e 742e 2054 6865 206f 7074 696f teInt. The optio │ │ │ │ -00020e60: 6e20 4d65 7468 6f64 2069 7320 6f6e 6c79 n Method is only │ │ │ │ -00020e70: 2075 7365 6420 6279 2074 6865 0a63 6f6d used by the.com │ │ │ │ -00020e80: 6d61 6e64 7320 2a6e 6f74 6520 4353 4d3a mands *note CSM: │ │ │ │ -00020e90: 2043 534d 2c20 616e 6420 2a6e 6f74 6520 CSM, and *note │ │ │ │ -00020ea0: 4575 6c65 723a 2045 756c 6572 2c20 616e Euler: Euler, an │ │ │ │ -00020eb0: 6420 6f6e 6c79 2069 6e20 636f 6d62 696e d only in combin │ │ │ │ -00020ec0: 6174 696f 6e20 7769 7468 0a2a 6e6f 7465 ation with.*note │ │ │ │ -00020ed0: 2043 6f6d 704d 6574 686f 643a 2043 6f6d CompMethod: Com │ │ │ │ -00020ee0: 704d 6574 686f 642c 3d3e 5072 6f6a 6563 pMethod,=>Projec │ │ │ │ -00020ef0: 7469 7665 4465 6772 6565 2e20 5468 6520 tiveDegree. The │ │ │ │ -00020f00: 4d65 7468 6f64 2049 6e63 6c75 7369 6f6e Method Inclusion │ │ │ │ -00020f10: 4578 636c 7573 696f 6e0a 7769 6c6c 2061 Exclusion.will a │ │ │ │ -00020f20: 6c77 6179 7320 6265 2075 7365 6420 7769 lways be used wi │ │ │ │ -00020f30: 7468 202a 6e6f 7465 2043 6f6d 704d 6574 th *note CompMet │ │ │ │ -00020f40: 686f 643a 2043 6f6d 704d 6574 686f 642c hod: CompMethod, │ │ │ │ -00020f50: 2050 6e52 6573 6964 7561 6c20 6f72 2062 PnResidual or b │ │ │ │ -00020f60: 6572 7469 6e69 2e0a 0a2b 2d2d 2d2d 2d2d ertini...+------ │ │ │ │ +00020b80: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00020b90: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00020ba0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00020bb0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00020bc0: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +00020bd0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +00020be0: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020fa0: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 5a5a --+.|i1 : R = ZZ │ │ │ │ -00020fb0: 2f33 3237 3439 5b78 5f30 2e2e 785f 365d /32749[x_0..x_6] │ │ │ │ -00020fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 │ │ │ │ -00021000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021010: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -00021020: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -00021030: 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 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00021040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021050: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 2020 7c0a |. │ │ │ │ -00021090: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c |o1 : Polynomial │ │ │ │ -000210a0: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ -000210b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000210c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00021070: 207c 0a7c 6f31 203a 2050 6f6c 796e 6f6d |.|o1 : Polynom │ │ │ │ +00021080: 6961 6c52 696e 6720 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│ │ │ │ -00021900: 6561 6c0a 6465 6669 6e65 2061 2073 6d6f eal.define a smo │ │ │ │ -00021910: 6f74 6820 7375 6273 6368 656d 6520 2869 oth subscheme (i │ │ │ │ -00021920: 6620 7468 6973 2069 7320 6b6e 6f77 6e29 f this is known) │ │ │ │ -00021930: 2c20 7365 6520 2a6e 6f74 6520 496e 6473 , see *note Inds │ │ │ │ -00021940: 4f66 536d 6f6f 7468 3a0a 496e 6473 4f66 OfSmooth:.IndsOf │ │ │ │ -00021950: 536d 6f6f 7468 2c2e 0a0a 4675 6e63 7469 Smooth,...Functi │ │ │ │ -00021960: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -00021970: 6c20 6172 6775 6d65 6e74 206e 616d 6564 l argument named │ │ │ │ -00021980: 204d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d Method:.======= │ │ │ │ -00021990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000219a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000219b0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4353 =======.. * "CS │ │ │ │ -000219c0: 4d28 2e2e 2e2c 4d65 7468 6f64 3d3e 2e2e M(...,Method=>.. │ │ │ │ -000219d0: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465 .)" -- see *note │ │ │ │ -000219e0: 2043 534d 3a20 4353 4d2c 202d 2d20 5468 CSM: CSM, -- Th │ │ │ │ -000219f0: 650a 2020 2020 4368 6572 6e2d 5363 6877 e. Chern-Schw │ │ │ │ -00021a00: 6172 747a 2d4d 6163 5068 6572 736f 6e20 artz-MacPherson │ │ │ │ -00021a10: 636c 6173 730a 2020 2a20 4575 6c65 7228 class. * Euler( │ │ │ │ -00021a20: 2e2e 2e2c 4d65 7468 6f64 3d3e 2e2e 2e29 ...,Method=>...) │ │ │ │ -00021a30: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ -00021a40: 6e74 6174 696f 6e29 0a0a 466f 7220 7468 ntation)..For th │ │ │ │ -00021a50: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00021a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00021a70: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00021a80: 6520 4d65 7468 6f64 3a20 4d65 7468 6f64 e Method: Method │ │ │ │ -00021a90: 2c20 6973 2061 202a 6e6f 7465 2073 796d , is a *note sym │ │ │ │ -00021aa0: 626f 6c3a 2028 4d61 6361 756c 6179 3244 bol: (Macaulay2D │ │ │ │ -00021ab0: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d oc)Symbol,...--- │ │ │ │ +00021840: 2d2d 2d2d 2d2d 2d2b 0a0a 5768 656e 2075 -------+..When u │ │ │ │ +00021850: 7369 6e67 2074 6865 2044 6972 6563 7443 sing the DirectC │ │ │ │ +00021860: 6f6d 706c 6574 6549 6e74 206d 6574 686f ompleteInt metho │ │ │ │ +00021870: 6420 6f6e 6520 6d61 7920 706f 7465 6e74 d one may potent │ │ │ │ +00021880: 6961 6c6c 7920 6675 7274 6865 7220 7370 ially further sp │ │ │ │ +00021890: 6565 6420 7570 0a63 6f6d 7075 7461 7469 eed up.computati │ │ │ │ +000218a0: 6f6e 2074 696d 6520 6279 2073 7065 6369 on time by speci │ │ │ │ +000218b0: 6679 696e 6720 7768 6174 2073 7562 7365 fying what subse │ │ │ │ +000218c0: 7420 6f66 2074 6865 2067 656e 6572 6174 t of the generat │ │ │ │ +000218d0: 6f72 7320 6f66 2074 6865 2069 6e70 7574 ors of the input │ │ │ │ +000218e0: 2069 6465 616c 0a64 6566 696e 6520 6120 ideal.define a │ │ │ │ +000218f0: 736d 6f6f 7468 2073 7562 7363 6865 6d65 smooth subscheme │ │ │ │ +00021900: 2028 6966 2074 6869 7320 6973 206b 6e6f (if this is kno │ │ │ │ +00021910: 776e 292c 2073 6565 202a 6e6f 7465 2049 wn), see *note I │ │ │ │ +00021920: 6e64 734f 6653 6d6f 6f74 683a 0a49 6e64 ndsOfSmooth:.Ind │ │ │ │ +00021930: 734f 6653 6d6f 6f74 682c 2e0a 0a46 756e sOfSmooth,...Fun │ │ │ │ +00021940: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ +00021950: 6f6e 616c 2061 7267 756d 656e 7420 6e61 onal argument na │ │ │ │ +00021960: 6d65 6420 4d65 7468 6f64 3a0a 3d3d 3d3d med Method:.==== │ │ │ │ +00021970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00021980: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00021990: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +000219a0: 2243 534d 282e 2e2e 2c4d 6574 686f 643d "CSM(...,Method= │ │ │ │ +000219b0: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ +000219c0: 6f74 6520 4353 4d3a 2043 534d 2c20 2d2d ote CSM: CSM, -- │ │ │ │ +000219d0: 2054 6865 0a20 2020 2043 6865 726e 2d53 The. Chern-S │ │ │ │ +000219e0: 6368 7761 7274 7a2d 4d61 6350 6865 7273 chwartz-MacPhers │ │ │ │ +000219f0: 6f6e 2063 6c61 7373 0a20 202a 2045 756c on class. * Eul │ │ │ │ +00021a00: 6572 282e 2e2e 2c4d 6574 686f 643d 3e2e er(...,Method=>. │ │ │ │ +00021a10: 2e2e 2920 286d 6973 7369 6e67 2064 6f63 ..) (missing doc │ │ │ │ +00021a20: 756d 656e 7461 7469 6f6e 290a 0a46 6f72 umentation)..For │ │ │ │ +00021a30: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00021a40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00021a50: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00021a60: 6e6f 7465 204d 6574 686f 643a 204d 6574 note Method: Met │ │ │ │ +00021a70: 686f 642c 2069 7320 6120 2a6e 6f74 6520 hod, is a *note │ │ │ │ +00021a80: 7379 6d62 6f6c 3a20 284d 6163 6175 6c61 symbol: (Macaula │ │ │ │ +00021a90: 7932 446f 6329 5379 6d62 6f6c 2c2e 0a0a y2Doc)Symbol,... │ │ │ │ +00021aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00021b10: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00021b20: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00021b30: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00021b40: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00021b50: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00021b60: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00021b70: 6765 732f 0a43 6861 7261 6374 6572 6973 ges/.Characteris │ │ │ │ -00021b80: 7469 6343 6c61 7373 6573 2e6d 323a 3234 ticClasses.m2:24 │ │ │ │ -00021b90: 3332 3a30 2e0a 1f0a 4669 6c65 3a20 4368 32:0....File: Ch │ │ │ │ -00021ba0: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ -00021bb0: 7365 732e 696e 666f 2c20 4e6f 6465 3a20 ses.info, Node: │ │ │ │ -00021bc0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ -00021bd0: 6e67 2c20 4e65 7874 3a20 4f75 7470 7574 ng, Next: Output │ │ │ │ -00021be0: 2c20 5072 6576 3a20 4d65 7468 6f64 2c20 , Prev: Method, │ │ │ │ -00021bf0: 5570 3a20 546f 700a 0a4d 756c 7469 5072 Up: Top..MultiPr │ │ │ │ -00021c00: 6f6a 436f 6f72 6452 696e 6720 2d2d 2041 ojCoordRing -- A │ │ │ │ -00021c10: 2071 7569 636b 2077 6179 2074 6f20 6275 quick way to bu │ │ │ │ -00021c20: 696c 6420 7468 6520 636f 6f72 6469 6e61 ild the coordina │ │ │ │ -00021c30: 7465 2072 696e 6720 6f66 2061 2070 726f te ring of a pro │ │ │ │ -00021c40: 6475 6374 206f 6620 7072 6f6a 6563 7469 duct of projecti │ │ │ │ -00021c50: 7665 2073 7061 6365 730a 2a2a 2a2a 2a2a ve spaces.****** │ │ │ │ +00021ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00021af0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00021b00: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00021b10: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00021b20: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00021b30: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +00021b40: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00021b50: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ +00021b60: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ +00021b70: 3a32 3433 323a 302e 0a1f 0a46 696c 653a :2432:0....File: │ │ │ │ +00021b80: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ +00021b90: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ +00021ba0: 653a 204d 756c 7469 5072 6f6a 436f 6f72 e: MultiProjCoor │ │ │ │ +00021bb0: 6452 696e 672c 204e 6578 743a 204f 7574 dRing, Next: Out │ │ │ │ +00021bc0: 7075 742c 2050 7265 763a 204d 6574 686f put, Prev: Metho │ │ │ │ +00021bd0: 642c 2055 703a 2054 6f70 0a0a 4d75 6c74 d, Up: Top..Mult │ │ │ │ +00021be0: 6950 726f 6a43 6f6f 7264 5269 6e67 202d iProjCoordRing - │ │ │ │ +00021bf0: 2d20 4120 7175 6963 6b20 7761 7920 746f - A quick way to │ │ │ │ +00021c00: 2062 7569 6c64 2074 6865 2063 6f6f 7264 build the coord │ │ │ │ +00021c10: 696e 6174 6520 7269 6e67 206f 6620 6120 inate ring of a │ │ │ │ +00021c20: 7072 6f64 7563 7420 6f66 2070 726f 6a65 product of proje │ │ │ │ +00021c30: 6374 6976 6520 7370 6163 6573 0a2a 2a2a ctive spaces.*** │ │ │ │ +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 2a2a 2a2a **************** │ │ │ │ -00021ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -00021cc0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -00021cd0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ -00021ce0: 6e67 2044 696d 730a 2020 2020 2020 2020 ng Dims. │ │ │ │ -00021cf0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ -00021d00: 6e67 2028 436f 6566 6652 696e 672c 4469 ng (CoeffRing,Di │ │ │ │ -00021d10: 6d73 290a 2020 2020 2020 2020 4d75 6c74 ms). Mult │ │ │ │ -00021d20: 6950 726f 6a43 6f6f 7264 5269 6e67 2028 iProjCoordRing ( │ │ │ │ -00021d30: 7661 722c 4469 6d73 290a 2020 2020 2020 var,Dims). │ │ │ │ -00021d40: 2020 4d75 6c74 6950 726f 6a43 6f6f 7264 MultiProjCoord │ │ │ │ -00021d50: 5269 6e67 2028 436f 6566 6652 696e 672c Ring (CoeffRing, │ │ │ │ -00021d60: 7661 722c 4469 6d73 290a 2020 2a20 496e var,Dims). * In │ │ │ │ -00021d70: 7075 7473 3a0a 2020 2020 2020 2a20 4469 puts:. * Di │ │ │ │ -00021d80: 6d73 2c20 6120 2a6e 6f74 6520 6c69 7374 ms, a *note list │ │ │ │ -00021d90: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00021da0: 4c69 7374 2c2c 2072 6570 7265 7365 6e74 List,, represent │ │ │ │ -00021db0: 696e 6720 7468 6520 6469 6d65 6e73 696f ing the dimensio │ │ │ │ -00021dc0: 6e73 206f 660a 2020 2020 2020 2020 7468 ns of. th │ │ │ │ -00021dd0: 6520 7072 6f6a 6563 7469 7665 2073 7061 e projective spa │ │ │ │ -00021de0: 6365 732c 2069 2e65 2e20 7b6e 5f31 2c2e ces, i.e. {n_1,. │ │ │ │ -00021df0: 2e2e 2c6e 5f6d 7d20 636f 7272 6573 706f ..,n_m} correspo │ │ │ │ -00021e00: 6e64 7320 746f 205c 5050 5e7b 6e5f 317d nds to \PP^{n_1} │ │ │ │ -00021e10: 0a20 2020 2020 2020 2078 2e2e 2e2e 2078 . x.... x │ │ │ │ -00021e20: 205c 5050 5e7b 6e5f 6d7d 0a20 2020 2020 \PP^{n_m}. │ │ │ │ -00021e30: 202a 2043 6f65 6666 5269 6e67 2c20 6120 * CoeffRing, a │ │ │ │ -00021e40: 2a6e 6f74 6520 7269 6e67 3a20 284d 6163 *note ring: (Mac │ │ │ │ -00021e50: 6175 6c61 7932 446f 6329 5269 6e67 2c2c aulay2Doc)Ring,, │ │ │ │ -00021e60: 2074 6865 2063 6f65 6666 6963 6965 6e74 the coefficient │ │ │ │ -00021e70: 2072 696e 6720 6f66 0a20 2020 2020 2020 ring of. │ │ │ │ -00021e80: 2074 6865 2067 7261 6465 6420 706f 6c79 the graded poly │ │ │ │ -00021e90: 6e6f 6d69 616c 2072 696e 6720 746f 2062 nomial ring to b │ │ │ │ -00021ea0: 6520 6275 696c 7420 6279 2074 6865 206d e built by the m │ │ │ │ -00021eb0: 6574 686f 642c 2062 7920 6465 6661 756c ethod, by defaul │ │ │ │ -00021ec0: 7420 7468 6973 0a20 2020 2020 2020 2069 t this. i │ │ │ │ -00021ed0: 7320 5c5a 5a2f 3332 3734 390a 2020 2020 s \ZZ/32749. │ │ │ │ -00021ee0: 2020 2a20 7661 722c 2061 202a 6e6f 7465 * var, a *note │ │ │ │ -00021ef0: 2073 796d 626f 6c3a 2028 4d61 6361 756c symbol: (Macaul │ │ │ │ -00021f00: 6179 3244 6f63 2953 796d 626f 6c2c 2c20 ay2Doc)Symbol,, │ │ │ │ -00021f10: 746f 2062 6520 7573 6564 2066 6f72 2074 to be used for t │ │ │ │ -00021f20: 6865 0a20 2020 2020 2020 2069 6e74 6572 he. inter │ │ │ │ -00021f30: 6d65 6469 6174 6573 206f 6620 7468 6520 mediates of the │ │ │ │ -00021f40: 6772 6164 6564 2070 6f6c 796e 6f6d 6961 graded polynomia │ │ │ │ -00021f50: 6c20 7269 6e67 2074 6f20 6265 2062 7569 l ring to be bui │ │ │ │ -00021f60: 6c74 2062 7920 7468 6520 6d65 7468 6f64 lt by the method │ │ │ │ -00021f70: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -00021f80: 2020 2020 2a20 6120 2a6e 6f74 6520 7269 * a *note ri │ │ │ │ -00021f90: 6e67 3a20 284d 6163 6175 6c61 7932 446f ng: (Macaulay2Do │ │ │ │ -00021fa0: 6329 5269 6e67 2c2c 2074 6865 2067 7261 c)Ring,, the gra │ │ │ │ -00021fb0: 6465 6420 636f 6f72 6469 6e61 7465 2072 ded coordinate r │ │ │ │ -00021fc0: 696e 6720 6f66 2074 6865 0a20 2020 2020 ing of the. │ │ │ │ -00021fd0: 2020 205c 5050 5e7b 6e5f 317d 2078 2e2e \PP^{n_1} x.. │ │ │ │ -00021fe0: 2e2e 2078 205c 5050 5e7b 6e5f 6d7d 2077 .. x \PP^{n_m} w │ │ │ │ -00021ff0: 6865 7265 207b 6e5f 312c 2e2e 2e2c 6e5f here {n_1,...,n_ │ │ │ │ -00022000: 6d7d 2069 7320 7468 6520 696e 7075 7420 m} is the input │ │ │ │ -00022010: 6c69 7374 206f 660a 2020 2020 2020 2020 list of. │ │ │ │ -00022020: 6469 6d65 6e73 696f 6e73 0a0a 4465 7363 dimensions..Desc │ │ │ │ -00022030: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -00022040: 3d3d 3d0a 0a43 6f6d 7075 7465 7320 7468 ===..Computes th │ │ │ │ -00022050: 6520 6772 6164 6564 2063 6f6f 7264 696e e graded coordin │ │ │ │ -00022060: 6174 6520 7269 6e67 206f 6620 7468 6520 ate ring of the │ │ │ │ -00022070: 5c50 505e 7b6e 5f31 7d20 782e 2e2e 2e20 \PP^{n_1} x.... │ │ │ │ -00022080: 7820 5c50 505e 7b6e 5f6d 7d20 7768 6572 x \PP^{n_m} wher │ │ │ │ -00022090: 650a 7b6e 5f31 2c2e 2e2e 2c6e 5f6d 7d20 e.{n_1,...,n_m} │ │ │ │ -000220a0: 6973 2074 6865 2069 6e70 7574 206c 6973 is the input lis │ │ │ │ -000220b0: 7420 6f66 2064 696d 656e 7369 6f6e 732e t of dimensions. │ │ │ │ -000220c0: 2054 6869 7320 6d65 7468 6f64 2069 7320 This method is │ │ │ │ -000220d0: 7573 6564 2074 6f20 7175 6963 6b6c 790a used to quickly. │ │ │ │ -000220e0: 6275 696c 6420 7468 6520 636f 6f72 6469 build the coordi │ │ │ │ -000220f0: 6e61 7465 2072 696e 6720 6f66 2061 2070 nate ring of a p │ │ │ │ -00022100: 726f 6475 6374 206f 6620 7072 6f6a 6563 roduct of projec │ │ │ │ -00022110: 7469 7665 2073 7061 6365 7320 666f 7220 tive spaces for │ │ │ │ -00022120: 7573 6520 696e 0a63 6f6d 7075 7461 7469 use in.computati │ │ │ │ -00022130: 6f6e 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ons...+--------- │ │ │ │ +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 │ │ │ │ +00021cf0: 2c44 696d 7329 0a20 2020 2020 2020 204d ,Dims). M │ │ │ │ +00021d00: 756c 7469 5072 6f6a 436f 6f72 6452 696e ultiProjCoordRin │ │ │ │ +00021d10: 6720 2876 6172 2c44 696d 7329 0a20 2020 g (var,Dims). │ │ │ │ +00021d20: 2020 2020 204d 756c 7469 5072 6f6a 436f MultiProjCo │ │ │ │ +00021d30: 6f72 6452 696e 6720 2843 6f65 6666 5269 ordRing (CoeffRi │ │ │ │ +00021d40: 6e67 2c76 6172 2c44 696d 7329 0a20 202a ng,var,Dims). * │ │ │ │ +00021d50: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00021d60: 2044 696d 732c 2061 202a 6e6f 7465 206c Dims, a *note l │ │ │ │ +00021d70: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +00021d80: 6f63 294c 6973 742c 2c20 7265 7072 6573 oc)List,, repres │ │ │ │ +00021d90: 656e 7469 6e67 2074 6865 2064 696d 656e enting the dimen │ │ │ │ +00021da0: 7369 6f6e 7320 6f66 0a20 2020 2020 2020 sions of. │ │ │ │ +00021db0: 2074 6865 2070 726f 6a65 6374 6976 6520 the projective │ │ │ │ +00021dc0: 7370 6163 6573 2c20 692e 652e 207b 6e5f spaces, i.e. {n_ │ │ │ │ +00021dd0: 312c 2e2e 2e2c 6e5f 6d7d 2063 6f72 7265 1,...,n_m} corre │ │ │ │ +00021de0: 7370 6f6e 6473 2074 6f20 5c50 505e 7b6e sponds to \PP^{n │ │ │ │ +00021df0: 5f31 7d0a 2020 2020 2020 2020 782e 2e2e _1}. x... │ │ │ │ +00021e00: 2e20 7820 5c50 505e 7b6e 5f6d 7d0a 2020 . x \PP^{n_m}. │ │ │ │ +00021e10: 2020 2020 2a20 436f 6566 6652 696e 672c * CoeffRing, │ │ │ │ +00021e20: 2061 202a 6e6f 7465 2072 696e 673a 2028 a *note ring: ( │ │ │ │ +00021e30: 4d61 6361 756c 6179 3244 6f63 2952 696e Macaulay2Doc)Rin │ │ │ │ +00021e40: 672c 2c20 7468 6520 636f 6566 6669 6369 g,, the coeffici │ │ │ │ +00021e50: 656e 7420 7269 6e67 206f 660a 2020 2020 ent ring of. │ │ │ │ +00021e60: 2020 2020 7468 6520 6772 6164 6564 2070 the graded p │ │ │ │ +00021e70: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2074 olynomial ring t │ │ │ │ +00021e80: 6f20 6265 2062 7569 6c74 2062 7920 7468 o be built by th │ │ │ │ +00021e90: 6520 6d65 7468 6f64 2c20 6279 2064 6566 e method, by def │ │ │ │ +00021ea0: 6175 6c74 2074 6869 730a 2020 2020 2020 ault this. │ │ │ │ +00021eb0: 2020 6973 205c 5a5a 2f33 3237 3439 0a20 is \ZZ/32749. │ │ │ │ +00021ec0: 2020 2020 202a 2076 6172 2c20 6120 2a6e * var, a *n │ │ │ │ +00021ed0: 6f74 6520 7379 6d62 6f6c 3a20 284d 6163 ote symbol: (Mac │ │ │ │ +00021ee0: 6175 6c61 7932 446f 6329 5379 6d62 6f6c aulay2Doc)Symbol │ │ │ │ +00021ef0: 2c2c 2074 6f20 6265 2075 7365 6420 666f ,, to be used fo │ │ │ │ +00021f00: 7220 7468 650a 2020 2020 2020 2020 696e r the. in │ │ │ │ +00021f10: 7465 726d 6564 6961 7465 7320 6f66 2074 termediates of t │ │ │ │ +00021f20: 6865 2067 7261 6465 6420 706f 6c79 6e6f he graded polyno │ │ │ │ +00021f30: 6d69 616c 2072 696e 6720 746f 2062 6520 mial ring to be │ │ │ │ +00021f40: 6275 696c 7420 6279 2074 6865 206d 6574 built by the met │ │ │ │ +00021f50: 686f 640a 2020 2a20 4f75 7470 7574 733a hod. * Outputs: │ │ │ │ +00021f60: 0a20 2020 2020 202a 2061 202a 6e6f 7465 . * a *note │ │ │ │ +00021f70: 2072 696e 673a 2028 4d61 6361 756c 6179 ring: (Macaulay │ │ │ │ +00021f80: 3244 6f63 2952 696e 672c 2c20 7468 6520 2Doc)Ring,, the │ │ │ │ +00021f90: 6772 6164 6564 2063 6f6f 7264 696e 6174 graded coordinat │ │ │ │ +00021fa0: 6520 7269 6e67 206f 6620 7468 650a 2020 e ring of the. │ │ │ │ +00021fb0: 2020 2020 2020 5c50 505e 7b6e 5f31 7d20 \PP^{n_1} │ │ │ │ +00021fc0: 782e 2e2e 2e20 7820 5c50 505e 7b6e 5f6d x.... x \PP^{n_m │ │ │ │ +00021fd0: 7d20 7768 6572 6520 7b6e 5f31 2c2e 2e2e } where {n_1,... │ │ │ │ +00021fe0: 2c6e 5f6d 7d20 6973 2074 6865 2069 6e70 ,n_m} is the inp │ │ │ │ +00021ff0: 7574 206c 6973 7420 6f66 0a20 2020 2020 ut list of. │ │ │ │ +00022000: 2020 2064 696d 656e 7369 6f6e 730a 0a44 dimensions..D │ │ │ │ +00022010: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00022020: 3d3d 3d3d 3d3d 0a0a 436f 6d70 7574 6573 ======..Computes │ │ │ │ +00022030: 2074 6865 2067 7261 6465 6420 636f 6f72 the graded coor │ │ │ │ +00022040: 6469 6e61 7465 2072 696e 6720 6f66 2074 dinate ring of t │ │ │ │ +00022050: 6865 205c 5050 5e7b 6e5f 317d 2078 2e2e he \PP^{n_1} x.. │ │ │ │ +00022060: 2e2e 2078 205c 5050 5e7b 6e5f 6d7d 2077 .. x \PP^{n_m} w │ │ │ │ +00022070: 6865 7265 0a7b 6e5f 312c 2e2e 2e2c 6e5f here.{n_1,...,n_ │ │ │ │ +00022080: 6d7d 2069 7320 7468 6520 696e 7075 7420 m} is the input │ │ │ │ +00022090: 6c69 7374 206f 6620 6469 6d65 6e73 696f list of dimensio │ │ │ │ +000220a0: 6e73 2e20 5468 6973 206d 6574 686f 6420 ns. This method │ │ │ │ +000220b0: 6973 2075 7365 6420 746f 2071 7569 636b is used to quick │ │ │ │ +000220c0: 6c79 0a62 7569 6c64 2074 6865 2063 6f6f ly.build the coo │ │ │ │ +000220d0: 7264 696e 6174 6520 7269 6e67 206f 6620 rdinate ring of │ │ │ │ +000220e0: 6120 7072 6f64 7563 7420 6f66 2070 726f a product of pro │ │ │ │ +000220f0: 6a65 6374 6976 6520 7370 6163 6573 2066 jective spaces f │ │ │ │ +00022100: 6f72 2075 7365 2069 6e0a 636f 6d70 7574 or use in.comput │ │ │ │ +00022110: 6174 696f 6e73 2e0a 0a2b 2d2d 2d2d 2d2d ations...+------ │ │ │ │ +00022120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022180: 2d2d 2d2d 2b0a 7c69 3120 3a20 533d 4d75 ----+.|i1 : S=Mu │ │ │ │ -00022190: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ -000221a0: 2851 512c 7379 6d62 6f6c 207a 2c7b 312c (QQ,symbol z,{1, │ │ │ │ -000221b0: 332c 337d 2920 2020 2020 2020 2020 2020 3,3}) │ │ │ │ +00022160: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 -------+.|i1 : S │ │ │ │ +00022170: 3d4d 756c 7469 5072 6f6a 436f 6f72 6452 =MultiProjCoordR │ │ │ │ +00022180: 696e 6728 5151 2c73 796d 626f 6c20 7a2c ing(QQ,symbol z, │ │ │ │ +00022190: 7b31 2c33 2c33 7d29 2020 2020 2020 2020 {1,3,3}) │ │ │ │ +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 7c0a 7c20 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 2020 2020 │ │ │ │ +00022200: 2020 2020 2020 207c 0a7c 6f31 203d 2053 |.|o1 = S │ │ │ │ 00022210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022220: 2020 2020 7c0a 7c6f 3120 3d20 5320 2020 |.|o1 = S │ │ │ │ +00022220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022250: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00022260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022270: 2020 2020 7c0a 7c20 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 2020 2020 │ │ │ │ -000222b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222c0: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79 |.|o1 : Poly │ │ │ │ -000222d0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +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 2020 2020 │ │ │ │ -00022300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022310: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000222f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00022300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022360: 2d2d 2d2d 2b0a 7c69 3220 3a20 6465 6772 ----+.|i2 : degr │ │ │ │ -00022370: 6565 7320 5320 2020 2020 2020 2020 2020 ees S │ │ │ │ +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 │ │ │ │ -00022390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022390: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000223a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000223b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022400: 2020 2020 7c0a 7c6f 3220 3d20 7b7b 312c |.|o2 = {{1, │ │ │ │ -00022410: 2030 2c20 307d 2c20 7b31 2c20 302c 2030 0, 0}, {1, 0, 0 │ │ │ │ -00022420: 7d2c 207b 302c 2031 2c20 307d 2c20 7b30 }, {0, 1, 0}, {0 │ │ │ │ -00022430: 2c20 312c 2030 7d2c 207b 302c 2031 2c20 , 1, 0}, {0, 1, │ │ │ │ -00022440: 307d 2c20 7b30 2c20 312c 2030 7d2c 207b 0}, {0, 1, 0}, { │ │ │ │ -00022450: 302c 2020 7c0a 7c20 2020 2020 2d2d 2d2d 0, |.| ---- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000224a0: 2d2d 2d2d 7c0a 7c20 2020 2020 302c 2031 ----|.| 0, 1 │ │ │ │ -000224b0: 7d2c 207b 302c 2030 2c20 317d 2c20 7b30 }, {0, 0, 1}, {0 │ │ │ │ -000224c0: 2c20 302c 2031 7d2c 207b 302c 2030 2c20 , 0, 1}, {0, 0, │ │ │ │ -000224d0: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ +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 7c0a 7c20 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 2020 2020 │ │ │ │ -00022530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022540: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374 |.|o2 : List │ │ │ │ +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 2020 2020 │ │ │ │ -00022580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022590: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225e0: 2d2d 2d2d 2b0a 7c69 3320 3a20 523d 4d75 ----+.|i3 : R=Mu │ │ │ │ -000225f0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ -00022600: 207b 322c 337d 2020 2020 2020 2020 2020 {2,3} │ │ │ │ -00022610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 7c0a 7c20 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 2020 2020 │ │ │ │ +00022660: 2020 2020 2020 207c 0a7c 6f33 203d 2052 |.|o3 = R │ │ │ │ 00022670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022680: 2020 2020 7c0a 7c6f 3320 3d20 5220 2020 |.|o3 = R │ │ │ │ +00022680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000226a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000226b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000226b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000226c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000226d0: 2020 2020 7c0a 7c20 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 2020 2020 │ │ │ │ -00022710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022720: 2020 2020 7c0a 7c6f 3320 3a20 506f 6c79 |.|o3 : Poly │ │ │ │ -00022730: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +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 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2020 2020 2020 2020 2020 │ │ │ │ +00023100: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00023110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023120: 2020 2020 7c0a 7c20 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 2020 2020 │ │ │ │ -00023160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023170: 2020 2020 7c0a 7c20 2020 2020 2020 2032 |.| 2 │ │ │ │ -00023180: 2033 2020 2020 2032 2032 2020 2020 2020 3 2 2 │ │ │ │ -00023190: 2033 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ -000231a0: 2032 2020 2020 3320 2020 2032 2020 2020 2 3 2 │ │ │ │ -000231b0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -000231c0: 2020 2020 7c0a 7c6f 3820 3d20 3130 6820 |.|o8 = 10h │ │ │ │ -000231d0: 6820 202d 2036 6820 6820 202d 2034 6820 h - 6h h - 4h │ │ │ │ -000231e0: 6820 202b 2033 6820 6820 202b 2033 6820 h + 3h h + 3h │ │ │ │ -000231f0: 6820 202b 2068 2020 2d20 6820 202d 2032 h + h - h - 2 │ │ │ │ -00023200: 6820 6820 202d 2068 2020 2b20 6820 202b h h - h + h + │ │ │ │ -00023210: 2068 2020 7c0a 7c20 2020 2020 2020 2031 h |.| 1 │ │ │ │ -00023220: 2032 2020 2020 2031 2032 2020 2020 2031 2 1 2 1 │ │ │ │ -00023230: 2032 2020 2020 2031 2032 2020 2020 2031 2 1 2 1 │ │ │ │ -00023240: 2032 2020 2020 3220 2020 2031 2020 2020 2 2 1 │ │ │ │ -00023250: 2031 2032 2020 2020 3220 2020 2031 2020 1 2 2 1 │ │ │ │ -00023260: 2020 3220 7c0a 7c20 2020 2020 2020 2020 2 |.| │ │ │ │ +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 │ │ │ │ +000231a0: 2020 2020 2020 207c 0a7c 6f38 203d 2031 |.|o8 = 1 │ │ │ │ +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 │ │ │ │ 00023270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023290: 2020 2020 2020 207c 0a7c 6f38 203a 2041 |.|o8 : A │ │ │ │ 000232a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232b0: 2020 2020 7c0a 7c6f 3820 3a20 4120 2020 |.|o8 : A │ │ │ │ +000232b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023300: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000232e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000232f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023350: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075 ----+..Ways to u │ │ │ │ -00023360: 7365 204d 756c 7469 5072 6f6a 436f 6f72 se MultiProjCoor │ │ │ │ -00023370: 6452 696e 673a 0a3d 3d3d 3d3d 3d3d 3d3d dRing:.========= │ │ │ │ -00023380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00023390: 3d3d 3d3d 3d3d 0a0a 2020 2a20 224d 756c ======.. * "Mul │ │ │ │ -000233a0: 7469 5072 6f6a 436f 6f72 6452 696e 6728 tiProjCoordRing( │ │ │ │ -000233b0: 4c69 7374 2922 0a20 202a 2022 4d75 6c74 List)". * "Mult │ │ │ │ -000233c0: 6950 726f 6a43 6f6f 7264 5269 6e67 2852 iProjCoordRing(R │ │ │ │ -000233d0: 696e 672c 4c69 7374 2922 0a20 202a 2022 ing,List)". * " │ │ │ │ -000233e0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ -000233f0: 6e67 2852 696e 672c 5379 6d62 6f6c 2c4c ng(Ring,Symbol,L │ │ │ │ -00023400: 6973 7429 220a 2020 2a20 224d 756c 7469 ist)". * "Multi │ │ │ │ -00023410: 5072 6f6a 436f 6f72 6452 696e 6728 5379 ProjCoordRing(Sy │ │ │ │ -00023420: 6d62 6f6c 2c4c 6973 7429 220a 0a46 6f72 mbol,List)"..For │ │ │ │ -00023430: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -00023440: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00023450: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -00023460: 6e6f 7465 204d 756c 7469 5072 6f6a 436f note MultiProjCo │ │ │ │ -00023470: 6f72 6452 696e 673a 204d 756c 7469 5072 ordRing: MultiPr │ │ │ │ -00023480: 6f6a 436f 6f72 6452 696e 672c 2069 7320 ojCoordRing, is │ │ │ │ -00023490: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66 a *note method.f │ │ │ │ -000234a0: 756e 6374 696f 6e3a 2028 4d61 6361 756c unction: (Macaul │ │ │ │ -000234b0: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ -000234c0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +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 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023510: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00023520: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00023530: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00023540: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00023550: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00023560: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -00023570: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00023580: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ -00023590: 6c61 7373 6573 2e6d 323a 3230 3530 3a30 lasses.m2:2050:0 │ │ │ │ -000235a0: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ -000235b0: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ -000235c0: 696e 666f 2c20 4e6f 6465 3a20 4f75 7470 info, Node: Outp │ │ │ │ -000235d0: 7574 2c20 4e65 7874 3a20 7072 6f62 6162 ut, Next: probab │ │ │ │ -000235e0: 696c 6973 7469 6320 616c 676f 7269 7468 ilistic algorith │ │ │ │ -000235f0: 6d2c 2050 7265 763a 204d 756c 7469 5072 m, Prev: MultiPr │ │ │ │ -00023600: 6f6a 436f 6f72 6452 696e 672c 2055 703a ojCoordRing, Up: │ │ │ │ -00023610: 2054 6f70 0a0a 4f75 7470 7574 0a2a 2a2a Top..Output.*** │ │ │ │ -00023620: 2a2a 2a0a 0a44 6573 6372 6970 7469 6f6e ***..Description │ │ │ │ -00023630: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -00023640: 6520 6f70 7469 6f6e 204f 7574 7075 7420 e option Output │ │ │ │ -00023650: 6973 206f 6e6c 7920 7573 6564 2062 7920 is only used by │ │ │ │ -00023660: 7468 6520 636f 6d6d 616e 6473 202a 6e6f the commands *no │ │ │ │ -00023670: 7465 2043 534d 3a20 4353 4d2c 2c20 2a6e te CSM: CSM,, *n │ │ │ │ -00023680: 6f74 6520 5365 6772 653a 0a53 6567 7265 ote Segre:.Segre │ │ │ │ -00023690: 2c2c 202a 6e6f 7465 2043 6865 726e 3a20 ,, *note Chern: │ │ │ │ -000236a0: 4368 6572 6e2c 2061 6e64 202a 6e6f 7465 Chern, and *note │ │ │ │ -000236b0: 2045 756c 6572 3a20 4575 6c65 722c 2074 Euler: Euler, t │ │ │ │ -000236c0: 6f20 7370 6563 6966 7920 7468 6520 7479 o specify the ty │ │ │ │ -000236d0: 7065 206f 660a 6f75 7470 7574 2074 6f20 pe of.output to │ │ │ │ -000236e0: 6265 2072 6574 7572 6e65 6420 746f 2074 be returned to t │ │ │ │ -000236f0: 6865 2075 7365 642e 2054 6869 7320 6f70 he used. This op │ │ │ │ -00023700: 7469 6f6e 2077 696c 6c20 6265 2069 676e tion will be ign │ │ │ │ -00023710: 6f72 6564 2077 6865 6e20 7573 6564 2077 ored when used w │ │ │ │ -00023720: 6974 680a 2a6e 6f74 6520 436f 6d70 4d65 ith.*note CompMe │ │ │ │ -00023730: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64 thod: CompMethod │ │ │ │ -00023740: 2c20 506e 5265 7369 6475 616c 206f 7220 , PnResidual or │ │ │ │ -00023750: 6265 7274 696e 692e 2054 6865 206f 7074 bertini. The opt │ │ │ │ -00023760: 696f 6e20 7769 6c6c 2061 6c73 6f20 6265 ion will also be │ │ │ │ -00023770: 0a69 676e 6f72 6520 7768 656e 202a 6e6f .ignore when *no │ │ │ │ -00023780: 7465 204d 6574 686f 643a 204d 6574 686f te Method: Metho │ │ │ │ -00023790: 642c 3d3e 4469 7265 6374 436f 6d70 6c65 d,=>DirectComple │ │ │ │ -000237a0: 7465 496e 7420 6973 2075 7365 642e 2054 teInt is used. T │ │ │ │ -000237b0: 6865 2064 6566 6175 6c74 0a6f 7574 7075 he default.outpu │ │ │ │ -000237c0: 7420 666f 7220 616c 6c20 7468 6573 6520 t for all these │ │ │ │ -000237d0: 6d65 7468 6f64 7320 6973 2043 686f 7752 methods is ChowR │ │ │ │ -000237e0: 696e 6745 6c65 6c6d 656e 7420 7768 6963 ingElelment whic │ │ │ │ -000237f0: 6820 7769 6c6c 2072 6574 7572 6e20 616e h will return an │ │ │ │ -00023800: 2065 6c65 6d65 6e74 0a6f 6620 7468 6520 element.of the │ │ │ │ -00023810: 6170 7072 6f70 7269 6174 6520 4368 6f77 appropriate Chow │ │ │ │ -00023820: 2072 696e 672e 2041 6c6c 206d 6574 686f ring. All metho │ │ │ │ -00023830: 6473 2061 6c73 6f20 6861 7665 2061 6e20 ds also have an │ │ │ │ -00023840: 6f70 7469 6f6e 2048 6173 6846 6f72 6d20 option HashForm │ │ │ │ -00023850: 7768 6963 680a 7265 7475 726e 7320 6164 which.returns ad │ │ │ │ -00023860: 6469 7469 6f6e 616c 2069 6e66 6f72 6d61 ditional informa │ │ │ │ -00023870: 7469 6f6e 2063 6f6d 7075 7465 6420 6279 tion computed by │ │ │ │ -00023880: 2074 6865 206d 6574 686f 6473 2064 7572 the methods dur │ │ │ │ -00023890: 696e 6720 7468 6569 7220 7374 616e 6461 ing their standa │ │ │ │ -000238a0: 7264 0a6f 7065 7261 7469 6f6e 2e0a 0a2b rd.operation...+ │ │ │ │ +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 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+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 ---------------- │ │ │ │ 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2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023990: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000239a0: 6f31 203d 2052 2020 2020 2020 2020 2020 o1 = R │ │ │ │ +00023980: 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 2020 2020 2020 2020 2020 │ │ │ │ -000239e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -00023a30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023a40: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ -00023a50: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00023a20: 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 2020 2020 2020 2020 2020 │ │ │ │ -00023a80: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00023ae0: 6932 203a 2041 3d43 686f 7752 696e 6728 i2 : A=ChowRing( │ │ │ │ -00023af0: 5229 2020 2020 2020 2020 2020 2020 2020 R) │ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -00023b70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023b80: 6f32 203d 2041 2020 2020 2020 2020 2020 o2 = A │ │ │ │ +00023b60: 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 2020 2020 2020 2020 2020 │ │ │ │ -00023bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -00023c10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023c20: 6f32 203a 2051 756f 7469 656e 7452 696e o2 : QuotientRin │ │ │ │ -00023c30: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +00023c00: 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 2020 2020 2020 2020 2020 │ │ │ │ -00023c60: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00023cc0: 6933 203a 2049 3d69 6465 616c 2872 616e i3 : I=ideal(ran │ │ │ │ -00023cd0: 646f 6d28 322c 5229 2c52 5f30 2a52 5f31 dom(2,R),R_0*R_1 │ │ │ │ -00023ce0: 2a52 5f36 2d52 5f30 5e33 293b 2020 2020 *R_6-R_0^3); │ │ │ │ -00023cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -00023d50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023d60: 6f33 203a 2049 6465 616c 206f 6620 5220 o3 : Ideal of R │ │ │ │ +00023d40: 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 2020 2020 2020 2020 2020 │ │ │ │ -00023da0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00023e00: 6934 203a 2063 736d 3d43 534d 2841 2c49 i4 : csm=CSM(A,I │ │ │ │ -00023e10: 2c4f 7574 7075 743d 3e48 6173 6846 6f72 ,Output=>HashFor │ │ │ │ -00023e20: 6d29 2020 2020 2020 2020 2020 2020 2020 m) │ │ │ │ -00023e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -00023e90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023ea0: 6f34 203d 204d 7574 6162 6c65 4861 7368 o4 = MutableHash │ │ │ │ -00023eb0: 5461 626c 657b 2e2e 2e34 2e2e 2e7d 2020 Table{...4...} │ │ │ │ +00023e80: 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 2020 2020 2020 2020 2020 │ │ │ │ -00023ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -00023f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023f40: 6f34 203a 204d 7574 6162 6c65 4861 7368 o4 : MutableHash │ │ │ │ -00023f50: 5461 626c 6520 2020 2020 2020 2020 2020 Table │ │ │ │ +00023f20: 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 2020 2020 2020 2020 2020 │ │ │ │ -00023f80: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00023fe0: 6935 203a 2070 6565 6b20 6373 6d20 2020 i5 : peek csm │ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024020: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -00024070: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000240a0: 2020 3620 2020 2020 2035 2020 2020 2020 6 5 │ │ │ │ -000240b0: 3420 2020 2020 2033 2020 2020 2020 3220 4 3 2 │ │ │ │ -000240c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000240d0: 6f35 203d 204d 7574 6162 6c65 4861 7368 o5 = MutableHash │ │ │ │ -000240e0: 5461 626c 657b 7b30 2c20 317d 203d 3e20 Table{{0, 1} => │ │ │ │ -000240f0: 3268 2020 2b20 3233 6820 202b 2033 3268 2h + 23h + 32h │ │ │ │ -00024100: 2020 2b20 3333 6820 202b 2031 3868 2020 + 33h + 18h │ │ │ │ -00024110: 2b20 3568 207d 2020 2020 2020 207c 0a7c + 5h } |.| │ │ │ │ -00024120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024140: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ -00024150: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ -00024160: 2020 2020 3120 2020 2020 2020 207c 0a7c 1 |.| │ │ │ │ -00024170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024190: 3620 2020 2020 2035 2020 2020 2020 3420 6 5 4 │ │ │ │ -000241a0: 2020 2020 2033 2020 2020 2032 2020 2020 3 2 │ │ │ │ -000241b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000241c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000241d0: 2020 2020 2020 4353 4d20 3d3e 2031 3068 CSM => 10h │ │ │ │ -000241e0: 2020 2b20 3132 6820 202b 2032 3268 2020 + 12h + 22h │ │ │ │ -000241f0: 2b20 3136 6820 202b 2036 6820 2020 2020 + 16h + 6h │ │ │ │ -00024200: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024230: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ -00024240: 2020 2020 2031 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00024250: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024270: 2020 2020 2020 2020 2020 2020 2020 2036 6 │ │ │ │ -00024280: 2020 2020 2020 3520 2020 2020 2034 2020 5 4 │ │ │ │ -00024290: 2020 2020 3320 2020 2020 2032 2020 2020 3 2 │ │ │ │ -000242a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000242b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000242c0: 2020 2020 2020 7b30 7d20 3d3e 2036 6820 {0} => 6h │ │ │ │ -000242d0: 202b 2031 3868 2020 2b20 3236 6820 202b + 18h + 26h + │ │ │ │ -000242e0: 2032 3268 2020 2b20 3130 6820 202b 2032 22h + 10h + 2 │ │ │ │ -000242f0: 6820 2020 2020 2020 2020 2020 207c 0a7c h |.| │ │ │ │ -00024300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024310: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -00024320: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ -00024330: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00024340: 2031 2020 2020 2020 2020 2020 207c 0a7c 1 |.| │ │ │ │ -00024350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024360: 2020 2020 2020 2020 2020 2020 2020 2036 6 │ │ │ │ -00024370: 2020 2020 2020 3520 2020 2020 2034 2020 5 4 │ │ │ │ -00024380: 2020 2020 3320 2020 2020 2032 2020 2020 3 2 │ │ │ │ -00024390: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000243a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000243b0: 2020 2020 2020 7b31 7d20 3d3e 2036 6820 {1} => 6h │ │ │ │ -000243c0: 202b 2031 3768 2020 2b20 3238 6820 202b + 17h + 28h + │ │ │ │ -000243d0: 2032 3768 2020 2b20 3134 6820 202b 2033 27h + 14h + 3 │ │ │ │ -000243e0: 6820 2020 2020 2020 2020 2020 207c 0a7c h |.| │ │ │ │ -000243f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024400: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -00024410: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ -00024420: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00024430: 2031 2020 2020 2020 2020 2020 207c 0a2b 1 |.+ │ │ │ │ +00024060: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024070: 2020 2020 2020 2020 2020 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 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 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7574 7075 743d 3e48 6173 6846 6f72 Output=>HashFor │ │ │ │ -00024730: 6d29 2020 2020 2020 2020 2020 2020 2020 m) │ │ │ │ -00024740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024750: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -000247a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 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2020 │ │ │ │ -00024d60: 2020 2020 2020 2020 2020 2020 2020 2036 6 │ │ │ │ -00024d70: 2020 2020 2020 3520 2020 2020 2034 2020 5 4 │ │ │ │ -00024d80: 2020 2020 3320 2020 2020 3220 2020 2020 3 2 │ │ │ │ -00024d90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024db0: 2020 2020 2020 4346 203d 3e20 3930 6820 CF => 90h │ │ │ │ -00024dc0: 202d 2031 3268 2020 2b20 3330 6820 202b - 12h + 30h + │ │ │ │ -00024dd0: 2031 3268 2020 2b20 3668 2020 2020 2020 12h + 6h │ │ │ │ -00024de0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e00: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -00024e10: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ -00024e20: 2020 2020 3120 2020 2020 3120 2020 2020 1 1 │ │ │ │ -00024e30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e50: 2020 2020 2020 2020 2020 2020 2036 2020 6 │ │ │ │ -00024e60: 2020 2035 2020 2020 2034 2020 2020 2033 5 4 3 │ │ │ │ -00024e70: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00024e80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ea0: 2020 2020 2020 4720 3d3e 2033 6820 202b G => 3h + │ │ │ │ -00024eb0: 2033 6820 202b 2033 6820 202b 2033 6820 3h + 3h + 3h │ │ │ │ -00024ec0: 202b 2033 6820 202b 2033 6820 202b 2031 + 3h + 3h + 1 │ │ │ │ -00024ed0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ef0: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -00024f00: 2020 2031 2020 2020 2031 2020 2020 2031 1 1 1 │ │ │ │ -00024f10: 2020 2020 2031 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00024f20: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00024970: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024990: 2020 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│ 000258b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258d0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00025930: 6931 3320 3a20 7065 656b 2065 7520 2020 i13 : peek eu │ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025970: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -000259c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000259d0: 6f31 3320 3d20 4d75 7461 626c 6548 6173 o13 = MutableHas │ │ │ │ -000259e0: 6854 6162 6c65 7b45 756c 6572 203d 3e20 hTable{Euler => │ │ │ │ -000259f0: 3130 2020 2020 2020 2020 2020 2020 2020 10 │ │ │ │ -00025a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025a10: 2020 2020 2020 7d20 2020 2020 207c 0a7c } |.| │ │ │ │ -00025a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025a40: 2020 2036 2020 2020 2020 3520 2020 2020 6 5 │ │ │ │ -00025a50: 2034 2020 2020 2020 3320 2020 2020 2032 4 3 2 │ │ │ │ -00025a60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00025a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025a80: 2020 2020 2020 207b 302c 2031 7d20 3d3e {0, 1} => │ │ │ │ -00025a90: 2032 6820 202b 2032 3368 2020 2b20 3332 2h + 23h + 32 │ │ │ │ -00025aa0: 6820 202b 2033 3368 2020 2b20 3138 6820 h + 33h + 18h │ │ │ │ -00025ab0: 202b 2035 6820 2020 2020 2020 207c 0a7c + 5h |.| │ │ │ │ -00025ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025ae0: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ -00025af0: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ -00025b00: 2020 2020 2031 2020 2020 2020 207c 0a7c 1 |.| │ │ │ │ -00025b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025b30: 2036 2020 2020 2020 3520 2020 2020 2034 6 5 4 │ │ │ │ -00025b40: 2020 2020 2020 3320 2020 2020 3220 2020 3 2 │ │ │ │ -00025b50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00025b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025b70: 2020 2020 2020 2043 534d 203d 3e20 3130 CSM => 10 │ │ │ │ -00025b80: 6820 202b 2031 3268 2020 2b20 3232 6820 h + 12h + 22h │ │ │ │ -00025b90: 202b 2031 3668 2020 2b20 3668 2020 2020 + 16h + 6h │ │ │ │ -00025ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00025bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025bd0: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ -00025be0: 2020 2020 2020 3120 2020 2020 3120 2020 1 1 │ │ │ │ -00025bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00025c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c20: 3620 2020 2020 2035 2020 2020 2020 3420 6 5 4 │ │ │ │ -00025c30: 2020 2020 2033 2020 2020 2020 3220 2020 3 2 │ │ │ │ -00025c40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00025c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025c60: 2020 2020 2020 207b 307d 203d 3e20 3668 {0} => 6h │ │ │ │ -00025c70: 2020 2b20 3138 6820 202b 2032 3668 2020 + 18h + 26h │ │ │ │ -00025c80: 2b20 3232 6820 202b 2031 3068 2020 2b20 + 22h + 10h + │ │ │ │ -00025c90: 3268 2020 2020 2020 2020 2020 207c 0a7c 2h |.| │ │ │ │ -00025ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025cc0: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ -00025cd0: 2020 2020 2031 2020 2020 2020 3120 2020 1 1 │ │ │ │ -00025ce0: 2020 3120 2020 2020 2020 2020 207c 0a7c 1 |.| │ │ │ │ -00025cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d10: 3620 2020 2020 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2020 2020 2020 => 10 │ │ │ │ +000259e0: 2020 2020 2020 2020 2020 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: 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 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 => │ │ │ │ +00025b60: 2031 3068 2020 2b20 3132 6820 202b 2032 10h + 12h + 2 │ │ │ │ +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: 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 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} => │ │ │ │ +00025c50: 2036 6820 202b 2031 3868 2020 2b20 3236 6h + 18h + 26 │ │ │ │ +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: 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 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: 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 ---------------- │ │ │ │ 00025de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -00025e30: 5468 6520 4d75 7461 626c 6548 6173 6854 The MutableHashT │ │ │ │ -00025e40: 6162 6c65 2072 6574 7572 6e65 6420 7769 able returned wi │ │ │ │ -00025e50: 7468 2074 6865 206f 7074 696f 6e20 4f75 th the option Ou │ │ │ │ -00025e60: 7470 7574 3d3e 4861 7368 466f 726d 2063 tput=>HashForm c │ │ │ │ -00025e70: 6f6e 7461 696e 730a 6469 6666 6572 656e ontains.differen │ │ │ │ -00025e80: 7420 696e 666f 726d 6174 696f 6e20 6465 t information de │ │ │ │ -00025e90: 7065 6e64 696e 6720 6f6e 2074 6865 206d pending on the m │ │ │ │ -00025ea0: 6574 686f 6420 7769 7468 2077 6869 6368 ethod with which │ │ │ │ -00025eb0: 2069 7420 6973 2075 7365 642e 0a41 6464 it is used..Add │ │ │ │ -00025ec0: 6974 696f 6e61 6c6c 7920 6966 2074 6865 itionally if the │ │ │ │ -00025ed0: 206f 7074 696f 6e20 2a6e 6f74 6520 496e option *note In │ │ │ │ -00025ee0: 7075 7449 7353 6d6f 6f74 683a 2049 6e70 putIsSmooth: Inp │ │ │ │ -00025ef0: 7574 4973 536d 6f6f 7468 2c20 6973 2075 utIsSmooth, is u │ │ │ │ -00025f00: 7365 6420 7468 656e 2074 6865 0a68 6173 sed then the.has │ │ │ │ -00025f10: 6820 7461 626c 6520 7265 7475 726e 6564 h table returned │ │ │ │ -00025f20: 2062 7920 7468 6520 6d65 7468 6f64 7320 by the methods │ │ │ │ -00025f30: 4575 6c65 7220 616e 6420 4353 4d20 7769 Euler and CSM wi │ │ │ │ -00025f40: 6c6c 2062 6520 7468 6520 7361 6d65 2061 ll be the same a │ │ │ │ -00025f50: 7320 7468 6174 0a72 6574 7572 6e65 6420 s that.returned │ │ │ │ -00025f60: 6279 2043 6865 726e 2e20 5768 656e 2075 by Chern. When u │ │ │ │ -00025f70: 7369 6e67 2074 6865 202a 6e6f 7465 2043 sing the *note C │ │ │ │ -00025f80: 534d 3a20 4353 4d2c 2020 636f 6d6d 616e SM: CSM, comman │ │ │ │ -00025f90: 6420 696e 2074 6865 2064 6566 6175 6c74 d in the default │ │ │ │ -00025fa0: 0a63 6f6e 6669 6775 7261 7469 6f6e 7320 .configurations │ │ │ │ -00025fb0: 2874 6861 7420 6973 202a 6e6f 7465 204d (that is *note M │ │ │ │ -00025fc0: 6574 686f 643a 204d 6574 686f 642c 3d3e ethod: Method,=> │ │ │ │ -00025fd0: 496e 636c 7573 696f 6e45 7863 6c75 7369 InclusionExclusi │ │ │ │ -00025fe0: 6f6e 2c20 2a6e 6f74 650a 436f 6d70 4d65 on, *note.CompMe │ │ │ │ -00025ff0: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64 thod: CompMethod │ │ │ │ -00026000: 2c3d 3e50 726f 6a65 6374 6976 6544 6567 ,=>ProjectiveDeg │ │ │ │ -00026010: 7265 6529 2074 6865 7265 2069 7320 7468 ree) there is th │ │ │ │ -00026020: 6520 6164 6469 7469 6f6e 616c 206f 7074 e additional opt │ │ │ │ -00026030: 696f 6e20 746f 0a73 6574 204f 7574 7075 ion to.set Outpu │ │ │ │ -00026040: 743d 3e48 6173 6846 6f72 6d58 4c2e 2054 t=>HashFormXL. T │ │ │ │ -00026050: 6869 7320 7265 7475 726e 7320 616c 6c20 his returns all │ │ │ │ -00026060: 7468 6520 7573 7561 6c20 696e 666f 726d the usual inform │ │ │ │ -00026070: 6174 696f 6e20 7468 6174 0a4f 7574 7075 ation that.Outpu │ │ │ │ -00026080: 743d 3e48 6173 6846 6f72 6d20 776f 756c t=>HashForm woul │ │ │ │ -00026090: 6420 666f 7220 7468 6973 2063 6f6e 6669 d for this confi │ │ │ │ -000260a0: 6775 7261 7469 6f6e 2077 6974 6820 7468 guration with th │ │ │ │ -000260b0: 6520 6164 6469 7469 6f6e 206f 6620 7468 e addition of th │ │ │ │ -000260c0: 650a 7072 6f6a 6563 7469 7665 2064 6567 e.projective deg │ │ │ │ -000260d0: 7265 6573 2061 6e64 2053 6567 7265 2063 rees and Segre c │ │ │ │ -000260e0: 6c61 7373 6573 206f 6620 7369 6e67 756c lasses of singul │ │ │ │ -000260f0: 6172 6974 7920 7375 6273 6368 656d 6573 arity subschemes │ │ │ │ -00026100: 2067 656e 6572 6174 6564 2062 7920 7468 generated by th │ │ │ │ -00026110: 650a 6879 7065 7273 7572 6661 6365 7320 e.hypersurfaces │ │ │ │ -00026120: 636f 6e73 6964 6572 6564 2069 6e20 7468 considered in th │ │ │ │ -00026130: 6520 696e 636c 7573 696f 6e2f 6578 636c e inclusion/excl │ │ │ │ -00026140: 7573 696f 6e20 7072 6f63 6564 7572 652c usion procedure, │ │ │ │ -00026150: 2074 6861 7420 6973 2069 6e0a 6669 6e64 that is in.find │ │ │ │ -00026160: 696e 6720 7468 6520 4353 4d20 636c 6173 ing the CSM clas │ │ │ │ -00026170: 7320 6f66 2061 6c6c 2068 7970 6572 7375 s of all hypersu │ │ │ │ -00026180: 7266 6163 6573 2067 656e 6572 6174 6564 rfaces generated │ │ │ │ -00026190: 2062 7920 7461 6b69 6e67 2061 2070 726f by taking a pro │ │ │ │ -000261a0: 6475 6374 206f 660a 736f 6d65 2073 7562 duct of.some sub │ │ │ │ -000261b0: 7365 7473 206f 6620 6765 6e65 7261 746f sets of generato │ │ │ │ -000261c0: 7273 206f 6620 7468 6520 696e 7075 7420 rs of the input │ │ │ │ -000261d0: 6964 6561 6c2e 204e 6f74 6520 7468 6174 ideal. Note that │ │ │ │ -000261e0: 2c20 7369 6e63 6520 7468 6520 4353 4d20 , since the CSM │ │ │ │ -000261f0: 636c 6173 730a 6f66 2061 2073 7562 7363 class.of a subsc │ │ │ │ -00026200: 6865 6d65 2065 7175 616c 7320 7468 6520 heme equals the │ │ │ │ -00026210: 4353 4d20 636c 6173 7320 6f66 2069 7473 CSM class of its │ │ │ │ -00026220: 2072 6564 7563 6564 2073 6368 656d 652c reduced scheme, │ │ │ │ -00026230: 206f 7220 6571 7569 7661 6c65 6e74 6c79 or equivalently │ │ │ │ -00026240: 2066 6f72 0a75 7320 7468 6520 4353 4d20 for.us the CSM │ │ │ │ -00026250: 636c 6173 7320 636f 7272 6573 706f 6e64 class correspond │ │ │ │ -00026260: 696e 6720 746f 2061 6e20 6964 6561 6c20 ing to an ideal │ │ │ │ -00026270: 4920 6571 7561 6c73 2074 6865 2043 534d I equals the CSM │ │ │ │ -00026280: 2063 6c61 7373 206f 6620 7468 650a 7261 class of the.ra │ │ │ │ -00026290: 6469 6361 6c20 6f66 2049 2c20 7468 656e dical of I, then │ │ │ │ -000262a0: 2069 6e74 6572 6e61 6c6c 7920 7765 2061 internally we a │ │ │ │ -000262b0: 6c77 6179 7320 776f 726b 2077 6974 6820 lways work with │ │ │ │ -000262c0: 7261 6469 6361 6c20 6964 6561 6c73 2028 radical ideals ( │ │ │ │ -000262d0: 666f 720a 6566 6669 6369 656e 6379 2072 for.efficiency r │ │ │ │ -000262e0: 6561 736f 6e73 292e 2048 656e 6365 2074 easons). Hence t │ │ │ │ -000262f0: 6865 2070 726f 6a65 6374 6976 6520 6465 he projective de │ │ │ │ -00026300: 6772 6565 7320 616e 6420 5365 6772 6520 grees and Segre │ │ │ │ -00026310: 636c 6173 7365 7320 636f 6d70 7574 6564 classes computed │ │ │ │ -00026320: 0a69 6e74 6572 6e61 6c6c 7920 7769 6c6c .internally will │ │ │ │ -00026330: 2062 6520 7468 6f73 6520 6f66 2074 6865 be those of the │ │ │ │ -00026340: 2072 6164 6963 616c 206f 6620 616e 2069 radical of an i │ │ │ │ -00026350: 6465 616c 2064 6566 696e 6564 2062 7920 deal defined by │ │ │ │ -00026360: 6120 706f 6c79 6e6f 6d69 616c 0a77 6869 a polynomial.whi │ │ │ │ -00026370: 6368 2069 7320 6120 7072 6f64 7563 7420 ch is a product │ │ │ │ -00026380: 6f66 2073 6f6d 6520 7375 6273 6574 206f of some subset o │ │ │ │ -00026390: 6620 7468 6520 6765 6e65 7261 746f 7273 f the generators │ │ │ │ -000263a0: 2e20 5765 2069 6c6c 7573 7472 6174 6520 . We illustrate │ │ │ │ -000263b0: 7468 6973 2077 6974 6820 616e 0a65 7861 this with an.exa │ │ │ │ -000263c0: 6d70 6c65 2062 656c 6f77 2e0a 0a2b 2d2d mple below...+-- │ │ │ │ +00025e10: 2b0a 0a54 6865 204d 7574 6162 6c65 4861 +..The MutableHa │ │ │ │ +00025e20: 7368 5461 626c 6520 7265 7475 726e 6564 shTable returned │ │ │ │ +00025e30: 2077 6974 6820 7468 6520 6f70 7469 6f6e with the option │ │ │ │ +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 2d2d ---------------- │ │ │ │ -00026400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00026420: 3420 3a20 6373 6d58 4c68 6173 683d 4353 4 : csmXLhash=CS │ │ │ │ -00026430: 4d28 412c 492c 4f75 7470 7574 3d3e 4861 M(A,I,Output=>Ha │ │ │ │ -00026440: 7368 466f 726d 584c 2920 2020 2020 2020 shFormXL) │ │ │ │ -00026450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026460: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 │ │ │ │ -000264a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000264b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000264c0: 3420 3d20 4d75 7461 626c 6548 6173 6854 4 = MutableHashT │ │ │ │ -000264d0: 6162 6c65 7b2e 2e2e 3130 2e2e 2e7d 2020 able{...10...} │ │ │ │ -000264e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000264f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026500: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026550: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00026560: 3420 3a20 4d75 7461 626c 6548 6173 6854 4 : MutableHashT │ │ │ │ -00026570: 6162 6c65 2020 2020 2020 2020 2020 2020 able │ │ │ │ -00026580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000265a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 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 2d2d ---------------- │ │ │ │ -000265e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00026600: 3520 3a20 7065 656b 2063 736d 584c 6861 5 : peek csmXLha │ │ │ │ -00026610: 7368 2020 2020 2020 2020 2020 2020 2020 sh │ │ │ │ -00026620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026640: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026690: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000266a0: 3520 3d20 4d75 7461 626c 6548 6173 6854 5 = MutableHashT │ │ │ │ -000266b0: 6162 6c65 7b47 284a 6163 6f62 6961 6e29 able{G(Jacobian) │ │ │ │ -000266c0: 7b30 7d20 3d3e 2030 2020 2020 2020 2020 {0} => 0 │ │ │ │ -000266d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000266e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000266f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026700: 2020 2020 2053 6567 7265 284a 6163 6f62 Segre(Jacob │ │ │ │ -00026710: 6961 6e29 7b30 7d20 3d3e 2030 2020 2020 ian){0} => 0 │ │ │ │ -00026720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026730: 2020 2020 2020 2020 2020 207c 0a7c 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 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 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207c 0a2b 2d2d |.+-- │ │ │ │ +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 2d2d ---------------- │ │ │ │ -000272b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000272c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -000272d0: 3620 3a20 4b3d 6964 6561 6c20 495f 302a 6 : K=ideal I_0* │ │ │ │ -000272e0: 495f 313b 2020 2020 2020 2020 2020 2020 I_1; │ │ │ │ -000272f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027310: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 │ │ │ │ 00027320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027360: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00027370: 3620 3a20 4964 6561 6c20 6f66 2052 2020 6 : Ideal of R │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -000273b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 2d2d ---------------- │ │ │ │ -000273f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00027410: 3720 3a20 4353 4d28 412c 7261 6469 6361 7 : CSM(A,radica │ │ │ │ -00027420: 6c20 4b29 3d3d 4353 4d28 412c 4b29 2020 l K)==CSM(A,K) │ │ │ │ -00027430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027450: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274a0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000274b0: 3720 3d20 7472 7565 2020 2020 2020 2020 7 = true │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 2d2d ---------------- │ │ │ │ -00027530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00027550: 3820 3a20 4a3d 6964 6561 6c20 6a61 636f 8 : J=ideal jaco │ │ │ │ -00027560: 6269 616e 2072 6164 6963 616c 204b 3b20 bian radical K; │ │ │ │ -00027570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027590: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -000275e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000275f0: 3820 3a20 4964 6561 6c20 6f66 2052 2020 8 : Ideal of R │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027630: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 2d2d ---------------- │ │ │ │ -00027670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00027690: 3920 3a20 7365 674a 3d53 6567 7265 2841 9 : segJ=Segre(A │ │ │ │ -000276a0: 2c4a 2c4f 7574 7075 743d 3e48 6173 6846 ,J,Output=>HashF │ │ │ │ -000276b0: 6f72 6d29 2020 2020 2020 2020 2020 2020 orm) │ │ │ │ -000276c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000276d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027720: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00027730: 3920 3d20 4d75 7461 626c 6548 6173 6854 9 = MutableHashT │ │ │ │ -00027740: 6162 6c65 7b2e 2e2e 342e 2e2e 7d20 2020 able{...4...} │ │ │ │ -00027750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027770: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000277d0: 3920 3a20 4d75 7461 626c 6548 6173 6854 9 : MutableHashT │ │ │ │ -000277e0: 6162 6c65 2020 2020 2020 2020 2020 2020 able │ │ │ │ -000277f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027810: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 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 2d2d ---------------- │ │ │ │ -00027850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00027870: 3020 3a20 6373 6d58 4c68 6173 6823 2822 0 : csmXLhash#(" │ │ │ │ -00027880: 4728 4a61 636f 6269 616e 2922 7c74 6f53 G(Jacobian)"|toS │ │ │ │ -00027890: 7472 696e 6728 7b30 2c31 7d29 293d 3d73 tring({0,1}))==s │ │ │ │ -000278a0: 6567 4a23 2247 2220 2020 2020 2020 2020 egJ#"G" │ │ │ │ -000278b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 │ │ │ │ -000278f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027900: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00027910: 3020 3d20 7472 7565 2020 2020 2020 2020 0 = true │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027950: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 2d2d ---------------- │ │ │ │ -00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -000279b0: 3120 3a20 6373 6d58 4c68 6173 6823 2822 1 : csmXLhash#(" │ │ │ │ -000279c0: 5365 6772 6528 4a61 636f 6269 616e 2922 Segre(Jacobian)" │ │ │ │ -000279d0: 7c74 6f53 7472 696e 6728 7b30 2c31 7d29 |toString({0,1}) │ │ │ │ -000279e0: 293d 3d73 6567 4a23 2253 6567 7265 2220 )==segJ#"Segre" │ │ │ │ -000279f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 │ │ │ │ -00027a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a40: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00027a50: 3120 3d20 7472 7565 2020 2020 2020 2020 1 = true │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00027a70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00027a80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00027a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4675 -----------+..Fu │ │ │ │ -00027af0: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ -00027b00: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ -00027b10: 616d 6564 204f 7574 7075 743a 0a3d 3d3d amed Output:.=== │ │ │ │ -00027b20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027b30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00027b50: 2022 4368 6572 6e28 2e2e 2e2c 4f75 7470 "Chern(...,Outp │ │ │ │ -00027b60: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565 ut=>...)" -- see │ │ │ │ -00027b70: 202a 6e6f 7465 2043 6865 726e 3a20 4368 *note Chern: Ch │ │ │ │ -00027b80: 6572 6e2c 202d 2d20 5468 6520 4368 6572 ern, -- The Cher │ │ │ │ -00027b90: 6e20 636c 6173 730a 2020 2a20 2243 534d n class. * "CSM │ │ │ │ -00027ba0: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e (...,Output=>... │ │ │ │ -00027bb0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -00027bc0: 4353 4d3a 2043 534d 2c20 2d2d 2054 6865 CSM: CSM, -- The │ │ │ │ -00027bd0: 0a20 2020 2043 6865 726e 2d53 6368 7761 . Chern-Schwa │ │ │ │ -00027be0: 7274 7a2d 4d61 6350 6865 7273 6f6e 2063 rtz-MacPherson c │ │ │ │ -00027bf0: 6c61 7373 0a20 202a 2022 4575 6c65 7228 lass. * "Euler( │ │ │ │ -00027c00: 2e2e 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 ...,Output=>...) │ │ │ │ -00027c10: 2220 2d2d 2073 6565 202a 6e6f 7465 2045 " -- see *note E │ │ │ │ -00027c20: 756c 6572 3a20 4575 6c65 722c 202d 2d20 uler: Euler, -- │ │ │ │ -00027c30: 5468 6520 4575 6c65 720a 2020 2020 4368 The Euler. Ch │ │ │ │ -00027c40: 6172 6163 7465 7269 7374 6963 0a20 202a aracteristic. * │ │ │ │ -00027c50: 2022 5365 6772 6528 2e2e 2e2c 4f75 7470 "Segre(...,Outp │ │ │ │ -00027c60: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565 ut=>...)" -- see │ │ │ │ -00027c70: 202a 6e6f 7465 2053 6567 7265 3a20 5365 *note Segre: Se │ │ │ │ -00027c80: 6772 652c 202d 2d20 5468 6520 5365 6772 gre, -- The Segr │ │ │ │ -00027c90: 6520 636c 6173 7320 6f66 2061 0a20 2020 e class of a. │ │ │ │ -00027ca0: 2073 7562 7363 6865 6d65 0a0a 466f 7220 subscheme..For │ │ │ │ -00027cb0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00027cc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027cd0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00027ce0: 6f74 6520 4f75 7470 7574 3a20 4f75 7470 ote Output: Outp │ │ │ │ -00027cf0: 7574 2c20 6973 2061 202a 6e6f 7465 2073 ut, is a *note s │ │ │ │ -00027d00: 796d 626f 6c3a 2028 4d61 6361 756c 6179 ymbol: (Macaulay │ │ │ │ -00027d10: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2Doc)Symbol,...- │ │ │ │ +00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00027ad0: 0a46 756e 6374 696f 6e73 2077 6974 6820 .Functions with │ │ │ │ +00027ae0: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ +00027af0: 7420 6e61 6d65 6420 4f75 7470 7574 3a0a t named Output:. │ │ │ │ +00027b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +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 │ │ │ │ +00027bc0: 6877 6172 747a 2d4d 6163 5068 6572 736f hwartz-MacPherso │ │ │ │ +00027bd0: 6e20 636c 6173 730a 2020 2a20 2245 756c n class. * "Eul │ │ │ │ +00027be0: 6572 282e 2e2e 2c4f 7574 7075 743d 3e2e er(...,Output=>. │ │ │ │ +00027bf0: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74 ..)" -- see *not │ │ │ │ +00027c00: 6520 4575 6c65 723a 2045 756c 6572 2c20 e Euler: Euler, │ │ │ │ +00027c10: 2d2d 2054 6865 2045 756c 6572 0a20 2020 -- The Euler. │ │ │ │ +00027c20: 2043 6861 7261 6374 6572 6973 7469 630a Characteristic. │ │ │ │ +00027c30: 2020 2a20 2253 6567 7265 282e 2e2e 2c4f * "Segre(...,O │ │ │ │ +00027c40: 7574 7075 743d 3e2e 2e2e 2922 202d 2d20 utput=>...)" -- │ │ │ │ +00027c50: 7365 6520 2a6e 6f74 6520 5365 6772 653a see *note Segre: │ │ │ │ +00027c60: 2053 6567 7265 2c20 2d2d 2054 6865 2053 Segre, -- The S │ │ │ │ +00027c70: 6567 7265 2063 6c61 7373 206f 6620 610a egre class of a. │ │ │ │ +00027c80: 2020 2020 7375 6273 6368 656d 650a 0a46 subscheme..F │ │ │ │ +00027c90: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +00027ca0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00027cb0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00027cc0: 202a 6e6f 7465 204f 7574 7075 743a 204f *note Output: O │ │ │ │ +00027cd0: 7574 7075 742c 2069 7320 6120 2a6e 6f74 utput, is a *not │ │ │ │ +00027ce0: 6520 7379 6d62 6f6c 3a20 284d 6163 6175 e symbol: (Macau │ │ │ │ +00027cf0: 6c61 7932 446f 6329 5379 6d62 6f6c 2c2e lay2Doc)Symbol,. │ │ │ │ +00027d00: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +00027d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -00027d70: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -00027d80: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -00027d90: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -00027da0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -00027db0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -00027dc0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00027dd0: 6b61 6765 732f 0a43 6861 7261 6374 6572 kages/.Character │ │ │ │ -00027de0: 6973 7469 6343 6c61 7373 6573 2e6d 323a isticClasses.m2: │ │ │ │ -00027df0: 3234 3639 3a30 2e0a 1f0a 4669 6c65 3a20 2469:0....File: │ │ │ │ -00027e00: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ -00027e10: 6173 7365 732e 696e 666f 2c20 4e6f 6465 asses.info, Node │ │ │ │ -00027e20: 3a20 7072 6f62 6162 696c 6973 7469 6320 : probabilistic │ │ │ │ -00027e30: 616c 676f 7269 7468 6d2c 204e 6578 743a algorithm, Next: │ │ │ │ -00027e40: 2053 6567 7265 2c20 5072 6576 3a20 4f75 Segre, Prev: Ou │ │ │ │ -00027e50: 7470 7574 2c20 5570 3a20 546f 700a 0a70 tput, Up: Top..p │ │ │ │ -00027e60: 726f 6261 6269 6c69 7374 6963 2061 6c67 robabilistic alg │ │ │ │ -00027e70: 6f72 6974 686d 0a2a 2a2a 2a2a 2a2a 2a2a orithm.********* │ │ │ │ -00027e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00027e90: 5468 6520 616c 676f 7269 7468 6d73 2075 The algorithms u │ │ │ │ -00027ea0: 7365 6420 666f 7220 7468 6520 636f 6d70 sed for the comp │ │ │ │ -00027eb0: 7574 6174 696f 6e20 6f66 2063 6861 7261 utation of chara │ │ │ │ -00027ec0: 6374 6572 6973 7469 6320 636c 6173 7365 cteristic classe │ │ │ │ -00027ed0: 7320 6172 650a 7072 6f62 6162 696c 6973 s are.probabilis │ │ │ │ -00027ee0: 7469 632e 2054 6865 6f72 6574 6963 616c tic. Theoretical │ │ │ │ -00027ef0: 6c79 2c20 7468 6579 2063 616c 6375 6c61 ly, they calcula │ │ │ │ -00027f00: 7465 2074 6865 2063 6c61 7373 6573 2063 te the classes c │ │ │ │ -00027f10: 6f72 7265 6374 6c79 2066 6f72 2061 0a67 orrectly for a.g │ │ │ │ -00027f20: 656e 6572 616c 2063 686f 6963 6520 6f66 eneral choice of │ │ │ │ -00027f30: 2063 6572 7461 696e 2070 6f6c 796e 6f6d certain polynom │ │ │ │ -00027f40: 6961 6c73 2e20 5468 6174 2069 732c 2074 ials. That is, t │ │ │ │ -00027f50: 6865 7265 2069 7320 616e 206f 7065 6e20 here is an open │ │ │ │ -00027f60: 6465 6e73 6520 5a61 7269 736b 690a 7365 dense Zariski.se │ │ │ │ -00027f70: 7420 666f 7220 7768 6963 6820 7468 6520 t for which the │ │ │ │ -00027f80: 616c 676f 7269 7468 6d20 7969 656c 6473 algorithm yields │ │ │ │ -00027f90: 2074 6865 2063 6f72 7265 6374 2063 6c61 the correct cla │ │ │ │ -00027fa0: 7373 2c20 692e 652e 2c20 7468 6520 636f ss, i.e., the co │ │ │ │ -00027fb0: 7272 6563 7420 636c 6173 730a 6973 2063 rrect class.is c │ │ │ │ -00027fc0: 616c 6375 6c61 7465 6420 7769 7468 2070 alculated with p │ │ │ │ -00027fd0: 726f 6261 6269 6c69 7479 2031 2e20 486f robability 1. Ho │ │ │ │ -00027fe0: 7765 7665 722c 2073 696e 6365 2074 6865 wever, since the │ │ │ │ -00027ff0: 2069 6d70 6c65 6d65 6e74 6174 696f 6e20 implementation │ │ │ │ -00028000: 776f 726b 7320 6f76 6572 0a61 2064 6973 works over.a dis │ │ │ │ -00028010: 6372 6574 6520 7072 6f62 6162 696c 6974 crete probabilit │ │ │ │ -00028020: 7920 7370 6163 6520 7468 6572 6520 6973 y space there is │ │ │ │ -00028030: 2061 2076 6572 7920 736d 616c 6c2c 2062 a very small, b │ │ │ │ -00028040: 7574 206e 6f6e 2d7a 6572 6f2c 2070 726f ut non-zero, pro │ │ │ │ -00028050: 6261 6269 6c69 7479 0a6f 6620 6e6f 7420 bability.of not │ │ │ │ -00028060: 636f 6d70 7574 696e 6720 7468 6520 636f computing the co │ │ │ │ -00028070: 7272 6563 7420 636c 6173 732e 2053 6b65 rrect class. Ske │ │ │ │ -00028080: 7074 6963 616c 2075 7365 7273 2073 686f ptical users sho │ │ │ │ -00028090: 756c 6420 7265 7065 6174 2063 616c 6375 uld repeat calcu │ │ │ │ -000280a0: 6c61 7469 6f6e 730a 7365 7665 7261 6c20 lations.several │ │ │ │ -000280b0: 7469 6d65 7320 746f 2069 6e63 7265 6173 times to increas │ │ │ │ -000280c0: 6520 7468 6520 7072 6f62 6162 696c 6974 e the probabilit │ │ │ │ -000280d0: 7920 6f66 2063 6f6d 7075 7469 6e67 2074 y of computing t │ │ │ │ -000280e0: 6865 2063 6f72 7265 6374 2063 6c61 7373 he correct class │ │ │ │ -000280f0: 2e0a 0a49 6e20 7468 6520 6361 7365 206f ...In the case o │ │ │ │ -00028100: 6620 7468 6520 7379 6d62 6f6c 6963 2069 f the symbolic i │ │ │ │ -00028110: 6d70 6c65 6d65 6e74 6174 696f 6e20 6f66 mplementation of │ │ │ │ -00028120: 2074 6865 2050 726f 6a65 6374 6976 6544 the ProjectiveD │ │ │ │ -00028130: 6567 7265 6520 6d65 7468 6f64 0a70 7261 egree method.pra │ │ │ │ -00028140: 6374 6963 616c 2065 7870 6572 6965 6e63 ctical experienc │ │ │ │ -00028150: 6520 616e 6420 616c 676f 7269 7468 6d20 e and algorithm │ │ │ │ -00028160: 7465 7374 696e 6720 696e 6469 6361 7465 testing indicate │ │ │ │ -00028170: 2074 6861 7420 6120 6669 6e69 7465 2066 that a finite f │ │ │ │ -00028180: 6965 6c64 2077 6974 680a 6f76 6572 2032 ield with.over 2 │ │ │ │ -00028190: 3530 3030 2065 6c65 6d65 6e74 7320 6973 5000 elements is │ │ │ │ -000281a0: 206d 6f72 6520 7468 616e 2073 7566 6669 more than suffi │ │ │ │ -000281b0: 6369 656e 7420 746f 2065 7870 6563 7420 cient to expect │ │ │ │ -000281c0: 6120 636f 7272 6563 7420 7265 7375 6c74 a correct result │ │ │ │ -000281d0: 2077 6974 680a 6869 6768 2070 726f 6261 with.high proba │ │ │ │ -000281e0: 6269 6c69 7479 2c20 692e 652e 2075 7369 bility, i.e. usi │ │ │ │ -000281f0: 6e67 2074 6865 2066 696e 6974 6520 6669 ng the finite fi │ │ │ │ -00028200: 656c 6420 6b6b 3d5a 5a2f 3235 3037 3320 eld kk=ZZ/25073 │ │ │ │ -00028210: 7468 6520 6578 7065 7269 6d65 6e74 616c the experimental │ │ │ │ -00028220: 0a63 6861 6e63 6520 6f66 2066 6169 6c75 .chance of failu │ │ │ │ -00028230: 7265 2077 6974 6820 7468 6520 5072 6f6a re with the Proj │ │ │ │ -00028240: 6563 7469 7665 4465 6772 6565 2061 6c67 ectiveDegree alg │ │ │ │ -00028250: 6f72 6974 686d 206f 6e20 6120 7661 7269 orithm on a vari │ │ │ │ -00028260: 6574 7920 6f66 2065 7861 6d70 6c65 730a ety of examples. │ │ │ │ -00028270: 7761 7320 6c65 7373 2074 6861 6e20 312f was less than 1/ │ │ │ │ -00028280: 3230 3030 2e20 5573 696e 6720 7468 6520 2000. Using the │ │ │ │ -00028290: 6669 6e69 7465 2066 6965 6c64 206b 6b3d finite field kk= │ │ │ │ -000282a0: 5a5a 2f33 3237 3439 2072 6573 756c 7465 ZZ/32749 resulte │ │ │ │ -000282b0: 6420 696e 206e 6f0a 6661 696c 7572 6573 d in no.failures │ │ │ │ -000282c0: 2069 6e20 6f76 6572 2031 3030 3030 2061 in over 10000 a │ │ │ │ -000282d0: 7474 656d 7074 7320 6f66 2073 6576 6572 ttempts of sever │ │ │ │ -000282e0: 616c 2064 6966 6665 7265 6e74 2065 7861 al different exa │ │ │ │ -000282f0: 6d70 6c65 732e 0a0a 5765 2069 6c6c 7573 mples...We illus │ │ │ │ -00028300: 7472 6174 6520 7468 6520 7072 6f62 6162 trate the probab │ │ │ │ -00028310: 696c 6973 7469 6320 6265 6861 7669 6f75 ilistic behaviou │ │ │ │ -00028320: 7220 7769 7468 2061 6e20 6578 616d 706c r with an exampl │ │ │ │ -00028330: 6520 7768 6572 6520 7468 6520 6368 6f73 e where the chos │ │ │ │ -00028340: 656e 0a72 616e 646f 6d20 7365 6564 206c en.random seed l │ │ │ │ -00028350: 6561 6473 2074 6f20 6120 7772 6f6e 6720 eads to a wrong │ │ │ │ -00028360: 7265 7375 6c74 2069 6e20 7468 6520 6669 result in the fi │ │ │ │ -00028370: 7273 7420 6361 6c63 756c 6174 696f 6e2e rst calculation. │ │ │ │ -00028380: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ -00028390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000283a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000283b0: 2d2d 2d2b 0a7c 6931 203a 2073 6574 5261 ---+.|i1 : setRa │ │ │ │ -000283c0: 6e64 6f6d 5365 6564 2031 3231 3b20 2020 ndomSeed 121; │ │ │ │ -000283d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000283e0: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574 |.| -- set │ │ │ │ -000283f0: 7469 6e67 2072 616e 646f 6d20 7365 6564 ting random seed │ │ │ │ -00028400: 2074 6f20 3132 3120 2020 2020 2020 2020 to 121 │ │ │ │ -00028410: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00028420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00028450: 3220 3a20 5220 3d20 5151 5b78 2c79 2c7a 2 : R = QQ[x,y,z │ │ │ │ -00028460: 2c77 5d20 2020 2020 2020 2020 2020 2020 ,w] │ │ │ │ -00028470: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028480: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00028490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027d50: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +00027d60: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +00027d70: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +00027d80: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +00027d90: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +00027da0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +00027db0: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ +00027dc0: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +00027dd0: 6d32 3a32 3436 393a 302e 0a1f 0a46 696c m2:2469:0....Fil │ │ │ │ +00027de0: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ +00027df0: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ +00027e00: 6f64 653a 2070 726f 6261 6269 6c69 7374 ode: probabilist │ │ │ │ +00027e10: 6963 2061 6c67 6f72 6974 686d 2c20 4e65 ic algorithm, Ne │ │ │ │ +00027e20: 7874 3a20 5365 6772 652c 2050 7265 763a xt: Segre, Prev: │ │ │ │ +00027e30: 204f 7574 7075 742c 2055 703a 2054 6f70 Output, Up: Top │ │ │ │ +00027e40: 0a0a 7072 6f62 6162 696c 6973 7469 6320 ..probabilistic │ │ │ │ +00027e50: 616c 676f 7269 7468 6d0a 2a2a 2a2a 2a2a algorithm.****** │ │ │ │ +00027e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00027e70: 2a0a 0a54 6865 2061 6c67 6f72 6974 686d *..The algorithm │ │ │ │ +00027e80: 7320 7573 6564 2066 6f72 2074 6865 2063 s used for the c │ │ │ │ +00027e90: 6f6d 7075 7461 7469 6f6e 206f 6620 6368 omputation of ch │ │ │ │ +00027ea0: 6172 6163 7465 7269 7374 6963 2063 6c61 aracteristic cla │ │ │ │ +00027eb0: 7373 6573 2061 7265 0a70 726f 6261 6269 sses are.probabi │ │ │ │ +00027ec0: 6c69 7374 6963 2e20 5468 656f 7265 7469 listic. Theoreti │ │ │ │ +00027ed0: 6361 6c6c 792c 2074 6865 7920 6361 6c63 cally, they calc │ │ │ │ +00027ee0: 756c 6174 6520 7468 6520 636c 6173 7365 ulate the classe │ │ │ │ +00027ef0: 7320 636f 7272 6563 746c 7920 666f 7220 s correctly for │ │ │ │ +00027f00: 610a 6765 6e65 7261 6c20 6368 6f69 6365 a.general choice │ │ │ │ +00027f10: 206f 6620 6365 7274 6169 6e20 706f 6c79 of certain poly │ │ │ │ +00027f20: 6e6f 6d69 616c 732e 2054 6861 7420 6973 nomials. That is │ │ │ │ +00027f30: 2c20 7468 6572 6520 6973 2061 6e20 6f70 , there is an op │ │ │ │ +00027f40: 656e 2064 656e 7365 205a 6172 6973 6b69 en dense Zariski │ │ │ │ +00027f50: 0a73 6574 2066 6f72 2077 6869 6368 2074 .set for which t │ │ │ │ +00027f60: 6865 2061 6c67 6f72 6974 686d 2079 6965 he algorithm yie │ │ │ │ +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 │ │ │ │ +00027f90: 2063 6f72 7265 6374 2063 6c61 7373 0a69 correct class.i │ │ │ │ +00027fa0: 7320 6361 6c63 756c 6174 6564 2077 6974 s calculated wit │ │ │ │ +00027fb0: 6820 7072 6f62 6162 696c 6974 7920 312e h probability 1. │ │ │ │ +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 7c0a 7c6f 3220 3d20 5220 2020 2020 |.|o2 = R │ │ │ │ -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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000284f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028510: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -00028520: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ -00028530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00028580: 7c69 3320 3a20 4920 3d20 6d69 6e6f 7273 |i3 : I = minors │ │ │ │ -00028590: 2832 2c6d 6174 7269 787b 7b78 2c79 2c7a (2,matrix{{x,y,z │ │ │ │ -000285a0: 7d2c 7b79 2c7a 2c77 7d7d 2920 2020 2020 },{y,z,w}}) │ │ │ │ -000285b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000285c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000285d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000285e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000285f0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00028600: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00028610: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ -00028620: 6465 616c 2028 2d20 7920 202b 2078 2a7a deal (- y + x*z │ │ │ │ -00028630: 2c20 2d20 792a 7a20 2b20 782a 772c 202d , - y*z + x*w, - │ │ │ │ -00028640: 207a 2020 2b20 792a 7729 7c0a 7c20 2020 z + y*w)|.| │ │ │ │ +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 2020 2020 2020 2020 │ │ │ │ 00028650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028670: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028680: 6f33 203a 2049 6465 616c 206f 6620 5220 o3 : Ideal of R │ │ │ │ -00028690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000286a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000286b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -000286c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000286d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000286e0: 2d2d 2d2b 0a7c 6934 203a 2043 6865 726e ---+.|i4 : Chern │ │ │ │ -000286f0: 2028 492c 436f 6d70 4d65 7468 6f64 3d3e (I,CompMethod=> │ │ │ │ -00028700: 506e 5265 7369 6475 616c 2920 2020 2020 PnResidual) │ │ │ │ -00028710: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00028720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028740: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00028750: 2020 2033 2020 2020 2032 2020 2020 2020 3 2 │ │ │ │ -00028760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028770: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00028780: 3420 3d20 3248 2020 2b20 3348 2020 2020 4 = 2H + 3H │ │ │ │ -00028790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000287a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000287b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000287c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000287d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000287e0: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20 |.| ZZ[H] │ │ │ │ -000287f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028810: 2020 2020 207c 0a7c 6f34 203a 202d 2d2d |.|o4 : --- │ │ │ │ -00028820: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ -00028830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028840: 2020 2020 2020 2020 7c0a 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reproducible-pat │ │ │ │ -00029000: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -00029010: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -00029020: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ -00029030: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ -00029040: 7365 732e 6d32 3a32 3337 383a 302e 0a1f ses.m2:2378:0... │ │ │ │ -00029050: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ -00029060: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ -00029070: 6f2c 204e 6f64 653a 2053 6567 7265 2c20 o, Node: Segre, │ │ │ │ -00029080: 4e65 7874 3a20 546f 7269 6343 686f 7752 Next: ToricChowR │ │ │ │ -00029090: 696e 672c 2050 7265 763a 2070 726f 6261 ing, Prev: proba │ │ │ │ -000290a0: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ -000290b0: 686d 2c20 5570 3a20 546f 700a 0a53 6567 hm, Up: Top..Seg │ │ │ │ -000290c0: 7265 202d 2d20 5468 6520 5365 6772 6520 re -- The Segre │ │ │ │ -000290d0: 636c 6173 7320 6f66 2061 2073 7562 7363 class of a subsc │ │ │ │ -000290e0: 6865 6d65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a heme.*********** │ │ │ │ -000290f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -00029110: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -00029120: 2020 5365 6772 6520 490a 2020 2020 2020 Segre I. │ │ │ │ -00029130: 2020 5365 6772 6528 412c 4929 0a20 2020 Segre(A,I). │ │ │ │ -00029140: 2020 2020 2053 6567 7265 2858 2c4a 290a Segre(X,J). │ │ │ │ -00029150: 2020 2020 2020 2020 5365 6772 6528 4368 Segre(Ch │ │ │ │ -00029160: 2c58 2c4a 290a 2020 2a20 496e 7075 7473 ,X,J). * Inputs │ │ │ │ -00029170: 3a0a 2020 2020 2020 2a20 492c 2061 6e20 :. * I, an │ │ │ │ -00029180: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ -00029190: 6361 756c 6179 3244 6f63 2949 6465 616c caulay2Doc)Ideal │ │ │ │ -000291a0: 2c2c 2061 206d 756c 7469 2d68 6f6d 6f67 ,, a multi-homog │ │ │ │ -000291b0: 656e 656f 7573 2069 6465 616c 2069 6e20 eneous ideal in │ │ │ │ -000291c0: 610a 2020 2020 2020 2020 6772 6164 6564 a. graded │ │ │ │ -000291d0: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ -000291e0: 206f 7665 7220 6120 6669 656c 6420 6465 over a field de │ │ │ │ -000291f0: 6669 6e69 6e67 2061 2063 6c6f 7365 6420 fining a closed │ │ │ │ -00029200: 7375 6273 6368 656d 6520 5620 6f66 0a20 subscheme V of. │ │ │ │ -00029210: 2020 2020 2020 205c 5050 5e7b 6e5f 317d \PP^{n_1} │ │ │ │ -00029220: 782e 2e2e 785c 5050 5e7b 6e5f 6d7d 0a20 x...x\PP^{n_m}. │ │ │ │ -00029230: 2020 2020 202a 2041 2c20 6120 2a6e 6f74 * A, a *not │ │ │ │ -00029240: 6520 7175 6f74 6965 6e74 2072 696e 673a e quotient ring: │ │ │ │ -00029250: 2028 4d61 6361 756c 6179 3244 6f63 2951 (Macaulay2Doc)Q │ │ │ │ -00029260: 756f 7469 656e 7452 696e 672c 2c0a 2020 uotientRing,,. │ │ │ │ -00029270: 2020 2020 2020 413d 5c5a 5a5b 685f 312c A=\ZZ[h_1, │ │ │ │ -00029280: 2e2e 2e2c 685f 6d5d 2f28 685f 315e 7b6e ...,h_m]/(h_1^{n │ │ │ │ -00029290: 5f31 2b31 7d2c 2e2e 2e2c 685f 6d5e 7b6e _1+1},...,h_m^{n │ │ │ │ -000292a0: 5f6d 2b31 7d29 2071 756f 7469 656e 7420 _m+1}) quotient │ │ │ │ -000292b0: 7269 6e67 0a20 2020 2020 2020 2072 6570 ring. rep │ │ │ │ -000292c0: 7265 7365 6e74 696e 6720 7468 6520 4368 resenting the Ch │ │ │ │ -000292d0: 6f77 2072 696e 6720 6f66 205c 5050 5e7b ow ring of \PP^{ │ │ │ │ -000292e0: 6e5f 317d 782e 2e2e 785c 5050 5e7b 6e5f n_1}x...x\PP^{n_ │ │ │ │ -000292f0: 6d7d 2c20 7468 6973 2072 696e 6720 7368 m}, this ring sh │ │ │ │ -00029300: 6f75 6c64 0a20 2020 2020 2020 2062 6520 ould. be │ │ │ │ -00029310: 6275 696c 7420 7573 696e 6720 7468 6520 built using the │ │ │ │ -00029320: 2a6e 6f74 6520 4368 6f77 5269 6e67 3a20 *note ChowRing: │ │ │ │ -00029330: 4368 6f77 5269 6e67 2c20 636f 6d6d 616e ChowRing, comman │ │ │ │ -00029340: 640a 2020 2020 2020 2a20 4a2c 2061 6e20 d. * J, an │ │ │ │ -00029350: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ -00029360: 6361 756c 6179 3244 6f63 2949 6465 616c caulay2Doc)Ideal │ │ │ │ -00029370: 2c2c 2069 6e20 7468 6520 6772 6164 6564 ,, in the graded │ │ │ │ -00029380: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ -00029390: 0a20 2020 2020 2020 2077 6869 6368 2069 . which i │ │ │ │ -000293a0: 7320 636f 6f72 6469 6e61 7465 2072 696e s coordinate rin │ │ │ │ -000293b0: 6720 6f66 2074 6865 204e 6f72 6d61 6c20 g of the Normal │ │ │ │ -000293c0: 546f 7269 6320 5661 7269 6574 7920 580a Toric Variety X. │ │ │ │ -000293d0: 2020 2020 2020 2a20 582c 2061 202a 6e6f * X, a *no │ │ │ │ -000293e0: 7465 206e 6f72 6d61 6c20 746f 7269 6320 te normal toric │ │ │ │ -000293f0: 7661 7269 6574 793a 0a20 2020 2020 2020 variety:. │ │ │ │ -00029400: 2028 4e6f 726d 616c 546f 7269 6356 6172 (NormalToricVar │ │ │ │ -00029410: 6965 7469 6573 294e 6f72 6d61 6c54 6f72 ieties)NormalTor │ │ │ │ -00029420: 6963 5661 7269 6574 792c 2c20 7768 6963 icVariety,, whic │ │ │ │ -00029430: 6820 6973 2074 6865 2061 6d62 6965 6e74 h is the ambient │ │ │ │ -00029440: 2073 7061 6365 0a20 2020 2020 2020 2077 space. w │ │ │ │ -00029450: 6869 6368 2063 6f6e 7461 696e 7320 5628 hich contains V( │ │ │ │ -00029460: 4a29 0a20 2020 2020 202a 2043 682c 2061 J). * Ch, a │ │ │ │ -00029470: 202a 6e6f 7465 2071 756f 7469 656e 7420 *note quotient │ │ │ │ -00029480: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ -00029490: 446f 6329 5175 6f74 6965 6e74 5269 6e67 Doc)QuotientRing │ │ │ │ -000294a0: 2c2c 2074 6865 2043 686f 7720 7269 6e67 ,, the Chow ring │ │ │ │ -000294b0: 0a20 2020 2020 2020 206f 6620 7468 6520 . of the │ │ │ │ -000294c0: 746f 7269 6320 7661 7269 6574 7920 582c toric variety X, │ │ │ │ -000294d0: 2043 683d 2872 696e 6720 4a29 2f28 5352 Ch=(ring J)/(SR │ │ │ │ -000294e0: 2b4c 5229 2077 6865 7265 2053 5220 6973 +LR) where SR is │ │ │ │ -000294f0: 2074 6865 0a20 2020 2020 2020 2053 7461 the. Sta │ │ │ │ -00029500: 6e6c 6579 2d52 6569 736e 6572 2069 6465 nley-Reisner ide │ │ │ │ -00029510: 616c 206f 6620 7468 6520 6661 6e20 6465 al of the fan de │ │ │ │ -00029520: 6669 6e69 6e67 2058 2061 6e64 204c 5220 fining X and LR │ │ │ │ -00029530: 6973 2074 6865 206c 696e 6561 720a 2020 is the linear. │ │ │ │ -00029540: 2020 2020 2020 7265 6c61 7469 6f6e 7320 relations │ │ │ │ -00029550: 6964 6561 6c2c 2074 6869 7320 7269 6e67 ideal, this ring │ │ │ │ -00029560: 2073 686f 756c 6420 6265 2062 7569 6c74 should be built │ │ │ │ -00029570: 2075 7369 6e67 2074 6865 202a 6e6f 7465 using the *note │ │ │ │ -00029580: 0a20 2020 2020 2020 2054 6f72 6963 4368 . ToricCh │ │ │ │ -00029590: 6f77 5269 6e67 3a20 546f 7269 6343 686f owRing: ToricCho │ │ │ │ -000295a0: 7752 696e 672c 2063 6f6d 6d61 6e64 0a20 wRing, command. │ │ │ │ -000295b0: 202a 202a 6e6f 7465 204f 7074 696f 6e61 * *note Optiona │ │ │ │ -000295c0: 6c20 696e 7075 7473 3a20 284d 6163 6175 l inputs: (Macau │ │ │ │ -000295d0: 6c61 7932 446f 6329 7573 696e 6720 6675 lay2Doc)using fu │ │ │ │ -000295e0: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ -000295f0: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20 ional inputs,:. │ │ │ │ -00029600: 2020 2020 202a 2043 6f6d 704d 6574 686f * CompMetho │ │ │ │ -00029610: 6420 286d 6973 7369 6e67 2064 6f63 756d d (missing docum │ │ │ │ -00029620: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ -00029630: 2c20 6465 6661 756c 7420 7661 6c75 650a , default value. │ │ │ │ -00029640: 2020 2020 2020 2020 5072 6f6a 6563 7469 Projecti │ │ │ │ -00029650: 7665 4465 6772 6565 2c20 5072 6f6a 6563 veDegree, Projec │ │ │ │ -00029660: 7469 7665 4465 6772 6565 2c20 7468 6973 tiveDegree, this │ │ │ │ -00029670: 2061 6c67 6f72 6974 686d 206d 6179 2062 algorithm may b │ │ │ │ -00029680: 6520 7573 6564 2066 6f72 0a20 2020 2020 e used for. │ │ │ │ -00029690: 2020 2073 7562 7363 6865 6d65 7320 6f66 subschemes of │ │ │ │ -000296a0: 2061 6e79 2061 7070 6c69 6361 626c 6520 any applicable │ │ │ │ -000296b0: 746f 7269 6320 7661 7269 6574 7920 2874 toric variety (t │ │ │ │ -000296c0: 6869 7320 6d61 7920 6265 2063 6865 636b his may be check │ │ │ │ -000296d0: 6564 2075 7369 6e67 0a20 2020 2020 2020 ed using. │ │ │ │ -000296e0: 2074 6865 202a 6e6f 7465 2043 6865 636b the *note Check │ │ │ │ -000296f0: 546f 7269 6356 6172 6965 7479 5661 6c69 ToricVarietyVali │ │ │ │ -00029700: 643a 2043 6865 636b 546f 7269 6356 6172 d: CheckToricVar │ │ │ │ -00029710: 6965 7479 5661 6c69 642c 2063 6f6d 6d61 ietyValid, comma │ │ │ │ -00029720: 6e64 290a 2020 2020 2020 2a20 436f 6d70 nd). * Comp │ │ │ │ -00029730: 4d65 7468 6f64 2028 6d69 7373 696e 6720 Method (missing │ │ │ │ -00029740: 646f 6375 6d65 6e74 6174 696f 6e29 203d documentation) = │ │ │ │ -00029750: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ -00029760: 616c 7565 0a20 2020 2020 2020 2050 726f alue. Pro │ │ │ │ -00029770: 6a65 6374 6976 6544 6567 7265 652c 2050 jectiveDegree, P │ │ │ │ -00029780: 6e52 6573 6964 7561 6c2c 2074 6869 7320 nResidual, this │ │ │ │ -00029790: 616c 676f 7269 7468 6d20 6d61 7920 6265 algorithm may be │ │ │ │ -000297a0: 2075 7365 6420 666f 7220 7375 6273 6368 used for subsch │ │ │ │ -000297b0: 656d 6573 0a20 2020 2020 2020 206f 6620 emes. of │ │ │ │ -000297c0: 5c50 505e 6e20 6f6e 6c79 0a20 2020 2020 \PP^n only. │ │ │ │ -000297d0: 202a 204f 7574 7075 7420 3d3e 202e 2e2e * Output => ... │ │ │ │ -000297e0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -000297f0: 4368 6f77 5269 6e67 456c 656d 656e 742c ChowRingElement, │ │ │ │ -00029800: 2043 686f 7752 696e 6745 6c65 6d65 6e74 ChowRingElement │ │ │ │ -00029810: 2c20 7265 7475 726e 730a 2020 2020 2020 , returns. │ │ │ │ -00029820: 2020 6120 5269 6e67 456c 656d 656e 7420 a RingElement │ │ │ │ -00029830: 696e 2074 6865 2043 686f 7720 7269 6e67 in the Chow ring │ │ │ │ -00029840: 206f 6620 7468 6520 6170 7072 6f70 7269 of the appropri │ │ │ │ -00029850: 6174 6520 616d 6269 656e 7420 7370 6163 ate ambient spac │ │ │ │ -00029860: 650a 2020 2020 2020 2a20 4f75 7470 7574 e. * Output │ │ │ │ -00029870: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -00029880: 2076 616c 7565 2043 686f 7752 696e 6745 value ChowRingE │ │ │ │ -00029890: 6c65 6d65 6e74 2c20 4861 7368 466f 726d lement, HashForm │ │ │ │ -000298a0: 2c20 4861 7368 466f 726d 0a20 2020 2020 , HashForm. │ │ │ │ -000298b0: 2020 2072 6574 7572 6e73 2061 204d 7574 returns a Mut │ │ │ │ -000298c0: 6162 6c65 4861 7368 5461 626c 6520 636f ableHashTable co │ │ │ │ -000298d0: 6e74 6169 6e69 6e67 2074 6865 2066 6f6c ntaining the fol │ │ │ │ -000298e0: 6c6f 7769 6e67 206b 6579 733a 2022 4722 lowing keys: "G" │ │ │ │ -000298f0: 2028 7468 650a 2020 2020 2020 2020 706f (the. po │ │ │ │ -00029900: 6c79 6e6f 6d69 616c 2077 6974 6820 636f lynomial with co │ │ │ │ -00029910: 6566 6669 6369 656e 7473 206f 6620 7468 efficients of th │ │ │ │ -00029920: 6520 6879 7065 7270 6c61 6e65 2063 6c61 e hyperplane cla │ │ │ │ -00029930: 7373 6573 2072 6570 7265 7365 6e74 696e sses representin │ │ │ │ -00029940: 6720 7468 650a 2020 2020 2020 2020 7072 g the. pr │ │ │ │ -00029950: 6f6a 6563 7469 7665 2064 6567 7265 6573 ojective degrees │ │ │ │ -00029960: 292c 2022 476c 6973 7422 2028 7468 6520 ), "Glist" (the │ │ │ │ -00029970: 6c69 7374 2066 6f72 6d20 6f66 2022 4722 list form of "G" │ │ │ │ -00029980: 2920 2c20 2253 6567 7265 2220 2874 6865 ) , "Segre" (the │ │ │ │ -00029990: 0a20 2020 2020 2020 2074 6f74 616c 2053 . total S │ │ │ │ -000299a0: 6567 7265 2063 6c61 7373 206f 6620 7468 egre class of th │ │ │ │ -000299b0: 6520 696e 7075 7429 2c22 5365 6772 654c e input),"SegreL │ │ │ │ -000299c0: 6973 7422 2028 7468 6520 6c69 7374 2066 ist" (the list f │ │ │ │ -000299d0: 6f72 6d20 6f66 2022 5365 6772 6522 290a orm of "Segre"). │ │ │ │ -000299e0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -000299f0: 2020 202a 2061 202a 6e6f 7465 2072 696e * a *note rin │ │ │ │ -00029a00: 6720 656c 656d 656e 743a 2028 4d61 6361 g element: (Maca │ │ │ │ -00029a10: 756c 6179 3244 6f63 2952 696e 6745 6c65 ulay2Doc)RingEle │ │ │ │ -00029a20: 6d65 6e74 2c2c 2074 6865 2070 7573 6866 ment,, the pushf │ │ │ │ -00029a30: 6f72 7761 7264 206f 660a 2020 2020 2020 orward of. │ │ │ │ -00029a40: 2020 7468 6520 746f 7461 6c20 5365 6772 the total Segr │ │ │ │ -00029a50: 6520 636c 6173 7320 6f66 2074 6865 2073 e class of the s │ │ │ │ -00029a60: 6368 656d 6520 5620 6465 6669 6e65 6420 cheme V defined │ │ │ │ -00029a70: 6279 2074 6865 2069 6e70 7574 2069 6465 by the input ide │ │ │ │ -00029a80: 616c 2074 6f20 7468 650a 2020 2020 2020 al to the. │ │ │ │ -00029a90: 2020 6170 7072 6f70 7269 6174 6520 4368 appropriate Ch │ │ │ │ -00029aa0: 6f77 2072 696e 670a 0a44 6573 6372 6970 ow ring..Descrip │ │ │ │ -00029ab0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00029ac0: 0a0a 466f 7220 6120 7375 6273 6368 656d ..For a subschem │ │ │ │ -00029ad0: 6520 5620 6f66 2061 6e20 6170 706c 6963 e V of an applic │ │ │ │ -00029ae0: 6162 6c65 2074 6f72 6963 2076 6172 6965 able toric varie │ │ │ │ -00029af0: 7479 2058 2074 6869 7320 636f 6d6d 616e ty X this comman │ │ │ │ -00029b00: 6420 636f 6d70 7574 6573 2074 6865 0a70 d computes the.p │ │ │ │ -00029b10: 7573 682d 666f 7277 6172 6420 6f66 2074 ush-forward of t │ │ │ │ -00029b20: 6865 2074 6f74 616c 2053 6567 7265 2063 he total Segre c │ │ │ │ -00029b30: 6c61 7373 2073 2856 2c58 2920 6f66 2056 lass s(V,X) of V │ │ │ │ -00029b40: 2069 6e20 5820 746f 2074 6865 2043 686f in X to the Cho │ │ │ │ -00029b50: 7720 7269 6e67 206f 6620 582e 0a0a 2b2d w ring of X...+- │ │ │ │ -00029b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00029b90: 3120 3a20 7365 7452 616e 646f 6d53 6565 1 : setRandomSee │ │ │ │ -00029ba0: 6420 3732 3b20 2020 2020 2020 2020 2020 d 72; │ │ │ │ -00029bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029bc0: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ -00029bd0: 6d20 7365 6564 2074 6f20 3732 2020 2020 m seed to 72 │ │ │ │ -00029be0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00029bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00029c20: 3220 3a20 5220 3d20 5a5a 2f33 3237 3439 2 : R = ZZ/32749 │ │ │ │ -00029c30: 5b77 2c79 2c7a 5d20 2020 2020 2020 2020 [w,y,z] │ │ │ │ -00029c40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029c80: 3220 3d20 5220 2020 2020 2020 2020 2020 2 = R │ │ │ │ -00029c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029ce0: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ -00029cf0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -00029d00: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00029d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00029d40: 3320 3a20 5365 6772 6528 6964 6561 6c28 3 : Segre(ideal( │ │ │ │ -00029d50: 772a 7929 2c43 6f6d 704d 6574 686f 643d w*y),CompMethod= │ │ │ │ -00029d60: 3e50 6e52 6573 6964 7561 6c29 7c0a 7c20 >PnResidual)|.| │ │ │ │ -00029d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029da0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -00029db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029dd0: 3320 3d20 2d20 3448 2020 2b20 3248 2020 3 = - 4H + 2H │ │ │ │ -00029de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029e30: 2020 2020 5a5a 5b48 5d20 2020 2020 2020 ZZ[H] │ │ │ │ -00029e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029e60: 3320 3a20 2d2d 2d2d 2d20 2020 2020 2020 3 : ----- │ │ │ │ -00029e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029e90: 2020 2020 2020 2033 2020 2020 2020 2020 3 │ │ │ │ -00029ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029ec0: 2020 2020 2020 4820 2020 2020 2020 2020 H │ │ │ │ -00029ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00029ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00029f20: 3420 3a20 413d 4368 6f77 5269 6e67 2852 4 : A=ChowRing(R │ │ │ │ -00029f30: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00029f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029f70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029f80: 3420 3d20 4120 2020 2020 2020 2020 2020 4 = A │ │ │ │ -00029f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029fe0: 3420 3a20 5175 6f74 6965 6e74 5269 6e67 4 : QuotientRing │ │ │ │ -00029ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a000: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002a040: 3520 3a20 5365 6772 6528 412c 6964 6561 5 : Segre(A,idea │ │ │ │ -0002a050: 6c28 775e 322a 792c 772a 795e 3229 2920 l(w^2*y,w*y^2)) │ │ │ │ -0002a060: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a090: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002a0a0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002a0d0: 3520 3d20 2d20 3368 2020 2b20 3268 2020 5 = - 3h + 2h │ │ │ │ -0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002a100: 2020 2020 2020 2020 3120 2020 2020 3120 1 1 │ │ │ │ -0002a110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a120: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a150: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002a160: 3520 3a20 4120 2020 2020 2020 2020 2020 5 : A │ │ │ │ -0002a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a180: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e ------------+..N │ │ │ │ -0002a1c0: 6f77 2063 6f6e 7369 6465 7220 616e 2065 ow consider an e │ │ │ │ -0002a1d0: 7861 6d70 6c65 2069 6e20 5c50 505e 3220 xample in \PP^2 │ │ │ │ -0002a1e0: 5c74 696d 6573 205c 5050 5e32 2c20 6966 \times \PP^2, if │ │ │ │ -0002a1f0: 2077 6520 696e 7075 7420 7468 6520 4368 we input the Ch │ │ │ │ -0002a200: 6f77 2072 696e 6720 4120 7468 650a 6f75 ow ring A the.ou │ │ │ │ -0002a210: 7470 7574 2077 696c 6c20 6265 2072 6574 tput will be ret │ │ │ │ -0002a220: 7572 6e65 6420 696e 2074 6865 2073 616d urned in the sam │ │ │ │ -0002a230: 6520 7269 6e67 2e20 546f 2065 6e73 7572 e ring. To ensur │ │ │ │ -0002a240: 6520 7072 6f70 6572 2066 756e 6374 696f e proper functio │ │ │ │ -0002a250: 6e20 6f66 2074 6865 0a6d 6574 686f 6473 n of the.methods │ │ │ │ -0002a260: 2077 6520 6275 696c 6420 7468 6520 4368 we build the Ch │ │ │ │ -0002a270: 6f77 2072 696e 6720 7573 696e 6720 7468 ow ring using th │ │ │ │ -0002a280: 6520 2a6e 6f74 6520 4368 6f77 5269 6e67 e *note ChowRing │ │ │ │ -0002a290: 3a20 4368 6f77 5269 6e67 2c20 636f 6d6d : ChowRing, comm │ │ │ │ -0002a2a0: 616e 642e 2057 650a 6d61 7920 616c 736f and. We.may also │ │ │ │ -0002a2b0: 2072 6574 7572 6e20 6120 4d75 7461 626c return a Mutabl │ │ │ │ -0002a2c0: 6548 6173 6854 6162 6c65 2e0a 0a2b 2d2d eHashTable...+-- │ │ │ │ +00028fa0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00028fb0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00028fc0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00028fd0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00028fe0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00028ff0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00029000: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00029010: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ +00029020: 6c61 7373 6573 2e6d 323a 3233 3738 3a30 lasses.m2:2378:0 │ │ │ │ +00029030: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ +00029040: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +00029050: 696e 666f 2c20 4e6f 6465 3a20 5365 6772 info, Node: Segr │ │ │ │ +00029060: 652c 204e 6578 743a 2054 6f72 6963 4368 e, Next: ToricCh │ │ │ │ +00029070: 6f77 5269 6e67 2c20 5072 6576 3a20 7072 owRing, Prev: pr │ │ │ │ +00029080: 6f62 6162 696c 6973 7469 6320 616c 676f obabilistic algo │ │ │ │ +00029090: 7269 7468 6d2c 2055 703a 2054 6f70 0a0a rithm, Up: Top.. │ │ │ │ +000290a0: 5365 6772 6520 2d2d 2054 6865 2053 6567 Segre -- The Seg │ │ │ │ +000290b0: 7265 2063 6c61 7373 206f 6620 6120 7375 re class of a su │ │ │ │ +000290c0: 6273 6368 656d 650a 2a2a 2a2a 2a2a 2a2a bscheme.******** │ │ │ │ +000290d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000290e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +000290f0: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +00029100: 2020 2020 2053 6567 7265 2049 0a20 2020 Segre I. │ │ │ │ +00029110: 2020 2020 2053 6567 7265 2841 2c49 290a Segre(A,I). │ │ │ │ +00029120: 2020 2020 2020 2020 5365 6772 6528 582c Segre(X, │ │ │ │ +00029130: 4a29 0a20 2020 2020 2020 2053 6567 7265 J). Segre │ │ │ │ +00029140: 2843 682c 582c 4a29 0a20 202a 2049 6e70 (Ch,X,J). * Inp │ │ │ │ +00029150: 7574 733a 0a20 2020 2020 202a 2049 2c20 uts:. * I, │ │ │ │ +00029160: 616e 202a 6e6f 7465 2069 6465 616c 3a20 an *note ideal: │ │ │ │ +00029170: 284d 6163 6175 6c61 7932 446f 6329 4964 (Macaulay2Doc)Id │ │ │ │ +00029180: 6561 6c2c 2c20 6120 6d75 6c74 692d 686f eal,, a multi-ho │ │ │ │ +00029190: 6d6f 6765 6e65 6f75 7320 6964 6561 6c20 mogeneous ideal │ │ │ │ +000291a0: 696e 2061 0a20 2020 2020 2020 2067 7261 in a. gra │ │ │ │ +000291b0: 6465 6420 706f 6c79 6e6f 6d69 616c 2072 ded polynomial r │ │ │ │ +000291c0: 696e 6720 6f76 6572 2061 2066 6965 6c64 ing over a field │ │ │ │ +000291d0: 2064 6566 696e 696e 6720 6120 636c 6f73 defining a clos │ │ │ │ +000291e0: 6564 2073 7562 7363 6865 6d65 2056 206f ed subscheme V o │ │ │ │ +000291f0: 660a 2020 2020 2020 2020 5c50 505e 7b6e f. \PP^{n │ │ │ │ +00029200: 5f31 7d78 2e2e 2e78 5c50 505e 7b6e 5f6d _1}x...x\PP^{n_m │ │ │ │ +00029210: 7d0a 2020 2020 2020 2a20 412c 2061 202a }. * A, a * │ │ │ │ +00029220: 6e6f 7465 2071 756f 7469 656e 7420 7269 note quotient ri │ │ │ │ +00029230: 6e67 3a20 284d 6163 6175 6c61 7932 446f ng: (Macaulay2Do │ │ │ │ +00029240: 6329 5175 6f74 6965 6e74 5269 6e67 2c2c c)QuotientRing,, │ │ │ │ +00029250: 0a20 2020 2020 2020 2041 3d5c 5a5a 5b68 . A=\ZZ[h │ │ │ │ +00029260: 5f31 2c2e 2e2e 2c68 5f6d 5d2f 2868 5f31 _1,...,h_m]/(h_1 │ │ │ │ +00029270: 5e7b 6e5f 312b 317d 2c2e 2e2e 2c68 5f6d ^{n_1+1},...,h_m │ │ │ │ +00029280: 5e7b 6e5f 6d2b 317d 2920 7175 6f74 6965 ^{n_m+1}) quotie │ │ │ │ +00029290: 6e74 2072 696e 670a 2020 2020 2020 2020 nt ring. │ │ │ │ +000292a0: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ +000292b0: 2043 686f 7720 7269 6e67 206f 6620 5c50 Chow ring of \P │ │ │ │ +000292c0: 505e 7b6e 5f31 7d78 2e2e 2e78 5c50 505e P^{n_1}x...x\PP^ │ │ │ │ +000292d0: 7b6e 5f6d 7d2c 2074 6869 7320 7269 6e67 {n_m}, this ring │ │ │ │ +000292e0: 2073 686f 756c 640a 2020 2020 2020 2020 should. │ │ │ │ +000292f0: 6265 2062 7569 6c74 2075 7369 6e67 2074 be built using t │ │ │ │ +00029300: 6865 202a 6e6f 7465 2043 686f 7752 696e he *note ChowRin │ │ │ │ +00029310: 673a 2043 686f 7752 696e 672c 2063 6f6d g: ChowRing, com │ │ │ │ +00029320: 6d61 6e64 0a20 2020 2020 202a 204a 2c20 mand. * J, │ │ │ │ +00029330: 616e 202a 6e6f 7465 2069 6465 616c 3a20 an *note ideal: │ │ │ │ +00029340: 284d 6163 6175 6c61 7932 446f 6329 4964 (Macaulay2Doc)Id │ │ │ │ +00029350: 6561 6c2c 2c20 696e 2074 6865 2067 7261 eal,, in the gra │ │ │ │ +00029360: 6465 6420 706f 6c79 6e6f 6d69 616c 2072 ded polynomial r │ │ │ │ +00029370: 696e 670a 2020 2020 2020 2020 7768 6963 ing. whic │ │ │ │ +00029380: 6820 6973 2063 6f6f 7264 696e 6174 6520 h is coordinate │ │ │ │ +00029390: 7269 6e67 206f 6620 7468 6520 4e6f 726d ring of the Norm │ │ │ │ +000293a0: 616c 2054 6f72 6963 2056 6172 6965 7479 al Toric Variety │ │ │ │ +000293b0: 2058 0a20 2020 2020 202a 2058 2c20 6120 X. * X, a │ │ │ │ +000293c0: 2a6e 6f74 6520 6e6f 726d 616c 2074 6f72 *note normal tor │ │ │ │ +000293d0: 6963 2076 6172 6965 7479 3a0a 2020 2020 ic variety:. │ │ │ │ +000293e0: 2020 2020 284e 6f72 6d61 6c54 6f72 6963 (NormalToric │ │ │ │ +000293f0: 5661 7269 6574 6965 7329 4e6f 726d 616c Varieties)Normal │ │ │ │ +00029400: 546f 7269 6356 6172 6965 7479 2c2c 2077 ToricVariety,, w │ │ │ │ +00029410: 6869 6368 2069 7320 7468 6520 616d 6269 hich is the ambi │ │ │ │ +00029420: 656e 7420 7370 6163 650a 2020 2020 2020 ent space. │ │ │ │ +00029430: 2020 7768 6963 6820 636f 6e74 6169 6e73 which contains │ │ │ │ +00029440: 2056 284a 290a 2020 2020 2020 2a20 4368 V(J). * Ch │ │ │ │ +00029450: 2c20 6120 2a6e 6f74 6520 7175 6f74 6965 , a *note quotie │ │ │ │ +00029460: 6e74 2072 696e 673a 2028 4d61 6361 756c nt ring: (Macaul │ │ │ │ +00029470: 6179 3244 6f63 2951 756f 7469 656e 7452 ay2Doc)QuotientR │ │ │ │ +00029480: 696e 672c 2c20 7468 6520 4368 6f77 2072 ing,, the Chow r │ │ │ │ +00029490: 696e 670a 2020 2020 2020 2020 6f66 2074 ing. of t │ │ │ │ +000294a0: 6865 2074 6f72 6963 2076 6172 6965 7479 he toric variety │ │ │ │ +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 │ │ │ │ +000294d0: 2069 7320 7468 650a 2020 2020 2020 2020 is the. │ │ │ │ +000294e0: 5374 616e 6c65 792d 5265 6973 6e65 7220 Stanley-Reisner │ │ │ │ +000294f0: 6964 6561 6c20 6f66 2074 6865 2066 616e ideal of the fan │ │ │ │ +00029500: 2064 6566 696e 696e 6720 5820 616e 6420 defining X and │ │ │ │ +00029510: 4c52 2069 7320 7468 6520 6c69 6e65 6172 LR is the linear │ │ │ │ +00029520: 0a20 2020 2020 2020 2072 656c 6174 696f . relatio │ │ │ │ +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 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 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 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 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 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 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 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 2020 2020 2020 2020 2020 207c | │ │ │ │ +00029f90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fb0: 2020 2020 2020 2020 2020 2020 2020 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 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 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 2d2d ---------------- │ │ │ │ -0002a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ -0002a320: 203a 2052 3d4d 756c 7469 5072 6f6a 436f : R=MultiProjCo │ │ │ │ -0002a330: 6f72 6452 696e 6728 7b32 2c32 7d29 2020 ordRing({2,2}) │ │ │ │ -0002a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a360: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3b0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0002a3c0: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a400: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a450: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0002a460: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -0002a470: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ -0002a480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a4a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 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 2d2d ---------------- │ │ │ │ -0002a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -0002a500: 203a 2072 3d67 656e 7320 5220 2020 2020 : r=gens R │ │ │ │ +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 2020 │ │ │ │ -0002a530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a540: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a590: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -0002a5a0: 203d 207b 7820 2c20 7820 2c20 7820 2c20 = {x , x , x , │ │ │ │ -0002a5b0: 7820 2c20 7820 2c20 7820 7d20 2020 2020 x , x , x } │ │ │ │ -0002a5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a5e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002a5f0: 2020 2020 2030 2020 2031 2020 2032 2020 0 1 2 │ │ │ │ -0002a600: 2033 2020 2034 2020 2035 2020 2020 2020 3 4 5 │ │ │ │ -0002a610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a630: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 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 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a680: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -0002a690: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a6d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 2d2d ---------------- │ │ │ │ -0002a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 2d2d ---------------- │ │ │ │ -0002bb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0002bb80: 3420 3a20 7331 3d3d 5365 6748 6173 6823 4 : s1==SegHash# │ │ │ │ -0002bb90: 2253 6567 7265 2220 2020 2020 2020 2020 "Segre" │ │ │ │ -0002bba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbc0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc10: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0002bc20: 3420 3d20 7472 7565 2020 2020 2020 2020 4 = true │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 2d2d ---------------- │ │ │ │ -0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 496e -----------+..In │ │ │ │ -0002bcc0: 2074 6865 2063 6173 6520 7768 6572 6520 the case where │ │ │ │ -0002bcd0: 7468 6520 616d 6269 656e 7420 7370 6163 the ambient spac │ │ │ │ -0002bce0: 6520 6973 2061 2074 6f72 6963 2076 6172 e is a toric var │ │ │ │ -0002bcf0: 6965 7479 2077 6869 6368 2069 7320 6e6f iety which is no │ │ │ │ -0002bd00: 7420 6120 7072 6f64 7563 740a 6f66 2070 t a product.of p │ │ │ │ -0002bd10: 726f 6a65 6374 6976 6520 7370 6163 6573 rojective spaces │ │ │ │ -0002bd20: 2077 6520 6d75 7374 206c 6f61 6420 7468 we must load th │ │ │ │ -0002bd30: 6520 4e6f 726d 616c 546f 7269 6356 6172 e NormalToricVar │ │ │ │ -0002bd40: 6965 7469 6573 2070 6163 6b61 6765 2061 ieties package a │ │ │ │ -0002bd50: 6e64 206d 7573 740a 616c 736f 2069 6e70 nd must.also inp │ │ │ │ -0002bd60: 7574 2074 6865 2074 6f72 6963 2076 6172 ut the toric var │ │ │ │ -0002bd70: 6965 7479 2e20 4966 2074 6865 2074 6f72 iety. If the tor │ │ │ │ -0002bd80: 6963 2076 6172 6965 7479 2069 7320 6120 ic variety is a │ │ │ │ -0002bd90: 7072 6f64 7563 7420 6f66 2070 726f 6a65 product of proje │ │ │ │ -0002bda0: 6374 6976 650a 7370 6163 6520 6974 2069 ctive.space it i │ │ │ │ -0002bdb0: 7320 7265 636f 6d6d 656e 6465 6420 746f s recommended to │ │ │ │ -0002bdc0: 2075 7365 2074 6865 2066 6f72 6d20 6162 use the form ab │ │ │ │ -0002bdd0: 6f76 6520 7261 7468 6572 2074 6861 6e20 ove rather than │ │ │ │ -0002bde0: 696e 7075 7474 696e 6720 7468 6520 746f inputting the to │ │ │ │ -0002bdf0: 7269 630a 7661 7269 6574 7920 666f 7220 ric.variety for │ │ │ │ -0002be00: 6566 6669 6369 656e 6379 2072 6561 736f efficiency reaso │ │ │ │ -0002be10: 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ns...+---------- │ │ │ │ +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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002be60: 7c69 3135 203a 206e 6565 6473 5061 636b |i15 : needsPack │ │ │ │ -0002be70: 6167 6520 224e 6f72 6d61 6c54 6f72 6963 age "NormalToric │ │ │ │ -0002be80: 5661 7269 6574 6965 7322 2020 2020 2020 Varieties" │ │ │ │ +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 207c 0a7c 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 2020 2020 2020 2020 2020 │ │ │ │ -0002bee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bef0: 2020 2020 7c0a 7c6f 3135 203d 204e 6f72 |.|o15 = Nor │ │ │ │ -0002bf00: 6d61 6c54 6f72 6963 5661 7269 6574 6965 malToricVarietie │ │ │ │ -0002bf10: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ -0002bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bf40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +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 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 2020 2020 2020 2020 │ │ │ │ -0002bf80: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -0002bf90: 203a 2050 6163 6b61 6765 2020 2020 2020 : Package │ │ │ │ +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 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfd0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c020: 2b0a 7c69 3136 203a 2052 686f 203d 207b +.|i16 : Rho = { │ │ │ │ -0002c030: 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 7d2c {1,0,0},{0,1,0}, │ │ │ │ -0002c040: 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 2c30 {0,0,1},{-1,-1,0 │ │ │ │ -0002c050: 7d2c 7b30 2c30 2c2d 317d 7d20 2020 2020 },{0,0,-1}} │ │ │ │ -0002c060: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +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 2020 2020 2020 2020 │ │ │ │ -0002c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c0b0: 2020 2020 2020 7c0a 7c6f 3136 203d 207b |.|o16 = { │ │ │ │ -0002c0c0: 7b31 2c20 302c 2030 7d2c 207b 302c 2031 {1, 0, 0}, {0, 1 │ │ │ │ -0002c0d0: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c , 0}, {0, 0, 1}, │ │ │ │ -0002c0e0: 207b 2d31 2c20 2d31 2c20 307d 2c20 7b30 {-1, -1, 0}, {0 │ │ │ │ -0002c0f0: 2c20 302c 202d 317d 7d20 2020 2020 2020 , 0, -1}} │ │ │ │ -0002c100: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 2020 │ │ │ │ -0002c130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c140: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002c150: 3136 203a 204c 6973 7420 2020 2020 2020 16 : List │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c190: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1e0: 2d2d 2b0a 7c69 3137 203a 2053 6967 6d61 --+.|i17 : Sigma │ │ │ │ -0002c1f0: 203d 207b 7b30 2c31 2c32 7d2c 7b31 2c32 = {{0,1,2},{1,2 │ │ │ │ -0002c200: 2c33 7d2c 7b30 2c32 2c33 7d2c 7b30 2c31 ,3},{0,2,3},{0,1 │ │ │ │ -0002c210: 2c34 7d2c 7b31 2c33 2c34 7d2c 7b30 2c33 ,4},{1,3,4},{0,3 │ │ │ │ -0002c220: 2c34 7d7d 2020 2020 2020 2020 207c 0a7c ,4}} |.| │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c270: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ -0002c280: 207b 7b30 2c20 312c 2032 7d2c 207b 312c {{0, 1, 2}, {1, │ │ │ │ -0002c290: 2032 2c20 337d 2c20 7b30 2c20 322c 2033 2, 3}, {0, 2, 3 │ │ │ │ -0002c2a0: 7d2c 207b 302c 2031 2c20 347d 2c20 7b31 }, {0, 1, 4}, {1 │ │ │ │ -0002c2b0: 2c20 332c 2034 7d2c 207b 302c 2033 2c20 , 3, 4}, {0, 3, │ │ │ │ -0002c2c0: 347d 7d7c 0a7c 2020 2020 2020 2020 2020 4}}|.| │ │ │ │ +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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c300: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c310: 7c6f 3137 203a 204c 6973 7420 2020 2020 |o17 : List │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -0002c350: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c3a0: 2d2d 2d2d 2b0a 7c69 3138 203a 2058 203d ----+.|i18 : X = │ │ │ │ -0002c3b0: 206e 6f72 6d61 6c54 6f72 6963 5661 7269 normalToricVari │ │ │ │ -0002c3c0: 6574 7928 5268 6f2c 5369 676d 612c 436f ety(Rho,Sigma,Co │ │ │ │ -0002c3d0: 6566 6669 6369 656e 7452 696e 6720 3d3e efficientRing => │ │ │ │ -0002c3e0: 5a5a 2f33 3237 3439 2920 2020 2020 207c ZZ/32749) | │ │ │ │ -0002c3f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +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: 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2020 2020 2020 |.| │ │ │ │ +0002c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c4d0: 7c0a 7c6f 3138 203a 204e 6f72 6d61 6c54 |.|o18 : NormalT │ │ │ │ -0002c4e0: 6f72 6963 5661 7269 6574 7920 2020 2020 oricVariety │ │ │ │ -0002c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c510: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +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 7c0a |. │ │ │ │ +0002c500: 2b2d 2d2d 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2020 2020 2020 2020 │ │ │ │ 0002c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c6d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 2020 2020 2020 2020 2020 │ │ │ │ -0002c720: 2020 2020 2020 2020 7c0a 7c6f 3230 203d |.|o20 = │ │ │ │ -0002c730: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +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 │ │ │ │ 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│ │ │ │ +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 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c850: 2d2d 2d2d 2b0a 7c69 3231 203a 2049 3d69 ----+.|i21 : I=i │ │ │ │ -0002c860: 6465 616c 2852 5f30 5e34 2a52 5f31 2c52 deal(R_0^4*R_1,R │ │ │ │ -0002c870: 5f30 2a52 5f33 2a52 5f34 2a52 5f32 2d52 _0*R_3*R_4*R_2-R │ │ │ │ -0002c880: 5f32 5e32 2a52 5f30 5e32 2920 2020 2020 _2^2*R_0^2) │ │ │ │ -0002c890: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c8a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +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 2020 2020 │ │ │ │ -0002c8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c8e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002c8f0: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ -0002c900: 2020 2032 2032 2020 2020 2020 2020 2020 2 2 │ │ │ │ -0002c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c930: 2020 2020 207c 0a7c 6f32 3120 3d20 6964 |.|o21 = id │ │ │ │ -0002c940: 6561 6c20 2878 2078 202c 202d 2078 2078 eal (x x , - x x │ │ │ │ -0002c950: 2020 2b20 7820 7820 7820 7820 2920 2020 + x x x x ) │ │ │ │ -0002c960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c980: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002c990: 2030 2031 2020 2020 2030 2032 2020 2020 0 1 0 2 │ │ │ │ -0002c9a0: 3020 3220 3320 3420 2020 2020 2020 2020 0 2 3 4 │ │ │ │ -0002c9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c9c0: 2020 2020 2020 2020 2020 207c 0a7c 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: 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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 2020 2020 │ │ │ │ -0002cc60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002cc70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002cc80: 2020 2020 2020 205a 5a5b 7820 2e2e 7820 ZZ[x ..x │ │ │ │ -0002cc90: 5d20 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 7c0a 7c20 |.| │ │ │ │ 0002cca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ccb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002ccb0: 2020 2020 2020 2020 2030 2020 2034 2020 0 4 │ │ │ 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-0002ce60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ce70: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002ce10: 2d2d 2d2b 0a7c 6932 3320 3a20 4368 3d54 ---+.|i23 : Ch=T │ │ │ │ +0002ce20: 6f72 6963 4368 6f77 5269 6e67 2858 2920 oricChowRing(X) │ │ │ │ +0002ce30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ce40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ce50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002ce60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002ce70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ce80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ce90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ceb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cec0: 2020 2020 2020 7c0a 7c6f 3233 203d 2043 |.|o23 = C │ │ │ │ -0002ced0: 6820 2020 2020 2020 2020 2020 2020 2020 h │ │ │ │ +0002cea0: 2020 2020 2020 2020 207c 0a7c 6f32 3320 |.|o23 │ │ 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5365 --+.|i24 : s3=Se │ │ │ │ -0002d000: 6772 6528 4368 2c58 2c49 2920 2020 2020 gre(Ch,X,I) │ │ │ │ +0002cfd0: 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20 7333 -----+.|i24 : s3 │ │ │ │ +0002cfe0: 3d53 6567 7265 2843 682c 582c 4929 2020 =Segre(Ch,X,I) │ │ │ │ +0002cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d030: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002d020: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002d030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d080: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0002d090: 2020 2020 2020 3220 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6564 2042 6572 7469 , provided Berti │ │ │ │ -0002d310: 6e69 2069 7320 202a 6e6f 7465 2069 6e73 ni is *note ins │ │ │ │ -0002d320: 7461 6c6c 6564 2061 6e64 2063 6f6e 6669 talled and confi │ │ │ │ -0002d330: 6775 7265 643a 0a63 6f6e 6669 6775 7269 gured:.configuri │ │ │ │ -0002d340: 6e67 2042 6572 7469 6e69 2c2e 204e 6f74 ng Bertini,. Not │ │ │ │ -0002d350: 6520 7468 6174 2074 6865 2062 6572 7469 e that the berti │ │ │ │ -0002d360: 6e69 206f 7074 696f 6e20 6973 206f 6e6c ni option is onl │ │ │ │ -0002d370: 7920 6176 6169 6c61 626c 6520 666f 720a y available for. │ │ │ │ -0002d380: 7375 6273 6368 656d 6573 206f 6620 5c50 subschemes of \P │ │ │ │ -0002d390: 505e 6e2e 0a0a 4f62 7365 7276 6520 7468 P^n...Observe th │ │ │ │ -0002d3a0: 6174 2074 6865 2061 6c67 6f72 6974 686d at the algorithm │ │ │ │ -0002d3b0: 2069 7320 6120 7072 6f62 6162 696c 6973 is a probabilis │ │ │ │ -0002d3c0: 7469 6320 616c 676f 7269 7468 6d20 616e tic algorithm an │ │ │ │ -0002d3d0: 6420 6d61 7920 6769 7665 2061 2077 726f d may give a wro │ │ │ │ -0002d3e0: 6e67 0a61 6e73 7765 7220 7769 7468 2061 ng.answer with a │ │ │ │ -0002d3f0: 2073 6d61 6c6c 2062 7574 206e 6f6e 7a65 small but nonze │ │ │ │ -0002d400: 726f 2070 726f 6261 6269 6c69 7479 2e20 ro probability. │ │ │ │ -0002d410: 5265 6164 206d 6f72 6520 756e 6465 7220 Read more under │ │ │ │ -0002d420: 2a6e 6f74 650a 7072 6f62 6162 696c 6973 *note.probabilis │ │ │ │ -0002d430: 7469 6320 616c 676f 7269 7468 6d3a 2070 tic algorithm: p │ │ │ │ -0002d440: 726f 6261 6269 6c69 7374 6963 2061 6c67 robabilistic alg │ │ │ │ -0002d450: 6f72 6974 686d 2c2e 0a0a 5761 7973 2074 orithm,...Ways t │ │ │ │ -0002d460: 6f20 7573 6520 5365 6772 653a 0a3d 3d3d o use Segre:.=== │ │ │ │ -0002d470: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0002d480: 0a20 202a 2022 5365 6772 6528 4964 6561 . * "Segre(Idea │ │ │ │ -0002d490: 6c29 220a 2020 2a20 2253 6567 7265 2849 l)". * "Segre(I │ │ │ │ -0002d4a0: 6465 616c 2c53 796d 626f 6c29 220a 2020 deal,Symbol)". │ │ │ │ -0002d4b0: 2a20 2253 6567 7265 2851 756f 7469 656e * "Segre(Quotien │ │ │ │ -0002d4c0: 7452 696e 672c 4964 6561 6c29 220a 0a46 tRing,Ideal)"..F │ │ │ │ -0002d4d0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -0002d4e0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -0002d4f0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -0002d500: 202a 6e6f 7465 2053 6567 7265 3a20 5365 *note Segre: Se │ │ │ │ -0002d510: 6772 652c 2069 7320 6120 2a6e 6f74 6520 gre, is a *note │ │ │ │ -0002d520: 6d65 7468 6f64 2066 756e 6374 696f 6e20 method function │ │ │ │ -0002d530: 7769 7468 206f 7074 696f 6e73 3a0a 284d with options:.(M │ │ │ │ -0002d540: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -0002d550: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70 odFunctionWithOp │ │ │ │ -0002d560: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d tions,...------- │ │ │ │ +0002d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0002d230: 416c 6c20 7468 6520 6578 616d 706c 6573 All the examples │ │ │ │ +0002d240: 2077 6572 6520 646f 6e65 2075 7369 6e67 were done using │ │ │ │ +0002d250: 2073 796d 626f 6c69 6320 636f 6d70 7574 symbolic comput │ │ │ │ +0002d260: 6174 696f 6e73 2077 6974 6820 4772 5c22 ations with Gr\" │ │ │ │ +0002d270: 6f62 6e65 7220 6261 7365 732e 0a43 6861 obner bases..Cha │ │ │ │ +0002d280: 6e67 696e 6720 7468 6520 6f70 7469 6f6e nging the option │ │ │ │ +0002d290: 202a 6e6f 7465 2043 6f6d 704d 6574 686f *note CompMetho │ │ │ │ +0002d2a0: 643a 2043 6f6d 704d 6574 686f 642c 2074 d: CompMethod, t │ │ │ │ +0002d2b0: 6f20 6265 7274 696e 6920 7769 6c6c 2064 o bertini will d │ │ │ │ +0002d2c0: 6f20 7468 6520 6d61 696e 0a63 6f6d 7075 o the main.compu │ │ │ │ +0002d2d0: 7461 7469 6f6e 7320 6e75 6d65 7269 6361 tations numerica │ │ │ │ +0002d2e0: 6c6c 792c 2070 726f 7669 6465 6420 4265 lly, provided Be │ │ │ │ +0002d2f0: 7274 696e 6920 6973 2020 2a6e 6f74 6520 rtini is *note │ │ │ │ +0002d300: 696e 7374 616c 6c65 6420 616e 6420 636f installed and co │ │ │ │ +0002d310: 6e66 6967 7572 6564 3a0a 636f 6e66 6967 nfigured:.config │ │ │ │ +0002d320: 7572 696e 6720 4265 7274 696e 692c 2e20 uring Bertini,. │ │ │ │ +0002d330: 4e6f 7465 2074 6861 7420 7468 6520 6265 Note that the be │ │ │ │ +0002d340: 7274 696e 6920 6f70 7469 6f6e 2069 7320 rtini option is │ │ │ │ +0002d350: 6f6e 6c79 2061 7661 696c 6162 6c65 2066 only available f │ │ │ │ +0002d360: 6f72 0a73 7562 7363 6865 6d65 7320 6f66 or.subschemes of │ │ │ │ +0002d370: 205c 5050 5e6e 2e0a 0a4f 6273 6572 7665 \PP^n...Observe │ │ │ │ +0002d380: 2074 6861 7420 7468 6520 616c 676f 7269 that the algori │ │ │ │ +0002d390: 7468 6d20 6973 2061 2070 726f 6261 6269 thm is a probabi │ │ │ │ +0002d3a0: 6c69 7374 6963 2061 6c67 6f72 6974 686d listic algorithm │ │ │ │ +0002d3b0: 2061 6e64 206d 6179 2067 6976 6520 6120 and may give a │ │ │ │ +0002d3c0: 7772 6f6e 670a 616e 7377 6572 2077 6974 wrong.answer wit │ │ │ │ +0002d3d0: 6820 6120 736d 616c 6c20 6275 7420 6e6f h a small but no │ │ │ │ +0002d3e0: 6e7a 6572 6f20 7072 6f62 6162 696c 6974 nzero probabilit │ │ │ │ +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 │ │ │ │ +0002d470: 6465 616c 2922 0a20 202a 2022 5365 6772 deal)". * "Segr │ │ │ │ +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 │ │ │ │ +0002d510: 6f6e 2077 6974 6820 6f70 7469 6f6e 733a on with options: │ │ │ │ +0002d520: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +0002d530: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ +0002d540: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ +0002d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d5b0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -0002d5c0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -0002d5d0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -0002d5e0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -0002d5f0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -0002d600: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -0002d610: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -0002d620: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ -0002d630: 6c61 7373 6573 2e6d 323a 3137 3633 3a30 lasses.m2:1763:0 │ │ │ │ -0002d640: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ -0002d650: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ -0002d660: 696e 666f 2c20 4e6f 6465 3a20 546f 7269 info, Node: Tori │ │ │ │ -0002d670: 6343 686f 7752 696e 672c 2050 7265 763a cChowRing, Prev: │ │ │ │ -0002d680: 2053 6567 7265 2c20 5570 3a20 546f 700a Segre, Up: Top. │ │ │ │ -0002d690: 0a54 6f72 6963 4368 6f77 5269 6e67 202d .ToricChowRing - │ │ │ │ -0002d6a0: 2d20 436f 6d70 7574 6573 2074 6865 2043 - Computes the C │ │ │ │ -0002d6b0: 686f 7720 7269 6e67 206f 6620 6120 6e6f how ring of a no │ │ │ │ -0002d6c0: 726d 616c 2074 6f72 6963 2076 6172 6965 rmal toric varie │ │ │ │ -0002d6d0: 7479 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ty.************* │ │ │ │ +0002d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +0002d5a0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +0002d5b0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +0002d5c0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +0002d5d0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +0002d5e0: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +0002d5f0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +0002d600: 6573 2f0a 4368 6172 6163 7465 7269 7374 es/.Characterist │ │ │ │ +0002d610: 6963 436c 6173 7365 732e 6d32 3a31 3736 icClasses.m2:176 │ │ │ │ +0002d620: 333a 302e 0a1f 0a46 696c 653a 2043 6861 3:0....File: Cha │ │ │ │ +0002d630: 7261 6374 6572 6973 7469 6343 6c61 7373 racteristicClass │ │ │ │ +0002d640: 6573 2e69 6e66 6f2c 204e 6f64 653a 2054 es.info, Node: T │ │ │ │ +0002d650: 6f72 6963 4368 6f77 5269 6e67 2c20 5072 oricChowRing, Pr │ │ │ │ +0002d660: 6576 3a20 5365 6772 652c 2055 703a 2054 ev: Segre, Up: T │ │ │ │ +0002d670: 6f70 0a0a 546f 7269 6343 686f 7752 696e op..ToricChowRin │ │ │ │ +0002d680: 6720 2d2d 2043 6f6d 7075 7465 7320 7468 g -- Computes th │ │ │ │ +0002d690: 6520 4368 6f77 2072 696e 6720 6f66 2061 e Chow ring of a │ │ │ │ +0002d6a0: 206e 6f72 6d61 6c20 746f 7269 6320 7661 normal toric va │ │ │ │ +0002d6b0: 7269 6574 790a 2a2a 2a2a 2a2a 2a2a 2a2a riety.********** │ │ │ │ +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 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d710: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -0002d720: 200a 2020 2020 2020 2020 546f 7269 6343 . ToricC │ │ │ │ -0002d730: 686f 7752 696e 6720 580a 2020 2a20 496e howRing X. * In │ │ │ │ -0002d740: 7075 7473 3a0a 2020 2020 2020 2a20 522c puts:. * R, │ │ │ │ -0002d750: 2061 202a 6e6f 7465 206e 6f72 6d61 6c20 a *note normal │ │ │ │ -0002d760: 746f 7269 6320 7661 7269 6574 793a 0a20 toric variety:. │ │ │ │ -0002d770: 2020 2020 2020 2028 4e6f 726d 616c 546f (NormalTo │ │ │ │ -0002d780: 7269 6356 6172 6965 7469 6573 294e 6f72 ricVarieties)Nor │ │ │ │ -0002d790: 6d61 6c54 6f72 6963 5661 7269 6574 792c malToricVariety, │ │ │ │ -0002d7a0: 2c20 4120 6e6f 726d 616c 2074 6f72 6963 , A normal toric │ │ │ │ -0002d7b0: 2076 6172 6965 7479 0a20 202a 204f 7574 variety. * Out │ │ │ │ -0002d7c0: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ -0002d7d0: 2a6e 6f74 6520 7175 6f74 6965 6e74 2072 *note quotient r │ │ │ │ -0002d7e0: 696e 673a 2028 4d61 6361 756c 6179 3244 ing: (Macaulay2D │ │ │ │ -0002d7f0: 6f63 2951 756f 7469 656e 7452 696e 672c oc)QuotientRing, │ │ │ │ -0002d800: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ -0002d810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4c 6574 ===========..Let │ │ │ │ -0002d820: 2058 2062 6520 6120 746f 7269 6320 7661 X be a toric va │ │ │ │ -0002d830: 7269 6574 7920 7769 7468 2074 6f74 616c riety with total │ │ │ │ -0002d840: 2063 6f6f 7264 696e 6174 6520 7269 6e67 coordinate ring │ │ │ │ -0002d850: 2028 436f 7820 7269 6e67 2920 522e 2054 (Cox ring) R. T │ │ │ │ -0002d860: 6869 7320 6d65 7468 6f64 0a63 6f6d 7075 his method.compu │ │ │ │ -0002d870: 7465 7320 7468 6520 4368 6f77 2072 696e tes the Chow rin │ │ │ │ -0002d880: 6720 2043 686f 7720 7269 6e67 2043 683d g Chow ring Ch= │ │ │ │ -0002d890: 522f 2853 522b 4c52 2920 6f66 2058 3b20 R/(SR+LR) of X; │ │ │ │ -0002d8a0: 6865 7265 2053 5220 6973 2074 6865 0a53 here SR is the.S │ │ │ │ -0002d8b0: 7461 6e6c 6579 2d52 6569 736e 6572 2069 tanley-Reisner i │ │ │ │ -0002d8c0: 6465 616c 206f 6620 7468 6520 636f 7272 deal of the corr │ │ │ │ -0002d8d0: 6573 706f 6e64 696e 6720 6661 6e20 616e esponding fan an │ │ │ │ -0002d8e0: 6420 4c52 2069 7320 7468 6520 6964 6561 d LR is the idea │ │ │ │ -0002d8f0: 6c20 6f66 206c 696e 6561 720a 7265 6c61 l of linear.rela │ │ │ │ -0002d900: 7469 6f6e 7320 616d 6f75 6e74 2074 6865 tions amount the │ │ │ │ -0002d910: 2072 6179 732e 2049 7420 6973 206e 6565 rays. It is nee │ │ │ │ -0002d920: 6465 6420 666f 7220 696e 7075 7420 696e ded for input in │ │ │ │ -0002d930: 746f 2074 6865 206d 6574 686f 6473 202a to the methods * │ │ │ │ -0002d940: 6e6f 7465 2053 6567 7265 3a0a 5365 6772 note Segre:.Segr │ │ │ │ -0002d950: 652c 2c20 2a6e 6f74 6520 4368 6572 6e3a e,, *note Chern: │ │ │ │ -0002d960: 2043 6865 726e 2c20 616e 6420 2a6e 6f74 Chern, and *not │ │ │ │ -0002d970: 6520 4353 4d3a 2043 534d 2c20 696e 2074 e CSM: CSM, in t │ │ │ │ -0002d980: 6865 2063 6173 6573 2077 6865 7265 2061 he cases where a │ │ │ │ -0002d990: 2074 6f72 6963 0a76 6172 6965 7479 2069 toric.variety i │ │ │ │ -0002d9a0: 7320 616c 736f 2069 6e70 7574 2074 6f20 s also input to │ │ │ │ -0002d9b0: 656e 7375 7265 2074 6861 7420 7468 6573 ensure that thes │ │ │ │ -0002d9c0: 6520 6d65 7468 6f64 7320 7265 7475 726e e methods return │ │ │ │ -0002d9d0: 2072 6573 756c 7473 2069 6e20 7468 6520 results in the │ │ │ │ -0002d9e0: 7361 6d65 0a72 696e 672e 2057 6520 6769 same.ring. We gi │ │ │ │ -0002d9f0: 7665 2061 6e20 6578 616d 706c 6520 6f66 ve an example of │ │ │ │ -0002da00: 2074 6865 2075 7365 206f 6620 7468 6973 the use of this │ │ │ │ -0002da10: 206d 6574 686f 6420 746f 2077 6f72 6b20 method to work │ │ │ │ -0002da20: 7769 7468 2065 6c65 6d65 6e74 7320 6f66 with elements of │ │ │ │ -0002da30: 2074 6865 0a43 686f 7720 7269 6e67 206f the.Chow ring o │ │ │ │ -0002da40: 6620 6120 746f 7269 6320 7661 7269 6574 f a toric variet │ │ │ │ -0002da50: 790a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d y..+------------ │ │ │ │ +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 │ │ │ │ +0002d740: 616c 2074 6f72 6963 2076 6172 6965 7479 al toric variety │ │ │ │ +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 │ │ │ │ +0002d780: 7479 2c2c 2041 206e 6f72 6d61 6c20 746f ty,, A normal to │ │ │ │ +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 │ │ │ │ +0002d7d0: 7932 446f 6329 5175 6f74 6965 6e74 5269 y2Doc)QuotientRi │ │ │ │ +0002d7e0: 6e67 2c2c 200a 0a44 6573 6372 6970 7469 ng,, ..Descripti │ │ │ │ +0002d7f0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0002d800: 4c65 7420 5820 6265 2061 2074 6f72 6963 Let X be a toric │ │ │ │ +0002d810: 2076 6172 6965 7479 2077 6974 6820 746f variety with to │ │ │ │ +0002d820: 7461 6c20 636f 6f72 6469 6e61 7465 2072 tal coordinate r │ │ │ │ +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 │ │ │ │ +0002d850: 6d70 7574 6573 2074 6865 2043 686f 7720 mputes the Chow │ │ │ │ +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 │ │ │ │ +0002d890: 650a 5374 616e 6c65 792d 5265 6973 6e65 e.Stanley-Reisne │ │ │ │ +0002d8a0: 7220 6964 6561 6c20 6f66 2074 6865 2063 r ideal of the c │ │ │ │ +0002d8b0: 6f72 7265 7370 6f6e 6469 6e67 2066 616e orresponding fan │ │ │ │ +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 │ │ │ │ +0002d900: 6e65 6564 6564 2066 6f72 2069 6e70 7574 needed for input │ │ │ │ +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 │ │ │ │ +0002d960: 6e20 7468 6520 6361 7365 7320 7768 6572 n the cases wher │ │ │ │ +0002d970: 6520 6120 746f 7269 630a 7661 7269 6574 e a toric.variet │ │ │ │ +0002d980: 7920 6973 2061 6c73 6f20 696e 7075 7420 y is also input │ │ │ │ +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 │ │ │ │ +0002d9b0: 7572 6e20 7265 7375 6c74 7320 696e 2074 urn results in t │ │ │ │ +0002d9c0: 6865 2073 616d 650a 7269 6e67 2e20 5765 he same.ring. We │ │ │ │ +0002d9d0: 2067 6976 6520 616e 2065 7861 6d70 6c65 give an example │ │ │ │ +0002d9e0: 206f 6620 7468 6520 7573 6520 6f66 2074 of the use of t │ │ │ │ +0002d9f0: 6869 7320 6d65 7468 6f64 2074 6f20 776f his method to wo │ │ │ │ +0002da00: 726b 2077 6974 6820 656c 656d 656e 7473 rk with elements │ │ │ │ +0002da10: 206f 6620 7468 650a 4368 6f77 2072 696e of the.Chow rin │ │ │ │ +0002da20: 6720 6f66 2061 2074 6f72 6963 2076 6172 g of a toric var │ │ │ │ +0002da30: 6965 7479 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d iety..+--------- │ │ │ │ +0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002daa0: 2d2b 0a7c 6931 203a 206e 6565 6473 5061 -+.|i1 : needsPa │ │ │ │ -0002dab0: 636b 6167 6520 224e 6f72 6d61 6c54 6f72 ckage "NormalTor │ │ │ │ -0002dac0: 6963 5661 7269 6574 6965 7322 2020 2020 icVarieties" │ │ │ │ -0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002da80: 2d2d 2d2d 2b0a 7c69 3120 3a20 6e65 6564 ----+.|i1 : need │ │ │ │ +0002da90: 7350 6163 6b61 6765 2022 4e6f 726d 616c sPackage "Normal │ │ │ │ +0002daa0: 546f 7269 6356 6172 6965 7469 6573 2220 ToricVarieties" │ │ │ │ +0002dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002daf0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db40: 207c 0a7c 6f31 203d 204e 6f72 6d61 6c54 |.|o1 = NormalT │ │ │ │ -0002db50: 6f72 6963 5661 7269 6574 6965 7320 2020 oricVarieties │ │ │ │ +0002db20: 2020 2020 7c0a 7c6f 3120 3d20 4e6f 726d |.|o1 = Norm │ │ │ │ +0002db30: 616c 546f 7269 6356 6172 6965 7469 6573 alToricVarieties │ │ │ │ +0002db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db70: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002db90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbe0: 207c 0a7c 6f31 203a 2050 6163 6b61 6765 |.|o1 : Package │ │ │ │ +0002dbc0: 2020 2020 7c0a 7c6f 3120 3a20 5061 636b |.|o1 : Pack │ │ │ │ +0002dbd0: 6167 6520 2020 2020 2020 2020 2020 2020 age │ │ │ │ +0002dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc80: 2d2b 0a7c 6932 203a 2052 686f 203d 207b -+.|i2 : Rho = { │ │ │ │ -0002dc90: 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 7d2c {1,0,0},{0,1,0}, │ │ │ │ -0002dca0: 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 2c30 {0,0,1},{-1,-1,0 │ │ │ │ -0002dcb0: 7d2c 7b30 2c30 2c2d 317d 7d20 2020 2020 },{0,0,-1}} │ │ │ │ +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: 207c 0a7c 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd20: 207c 0a7c 6f32 203d 207b 7b31 2c20 302c |.|o2 = {{1, 0, │ │ │ │ -0002dd30: 2030 7d2c 207b 302c 2031 2c20 307d 2c20 0}, {0, 1, 0}, │ │ │ │ -0002dd40: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20 {0, 0, 1}, {-1, │ │ │ │ -0002dd50: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d -1, 0}, {0, 0, - │ │ │ │ -0002dd60: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ -0002dd70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 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: 207c 0a7c 6f32 203a 204c 6973 7420 2020 |.|o2 : List │ │ │ │ +0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de60: 2d2b 0a7c 6933 203a 2053 6967 6d61 203d -+.|i3 : Sigma = │ │ │ │ -0002de70: 207b 7b30 2c31 2c32 7d2c 7b31 2c32 2c33 {{0,1,2},{1,2,3 │ │ │ │ -0002de80: 7d2c 7b30 2c32 2c33 7d2c 7b30 2c31 2c34 },{0,2,3},{0,1,4 │ │ │ │ -0002de90: 7d2c 7b31 2c33 2c34 7d2c 7b30 2c33 2c34 },{1,3,4},{0,3,4 │ │ │ │ -0002dea0: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ -0002deb0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df00: 207c 0a7c 6f33 203d 207b 7b30 2c20 312c |.|o3 = {{0, 1, │ │ │ │ -0002df10: 2032 7d2c 207b 312c 2032 2c20 337d 2c20 2}, {1, 2, 3}, │ │ │ │ -0002df20: 7b30 2c20 322c 2033 7d2c 207b 302c 2031 {0, 2, 3}, {0, 1 │ │ │ │ -0002df30: 2c20 347d 2c20 7b31 2c20 332c 2034 7d2c , 4}, {1, 3, 4}, │ │ │ │ -0002df40: 207b 302c 2033 2c20 347d 7d20 2020 2020 {0, 3, 4}} │ │ │ │ -0002df50: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 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: 207c 0a7c 6f33 203a 204c 6973 7420 2020 |.|o3 : List │ │ │ │ +0002dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dff0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e040: 2d2b 0a7c 6934 203a 2058 203d 206e 6f72 -+.|i4 : X = nor │ │ │ │ -0002e050: 6d61 6c54 6f72 6963 5661 7269 6574 7928 malToricVariety( │ │ │ │ -0002e060: 5268 6f2c 5369 676d 612c 436f 6566 6669 Rho,Sigma,Coeffi │ │ │ │ -0002e070: 6369 656e 7452 696e 6720 3d3e 5a5a 2f33 cientRing =>ZZ/3 │ │ │ │ -0002e080: 3237 3439 2920 2020 2020 2020 2020 2020 2749) │ │ │ │ -0002e090: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 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: 207c 0a7c 6f34 203d 2058 2020 2020 2020 |.|o4 = X │ │ │ │ +0002e0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e110: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002e120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e130: 207c 0a7c 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 2020 2020 2020 │ │ │ │ -0002e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e180: 207c 0a7c 6f34 203a 204e 6f72 6d61 6c54 |.|o4 : NormalT │ │ │ │ -0002e190: 6f72 6963 5661 7269 6574 7920 2020 2020 oricVariety │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e220: 2d2b 0a7c 6935 203a 2052 3d72 696e 6720 -+.|i5 : R=ring │ │ │ │ -0002e230: 5820 2020 2020 2020 2020 2020 2020 2020 X │ │ │ │ +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 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: 207c 0a7c 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 2020 2020 2020 │ │ │ │ +0002e2a0: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020 |.|o5 = R │ │ │ │ 0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2c0: 207c 0a7c 6f35 203d 2052 2020 2020 2020 |.|o5 = R │ │ │ │ +0002e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e2f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e310: 207c 0a7c 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: 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│ -0002e450: 207c 0a7c 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 2020 2020 2020 │ │ │ │ +0002e480: 2020 2020 7c0a 7c6f 3620 3d20 4368 2020 |.|o6 = Ch │ │ │ │ 0002e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4a0: 207c 0a7c 6f36 203d 2043 6820 2020 2020 |.|o6 = Ch │ │ │ │ +0002e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e4d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4f0: 207c 0a7c 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 2020 2020 2020 │ │ │ │ -0002e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e540: 207c 0a7c 6f36 203a 2051 756f 7469 656e |.|o6 : Quotien │ │ │ │ -0002e550: 7452 696e 6720 2020 2020 2020 2020 2020 tRing │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e590: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5e0: 2d2b 0a7c 6937 203a 2064 6573 6372 6962 -+.|i7 : describ │ │ │ │ -0002e5f0: 6520 4368 2020 2020 2020 2020 2020 2020 e Ch │ │ │ │ +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 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: 207c 0a7c 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 2020 2020 2020 │ │ │ │ -0002e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e680: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002e690: 2020 2020 2020 2020 205a 5a5b 7820 2e2e ZZ[x .. │ │ │ │ -0002e6a0: 7820 5d20 2020 2020 2020 2020 2020 2020 x ] │ │ │ │ -0002e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ 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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 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 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e860: 2d2b 0a7c 6938 203a 2072 3d67 656e 7320 -+.|i8 : r=gens │ │ │ │ -0002e870: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +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 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: 207c 0a7c 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 2020 2020 2020 │ │ │ │ -0002e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e900: 207c 0a7c 6f38 203d 207b 7820 2c20 7820 |.|o8 = {x , x │ │ │ │ -0002e910: 2c20 7820 2c20 7820 2c20 7820 7d20 2020 , x , x , x } │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e950: 207c 0a7c 2020 2020 2020 2030 2020 2031 |.| 0 1 │ │ │ │ -0002e960: 2020 2032 2020 2033 2020 2034 2020 2020 2 3 4 │ │ │ │ +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 2020 2020 2020 │ │ │ │ +0002e980: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e9a0: 207c 0a7c 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 2020 2020 2020 │ │ │ │ +0002e9d0: 2020 2020 7c0a 7c6f 3820 3a20 4c69 7374 |.|o8 : List │ │ │ │ 0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e9f0: 207c 0a7c 6f38 203a 204c 6973 7420 2020 |.|o8 : List │ │ │ │ +0002e9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ea00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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│ │ │ -0002eb50: 3633 3136 7820 2920 2020 2020 2020 2020 6316x ) │ │ │ │ -0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb80: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002eb90: 2020 2030 2020 2020 2020 2020 3120 2020 0 1 │ │ │ │ -0002eba0: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ -0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebd0: 207c 0a7c 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 2020 2020 2020 │ │ │ │ -0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec20: 207c 0a7c 6f39 203a 2049 6465 616c 206f |.|o9 : Ideal o │ │ │ │ -0002ec30: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ +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 2020 2020 2020 │ │ │ │ -0002ec60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ecc0: 2d2b 0a7c 6931 3020 3a20 4b3d 6964 6561 -+.|i10 : K=idea │ │ │ │ -0002ecd0: 6c28 7261 6e64 6f6d 287b 312c 317d 2c52 l(random({1,1},R │ │ │ │ -0002ece0: 2929 2020 2020 2020 2020 2020 2020 2020 )) │ │ │ │ -0002ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 2020 │ │ │ │ -0002f750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f760: 207c 0a7c 6f31 3420 3d20 3278 2078 2020 |.|o14 = 2x x │ │ │ │ -0002f770: 2b20 7820 202b 2078 2078 2020 2020 2020 + x + x x │ │ │ │ +0002f740: 2020 2020 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 2020 2020 2020 │ │ │ │ -0002f7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7b0: 207c 0a7c 2020 2020 2020 2020 3320 3420 |.| 3 4 │ │ │ │ -0002f7c0: 2020 2033 2020 2020 3320 3420 2020 2020 3 3 4 │ │ │ │ +0002f790: 2020 2020 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: 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2020 2020 │ │ │ │ -0002f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002f880: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002f890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f8f0: 2d2b 0a0a 466f 7220 7468 6520 7072 6f67 -+..For the prog │ │ │ │ -0002f900: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -0002f910: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -0002f920: 626a 6563 7420 2a6e 6f74 6520 546f 7269 bject *note Tori │ │ │ │ -0002f930: 6343 686f 7752 696e 673a 2054 6f72 6963 cChowRing: Toric │ │ │ │ -0002f940: 4368 6f77 5269 6e67 2c20 6973 2061 202a ChowRing, is a * │ │ │ │ -0002f950: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ -0002f960: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ -0002f970: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -0002f980: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0002f8d0: 2d2d 2d2d 2b0a 0a46 6f72 2074 6865 2070 ----+..For the p │ │ │ │ +0002f8e0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +0002f8f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +0002f900: 6520 6f62 6a65 6374 202a 6e6f 7465 2054 e object *note T │ │ │ │ +0002f910: 6f72 6963 4368 6f77 5269 6e67 3a20 546f oricChowRing: To │ │ │ │ +0002f920: 7269 6343 686f 7752 696e 672c 2069 7320 ricChowRing, is │ │ │ │ +0002f930: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +0002f940: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +0002f950: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +0002f960: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +0002f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 2d2d 2d2d ---------------- │ │ │ │ -0002f9d0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -0002f9e0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0002f9f0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -0002fa00: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -0002fa10: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -0002fa20: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -0002fa30: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ -0002fa40: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ -0002fa50: 7365 732e 6d32 3a31 3935 313a 302e 0a1f ses.m2:1951:0... │ │ │ │ -0002fa60: 0a54 6167 2054 6162 6c65 3a0a 4e6f 6465 .Tag Table:.Node │ │ │ │ -0002fa70: 3a20 546f 707f 3239 310a 4e6f 6465 3a20 : Top.291.Node: │ │ │ │ -0002fa80: 6265 7274 696e 6943 6865 636b 7f31 3637 bertiniCheck.167 │ │ │ │ -0002fa90: 3035 0a4e 6f64 653a 2043 6865 636b 536d 05.Node: CheckSm │ │ │ │ -0002faa0: 6f6f 7468 7f31 3739 3232 0a4e 6f64 653a ooth.17922.Node: │ │ │ │ -0002fab0: 2043 6865 636b 546f 7269 6356 6172 6965 CheckToricVarie │ │ │ │ -0002fac0: 7479 5661 6c69 647f 3232 3731 380a 4e6f tyValid.22718.No │ │ │ │ -0002fad0: 6465 3a20 4368 6572 6e7f 3333 3730 390a de: Chern.33709. │ │ │ │ -0002fae0: 4e6f 6465 3a20 4368 6f77 5269 6e67 7f35 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ToricChowRing.18 │ │ │ │ -0002fc90: 3539 3232 0a1f 0a45 6e64 2054 6167 2054 5922...End Tag T │ │ │ │ -0002fca0: 6162 6c65 0a able. │ │ │ │ +0002f9b0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0002f9c0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0002f9d0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0002f9e0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0002f9f0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0002fa00: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +0002fa10: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0002fa20: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ +0002fa30: 6c61 7373 6573 2e6d 323a 3139 3531 3a30 lasses.m2:1951:0 │ │ │ │ +0002fa40: 2e0a 1f0a 5461 6720 5461 626c 653a 0a4e ....Tag Table:.N │ │ │ │ +0002fa50: 6f64 653a 2054 6f70 7f32 3931 0a4e 6f64 ode: Top.291.Nod │ │ │ │ +0002fa60: 653a 2062 6572 7469 6e69 4368 6563 6b7f e: bertiniCheck. │ │ │ │ +0002fa70: 3136 3730 350a 4e6f 6465 3a20 4368 6563 16705.Node: Chec │ │ │ │ +0002fa80: 6b53 6d6f 6f74 687f 3137 3932 320a 4e6f kSmooth.17922.No │ │ │ │ +0002fa90: 6465 3a20 4368 6563 6b54 6f72 6963 5661 de: CheckToricVa │ │ │ │ +0002faa0: 7269 6574 7956 616c 6964 7f32 3237 3138 rietyValid.22718 │ │ │ │ +0002fab0: 0a4e 6f64 653a 2043 6865 726e 7f33 3337 .Node: Chern.337 │ │ │ │ +0002fac0: 3039 0a4e 6f64 653a 2043 686f 7752 696e 09.Node: ChowRin │ │ │ │ +0002fad0: 677f 3531 3231 370a 4e6f 6465 3a20 436c g.51217.Node: Cl │ │ │ │ +0002fae0: 6173 7349 6e43 686f 7752 696e 677f 3538 assInChowRing.58 │ │ │ │ +0002faf0: 3934 380a 4e6f 6465 3a20 436c 6173 7349 948.Node: ClassI │ │ │ │ +0002fb00: 6e54 6f72 6963 4368 6f77 5269 6e67 7f36 nToricChowRing.6 │ │ │ │ +0002fb10: 3133 3038 0a4e 6f64 653a 2043 6f6d 704d 1308.Node: CompM │ │ │ │ +0002fb20: 6574 686f 647f 3636 3535 310a 4e6f 6465 ethod.66551.Node │ │ │ │ +0002fb30: 3a20 636f 6e66 6967 7572 696e 6720 4265 : configuring Be │ │ │ │ +0002fb40: 7274 696e 697f 3736 3437 380a 4e6f 6465 rtini.76478.Node │ │ │ │ +0002fb50: 3a20 4353 4d7f 3738 3131 330a 4e6f 6465 : CSM.78113.Node │ │ │ │ +0002fb60: 3a20 4575 6c65 727f 3130 3137 3531 0a4e : Euler.101751.N │ │ │ │ +0002fb70: 6f64 653a 2045 756c 6572 4166 6669 6e65 ode: EulerAffine │ │ │ │ +0002fb80: 7f31 3139 3632 380a 4e6f 6465 3a20 496e .119628.Node: In │ │ │ │ +0002fb90: 6473 4f66 536d 6f6f 7468 7f31 3232 3134 dsOfSmooth.12214 │ │ │ │ +0002fba0: 300a 4e6f 6465 3a20 496e 7075 7449 7353 0.Node: InputIsS │ │ │ │ +0002fbb0: 6d6f 6f74 687f 3132 3630 3935 0a4e 6f64 mooth.126095.Nod │ │ │ │ +0002fbc0: 653a 2069 734d 756c 7469 486f 6d6f 6765 e: isMultiHomoge │ │ │ │ +0002fbd0: 6e65 6f75 737f 3133 3030 3738 0a4e 6f64 neous.130078.Nod │ │ │ │ +0002fbe0: 653a 204d 6574 686f 647f 3133 3431 3536 e: Method.134156 │ │ │ │ +0002fbf0: 0a4e 6f64 653a 204d 756c 7469 5072 6f6a .Node: MultiProj │ │ │ │ +0002fc00: 436f 6f72 6452 696e 677f 3133 3831 3035 CoordRing.138105 │ │ │ │ +0002fc10: 0a4e 6f64 653a 204f 7574 7075 747f 3134 .Node: Output.14 │ │ │ │ +0002fc20: 3437 3733 0a4e 6f64 653a 2070 726f 6261 4773.Node: proba │ │ │ │ +0002fc30: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ +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/CohomCalg.info.gz │ │ │ ├── CohomCalg.info │ │ │ │ @@ -1042,15 +1042,15 @@ │ │ │ │ 00004110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00004130: 7c69 3230 203a 2065 6c61 7073 6564 5469 |i20 : elapsedTi │ │ │ │ 00004140: 6d65 2068 7665 6373 203d 2063 6f68 6f6d me hvecs = cohom │ │ │ │ 00004150: 4361 6c67 2858 2c20 4432 2920 2020 2020 Calg(X, D2) │ │ │ │ 00004160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004170: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00004180: 7c20 2d2d 2033 2e30 3931 3431 7320 656c | -- 3.09141s el │ │ │ │ +00004180: 7c20 2d2d 2032 2e37 3938 3937 7320 656c | -- 2.79897s 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 │ │ │ │ @@ -1677,16 +1677,16 @@ │ │ │ │ 000068c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000068d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000068e0: 7c69 3233 203a 2065 6c61 7073 6564 5469 |i23 : elapsedTi │ │ │ │ 000068f0: 6d65 2063 6f68 6f6d 7665 6331 203d 2063 me cohomvec1 = c │ │ │ │ 00006900: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058 ohomCalg(X_3 + X │ │ │ │ 00006910: 5f37 202b 2058 5f38 2920 2020 2020 2020 _7 + X_8) │ │ │ │ 00006920: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00006930: 7c20 2d2d 202e 3338 3039 7320 656c 6170 | -- .3809s elap │ │ │ │ -00006940: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ +00006930: 7c20 2d2d 202e 3433 3630 3373 2065 6c61 | -- .43603s ela │ │ │ │ +00006940: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00006950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006970: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00006980: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00006990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000069a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000069b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1712,15 +1712,15 @@ │ │ │ │ 00006af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00006b10: 7c69 3234 203a 2065 6c61 7073 6564 5469 |i24 : elapsedTi │ │ │ │ 00006b20: 6d65 2063 6f68 6f6d 7665 6332 203d 2066 me cohomvec2 = f │ │ │ │ 00006b30: 6f72 206a 2066 726f 6d20 3020 746f 2064 or j from 0 to d │ │ │ │ 00006b40: 696d 2058 206c 6973 7420 7261 6e6b 2048 im X list rank H │ │ │ │ 00006b50: 485e 6a28 582c 2020 2020 2020 2020 7c0a H^j(X, |. │ │ │ │ -00006b60: 7c20 2d2d 2031 312e 3531 3735 7320 656c | -- 11.5175s el │ │ │ │ +00006b60: 7c20 2d2d 2038 2e31 3230 3239 7320 656c | -- 8.12029s el │ │ │ │ 00006b70: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00006b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00006bb0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00006bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1797,15 +1797,15 @@ │ │ │ │ 00007040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00007060: 7c69 3237 203a 2065 6c61 7073 6564 5469 |i27 : elapsedTi │ │ │ │ 00007070: 6d65 2063 6f68 6f6d 7665 6331 203d 2063 me cohomvec1 = c │ │ │ │ 00007080: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058 ohomCalg(X_3 + X │ │ │ │ 00007090: 5f37 202d 2058 5f38 2920 2020 2020 2020 _7 - X_8) │ │ │ │ 000070a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000070b0: 7c20 2d2d 202e 3339 3739 3238 7320 656c | -- .397928s el │ │ │ │ +000070b0: 7c20 2d2d 202e 3432 3837 3037 7320 656c | -- .428707s el │ │ │ │ 000070c0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 000070d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000070e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000070f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00007100: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00007110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1832,20 +1832,20 @@ │ │ │ │ 00007270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00007290: 7c69 3238 203a 2065 6c61 7073 6564 5469 |i28 : elapsedTi │ │ │ │ 000072a0: 6d65 2063 6f68 6f6d 7665 6332 203d 2065 me cohomvec2 = e │ │ │ │ 000072b0: 6c61 7073 6564 5469 6d65 2066 6f72 206a lapsedTime for j │ │ │ │ 000072c0: 2066 726f 6d20 3020 746f 2064 696d 2058 from 0 to dim X │ │ │ │ 000072d0: 206c 6973 7420 7261 6e6b 2020 2020 7c0a list rank |. │ │ │ │ -000072e0: 7c20 2d2d 202e 3535 3339 3738 7320 656c | -- .553978s el │ │ │ │ +000072e0: 7c20 2d2d 202e 3338 3037 3033 7320 656c | -- .380703s el │ │ │ │ 000072f0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00007300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007320: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00007330: 7c20 2d2d 202e 3535 3430 3039 7320 656c | -- .554009s el │ │ │ │ +00007330: 7c20 2d2d 202e 3338 3037 3239 7320 656c | -- .380729s el │ │ │ │ 00007340: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00007350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007370: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00007380: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00007390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000073a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/CompleteIntersectionResolutions.info.gz │ │ │ ├── CompleteIntersectionResolutions.info │ │ │ │ @@ -4344,16 +4344,16 @@ │ │ │ │ 00010f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010f80: 2d2d 2d2b 0a7c 6937 203a 2074 696d 6520 ---+.|i7 : time │ │ │ │ 00010f90: 4720 3d20 4569 7365 6e62 7564 5368 616d G = EisenbudSham │ │ │ │ 00010fa0: 6173 6828 6666 2c46 2c6c 656e 2920 2020 ash(ff,F,len) │ │ │ │ 00010fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fd0: 2020 207c 0a7c 202d 2d20 7573 6564 2036 |.| -- used 6 │ │ │ │ -00010fe0: 2e36 3833 3932 7320 2863 7075 293b 2035 .68392s (cpu); 5 │ │ │ │ -00010ff0: 2e30 3138 3533 7320 2874 6872 6561 6429 .01853s (thread) │ │ │ │ +00010fe0: 2e34 3538 3836 7320 2863 7075 293b 2035 .45886s (cpu); 5 │ │ │ │ +00010ff0: 2e30 3232 3738 7320 2874 6872 6561 6429 .02278s (thread) │ │ │ │ 00011000: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00011010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011020: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4884,16 +4884,16 @@ │ │ │ │ 00013130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013150: 2d2b 0a7c 6932 3020 3a20 4646 203d 2074 -+.|i20 : FF = t │ │ │ │ 00013160: 696d 6520 5368 616d 6173 6828 5231 2c46 ime Shamash(R1,F │ │ │ │ 00013170: 2c34 2920 2020 2020 2020 2020 2020 2020 ,4) │ │ │ │ 00013180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013190: 2020 207c 0a7c 202d 2d20 7573 6564 2030 |.| -- used 0 │ │ │ │ -000131a0: 2e32 3332 3835 3873 2028 6370 7529 3b20 .232858s (cpu); │ │ │ │ -000131b0: 302e 3135 3631 3436 7320 2874 6872 6561 0.156146s (threa │ │ │ │ +000131a0: 2e31 3733 3238 3773 2028 6370 7529 3b20 .173287s (cpu); │ │ │ │ +000131b0: 302e 3130 3635 3131 7320 2874 6872 6561 0.106511s (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 │ │ │ │ @@ -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 3830 3632 3273 2028 6370 ed 0.980622s (cp │ │ │ │ -00013440: 7529 3b20 302e 3735 3432 3438 7320 2874 u); 0.754248s (t │ │ │ │ -00013450: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00013430: 6564 2030 2e39 3339 3532 3773 2028 6370 ed 0.939527s (cp │ │ │ │ +00013440: 7529 3b20 302e 3734 3233 3573 2028 7468 u); 0.74235s (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 2d2d ---------------- │ │ │ │ 00013780: 2b0a 7c69 3232 203a 2047 4720 3d20 7469 +.|i22 : GG = ti │ │ │ │ 00013790: 6d65 2045 6973 656e 6275 6453 6861 6d61 me EisenbudShama │ │ │ │ 000137a0: 7368 2852 312c 465b 325d 2c34 2920 2020 sh(R1,F[2],4) │ │ │ │ 000137b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000137c0: 7365 6420 302e 3939 3436 3636 7320 2863 sed 0.994666s (c │ │ │ │ -000137d0: 7075 293b 2030 2e37 3334 3738 3373 2028 pu); 0.734783s ( │ │ │ │ +000137c0: 7365 6420 302e 3935 3432 3437 7320 2863 sed 0.954247s (c │ │ │ │ +000137d0: 7075 293b 2030 2e37 3331 3730 3773 2028 pu); 0.731707s ( │ │ │ │ 000137e0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 000137f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00013800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013820: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00013830: 2020 2031 2020 2020 2020 2036 2020 2020 1 6 │ │ │ │ 00013840: 2020 2031 3820 2020 2020 2020 3338 2020 18 38 │ │ │ │ @@ -28228,45 +28228,45 @@ │ │ │ │ 0006e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ 0006e460: 7375 6d54 776f 4d6f 6e6f 6d69 616c 7328 sumTwoMonomials( │ │ │ │ 0006e470: 322c 3329 2020 2020 2020 2020 2020 2020 2,3) │ │ │ │ 0006e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e490: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -0006e4a0: 3534 3735 3131 7320 2863 7075 293b 2030 547511s (cpu); 0 │ │ │ │ -0006e4b0: 2e34 3130 3232 7320 2874 6872 6561 6429 .41022s (thread) │ │ │ │ -0006e4c0: 3b20 3073 2028 6763 2920 2020 7c0a 7c32 ; 0s (gc) |.|2 │ │ │ │ +0006e4a0: 3634 3733 3333 7320 2863 7075 293b 2030 647333s (cpu); 0 │ │ │ │ +0006e4b0: 2e34 3033 3332 3373 2028 7468 7265 6164 .403323s (thread │ │ │ │ +0006e4c0: 293b 2030 7320 2867 6329 2020 7c0a 7c32 ); 0s (gc) |.|2 │ │ │ │ 0006e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e500: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b |.|Tally{{ │ │ │ │ 0006e510: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20 {2, 2}, {1, 2}} │ │ │ │ 0006e520: 3d3e 2033 7d20 2020 2020 2020 2020 2020 => 3} │ │ │ │ 0006e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e540: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e570: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0006e580: 2075 7365 6420 302e 3130 3638 3434 7320 used 0.106844s │ │ │ │ -0006e590: 2863 7075 293b 2030 2e31 3036 3431 7320 (cpu); 0.10641s │ │ │ │ -0006e5a0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0006e5b0: 2920 2020 7c0a 7c33 2020 2020 2020 2020 ) |.|3 │ │ │ │ +0006e580: 2075 7365 6420 302e 3133 3433 3831 7320 used 0.134381s │ │ │ │ +0006e590: 2863 7075 293b 2030 2e31 3131 3437 3673 (cpu); 0.111476s │ │ │ │ +0006e5a0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +0006e5b0: 6329 2020 7c0a 7c33 2020 2020 2020 2020 c) |.|3 │ │ │ │ 0006e5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e5e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0006e5f0: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20 |Tally{{{2, 2}, │ │ │ │ 0006e600: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020 {1, 2}} => 1} │ │ │ │ 0006e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e620: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0006e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e660: 2020 7c0a 7c20 2d2d 2075 7365 6420 342e |.| -- used 4. │ │ │ │ -0006e670: 3035 3865 2d30 3673 2028 6370 7529 3b20 058e-06s (cpu); │ │ │ │ -0006e680: 332e 3438 3665 2d30 3673 2028 7468 7265 3.486e-06s (thre │ │ │ │ +0006e660: 2020 7c0a 7c20 2d2d 2075 7365 6420 332e |.| -- used 3. │ │ │ │ +0006e670: 3535 3265 2d30 3673 2028 6370 7529 3b20 552e-06s (cpu); │ │ │ │ +0006e680: 322e 3432 3565 2d30 3673 2028 7468 7265 2.425e-06s (thre │ │ │ │ 0006e690: 6164 293b 2030 7320 2867 6329 7c0a 7c34 ad); 0s (gc)|.|4 │ │ │ │ 0006e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e6d0: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7d |.|Tally{} │ │ │ │ 0006e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -28810,17 +28810,17 @@ │ │ │ │ 00070890: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000708a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000708b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000708c0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7477 ------+.|i2 : tw │ │ │ │ 000708d0: 6f4d 6f6e 6f6d 6961 6c73 2832 2c33 2920 oMonomials(2,3) │ │ │ │ 000708e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000708f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00070900: 7c20 2d2d 2075 7365 6420 312e 3034 3038 | -- used 1.0408 │ │ │ │ -00070910: 3973 2028 6370 7529 3b20 302e 3833 3934 9s (cpu); 0.8394 │ │ │ │ -00070920: 3837 7320 2874 6872 6561 6429 3b20 3073 87s (thread); 0s │ │ │ │ +00070900: 7c20 2d2d 2075 7365 6420 312e 3031 3934 | -- used 1.0194 │ │ │ │ +00070910: 3373 2028 6370 7529 3b20 302e 3632 3838 3s (cpu); 0.6288 │ │ │ │ +00070920: 3138 7320 2874 6872 6561 6429 3b20 3073 18s (thread); 0s │ │ │ │ 00070930: 2028 6763 2920 7c0a 7c32 2020 2020 2020 (gc) |.|2 │ │ │ │ 00070940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070960: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00070970: 7c54 616c 6c79 7b7b 7b31 2c20 317d 7d20 |Tally{{{1, 1}} │ │ │ │ 00070980: 3d3e 2032 2020 2020 2020 2020 7d20 2020 => 2 } │ │ │ │ 00070990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -28828,16 +28828,16 @@ │ │ │ │ 000709b0: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20 {2, 2}, {1, 2}} │ │ │ │ 000709c0: 3d3e 2034 2020 2020 2020 2020 2020 2020 => 4 │ │ │ │ 000709d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000709e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000709f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070a10: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00070a20: 6420 302e 3535 3236 3739 7320 2863 7075 d 0.552679s (cpu │ │ │ │ -00070a30: 293b 2030 2e33 3935 3239 3373 2028 7468 ); 0.395293s (th │ │ │ │ +00070a20: 6420 302e 3536 3139 3131 7320 2863 7075 d 0.561911s (cpu │ │ │ │ +00070a30: 293b 2030 2e33 3734 3732 3273 2028 7468 ); 0.374722s (th │ │ │ │ 00070a40: 7265 6164 293b 2030 7320 2867 6329 7c0a read); 0s (gc)|. │ │ │ │ 00070a50: 7c33 2020 2020 2020 2020 2020 2020 2020 |3 │ │ │ │ 00070a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070a80: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b |.|Tally{{ │ │ │ │ 00070a90: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20 {2, 2}, {1, 2}} │ │ │ │ 00070aa0: 3d3e 2032 7d20 2020 2020 2020 2020 2020 => 2} │ │ │ │ @@ -28845,18 +28845,18 @@ │ │ │ │ 00070ac0: 7c20 2020 2020 207b 7b33 2c20 337d 2c20 | {{3, 3}, │ │ │ │ 00070ad0: 7b32 2c20 337d 7d20 3d3e 2031 2020 2020 {2, 3}} => 1 │ │ │ │ 00070ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00070b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070b20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00070b30: 7c20 2d2d 2075 7365 6420 302e 3130 3430 | -- used 0.1040 │ │ │ │ -00070b40: 3773 2028 6370 7529 3b20 302e 3130 3337 7s (cpu); 0.1037 │ │ │ │ -00070b50: 3934 7320 2874 6872 6561 6429 3b20 3073 94s (thread); 0s │ │ │ │ -00070b60: 2028 6763 2920 7c0a 7c34 2020 2020 2020 (gc) |.|4 │ │ │ │ +00070b30: 7c20 2d2d 2075 7365 6420 302e 3230 3232 | -- used 0.2022 │ │ │ │ +00070b40: 3438 7320 2863 7075 293b 2030 2e31 3435 48s (cpu); 0.145 │ │ │ │ +00070b50: 3239 3373 2028 7468 7265 6164 293b 2030 293s (thread); 0 │ │ │ │ +00070b60: 7320 2867 6329 7c0a 7c34 2020 2020 2020 s (gc)|.|4 │ │ │ │ 00070b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070b90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00070ba0: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20 |Tally{{{2, 2}, │ │ │ │ 00070bb0: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020 {1, 2}} => 1} │ │ │ │ 00070bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070bd0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ ├── ./usr/share/info/ConnectionMatrices.info.gz │ │ │ ├── ConnectionMatrices.info │ │ │ │ @@ -2993,30 +2993,30 @@ │ │ │ │ 0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0000bb20: 3920 3a20 656c 6170 7365 6454 696d 6520 9 : elapsedTime │ │ │ │ 0000bb30: 4120 3d20 7066 6166 6669 616e 5379 7374 A = pfaffianSyst │ │ │ │ 0000bb40: 656d 2049 3b20 2020 2020 2020 2020 2020 em I; │ │ │ │ 0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0000bb70: 2d2d 2034 2e33 3936 3431 7320 656c 6170 -- 4.39641s elap │ │ │ │ +0000bb70: 2d2d 2032 2e35 3339 3136 7320 656c 6170 -- 2.53916s elap │ │ │ │ 0000bb80: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0000bb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0000bc10: 3130 203a 2065 6c61 7073 6564 5469 6d65 10 : elapsedTime │ │ │ │ 0000bc20: 2061 7373 6572 7420 6973 496e 7465 6772 assert isIntegr │ │ │ │ 0000bc30: 6162 6c65 2041 2020 2020 2020 2020 2020 able A │ │ │ │ 0000bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0000bc60: 2d2d 2035 2e34 3638 3832 7320 656c 6170 -- 5.46882s elap │ │ │ │ +0000bc60: 2d2d 2033 2e35 3437 3334 7320 656c 6170 -- 3.54734s elap │ │ │ │ 0000bc70: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0000bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bca0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5137,16 +5137,16 @@ │ │ │ │ 00014100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00014120: 7c69 3134 203a 2065 6c61 7073 6564 5469 |i14 : elapsedTi │ │ │ │ 00014130: 6d65 2067 203d 2067 6175 6765 4d61 7472 me g = gaugeMatr │ │ │ │ 00014140: 6978 2849 2c20 4229 3b20 2020 2020 2020 ix(I, B); │ │ │ │ 00014150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014170: 7c20 2d2d 202e 3630 3639 3734 7320 656c | -- .606974s el │ │ │ │ -00014180: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +00014170: 7c20 2d2d 202e 3437 3031 3373 2065 6c61 | -- .47013s ela │ │ │ │ +00014180: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00014190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000141a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000141b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000141c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000141d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000141e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000141f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5167,30 +5167,30 @@ │ │ │ │ 000142e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000142f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00014300: 7c69 3135 203a 2065 6c61 7073 6564 5469 |i15 : elapsedTi │ │ │ │ 00014310: 6d65 2041 3120 3d20 6761 7567 6554 7261 me A1 = gaugeTra │ │ │ │ 00014320: 6e73 666f 726d 2867 2c20 4129 3b20 2020 nsform(g, A); │ │ │ │ 00014330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014340: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014350: 7c20 2d2d 2031 2e33 3030 3333 7320 656c | -- 1.30033s el │ │ │ │ +00014350: 7c20 2d2d 202e 3939 3234 3838 7320 656c | -- .992488s el │ │ │ │ 00014360: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00014370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014390: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000143a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000143b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000143c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000143d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000143e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000143f0: 7c69 3136 203a 2065 6c61 7073 6564 5469 |i16 : elapsedTi │ │ │ │ 00014400: 6d65 2061 7373 6572 7420 6973 496e 7465 me assert isInte │ │ │ │ 00014410: 6772 6162 6c65 2041 3120 2020 2020 2020 grable A1 │ │ │ │ 00014420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014430: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014440: 7c20 2d2d 202e 3835 3739 3433 7320 656c | -- .857943s el │ │ │ │ +00014440: 7c20 2d2d 202e 3734 3131 3738 7320 656c | -- .741178s el │ │ │ │ 00014450: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00014460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014480: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00014490: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000144a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000144b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5608,31 +5608,31 @@ │ │ │ │ 00015e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015e90: 2d2d 2d2b 0a7c 6931 3920 3a20 656c 6170 ---+.|i19 : elap │ │ │ │ 00015ea0: 7365 6454 696d 6520 4132 203d 2067 6175 sedTime A2 = gau │ │ │ │ 00015eb0: 6765 5472 616e 7366 6f72 6d28 6368 616e geTransform(chan │ │ │ │ 00015ec0: 6765 4570 732c 2041 3129 3b20 2020 2020 geEps, A1); │ │ │ │ 00015ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015ee0: 2020 207c 0a7c 202d 2d20 2e33 3933 3538 |.| -- .39358 │ │ │ │ -00015ef0: 3773 2065 6c61 7073 6564 2020 2020 2020 7s elapsed │ │ │ │ +00015ee0: 2020 207c 0a7c 202d 2d20 2e32 3933 3334 |.| -- .29334 │ │ │ │ +00015ef0: 3573 2065 6c61 7073 6564 2020 2020 2020 5s elapsed │ │ │ │ 00015f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015f30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00015f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015f80: 2d2d 2d2b 0a7c 6932 3020 3a20 656c 6170 ---+.|i20 : elap │ │ │ │ 00015f90: 7365 6454 696d 6520 6173 7365 7274 2069 sedTime assert i │ │ │ │ 00015fa0: 7349 6e74 6567 7261 626c 6520 4132 2020 sIntegrable A2 │ │ │ │ 00015fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015fd0: 2020 207c 0a7c 202d 2d20 312e 3036 3132 |.| -- 1.0612 │ │ │ │ -00015fe0: 3473 2065 6c61 7073 6564 2020 2020 2020 4s elapsed │ │ │ │ +00015fd0: 2020 207c 0a7c 202d 2d20 2e35 3931 3632 |.| -- .59162 │ │ │ │ +00015fe0: 3973 2065 6c61 7073 6564 2020 2020 2020 9s elapsed │ │ │ │ 00015ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016020: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00016030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -6018,30 +6018,30 @@ │ │ │ │ 00017810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ 00017830: 203a 2065 6c61 7073 6564 5469 6d65 2041 : elapsedTime A │ │ │ │ 00017840: 203d 2070 6661 6666 6961 6e53 7973 7465 = pfaffianSyste │ │ │ │ 00017850: 6d20 493b 2020 2020 2020 2020 2020 2020 m I; │ │ │ │ 00017860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017870: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00017880: 2d20 2e33 3239 3332 7320 656c 6170 7365 - .32932s elapse │ │ │ │ +00017880: 2d20 2e31 3934 3735 7320 656c 6170 7365 - .19475s elapse │ │ │ │ 00017890: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 000178a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000178b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000178c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 000178d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000178e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000178f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ 00017920: 203a 2065 6c61 7073 6564 5469 6d65 2061 : elapsedTime a │ │ │ │ 00017930: 7373 6572 7420 6973 496e 7465 6772 6162 ssert isIntegrab │ │ │ │ 00017940: 6c65 2041 2020 2020 2020 2020 2020 2020 le A │ │ │ │ 00017950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017960: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00017970: 2d20 2e31 3638 3736 3873 2065 6c61 7073 - .168768s elaps │ │ │ │ +00017970: 2d20 2e31 3436 3238 3173 2065 6c61 7073 - .146281s elaps │ │ │ │ 00017980: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ 00017990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000179a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000179b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 000179c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000179d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000179e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/Cremona.info.gz │ │ │ ├── Cremona.info │ │ │ │ @@ -147,16 +147,16 @@ │ │ │ │ 00000920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000930: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2074 -------+.|i2 : t │ │ │ │ 00000940: 696d 6520 7068 6920 3d20 746f 4d61 7020 ime phi = toMap │ │ │ │ 00000950: 6d69 6e6f 7273 2833 2c6d 6174 7269 787b minors(3,matrix{ │ │ │ │ 00000960: 7b74 5f30 2e2e 745f 347d 2c7b 745f 312e {t_0..t_4},{t_1. │ │ │ │ 00000970: 2e74 5f35 7d2c 7b74 5f32 2e2e 745f 367d .t_5},{t_2..t_6} │ │ │ │ 00000980: 7d29 2020 2020 207c 0a7c 202d 2d20 7573 }) |.| -- us │ │ │ │ -00000990: 6564 2030 2e30 3035 3538 3539 3373 2028 ed 0.00558593s ( │ │ │ │ -000009a0: 6370 7529 3b20 302e 3030 3535 3831 3132 cpu); 0.00558112 │ │ │ │ +00000990: 6564 2030 2e30 3034 3930 3936 3473 2028 ed 0.00490964s ( │ │ │ │ +000009a0: 6370 7529 3b20 302e 3030 3439 3037 3333 cpu); 0.00490733 │ │ │ │ 000009b0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 000009c0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 000009d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000009e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000009f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -322,16 +322,16 @@ │ │ │ │ 00001410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001420: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 -------+.|i3 : t │ │ │ │ 00001430: 696d 6520 4a20 3d20 6b65 726e 656c 2870 ime J = kernel(p │ │ │ │ 00001440: 6869 2c32 2920 2020 2020 2020 2020 2020 hi,2) │ │ │ │ 00001450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001470: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001480: 6564 2030 2e30 3539 3136 3433 7320 2863 ed 0.0591643s (c │ │ │ │ -00001490: 7075 293b 2030 2e30 3539 3137 3432 7320 pu); 0.0591742s │ │ │ │ +00001480: 6564 2030 2e30 3436 3035 3636 7320 2863 ed 0.0460566s (c │ │ │ │ +00001490: 7075 293b 2030 2e30 3436 3035 3839 7320 pu); 0.0460589s │ │ │ │ 000014a0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 000014b0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 000014c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000014d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000014e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000014f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -387,16 +387,16 @@ │ │ │ │ 00001820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001830: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 -------+.|i4 : t │ │ │ │ 00001840: 696d 6520 6465 6772 6565 4d61 7020 7068 ime degreeMap ph │ │ │ │ 00001850: 6920 2020 2020 2020 2020 2020 2020 2020 i │ │ │ │ 00001860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001880: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001890: 6564 2030 2e30 3335 3431 3336 7320 2863 ed 0.0354136s (c │ │ │ │ -000018a0: 7075 293b 2030 2e30 3335 3432 3033 7320 pu); 0.0354203s │ │ │ │ +00001890: 6564 2030 2e30 3238 3432 3232 7320 2863 ed 0.0284222s (c │ │ │ │ +000018a0: 7075 293b 2030 2e30 3238 3432 3533 7320 pu); 0.0284253s │ │ │ │ 000018b0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 000018c0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 000018d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000018e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000018f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -412,17 +412,17 @@ │ │ │ │ 000019b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000019c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 -------+.|i5 : t │ │ │ │ 000019d0: 696d 6520 7072 6f6a 6563 7469 7665 4465 ime projectiveDe │ │ │ │ 000019e0: 6772 6565 7320 7068 6920 2020 2020 2020 grees phi │ │ │ │ 000019f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a10: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001a20: 6564 2030 2e36 3833 3933 3873 2028 6370 ed 0.683938s (cp │ │ │ │ -00001a30: 7529 3b20 302e 3533 3731 3236 7320 2874 u); 0.537126s (t │ │ │ │ -00001a40: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00001a20: 6564 2030 2e34 3635 3831 7320 2863 7075 ed 0.46581s (cpu │ │ │ │ +00001a30: 293b 2030 2e33 3937 3036 3373 2028 7468 ); 0.397063s (th │ │ │ │ +00001a40: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00001a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ab0: 2020 2020 2020 207c 0a7c 6f35 203d 207b |.|o5 = { │ │ │ │ @@ -447,18 +447,18 @@ │ │ │ │ 00001be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001bf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2074 -------+.|i6 : t │ │ │ │ 00001c00: 696d 6520 7072 6f6a 6563 7469 7665 4465 ime projectiveDe │ │ │ │ 00001c10: 6772 6565 7328 7068 692c 4e75 6d44 6567 grees(phi,NumDeg │ │ │ │ 00001c20: 7265 6573 3d3e 3029 2020 2020 2020 2020 rees=>0) │ │ │ │ 00001c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001c40: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001c50: 6564 2030 2e32 3233 3032 3373 2028 6370 ed 0.223023s (cp │ │ │ │ -00001c60: 7529 3b20 302e 3133 3734 3333 7320 2874 u); 0.137433s (t │ │ │ │ -00001c70: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ -00001c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00001c50: 6564 2030 2e30 3539 3239 3737 7320 2863 ed 0.0592977s (c │ │ │ │ +00001c60: 7075 293b 2030 2e30 3539 3330 3336 7320 pu); 0.0593036s │ │ │ │ +00001c70: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +00001c80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00001c90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ce0: 2020 2020 2020 207c 0a7c 6f36 203d 207b |.|o6 = { │ │ │ │ 00001cf0: 357d 2020 2020 2020 2020 2020 2020 2020 5} │ │ │ │ @@ -482,15 +482,15 @@ │ │ │ │ 00001e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001e20: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2074 -------+.|i7 : t │ │ │ │ 00001e30: 696d 6520 7068 6920 3d20 746f 4d61 7028 ime phi = toMap( │ │ │ │ 00001e40: 7068 6920 2020 2020 2020 2020 2020 2020 phi │ │ │ │ 00001e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001e70: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001e80: 6564 2030 2e30 3033 3534 3130 3973 2028 ed 0.00354109s ( │ │ │ │ +00001e80: 6564 2030 2e30 3032 3233 3636 3973 2028 ed 0.00223669s ( │ │ │ │ 00001e90: 6370 7520 2020 2020 2020 2020 2020 2020 cpu │ │ │ │ 00001ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -567,16 +567,16 @@ │ │ │ │ 00002360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 2c44 6f6d 696e -------|.|,Domin │ │ │ │ 00002380: 616e 743d 3e4a 2920 2020 2020 2020 2020 ant=>J) │ │ │ │ 00002390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023c0: 2020 2020 2020 207c 0a7c 293b 2030 2e30 |.|); 0.0 │ │ │ │ -000023d0: 3033 3534 3233 3573 2028 7468 7265 6164 0354235s (thread │ │ │ │ -000023e0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +000023d0: 3032 3233 3937 7320 2874 6872 6561 6429 022397s (thread) │ │ │ │ +000023e0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 000023f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002410: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00002420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -832,16 +832,16 @@ │ │ │ │ 000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003400: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 -------+.|i8 : t │ │ │ │ 00003410: 696d 6520 7073 6920 3d20 696e 7665 7273 ime psi = invers │ │ │ │ 00003420: 654d 6170 2070 6869 2020 2020 2020 2020 eMap phi │ │ │ │ 00003430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003450: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00003460: 6564 2030 2e37 3433 3138 3373 2028 6370 ed 0.743183s (cp │ │ │ │ -00003470: 7529 3b20 302e 3636 3231 3035 7320 2874 u); 0.662105s (t │ │ │ │ +00003460: 6564 2030 2e35 3035 3330 3273 2028 6370 ed 0.505302s (cp │ │ │ │ +00003470: 7529 3b20 302e 3430 3635 3939 7320 2874 u); 0.406599s (t │ │ │ │ 00003480: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00003490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000034b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000034e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1117,18 +1117,18 @@ │ │ │ │ 000045c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000045d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 -------+.|i9 : t │ │ │ │ 000045e0: 696d 6520 6973 496e 7665 7273 654d 6170 ime isInverseMap │ │ │ │ 000045f0: 2870 6869 2c70 7369 2920 2020 2020 2020 (phi,psi) │ │ │ │ 00004600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004620: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00004630: 6564 2030 2e30 3136 3435 3673 2028 6370 ed 0.016456s (cp │ │ │ │ -00004640: 7529 3b20 302e 3031 3634 3537 3973 2028 u); 0.0164579s ( │ │ │ │ -00004650: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -00004660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00004630: 6564 2030 2e30 3130 3137 3535 7320 2863 ed 0.0101755s (c │ │ │ │ +00004640: 7075 293b 2030 2e30 3130 3138 3834 7320 pu); 0.0101884s │ │ │ │ +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 2e33 3736 3530 3773 2028 6370 ed 0.376507s (cp │ │ │ │ -000047d0: 7529 3b20 302e 3237 3433 3137 7320 2874 u); 0.274317s (t │ │ │ │ +000047c0: 6564 2030 2e31 3533 3934 3973 2028 6370 ed 0.153949s (cp │ │ │ │ +000047d0: 7529 3b20 302e 3135 3339 3536 7320 2874 u); 0.153956s (t │ │ │ │ 000047e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000047f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004800: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00004810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1167,16 +1167,16 @@ │ │ │ │ 000048e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000048f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ 00004900: 7469 6d65 2070 726f 6a65 6374 6976 6544 time projectiveD │ │ │ │ 00004910: 6567 7265 6573 2070 7369 2020 2020 2020 egrees psi │ │ │ │ 00004920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004940: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00004950: 6564 2035 2e36 3532 3231 7320 2863 7075 ed 5.65221s (cpu │ │ │ │ -00004960: 293b 2034 2e37 3931 3539 7320 2874 6872 ); 4.79159s (thr │ │ │ │ +00004950: 6564 2034 2e37 3935 3638 7320 2863 7075 ed 4.79568s (cpu │ │ │ │ +00004960: 293b 2034 2e33 3035 3837 7320 2874 6872 ); 4.30587s (thr │ │ │ │ 00004970: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00004980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004990: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000049a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000049d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1214,16 +1214,16 @@ │ │ │ │ 00004bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00004be0: 0a7c 6931 3220 3a20 7469 6d65 2070 6869 .|i12 : time phi │ │ │ │ 00004bf0: 203d 2072 6174 696f 6e61 6c4d 6170 206d = rationalMap m │ │ │ │ 00004c00: 696e 6f72 7328 332c 6d61 7472 6978 7b7b inors(3,matrix{{ │ │ │ │ 00004c10: 745f 302e 2e74 5f34 7d2c 7b74 5f31 2e2e t_0..t_4},{t_1.. │ │ │ │ 00004c20: 745f 357d 2c7b 745f 322e 2e74 5f36 207c t_5},{t_2..t_6 | │ │ │ │ 00004c30: 0a7c 202d 2d20 7573 6564 2030 2e30 3032 .| -- used 0.002 │ │ │ │ -00004c40: 3234 3034 3973 2028 6370 7529 3b20 302e 24049s (cpu); 0. │ │ │ │ -00004c50: 3030 3232 3431 3536 7320 2874 6872 6561 00224156s (threa │ │ │ │ +00004c40: 3339 3534 3273 2028 6370 7529 3b20 302e 39542s (cpu); 0. │ │ │ │ +00004c50: 3030 3233 3939 3438 7320 2874 6872 6561 00239948s (threa │ │ │ │ 00004c60: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00004c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00004c80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00004c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -1493,17 +1493,17 @@ │ │ │ │ 00005d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00005d60: 0a7c 6931 3320 3a20 7469 6d65 2070 6869 .|i13 : time phi │ │ │ │ 00005d70: 203d 2072 6174 696f 6e61 6c4d 6170 2870 = rationalMap(p │ │ │ │ 00005d80: 6869 2c44 6f6d 696e 616e 743d 3e32 2920 hi,Dominant=>2) │ │ │ │ 00005d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005da0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3532 .| -- used 0.052 │ │ │ │ -00005dc0: 3338 3737 7320 2863 7075 293b 2030 2e30 3877s (cpu); 0.0 │ │ │ │ -00005dd0: 3532 3339 3733 7320 2874 6872 6561 6429 523973s (thread) │ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3531 .| -- used 0.051 │ │ │ │ +00005dc0: 3336 3835 7320 2863 7075 293b 2030 2e30 3685s (cpu); 0.0 │ │ │ │ +00005dd0: 3531 3337 3135 7320 2874 6872 6561 6429 513715s (thread) │ │ │ │ 00005de0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00005df0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00005e00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00005e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2153,17 +2153,17 @@ │ │ │ │ 00008680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 000086a0: 0a7c 6931 3420 3a20 7469 6d65 2070 6869 .|i14 : time phi │ │ │ │ 000086b0: 5e28 2d31 2920 2020 2020 2020 2020 2020 ^(-1) │ │ │ │ 000086c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000086d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000086e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000086f0: 0a7c 202d 2d20 7573 6564 2030 2e35 3836 .| -- used 0.586 │ │ │ │ -00008700: 3234 3173 2028 6370 7529 3b20 302e 3434 241s (cpu); 0.44 │ │ │ │ -00008710: 3634 3031 7320 2874 6872 6561 6429 3b20 6401s (thread); │ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e35 3131 .| -- used 0.511 │ │ │ │ +00008700: 3832 3673 2028 6370 7529 3b20 302e 3431 826s (cpu); 0.41 │ │ │ │ +00008710: 3832 3534 7320 2874 6872 6561 6429 3b20 8254s (thread); │ │ │ │ 00008720: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00008730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00008740: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00008750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008780: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2708,17 +2708,17 @@ │ │ │ │ 0000a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000a950: 0a7c 6931 3520 3a20 7469 6d65 2064 6567 .|i15 : time deg │ │ │ │ 0000a960: 7265 6573 2070 6869 5e28 2d31 2920 2020 rees phi^(-1) │ │ │ │ 0000a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a990: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e36 3234 .| -- used 0.624 │ │ │ │ -0000a9b0: 3830 3973 2028 6370 7529 3b20 302e 3436 809s (cpu); 0.46 │ │ │ │ -0000a9c0: 3030 3832 7320 2874 6872 6561 6429 3b20 0082s (thread); │ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e32 3739 .| -- used 0.279 │ │ │ │ +0000a9b0: 3236 3173 2028 6370 7529 3b20 302e 3233 261s (cpu); 0.23 │ │ │ │ +0000a9c0: 3937 3139 7320 2874 6872 6561 6429 3b20 9719s (thread); │ │ │ │ 0000a9d0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 0000a9e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000a9f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000aa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2743,17 +2743,17 @@ │ │ │ │ 0000ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000ab80: 0a7c 6931 3620 3a20 7469 6d65 2064 6567 .|i16 : time deg │ │ │ │ 0000ab90: 7265 6573 2070 6869 2020 2020 2020 2020 rees phi │ │ │ │ 0000aba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000abb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000abc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3235 .| -- used 0.025 │ │ │ │ -0000abe0: 3737 3536 7320 2863 7075 293b 2030 2e30 7756s (cpu); 0.0 │ │ │ │ -0000abf0: 3235 3232 3638 7320 2874 6872 6561 6429 252268s (thread) │ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3238 .| -- used 0.028 │ │ │ │ +0000abe0: 3137 3138 7320 2863 7075 293b 2030 2e30 1718s (cpu); 0.0 │ │ │ │ +0000abf0: 3137 3332 3637 7320 2874 6872 6561 6429 173267s (thread) │ │ │ │ 0000ac00: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0000ac10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000ac20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2778,18 +2778,18 @@ │ │ │ │ 0000ad90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ada0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000adb0: 0a7c 6931 3720 3a20 7469 6d65 2064 6573 .|i17 : time des │ │ │ │ 0000adc0: 6372 6962 6520 7068 6920 2020 2020 2020 cribe phi │ │ │ │ 0000add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000adf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3034 .| -- used 0.004 │ │ │ │ -0000ae10: 3539 3032 3373 2028 6370 7529 3b20 302e 59023s (cpu); 0. │ │ │ │ -0000ae20: 3030 3435 3930 3732 7320 2874 6872 6561 00459072s (threa │ │ │ │ -0000ae30: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3033 .| -- used 0.003 │ │ │ │ +0000ae10: 3836 3736 7320 2863 7075 293b 2030 2e30 8676s (cpu); 0.0 │ │ │ │ +0000ae20: 3033 3837 3332 3873 2028 7468 7265 6164 0387328s (thread │ │ │ │ +0000ae30: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0000ae40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000ae50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000aea0: 0a7c 6f31 3720 3d20 7261 7469 6f6e 616c .|o17 = rational │ │ │ │ @@ -2843,18 +2843,18 @@ │ │ │ │ 0000b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b1c0: 0a7c 6931 3820 3a20 7469 6d65 2064 6573 .|i18 : time des │ │ │ │ 0000b1d0: 6372 6962 6520 7068 695e 282d 3129 2020 cribe phi^(-1) │ │ │ │ 0000b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b200: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3133 .| -- used 0.013 │ │ │ │ -0000b220: 3830 3435 7320 2863 7075 293b 2030 2e30 8045s (cpu); 0.0 │ │ │ │ -0000b230: 3133 3830 3739 7320 2874 6872 6561 6429 138079s (thread) │ │ │ │ -0000b240: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3130 .| -- used 0.010 │ │ │ │ +0000b220: 3033 3473 2028 6370 7529 3b20 302e 3031 034s (cpu); 0.01 │ │ │ │ +0000b230: 3030 3339 3373 2028 7468 7265 6164 293b 00393s (thread); │ │ │ │ +0000b240: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0000b250: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b260: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b2a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b2b0: 0a7c 6f31 3820 3d20 7261 7469 6f6e 616c .|o18 = rational │ │ │ │ @@ -2923,17 +2923,17 @@ │ │ │ │ 0000b6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b6c0: 0a7c 6931 3920 3a20 7469 6d65 2028 662c .|i19 : time (f, │ │ │ │ 0000b6d0: 6729 203d 2067 7261 7068 2070 6869 5e2d g) = graph phi^- │ │ │ │ 0000b6e0: 313b 2066 3b20 2020 2020 2020 2020 2020 1; f; │ │ │ │ 0000b6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b700: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3133 .| -- used 0.013 │ │ │ │ -0000b720: 3231 3636 7320 2863 7075 293b 2030 2e30 2166s (cpu); 0.0 │ │ │ │ -0000b730: 3133 3139 3036 7320 2874 6872 6561 6429 131906s (thread) │ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3130 .| -- used 0.010 │ │ │ │ +0000b720: 3131 3539 7320 2863 7075 293b 2030 2e30 1159s (cpu); 0.0 │ │ │ │ +0000b730: 3130 3132 3036 7320 2874 6872 6561 6429 101206s (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 | │ │ │ │ @@ -2958,18 +2958,18 @@ │ │ │ │ 0000b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b8f0: 0a7c 6932 3120 3a20 7469 6d65 2064 6567 .|i21 : time deg │ │ │ │ 0000b900: 7265 6573 2066 2020 2020 2020 2020 2020 rees f │ │ │ │ 0000b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b930: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b940: 0a7c 202d 2d20 7573 6564 2031 2e36 3033 .| -- used 1.603 │ │ │ │ -0000b950: 3732 7320 2863 7075 293b 2031 2e31 3831 72s (cpu); 1.181 │ │ │ │ -0000b960: 3134 7320 2874 6872 6561 6429 3b20 3073 14s (thread); 0s │ │ │ │ -0000b970: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e31 3232 .| -- used 1.122 │ │ │ │ +0000b950: 3539 7320 2863 7075 293b 2030 2e38 3337 59s (cpu); 0.837 │ │ │ │ +0000b960: 3937 3373 2028 7468 7265 6164 293b 2030 973s (thread); 0 │ │ │ │ +0000b970: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000b980: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b990: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b9e0: 0a7c 6f32 3120 3d20 7b39 3034 2c20 3530 .|o21 = {904, 50 │ │ │ │ @@ -2993,17 +2993,17 @@ │ │ │ │ 0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000bb20: 0a7c 6932 3220 3a20 7469 6d65 2064 6567 .|i22 : time deg │ │ │ │ 0000bb30: 7265 6520 6620 2020 2020 2020 2020 2020 ree f │ │ │ │ 0000bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000bb70: 0a7c 202d 2d20 7573 6564 2032 2e30 3136 .| -- used 2.016 │ │ │ │ -0000bb80: 3865 2d30 3573 2028 6370 7529 3b20 312e 8e-05s (cpu); 1. │ │ │ │ -0000bb90: 3936 3036 652d 3035 7320 2874 6872 6561 9606e-05s (threa │ │ │ │ +0000bb70: 0a7c 202d 2d20 7573 6564 2031 2e35 3437 .| -- used 1.547 │ │ │ │ +0000bb80: 3465 2d30 3573 2028 6370 7529 3b20 312e 4e-05s (cpu); 1. │ │ │ │ +0000bb90: 3437 3136 652d 3035 7320 2874 6872 6561 4716e-05s (threa │ │ │ │ 0000bba0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0000bbb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -3019,16 +3019,16 @@ │ │ │ │ 0000bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000bcb0: 0a7c 6932 3320 3a20 7469 6d65 2064 6573 .|i23 : time des │ │ │ │ 0000bcc0: 6372 6962 6520 6620 2020 2020 2020 2020 cribe f │ │ │ │ 0000bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bcf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bd00: 0a7c 202d 2d20 7573 6564 2030 2e30 3031 .| -- used 0.001 │ │ │ │ -0000bd10: 3731 3638 3673 2028 6370 7529 3b20 302e 71686s (cpu); 0. │ │ │ │ -0000bd20: 3030 3137 3137 3935 7320 2874 6872 6561 00171795s (threa │ │ │ │ +0000bd10: 3332 3039 3573 2028 6370 7529 3b20 302e 32095s (cpu); 0. │ │ │ │ +0000bd20: 3030 3133 3234 3436 7320 2874 6872 6561 00132446s (threa │ │ │ │ 0000bd30: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0000bd40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bd50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bd90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -4676,16 +4676,16 @@ │ │ │ │ 00012430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012440: 2b0a 7c69 3420 3a20 7469 6d65 2070 7369 +.|i4 : time psi │ │ │ │ 00012450: 203d 2061 6273 7472 6163 7452 6174 696f = abstractRatio │ │ │ │ 00012460: 6e61 6c4d 6170 2850 342c 5035 2c66 2920 nalMap(P4,P5,f) │ │ │ │ 00012470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012490: 7c0a 7c20 2d2d 2075 7365 6420 302e 3030 |.| -- used 0.00 │ │ │ │ -000124a0: 3034 3736 3832 3573 2028 6370 7529 3b20 0476825s (cpu); │ │ │ │ -000124b0: 302e 3030 3034 3732 3530 3673 2028 7468 0.000472506s (th │ │ │ │ +000124a0: 3033 3935 3733 3673 2028 6370 7529 3b20 0395736s (cpu); │ │ │ │ +000124b0: 302e 3030 3033 3931 3735 3373 2028 7468 0.000391753s (th │ │ │ │ 000124c0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000124d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000124e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000124f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4746,18 +4746,18 @@ │ │ │ │ 00012890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000128a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000128b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000128c0: 6935 203a 2074 696d 6520 7072 6f6a 6563 i5 : time projec │ │ │ │ 000128d0: 7469 7665 4465 6772 6565 7328 7073 692c tiveDegrees(psi, │ │ │ │ 000128e0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 000128f0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00012900: 2d20 7573 6564 2030 2e31 3139 3473 2028 - used 0.1194s ( │ │ │ │ -00012910: 6370 7529 3b20 302e 3131 3934 3233 7320 cpu); 0.119423s │ │ │ │ -00012920: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00012930: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +00012900: 2d20 7573 6564 2030 2e31 3139 3034 3173 - used 0.119041s │ │ │ │ +00012910: 2028 6370 7529 3b20 302e 3131 3930 3436 (cpu); 0.119046 │ │ │ │ +00012920: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00012930: 6763 2920 2020 2020 207c 0a7c 2020 2020 gc) |.| │ │ │ │ 00012940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012970: 2020 2020 2020 207c 0a7c 6f35 203d 2032 |.|o5 = 2 │ │ │ │ 00012980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000129a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4765,17 +4765,17 @@ │ │ │ │ 000129c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000129d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000129e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000129f0: 2d2d 2d2b 0a7c 6936 203a 2074 696d 6520 ---+.|i6 : time │ │ │ │ 00012a00: 7261 7469 6f6e 616c 4d61 7020 7073 6920 rationalMap psi │ │ │ │ 00012a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e35 |.| -- used 0.5 │ │ │ │ -00012a40: 3332 3736 3973 2028 6370 7529 3b20 302e 32769s (cpu); 0. │ │ │ │ -00012a50: 3430 3139 3233 7320 2874 6872 6561 6429 401923s (thread) │ │ │ │ +00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e34 |.| -- used 0.4 │ │ │ │ +00012a40: 3234 3430 3273 2028 6370 7529 3b20 302e 24402s (cpu); 0. │ │ │ │ +00012a50: 3333 3837 3934 7320 2874 6872 6561 6429 338794s (thread) │ │ │ │ 00012a60: 3b20 3073 2028 6763 2920 2020 2020 207c ; 0s (gc) | │ │ │ │ 00012a70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00012a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012aa0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00012ab0: 6f36 203d 202d 2d20 7261 7469 6f6e 616c o6 = -- rational │ │ │ │ 00012ac0: 206d 6170 202d 2d20 2020 2020 2020 2020 map -- │ │ │ │ @@ -5189,17 +5189,17 @@ │ │ │ │ 00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a --------+.|i14 : │ │ │ │ 00014460: 2074 696d 6520 5420 3d20 6162 7374 7261 time T = abstra │ │ │ │ 00014470: 6374 5261 7469 6f6e 616c 4d61 7028 492c ctRationalMap(I, │ │ │ │ 00014480: 224f 4144 5022 2920 2020 2020 2020 2020 "OADP") │ │ │ │ 00014490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144a0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -000144b0: 6420 302e 3033 3833 3137 3673 2028 6370 d 0.0383176s (cp │ │ │ │ -000144c0: 7529 3b20 302e 3033 3833 3230 3273 2028 u); 0.0383202s ( │ │ │ │ -000144d0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000144b0: 6420 302e 3034 3036 3332 7320 2863 7075 d 0.040632s (cpu │ │ │ │ +000144c0: 293b 2030 2e30 3430 3631 3632 7320 2874 ); 0.0406162s (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 │ │ │ │ 00014540: 2020 7c0a 7c6f 3134 203d 202d 2d20 7261 |.|o14 = -- ra │ │ │ │ @@ -5265,16 +5265,16 @@ │ │ │ │ 00014900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014930: 2d2b 0a7c 6931 3520 3a20 7469 6d65 2070 -+.|i15 : time p │ │ │ │ 00014940: 726f 6a65 6374 6976 6544 6567 7265 6573 rojectiveDegrees │ │ │ │ 00014950: 2854 2c32 2920 2020 2020 2020 2020 2020 (T,2) │ │ │ │ 00014960: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00014970: 6420 322e 3138 3636 3373 2028 6370 7529 d 2.18663s (cpu) │ │ │ │ -00014980: 3b20 312e 3538 3437 7320 2874 6872 6561 ; 1.5847s (threa │ │ │ │ +00014970: 6420 322e 3237 3636 7320 2863 7075 293b d 2.2766s (cpu); │ │ │ │ +00014980: 2031 2e35 3639 3431 7320 2874 6872 6561 1.56941s (threa │ │ │ │ 00014990: 6429 3b20 3073 2028 6763 297c 0a7c 2020 d); 0s (gc)|.| │ │ │ │ 000149a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149d0: 7c0a 7c6f 3135 203d 2033 2020 2020 2020 |.|o15 = 3 │ │ │ │ 000149e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5290,17 +5290,17 @@ │ │ │ │ 00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ac0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 7469 -----+.|i16 : ti │ │ │ │ 00014ad0: 6d65 2054 3220 3d20 5420 2a20 5420 2020 me T2 = T * T │ │ │ │ 00014ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b00: 2020 7c0a 7c20 2d2d 2075 7365 6420 332e |.| -- used 3. │ │ │ │ -00014b10: 3030 3037 652d 3035 7320 2863 7075 293b 0007e-05s (cpu); │ │ │ │ -00014b20: 2032 2e39 3730 3565 2d30 3573 2028 7468 2.9705e-05s (th │ │ │ │ +00014b00: 2020 7c0a 7c20 2d2d 2075 7365 6420 322e |.| -- used 2. │ │ │ │ +00014b10: 3635 3332 652d 3035 7320 2863 7075 293b 6532e-05s (cpu); │ │ │ │ +00014b20: 2032 2e34 3336 3665 2d30 3573 2028 7468 2.4366e-05s (th │ │ │ │ 00014b30: 7265 6164 293b 2030 7320 2867 6329 207c read); 0s (gc) | │ │ │ │ 00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00014b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 00014b80: 3136 203d 202d 2d20 7261 7469 6f6e 616c 16 = -- rational │ │ │ │ 00014b90: 206d 6170 202d 2d20 2020 2020 2020 2020 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 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 00014e20: 3720 3a20 7469 6d65 2070 726f 6a65 6374 7 : time project │ │ │ │ 00014e30: 6976 6544 6567 7265 6573 2854 322c 3229 iveDegrees(T2,2) │ │ │ │ 00014e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e50: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00014e60: 7365 6420 332e 3637 3932 3873 2028 6370 sed 3.67928s (cp │ │ │ │ -00014e70: 7529 3b20 322e 3630 3139 7320 2874 6872 u); 2.6019s (thr │ │ │ │ -00014e80: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00014e60: 7365 6420 332e 3531 3338 3573 2028 6370 sed 3.51385s (cp │ │ │ │ +00014e70: 7529 3b20 322e 3439 3836 3673 2028 7468 u); 2.49866s (th │ │ │ │ +00014e80: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00014e90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ed0: 2020 7c0a 7c6f 3137 203d 2031 2020 2020 |.|o17 = 1 │ │ │ │ 00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5430,17 +5430,17 @@ │ │ │ │ 00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015370: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 7469 -----+.|i21 : ti │ │ │ │ 00015380: 6d65 2066 203d 2072 6174 696f 6e61 6c4d me f = rationalM │ │ │ │ 00015390: 6170 2054 2020 2020 2020 2020 2020 2020 ap T │ │ │ │ 000153a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000153c0: 7573 6564 2033 2e31 3236 3536 7320 2863 used 3.12656s (c │ │ │ │ -000153d0: 7075 293b 2032 2e32 3137 7320 2874 6872 pu); 2.217s (thr │ │ │ │ -000153e0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +000153c0: 7573 6564 2032 2e39 3838 3738 7320 2863 used 2.98878s (c │ │ │ │ +000153d0: 7075 293b 2032 2e31 3132 3339 7320 2874 pu); 2.11239s (t │ │ │ │ +000153e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000153f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00015400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015440: 207c 0a7c 6f32 3120 3d20 2d2d 2072 6174 |.|o21 = -- rat │ │ │ │ 00015450: 696f 6e61 6c20 6d61 7020 2d2d 2020 2020 ional map -- │ │ │ │ @@ -6678,16 +6678,16 @@ │ │ │ │ 0001a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a160: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ 0001a170: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ 0001a180: 703a 2073 7465 7020 3130 206f 6620 3130 p: step 10 of 10 │ │ │ │ 0001a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a1b0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -0001a1c0: 6420 302e 3535 3931 3734 7320 2863 7075 d 0.559174s (cpu │ │ │ │ -0001a1d0: 293b 2030 2e33 3734 3838 3773 2028 7468 ); 0.374887s (th │ │ │ │ +0001a1c0: 6420 302e 3434 3431 3339 7320 2863 7075 d 0.444139s (cpu │ │ │ │ +0001a1d0: 293b 2030 2e33 3030 3739 3173 2028 7468 ); 0.300791s (th │ │ │ │ 0001a1e0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a200: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 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 │ │ │ │ @@ -8043,16 +8043,16 @@ │ │ │ │ 0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6b0: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ 0001f6c0: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ 0001f6d0: 703a 2073 7465 7020 3320 6f66 2033 2020 p: step 3 of 3 │ │ │ │ 0001f6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f700: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -0001f710: 6420 302e 3336 3739 3339 7320 2863 7075 d 0.367939s (cpu │ │ │ │ -0001f720: 293b 2030 2e32 3830 3934 3273 2028 7468 ); 0.280942s (th │ │ │ │ +0001f710: 6420 302e 3234 3932 3331 7320 2863 7075 d 0.249231s (cpu │ │ │ │ +0001f720: 293b 2030 2e31 3634 3530 3773 2028 7468 ); 0.164507s (th │ │ │ │ 0001f730: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 0001f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f750: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -10404,18 +10404,18 @@ │ │ │ │ 00028a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00028a50: 0a7c 2d2d 2061 7070 726f 7869 6d61 7465 .|-- approximate │ │ │ │ 00028a60: 496e 7665 7273 654d 6170 3a20 7374 6570 InverseMap: step │ │ │ │ 00028a70: 2033 206f 6620 3320 2020 2020 2020 2020 3 of 3 │ │ │ │ 00028a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028aa0: 0a7c 202d 2d20 7573 6564 2032 2e37 3631 .| -- used 2.761 │ │ │ │ -00028ab0: 3373 2028 6370 7529 3b20 312e 3839 3330 3s (cpu); 1.8930 │ │ │ │ -00028ac0: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ -00028ad0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00028aa0: 0a7c 202d 2d20 7573 6564 2031 2e39 3932 .| -- used 1.992 │ │ │ │ +00028ab0: 3837 7320 2863 7075 293b 2031 2e36 3236 87s (cpu); 1.626 │ │ │ │ +00028ac0: 3932 7320 2874 6872 6561 6429 3b20 3073 92s (thread); 0s │ │ │ │ +00028ad0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00028ae0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00028af0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00028b40: 0a7c 6f38 203d 202d 2d20 7261 7469 6f6e .|o8 = -- ration │ │ │ │ @@ -11709,18 +11709,18 @@ │ │ │ │ 0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002dbe0: 0a7c 4365 7274 6966 793a 206f 7574 7075 .|Certify: outpu │ │ │ │ 0002dbf0: 7420 6365 7274 6966 6965 6421 2020 2020 t certified! │ │ │ │ 0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002dc30: 0a7c 202d 2d20 7573 6564 2033 2e38 3935 .| -- used 3.895 │ │ │ │ -0002dc40: 3535 7320 2863 7075 293b 2032 2e39 3232 55s (cpu); 2.922 │ │ │ │ -0002dc50: 3338 7320 2874 6872 6561 6429 3b20 3073 38s (thread); 0s │ │ │ │ -0002dc60: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0002dc30: 0a7c 202d 2d20 7573 6564 2032 2e38 3433 .| -- used 2.843 │ │ │ │ +0002dc40: 3273 2028 6370 7529 3b20 322e 3339 3137 2s (cpu); 2.3917 │ │ │ │ +0002dc50: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ +0002dc60: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0002dc70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002dc80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dcc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002dcd0: 0a7c 6f31 3020 3d20 2d2d 2072 6174 696f .|o10 = -- ratio │ │ │ │ @@ -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 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ 00034380: 3a20 7469 6d65 2043 6865 726e 5363 6877 : time ChernSchw │ │ │ │ 00034390: 6172 747a 4d61 6350 6865 7273 6f6e 2043 artzMacPherson C │ │ │ │ 000343a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000343b0: 2020 207c 0a7c 202d 2d20 7573 6564 2031 |.| -- used 1 │ │ │ │ -000343c0: 2e32 3639 3536 7320 2863 7075 293b 2030 .26956s (cpu); 0 │ │ │ │ -000343d0: 2e38 3934 3033 3173 2028 7468 7265 6164 .894031s (thread │ │ │ │ +000343c0: 2e31 3235 3936 7320 2863 7075 293b 2030 .12596s (cpu); 0 │ │ │ │ +000343d0: 2e38 3534 3738 3973 2028 7468 7265 6164 .854789s (thread │ │ │ │ 000343e0: 293b 2030 7320 2867 6329 2020 7c0a 7c20 ); 0s (gc) |.| │ │ │ │ 000343f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034420: 2020 2020 207c 0a7c 2020 2020 2020 2034 |.| 4 │ │ │ │ 00034430: 2020 2020 2033 2020 2020 2032 2020 2020 3 2 │ │ │ │ 00034440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -13408,18 +13408,18 @@ │ │ │ │ 000345f0: 6934 203a 2074 696d 6520 4368 6572 6e53 i4 : time ChernS │ │ │ │ 00034600: 6368 7761 7274 7a4d 6163 5068 6572 736f chwartzMacPherso │ │ │ │ 00034610: 6e28 432c 4365 7274 6966 793d 3e74 7275 n(C,Certify=>tru │ │ │ │ 00034620: 6529 2020 2020 7c0a 7c43 6572 7469 6679 e) |.|Certify │ │ │ │ 00034630: 3a20 6f75 7470 7574 2063 6572 7469 6669 : output certifi │ │ │ │ 00034640: 6564 2120 2020 2020 2020 2020 2020 2020 ed! │ │ │ │ 00034650: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00034660: 0a7c 202d 2d20 7573 6564 2031 2e34 3639 .| -- used 1.469 │ │ │ │ -00034670: 3032 7320 2863 7075 293b 2031 2e30 3033 02s (cpu); 1.003 │ │ │ │ -00034680: 3039 7320 2874 6872 6561 6429 3b20 3073 09s (thread); 0s │ │ │ │ -00034690: 2028 6763 2920 2020 7c0a 7c20 2020 2020 (gc) |.| │ │ │ │ +00034660: 0a7c 202d 2d20 7573 6564 2031 2e31 3638 .| -- used 1.168 │ │ │ │ +00034670: 3937 7320 2863 7075 293b 2030 2e38 3139 97s (cpu); 0.819 │ │ │ │ +00034680: 3834 3673 2028 7468 7265 6164 293b 2030 846s (thread); 0 │ │ │ │ +00034690: 7320 2867 6329 2020 7c0a 7c20 2020 2020 s (gc) |.| │ │ │ │ 000346a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000346b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000346c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000346d0: 207c 0a7c 2020 2020 2020 2034 2020 2020 |.| 4 │ │ │ │ 000346e0: 2033 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ 000346f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034700: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ @@ -13619,16 +13619,16 @@ │ │ │ │ 00035320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035330: 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 696d -----+.|i9 : tim │ │ │ │ 00035340: 6520 4368 6572 6e43 6c61 7373 2047 2020 e ChernClass G │ │ │ │ 00035350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035380: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00035390: 2030 2e31 3130 3239 3873 2028 6370 7529 0.110298s (cpu) │ │ │ │ -000353a0: 3b20 302e 3131 3033 3035 7320 2874 6872 ; 0.110305s (thr │ │ │ │ +00035390: 2030 2e31 3139 3336 3273 2028 6370 7529 0.119362s (cpu) │ │ │ │ +000353a0: 3b20 302e 3131 3933 3634 7320 2874 6872 ; 0.119364s (thr │ │ │ │ 000353b0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 000353c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000353d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 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 │ │ │ │ @@ -13679,16 +13679,16 @@ │ │ │ │ 000356e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000356f0: 2020 2020 207c 0a7c 4365 7274 6966 793a |.|Certify: │ │ │ │ 00035700: 206f 7574 7075 7420 6365 7274 6966 6965 output certifie │ │ │ │ 00035710: 6421 2020 2020 2020 2020 2020 2020 2020 d! │ │ │ │ 00035720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035740: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00035750: 2030 2e30 3131 3333 3539 7320 2863 7075 0.0113359s (cpu │ │ │ │ -00035760: 293b 2030 2e30 3130 3732 3335 7320 2874 ); 0.0107235s (t │ │ │ │ +00035750: 2030 2e30 3233 3331 3932 7320 2863 7075 0.0233192s (cpu │ │ │ │ +00035760: 293b 2030 2e30 3131 3730 3532 7320 2874 ); 0.0117052s (t │ │ │ │ 00035770: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00035780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035790: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000357a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000357d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -16335,17 +16335,17 @@ │ │ │ │ 0003fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 0003fd00: 203a 2074 696d 6520 6465 6772 6565 4d61 : time degreeMa │ │ │ │ 0003fd10: 7020 7068 6920 2020 2020 2020 2020 2020 p 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 207c 0a7c 202d |.| - │ │ │ │ -0003fd50: 2d20 7573 6564 2030 2e30 3439 3732 3438 - used 0.0497248 │ │ │ │ -0003fd60: 7320 2863 7075 293b 2030 2e30 3439 3732 s (cpu); 0.04972 │ │ │ │ -0003fd70: 3538 7320 2874 6872 6561 6429 3b20 3073 58s (thread); 0s │ │ │ │ +0003fd50: 2d20 7573 6564 2030 2e30 3437 3935 3039 - used 0.0479509 │ │ │ │ +0003fd60: 7320 2863 7075 293b 2030 2e30 3437 3935 s (cpu); 0.04795 │ │ │ │ +0003fd70: 3131 7320 2874 6872 6561 6429 3b20 3073 11s (thread); 0s │ │ │ │ 0003fd80: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0003fd90: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0003fda0: 2020 2020 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 207c 0a7c 6f35 |.|o5 │ │ │ │ @@ -17510,18 +17510,18 @@ │ │ │ │ 00044650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ 00044670: 203a 2074 696d 6520 6465 6772 6565 4d61 : time degreeMa │ │ │ │ 00044680: 7020 7068 6927 2020 2020 2020 2020 2020 p 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 207c 0a7c 202d |.| - │ │ │ │ -000446c0: 2d20 7573 6564 2031 2e34 3632 3738 7320 - used 1.46278s │ │ │ │ -000446d0: 2863 7075 293b 2030 2e38 3433 3731 3473 (cpu); 0.843714s │ │ │ │ -000446e0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -000446f0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +000446c0: 2d20 7573 6564 2031 2e32 3634 3173 2028 - used 1.2641s ( │ │ │ │ +000446d0: 6370 7529 3b20 302e 3638 3034 3039 7320 cpu); 0.680409s │ │ │ │ +000446e0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +000446f0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00044700: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00044710: 2020 2020 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 207c 0a7c 6f37 |.|o7 │ │ │ │ 00044760: 203d 2031 3420 2020 2020 2020 2020 2020 = 14 │ │ │ │ @@ -18324,17 +18324,17 @@ │ │ │ │ 00047930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00047940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00047950: 3220 3a20 7469 6d65 2045 756c 6572 4368 2 : time EulerCh │ │ │ │ 00047960: 6172 6163 7465 7269 7374 6963 2049 2020 aracteristic I │ │ │ │ 00047970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047990: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000479a0: 2d2d 2075 7365 6420 302e 3539 3032 3432 -- used 0.590242 │ │ │ │ -000479b0: 7320 2863 7075 293b 2030 2e32 3431 3333 s (cpu); 0.24133 │ │ │ │ -000479c0: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ +000479a0: 2d2d 2075 7365 6420 302e 3437 3335 3435 -- used 0.473545 │ │ │ │ +000479b0: 7320 2863 7075 293b 2030 2e31 3939 3433 s (cpu); 0.19943 │ │ │ │ +000479c0: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ 000479d0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000479e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000479f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -18354,17 +18354,17 @@ │ │ │ │ 00047b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b20: 2020 2020 2020 2020 2020 2020 7c0a 7c43 |.|C │ │ │ │ 00047b30: 6572 7469 6679 3a20 6f75 7470 7574 2063 ertify: output c │ │ │ │ 00047b40: 6572 7469 6669 6564 2120 2020 2020 2020 ertified! │ │ │ │ 00047b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00047b80: 2d2d 2075 7365 6420 302e 3031 3237 3437 -- used 0.012747 │ │ │ │ -00047b90: 3373 2028 6370 7529 3b20 302e 3031 3232 3s (cpu); 0.0122 │ │ │ │ -00047ba0: 3936 3773 2028 7468 7265 6164 293b 2030 967s (thread); 0 │ │ │ │ +00047b80: 2d2d 2075 7365 6420 302e 3037 3030 3030 -- used 0.070000 │ │ │ │ +00047b90: 3473 2028 6370 7529 3b20 302e 3031 3830 4s (cpu); 0.0180 │ │ │ │ +00047ba0: 3639 3373 2028 7468 7265 6164 293b 2030 693s (thread); 0 │ │ │ │ 00047bb0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00047bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00047bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047c10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -19033,18 +19033,18 @@ │ │ │ │ 0004a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a590: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7469 ------+.|i4 : ti │ │ │ │ 0004a5a0: 6d65 2066 6f72 6365 496d 6167 6528 5068 me forceImage(Ph │ │ │ │ 0004a5b0: 692c 6964 6561 6c20 305f 2874 6172 6765 i,ideal 0_(targe │ │ │ │ 0004a5c0: 7420 5068 6929 2920 2020 2020 2020 2020 t Phi)) │ │ │ │ 0004a5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a5e0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -0004a5f0: 6420 302e 3030 3036 3637 3736 3273 2028 d 0.000667762s ( │ │ │ │ -0004a600: 6370 7529 3b20 302e 3030 3036 3536 3037 cpu); 0.00065607 │ │ │ │ -0004a610: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -0004a620: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +0004a5f0: 6420 302e 3030 3036 3237 3735 3173 2028 d 0.000627751s ( │ │ │ │ +0004a600: 6370 7529 3b20 302e 3030 3036 3235 3433 cpu); 0.00062543 │ │ │ │ +0004a610: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ +0004a620: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0004a630: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0004a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a680: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 5068 ------+.|i5 : Ph │ │ │ │ 0004a690: 693b 2020 2020 2020 2020 2020 2020 2020 i; │ │ │ │ @@ -19645,16 +19645,16 @@ │ │ │ │ 0004cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004cbd0: 2d2b 0a7c 6933 203a 2074 696d 6520 2870 -+.|i3 : time (p │ │ │ │ 0004cbe0: 312c 7032 2920 3d20 6772 6170 6820 7068 1,p2) = graph ph │ │ │ │ 0004cbf0: 693b 2020 2020 2020 2020 2020 2020 2020 i; │ │ │ │ 0004cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004cc20: 207c 0a7c 202d 2d20 7573 6564 2030 2e30 |.| -- used 0.0 │ │ │ │ -0004cc30: 3136 3632 3038 7320 2863 7075 293b 2030 166208s (cpu); 0 │ │ │ │ -0004cc40: 2e30 3136 3333 3839 7320 2874 6872 6561 .0163389s (threa │ │ │ │ +0004cc30: 3830 3934 3338 7320 2863 7075 293b 2030 809438s (cpu); 0 │ │ │ │ +0004cc40: 2e30 3233 3434 3339 7320 2874 6872 6561 .0234439s (threa │ │ │ │ 0004cc50: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0004cc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004cc70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0004cc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004cc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004cca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -20941,18 +20941,18 @@ │ │ │ │ 00051cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00051ce0: 7c69 3920 3a20 7469 6d65 2067 203d 2067 |i9 : time g = g │ │ │ │ 00051cf0: 7261 7068 2070 323b 2020 2020 2020 2020 raph p2; │ │ │ │ 00051d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00051d30: 7c20 2d2d 2075 7365 6420 302e 3033 3139 | -- used 0.0319 │ │ │ │ -00051d40: 3939 3773 2028 6370 7529 3b20 302e 3033 997s (cpu); 0.03 │ │ │ │ -00051d50: 3136 3731 3573 2028 7468 7265 6164 293b 16715s (thread); │ │ │ │ -00051d60: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +00051d30: 7c20 2d2d 2075 7365 6420 302e 3039 3734 | -- used 0.0974 │ │ │ │ +00051d40: 3137 7320 2863 7075 293b 2030 2e30 3432 17s (cpu); 0.042 │ │ │ │ +00051d50: 3332 3137 7320 2874 6872 6561 6429 3b20 3217s (thread); │ │ │ │ +00051d60: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00051d70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00051d80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00051d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00051dd0: 7c69 3130 203a 2067 5f30 3b20 2020 2020 |i10 : g_0; │ │ │ │ @@ -21662,16 +21662,16 @@ │ │ │ │ 000549d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000549e0: 2d2d 2b0a 7c69 3320 3a20 7469 6d65 2069 --+.|i3 : time i │ │ │ │ 000549f0: 6465 616c 2070 6869 2020 2020 2020 2020 deal phi │ │ │ │ 00054a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a30: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00054a40: 3030 3335 3237 3832 7320 2863 7075 293b 00352782s (cpu); │ │ │ │ -00054a50: 2030 2e30 3033 3532 3237 3673 2028 7468 0.00352276s (th │ │ │ │ +00054a40: 3030 3337 3639 3232 7320 2863 7075 293b 00376922s (cpu); │ │ │ │ +00054a50: 2030 2e30 3033 3736 3931 3573 2028 7468 0.00376915s (th │ │ │ │ 00054a60: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00054a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a80: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00054a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -22297,16 +22297,16 @@ │ │ │ │ 00057180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00057190: 2d2d 2b0a 7c69 3620 3a20 7469 6d65 2069 --+.|i6 : time i │ │ │ │ 000571a0: 6465 616c 2070 6869 2720 2020 2020 2020 deal phi' │ │ │ │ 000571b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000571e0: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -000571f0: 3238 3138 3437 7320 2863 7075 293b 2030 281847s (cpu); 0 │ │ │ │ -00057200: 2e31 3530 3730 3673 2028 7468 7265 6164 .150706s (thread │ │ │ │ +000571f0: 3235 3330 3536 7320 2863 7075 293b 2030 253056s (cpu); 0 │ │ │ │ +00057200: 2e31 3433 3934 3773 2028 7468 7265 6164 .143947s (thread │ │ │ │ 00057210: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00057220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057230: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00057240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -24855,18 +24855,18 @@ │ │ │ │ 00061160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00061180: 0a7c 6933 203a 2074 696d 6520 696e 7665 .|i3 : time inve │ │ │ │ 00061190: 7273 6520 7068 6920 2020 2020 2020 2020 rse 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 207c | │ │ │ │ -000611d0: 0a7c 202d 2d20 7573 6564 2030 2e30 3537 .| -- used 0.057 │ │ │ │ -000611e0: 3835 3937 7320 2863 7075 293b 2030 2e30 8597s (cpu); 0.0 │ │ │ │ -000611f0: 3537 3836 3435 7320 2874 6872 6561 6429 578645s (thread) │ │ │ │ -00061200: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +000611d0: 0a7c 202d 2d20 7573 6564 2030 2e30 3536 .| -- used 0.056 │ │ │ │ +000611e0: 3335 3832 7320 2863 7075 293b 2030 2e30 3582s (cpu); 0.0 │ │ │ │ +000611f0: 3536 3336 7320 2874 6872 6561 6429 3b20 5636s (thread); │ │ │ │ +00061200: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00061210: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00061220: 0a7c 2020 2020 2020 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 207c | │ │ │ │ 00061270: 0a7c 6f33 203d 202d 2d20 7261 7469 6f6e .|o3 = -- ration │ │ │ │ @@ -27855,16 +27855,16 @@ │ │ │ │ 0006cce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ccf0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2074 696d -----+.|i2 : tim │ │ │ │ 0006cd00: 6520 7073 6920 3d20 696e 7665 7273 654d e psi = inverseM │ │ │ │ 0006cd10: 6170 2070 6869 2020 2020 2020 2020 2020 ap phi │ │ │ │ 0006cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd40: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0006cd50: 2030 2e30 3833 3331 3236 7320 2863 7075 0.0833126s (cpu │ │ │ │ -0006cd60: 293b 2030 2e30 3833 3333 3135 7320 2874 ); 0.0833315s (t │ │ │ │ +0006cd50: 2030 2e30 3832 3039 3436 7320 2863 7075 0.0820946s (cpu │ │ │ │ +0006cd60: 293b 2030 2e30 3832 3039 3533 7320 2874 ); 0.0820953s (t │ │ │ │ 0006cd70: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0006cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd90: 2020 2020 207c 0a7c 2020 2020 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 │ │ │ │ @@ -28539,18 +28539,18 @@ │ │ │ │ 0006f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0006f7c0: 6935 203a 2074 696d 6520 7073 6920 3d20 i5 : time psi = │ │ │ │ 0006f7d0: 696e 7665 7273 654d 6170 2070 6869 2020 inverseMap phi │ │ │ │ 0006f7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f800: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006f810: 202d 2d20 7573 6564 2030 2e33 3030 3339 -- used 0.30039 │ │ │ │ -0006f820: 3173 2028 6370 7529 3b20 302e 3230 3830 1s (cpu); 0.2080 │ │ │ │ -0006f830: 3538 7320 2874 6872 6561 6429 3b20 3073 58s (thread); 0s │ │ │ │ -0006f840: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0006f810: 202d 2d20 7573 6564 2030 2e32 3830 3231 -- used 0.28021 │ │ │ │ +0006f820: 3173 2028 6370 7529 3b20 302e 3138 3531 1s (cpu); 0.1851 │ │ │ │ +0006f830: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +0006f840: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 0006f850: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0006f860: 2020 2020 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 207c 0a7c |.| │ │ │ │ 0006f8b0: 6f35 203d 206d 6170 2028 5151 5b77 202e o5 = map (QQ[w . │ │ │ │ @@ -29536,16 +29536,16 @@ │ │ │ │ 000735f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00073600: 2d2d 2d2d 2b0a 7c69 3320 3a20 7469 6d65 ----+.|i3 : time │ │ │ │ 00073610: 2069 7342 6972 6174 696f 6e61 6c20 7068 isBirational ph │ │ │ │ 00073620: 6920 2020 2020 2020 2020 2020 2020 2020 i │ │ │ │ 00073630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073650: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00073660: 302e 3032 3032 3930 3173 2028 6370 7529 0.0202901s (cpu) │ │ │ │ -00073670: 3b20 302e 3032 3032 3839 3473 2028 7468 ; 0.0202894s (th │ │ │ │ +00073660: 302e 3031 3931 3132 3173 2028 6370 7529 0.0191121s (cpu) │ │ │ │ +00073670: 3b20 302e 3031 3931 3134 3373 2028 7468 ; 0.0191143s (th │ │ │ │ 00073680: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00073690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000736a0: 2020 2020 7c0a 7c20 2020 2020 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 │ │ │ │ @@ -29566,17 +29566,17 @@ │ │ │ │ 000737d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000737e0: 2020 2020 7c0a 7c43 6572 7469 6679 3a20 |.|Certify: │ │ │ │ 000737f0: 6f75 7470 7574 2063 6572 7469 6669 6564 output certified │ │ │ │ 00073800: 2120 2020 2020 2020 2020 2020 2020 2020 ! │ │ │ │ 00073810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073830: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00073840: 302e 3031 3435 3735 3273 2028 6370 7529 0.0145752s (cpu) │ │ │ │ -00073850: 3b20 302e 3031 3431 3536 7320 2874 6872 ; 0.014156s (thr │ │ │ │ -00073860: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00073840: 302e 3036 3335 3339 3473 2028 6370 7529 0.0635394s (cpu) │ │ │ │ +00073850: 3b20 302e 3031 3937 3539 3673 2028 7468 ; 0.0197596s (th │ │ │ │ +00073860: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00073870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00073880: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00073890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000738d0: 2020 2020 7c0a 7c6f 3420 3d20 7472 7565 |.|o4 = true │ │ │ │ @@ -29739,16 +29739,16 @@ │ │ │ │ 000742a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000742b0: 2020 2020 2020 2020 2020 207c 0a7c 4365 |.|Ce │ │ │ │ 000742c0: 7274 6966 793a 206f 7574 7075 7420 6365 rtify: output ce │ │ │ │ 000742d0: 7274 6966 6965 6421 2020 2020 2020 2020 rtified! │ │ │ │ 000742e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000742f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074300: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00074310: 2d20 7573 6564 2033 2e33 3335 3736 7320 - used 3.33576s │ │ │ │ -00074320: 2863 7075 293b 2032 2e35 3932 3732 7320 (cpu); 2.59272s │ │ │ │ +00074310: 2d20 7573 6564 2032 2e32 3033 3934 7320 - used 2.20394s │ │ │ │ +00074320: 2863 7075 293b 2031 2e39 3033 3339 7320 (cpu); 1.90339s │ │ │ │ 00074330: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00074340: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00074350: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00074360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00074390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -30174,16 +30174,16 @@ │ │ │ │ 00075dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075de0: 2020 2020 2020 2020 2020 207c 0a7c 4365 |.|Ce │ │ │ │ 00075df0: 7274 6966 793a 206f 7574 7075 7420 6365 rtify: output ce │ │ │ │ 00075e00: 7274 6966 6965 6421 2020 2020 2020 2020 rtified! │ │ │ │ 00075e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00075e30: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00075e40: 2d20 7573 6564 2034 2e30 3537 3335 7320 - used 4.05735s │ │ │ │ -00075e50: 2863 7075 293b 2032 2e35 3536 3838 7320 (cpu); 2.55688s │ │ │ │ +00075e40: 2d20 7573 6564 2033 2e35 3438 3531 7320 - used 3.54851s │ │ │ │ +00075e50: 2863 7075 293b 2032 2e34 3632 3638 7320 (cpu); 2.46268s │ │ │ │ 00075e60: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00075e70: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00075e80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00075e90: 2020 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 │ │ │ │ @@ -31490,16 +31490,16 @@ │ │ │ │ 0007b010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007b020: 2d2b 0a7c 6932 203a 2074 696d 6520 6b65 -+.|i2 : time ke │ │ │ │ 0007b030: 726e 656c 2870 6869 2c31 2920 2020 2020 rnel(phi,1) │ │ │ │ 0007b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b070: 207c 0a7c 202d 2d20 7573 6564 2030 2e30 |.| -- used 0.0 │ │ │ │ -0007b080: 3138 3031 3439 7320 2863 7075 293b 2030 180149s (cpu); 0 │ │ │ │ -0007b090: 2e30 3138 3031 3237 7320 2874 6872 6561 .0180127s (threa │ │ │ │ +0007b080: 3139 3336 3632 7320 2863 7075 293b 2030 193662s (cpu); 0 │ │ │ │ +0007b090: 2e30 3139 3336 3735 7320 2874 6872 6561 .0193675s (threa │ │ │ │ 0007b0a0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0007b0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b0c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0007b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -31529,18 +31529,18 @@ │ │ │ │ 0007b280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007b290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007b2a0: 2d2b 0a7c 6933 203a 2074 696d 6520 6b65 -+.|i3 : time ke │ │ │ │ 0007b2b0: 726e 656c 2870 6869 2c32 2920 2020 2020 rnel(phi,2) │ │ │ │ 0007b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0007b2f0: 207c 0a7c 202d 2d20 7573 6564 2031 2e30 |.| -- used 1.0 │ │ │ │ -0007b300: 3634 3531 7320 2863 7075 293b 2030 2e35 6451s (cpu); 0.5 │ │ │ │ -0007b310: 3535 3531 3273 2028 7468 7265 6164 293b 55512s (thread); │ │ │ │ -0007b320: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +0007b2f0: 207c 0a7c 202d 2d20 7573 6564 2030 2e39 |.| -- used 0.9 │ │ │ │ +0007b300: 3236 3833 3273 2028 6370 7529 3b20 302e 26832s (cpu); 0. │ │ │ │ +0007b310: 3433 3738 3335 7320 2874 6872 6561 6429 437835s (thread) │ │ │ │ +0007b320: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0007b330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b340: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0007b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b390: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ @@ -32431,17 +32431,17 @@ │ │ │ │ 0007eae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0007eaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0007eb00: 0a7c 6933 203a 2074 696d 6520 7061 7261 .|i3 : time para │ │ │ │ 0007eb10: 6d65 7472 697a 6520 4c20 2020 2020 2020 metrize L │ │ │ │ 0007eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007eb40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0007eb50: 0a7c 202d 2d20 7573 6564 2030 2e30 3034 .| -- used 0.004 │ │ │ │ -0007eb60: 3433 3937 3373 2028 6370 7529 3b20 302e 43973s (cpu); 0. │ │ │ │ -0007eb70: 3030 3434 3335 3235 7320 2874 6872 6561 00443525s (threa │ │ │ │ +0007eb50: 0a7c 202d 2d20 7573 6564 2030 2e30 3035 .| -- used 0.005 │ │ │ │ +0007eb60: 3031 3933 3573 2028 6370 7529 3b20 302e 01935s (cpu); 0. │ │ │ │ +0007eb70: 3030 3530 3137 3833 7320 2874 6872 6561 00501783s (threa │ │ │ │ 0007eb80: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0007eb90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0007eba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0007ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007ebd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007ebe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -32941,18 +32941,18 @@ │ │ │ │ 00080ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00080ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00080ae0: 0a7c 6935 203a 2074 696d 6520 7061 7261 .|i5 : time para │ │ │ │ 00080af0: 6d65 7472 697a 6520 5120 2020 2020 2020 metrize Q │ │ │ │ 00080b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080b20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00080b30: 0a7c 202d 2d20 7573 6564 2030 2e37 3637 .| -- used 0.767 │ │ │ │ -00080b40: 3534 7320 2863 7075 293b 2030 2e34 3630 54s (cpu); 0.460 │ │ │ │ -00080b50: 3238 3273 2028 7468 7265 6164 293b 2030 282s (thread); 0 │ │ │ │ -00080b60: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +00080b30: 0a7c 202d 2d20 7573 6564 2030 2e35 3438 .| -- used 0.548 │ │ │ │ +00080b40: 3431 3273 2028 6370 7529 3b20 302e 3338 412s (cpu); 0.38 │ │ │ │ +00080b50: 3535 3739 7320 2874 6872 6561 6429 3b20 5579s (thread); │ │ │ │ +00080b60: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00080b70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00080b80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00080b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00080bc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00080bd0: 0a7c 6f35 203d 202d 2d20 7261 7469 6f6e .|o5 = -- ration │ │ │ │ @@ -34403,17 +34403,17 @@ │ │ │ │ 00086620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00086630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00086640: 3220 3a20 7469 6d65 2070 203d 2070 6f69 2 : time p = poi │ │ │ │ 00086650: 6e74 2073 6f75 7263 6520 6620 2020 2020 nt source f │ │ │ │ 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 7c0a 7c20 |.| │ │ │ │ -00086690: 2d2d 2075 7365 6420 302e 3239 3836 3233 -- used 0.298623 │ │ │ │ -000866a0: 7320 2863 7075 293b 2030 2e31 3938 3135 s (cpu); 0.19815 │ │ │ │ -000866b0: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ +00086690: 2d2d 2075 7365 6420 302e 3236 3035 3636 -- used 0.260566 │ │ │ │ +000866a0: 7320 2863 7075 293b 2030 2e31 3534 3439 s (cpu); 0.15449 │ │ │ │ +000866b0: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ 000866c0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000866d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000866e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000866f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086720: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -34518,18 +34518,18 @@ │ │ │ │ 00086d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00086d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00086d70: 3320 3a20 7469 6d65 2070 203d 3d20 665e 3 : time p == f^ │ │ │ │ 00086d80: 2a20 6620 7020 2020 2020 2020 2020 2020 * f p │ │ │ │ 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 7c0a 7c20 |.| │ │ │ │ -00086dc0: 2d2d 2075 7365 6420 302e 3137 3639 3434 -- used 0.176944 │ │ │ │ -00086dd0: 7320 2863 7075 293b 2030 2e31 3736 3934 s (cpu); 0.17694 │ │ │ │ -00086de0: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ -00086df0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00086dc0: 2d2d 2075 7365 6420 302e 3039 3535 3233 -- used 0.095523 │ │ │ │ +00086dd0: 3273 2028 6370 7529 3b20 302e 3039 3535 2s (cpu); 0.0955 │ │ │ │ +00086de0: 3236 3173 2028 7468 7265 6164 293b 2030 261s (thread); 0 │ │ │ │ +00086df0: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 00086e00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00086e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00086e50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 00086e60: 3320 3d20 7472 7565 2020 2020 2020 2020 3 = true │ │ │ │ @@ -34850,16 +34850,16 @@ │ │ │ │ 00088210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088220: 2020 7c0a 7c43 6572 7469 6679 3a20 6f75 |.|Certify: ou │ │ │ │ 00088230: 7470 7574 2063 6572 7469 6669 6564 2120 tput certified! │ │ │ │ 00088240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088270: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00088280: 3032 3738 3430 3173 2028 6370 7529 3b20 0278401s (cpu); │ │ │ │ -00088290: 302e 3032 3732 3532 3673 2028 7468 7265 0.0272526s (thre │ │ │ │ +00088280: 3032 3635 3337 3273 2028 6370 7529 3b20 0265372s (cpu); │ │ │ │ +00088290: 302e 3031 3532 3938 3673 2028 7468 7265 0.0152986s (thre │ │ │ │ 000882a0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000882b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000882c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000882d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000882e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000882f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -34980,17 +34980,17 @@ │ │ │ │ 00088a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088a40: 2020 7c0a 7c43 6572 7469 6679 3a20 6f75 |.|Certify: ou │ │ │ │ 00088a50: 7470 7574 2063 6572 7469 6669 6564 2120 tput certified! │ │ │ │ 00088a60: 2020 2020 2020 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 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00088aa0: 3031 3933 3533 7320 2863 7075 293b 2030 019353s (cpu); 0 │ │ │ │ -00088ab0: 2e30 3138 3734 3336 7320 2874 6872 6561 .0187436s (threa │ │ │ │ -00088ac0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +00088aa0: 3037 3138 3331 3773 2028 6370 7529 3b20 0718317s (cpu); │ │ │ │ +00088ab0: 302e 3031 3832 3037 3173 2028 7468 7265 0.0182071s (thre │ │ │ │ +00088ac0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00088ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088ae0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00088af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00088b30: 2020 7c0a 7c6f 3520 3d20 7b32 2c20 342c |.|o5 = {2, 4, │ │ │ │ @@ -35305,16 +35305,16 @@ │ │ │ │ 00089e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00089e90: 2d2d 2b0a 7c69 3720 3a20 7469 6d65 2070 --+.|i7 : time p │ │ │ │ 00089ea0: 726f 6a65 6374 6976 6544 6567 7265 6573 rojectiveDegrees │ │ │ │ 00089eb0: 2070 6869 2020 2020 2020 2020 2020 2020 phi │ │ │ │ 00089ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089ee0: 2020 7c0a 7c20 2d2d 2075 7365 6420 352e |.| -- used 5. │ │ │ │ -00089ef0: 3234 3839 652d 3035 7320 2863 7075 293b 2489e-05s (cpu); │ │ │ │ -00089f00: 2034 2e37 3134 3865 2d30 3573 2028 7468 4.7148e-05s (th │ │ │ │ +00089ef0: 3632 3839 652d 3035 7320 2863 7075 293b 6289e-05s (cpu); │ │ │ │ +00089f00: 2035 2e30 3432 3865 2d30 3573 2028 7468 5.0428e-05s (th │ │ │ │ 00089f10: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00089f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089f30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00089f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00089f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -35340,17 +35340,17 @@ │ │ │ │ 0008a0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0008a0c0: 2d2d 2b0a 7c69 3820 3a20 7469 6d65 2070 --+.|i8 : time p │ │ │ │ 0008a0d0: 726f 6a65 6374 6976 6544 6567 7265 6573 rojectiveDegrees │ │ │ │ 0008a0e0: 2870 6869 2c4e 756d 4465 6772 6565 733d (phi,NumDegrees= │ │ │ │ 0008a0f0: 3e31 2920 2020 2020 2020 2020 2020 2020 >1) │ │ │ │ 0008a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a110: 2020 7c0a 7c20 2d2d 2075 7365 6420 322e |.| -- used 2. │ │ │ │ -0008a120: 3632 3665 2d30 3573 2028 6370 7529 3b20 626e-05s (cpu); │ │ │ │ -0008a130: 322e 3632 3139 652d 3035 7320 2874 6872 2.6219e-05s (thr │ │ │ │ -0008a140: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +0008a120: 3230 3035 652d 3035 7320 2863 7075 293b 2005e-05s (cpu); │ │ │ │ +0008a130: 2032 2e31 3639 3965 2d30 3573 2028 7468 2.1699e-05s (th │ │ │ │ +0008a140: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 0008a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a160: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0008a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0008a1b0: 2020 7c0a 7c6f 3820 3d20 7b34 2c20 317d |.|o8 = {4, 1} │ │ │ │ @@ -37833,18 +37833,18 @@ │ │ │ │ 00093c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00093c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00093ca0: 6934 203a 2074 696d 6520 7068 6921 203b i4 : time phi! ; │ │ │ │ 00093cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093ce0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00093cf0: 202d 2d20 7573 6564 2030 2e30 3537 3233 -- used 0.05723 │ │ │ │ -00093d00: 3935 7320 2863 7075 293b 2030 2e30 3536 95s (cpu); 0.056 │ │ │ │ -00093d10: 3836 3339 7320 2874 6872 6561 6429 3b20 8639s (thread); │ │ │ │ -00093d20: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +00093cf0: 202d 2d20 7573 6564 2030 2e31 3134 3737 -- used 0.11477 │ │ │ │ +00093d00: 3973 2028 6370 7529 3b20 302e 3036 3335 9s (cpu); 0.0635 │ │ │ │ +00093d10: 3435 7320 2874 6872 6561 6429 3b20 3073 45s (thread); 0s │ │ │ │ +00093d20: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00093d30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00093d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00093d80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00093d90: 6f34 203a 2052 6174 696f 6e61 6c4d 6170 o4 : RationalMap │ │ │ │ @@ -38008,18 +38008,18 @@ │ │ │ │ 00094770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00094780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00094790: 6939 203a 2074 696d 6520 7068 6921 203b i9 : time phi! ; │ │ │ │ 000947a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000947d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000947e0: 202d 2d20 7573 6564 2030 2e30 3430 3135 -- used 0.04015 │ │ │ │ -000947f0: 3237 7320 2863 7075 293b 2030 2e30 3339 27s (cpu); 0.039 │ │ │ │ -00094800: 3737 7320 2874 6872 6561 6429 3b20 3073 77s (thread); 0s │ │ │ │ -00094810: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +000947e0: 202d 2d20 7573 6564 2030 2e30 3935 3436 -- used 0.09546 │ │ │ │ +000947f0: 3939 7320 2863 7075 293b 2030 2e30 3431 99s (cpu); 0.041 │ │ │ │ +00094800: 3437 3139 7320 2874 6872 6561 6429 3b20 4719s (thread); │ │ │ │ +00094810: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00094820: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00094830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00094840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00094850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00094860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00094870: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00094880: 6f39 203a 2052 6174 696f 6e61 6c4d 6170 o9 : RationalMap │ │ │ │ @@ -40052,16 +40052,16 @@ │ │ │ │ 0009c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ 0009c750: 203a 2074 696d 6520 7068 695e 2a2a 2071 : time phi^** q │ │ │ │ 0009c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c790: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -0009c7a0: 2d20 7573 6564 2030 2e33 3638 3432 3873 - used 0.368428s │ │ │ │ -0009c7b0: 2028 6370 7529 3b20 302e 3230 3734 3539 (cpu); 0.207459 │ │ │ │ +0009c7a0: 2d20 7573 6564 2030 2e33 3235 3131 3973 - used 0.325119s │ │ │ │ +0009c7b0: 2028 6370 7529 3b20 302e 3138 3831 3736 (cpu); 0.188176 │ │ │ │ 0009c7c0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 0009c7d0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 0009c7e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0009c7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -42172,17 +42172,17 @@ │ │ │ │ 000a4bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000a4bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000a4bd0: 7c69 3320 3a20 7469 6d65 2070 6869 203d |i3 : time phi = │ │ │ │ 000a4be0: 2072 6174 696f 6e61 6c4d 6170 2856 2c33 rationalMap(V,3 │ │ │ │ 000a4bf0: 2c32 2920 2020 2020 2020 2020 2020 2020 ,2) │ │ │ │ 000a4c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000a4c20: 7c20 2d2d 2075 7365 6420 302e 3131 3737 | -- used 0.1177 │ │ │ │ -000a4c30: 3334 7320 2863 7075 293b 2030 2e31 3137 34s (cpu); 0.117 │ │ │ │ -000a4c40: 3730 3973 2028 7468 7265 6164 293b 2030 709s (thread); 0 │ │ │ │ +000a4c20: 7c20 2d2d 2075 7365 6420 302e 3130 3138 | -- used 0.1018 │ │ │ │ +000a4c30: 3236 7320 2863 7075 293b 2030 2e31 3031 26s (cpu); 0.101 │ │ │ │ +000a4c40: 3832 3773 2028 7468 7265 6164 293b 2030 827s (thread); 0 │ │ │ │ 000a4c50: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 000a4c60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000a4c70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000a4c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000a4cb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -43705,17 +43705,17 @@ │ │ │ │ 000aab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000aab90: 2d2d 2d2d 2d2b 0a7c 6936 203a 2074 696d -----+.|i6 : tim │ │ │ │ 000aaba0: 6520 7068 6920 3d20 7261 7469 6f6e 616c e phi = rational │ │ │ │ 000aabb0: 4d61 7020 4420 2020 2020 2020 2020 2020 Map D │ │ │ │ 000aabc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aabd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aabe0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000aabf0: 2030 2e30 3239 3132 3373 2028 6370 7529 0.029123s (cpu) │ │ │ │ -000aac00: 3b20 302e 3032 3931 3233 7320 2874 6872 ; 0.029123s (thr │ │ │ │ -000aac10: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +000aabf0: 2030 2e30 3331 3733 3034 7320 2863 7075 0.0317304s (cpu │ │ │ │ +000aac00: 293b 2030 2e30 3331 3732 3636 7320 2874 ); 0.0317266s (t │ │ │ │ +000aac10: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000aac20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aac30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000aac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aac60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aac70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aac80: 2020 2020 207c 0a7c 6f36 203d 202d 2d20 |.|o6 = -- │ │ │ │ @@ -44715,17 +44715,17 @@ │ │ │ │ 000aeaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000aeab0: 2d2d 2d2d 2d2b 0a7c 6937 203a 2074 696d -----+.|i7 : tim │ │ │ │ 000aeac0: 6520 3f20 696d 6167 6528 7068 692c 2246 e ? image(phi,"F │ │ │ │ 000aead0: 3422 2920 2020 2020 2020 2020 2020 2020 4") │ │ │ │ 000aeae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeb00: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000aeb10: 2031 2e32 3331 3637 7320 2863 7075 293b 1.23167s (cpu); │ │ │ │ -000aeb20: 2030 2e37 3737 3236 7320 2874 6872 6561 0.77726s (threa │ │ │ │ -000aeb30: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +000aeb10: 2031 2e34 3639 3634 7320 2863 7075 293b 1.46964s (cpu); │ │ │ │ +000aeb20: 2030 2e35 3936 3738 3273 2028 7468 7265 0.596782s (thre │ │ │ │ +000aeb30: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000aeb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeb50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000aeb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000aeba0: 2020 2020 207c 0a7c 6f37 203d 2073 7572 |.|o7 = sur │ │ │ │ @@ -46253,17 +46253,17 @@ │ │ │ │ 000b4ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b4ad0: 2d2b 0a7c 6934 203a 2074 696d 6520 5365 -+.|i4 : time Se │ │ │ │ 000b4ae0: 6772 6543 6c61 7373 2058 2020 2020 2020 greClass X │ │ │ │ 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: 207c 0a7c 202d 2d20 7573 6564 2030 2e38 |.| -- used 0.8 │ │ │ │ -000b4b30: 3835 3032 3773 2028 6370 7529 3b20 302e 85027s (cpu); 0. │ │ │ │ -000b4b40: 3531 3139 3273 2028 7468 7265 6164 293b 51192s (thread); │ │ │ │ -000b4b50: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +000b4b30: 3734 3237 3773 2028 6370 7529 3b20 302e 74277s (cpu); 0. │ │ │ │ +000b4b40: 3531 3337 3433 7320 2874 6872 6561 6429 513743s (thread) │ │ │ │ +000b4b50: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 000b4b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4b70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000b4b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4bc0: 207c 0a7c 2020 2020 2020 2020 2020 3720 |.| 7 │ │ │ │ @@ -46307,17 +46307,17 @@ │ │ │ │ 000b4e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b4e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b4e40: 2d2b 0a7c 6935 203a 2074 696d 6520 5365 -+.|i5 : time Se │ │ │ │ 000b4e50: 6772 6543 6c61 7373 206c 6966 7428 582c greClass lift(X, │ │ │ │ 000b4e60: 5037 2920 2020 2020 2020 2020 2020 2020 P7) │ │ │ │ 000b4e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000b4e90: 207c 0a7c 202d 2d20 7573 6564 2030 2e35 |.| -- used 0.5 │ │ │ │ -000b4ea0: 3638 3533 3573 2028 6370 7529 3b20 302e 68535s (cpu); 0. │ │ │ │ -000b4eb0: 3334 3333 3331 7320 2874 6872 6561 6429 343331s (thread) │ │ │ │ +000b4e90: 207c 0a7c 202d 2d20 7573 6564 2030 2e34 |.| -- used 0.4 │ │ │ │ +000b4ea0: 3135 3830 3173 2028 6370 7529 3b20 302e 15801s (cpu); 0. │ │ │ │ +000b4eb0: 3238 3633 3036 7320 2874 6872 6561 6429 286306s (thread) │ │ │ │ 000b4ec0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 000b4ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4ee0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000b4ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b4f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46368,16 +46368,16 @@ │ │ │ │ 000b51f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5200: 207c 0a7c 4365 7274 6966 793a 206f 7574 |.|Certify: out │ │ │ │ 000b5210: 7075 7420 6365 7274 6966 6965 6421 2020 put certified! │ │ │ │ 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: 207c 0a7c 202d 2d20 7573 6564 2030 2e30 |.| -- used 0.0 │ │ │ │ -000b5260: 3231 3635 3239 7320 2863 7075 293b 2030 216529s (cpu); 0 │ │ │ │ -000b5270: 2e30 3231 3335 3237 7320 2874 6872 6561 .0213527s (threa │ │ │ │ +000b5260: 3439 3937 3938 7320 2863 7075 293b 2030 499798s (cpu); 0 │ │ │ │ +000b5270: 2e30 3232 3239 3834 7320 2874 6872 6561 .0222984s (threa │ │ │ │ 000b5280: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 000b5290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b52a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000b52b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b52c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b52d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b52e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46427,18 +46427,18 @@ │ │ │ │ 000b55a0: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ 000b55b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b55c0: 207c 0a7c 4365 7274 6966 793a 206f 7574 |.|Certify: out │ │ │ │ 000b55d0: 7075 7420 6365 7274 6966 6965 6421 2020 put certified! │ │ │ │ 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: 207c 0a7c 202d 2d20 7573 6564 2030 2e30 |.| -- used 0.0 │ │ │ │ -000b5620: 3937 3330 3433 7320 2863 7075 293b 2030 973043s (cpu); 0 │ │ │ │ -000b5630: 2e30 3937 3034 3632 7320 2874 6872 6561 .0970462s (threa │ │ │ │ -000b5640: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +000b5610: 207c 0a7c 202d 2d20 7573 6564 2030 2e32 |.| -- used 0.2 │ │ │ │ +000b5620: 3830 3237 3173 2028 6370 7529 3b20 302e 80271s (cpu); 0. │ │ │ │ +000b5630: 3135 3133 3231 7320 2874 6872 6561 6429 151321s (thread) │ │ │ │ +000b5640: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 000b5650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5660: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000b5670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b56a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b56b0: 207c 0a7c 2020 2020 2020 2020 2020 3720 |.| 7 │ │ │ │ @@ -46544,25 +46544,25 @@ │ │ │ │ 000b5cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b5d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d50: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ -000b5d60: 2020 2020 5a5a 2020 2020 2020 2020 2020 ZZ │ │ │ │ +000b5d60: 3d20 2020 5a5a 2020 2020 2020 2020 2020 = ZZ │ │ │ │ 000b5d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000b5da0: 2020 2020 2020 2020 2020 7c0a 7c3d 202d |.|= - │ │ │ │ -000b5db0: 2d2d 2d2d 2d5b 7820 2e2e 7820 5d20 2020 -----[x ..x ] │ │ │ │ +000b5da0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ +000b5db0: 2d2d 2d2d 5b78 202e 2e78 205d 2020 2020 ----[x ..x ] │ │ │ │ 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 7c0a 7c20 2031 |.| 1 │ │ │ │ -000b5e00: 3030 3030 3320 2030 2020 2036 2020 2020 00003 0 6 │ │ │ │ +000b5df0: 2020 2020 2020 2020 2020 7c0a 7c20 3130 |.| 10 │ │ │ │ +000b5e00: 3030 3033 2020 3020 2020 3620 2020 2020 0003 0 6 │ │ │ │ 000b5e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5e40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b5e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46579,18 +46579,18 @@ │ │ │ │ 000b5f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b5f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 ----------+.|i10 │ │ │ │ 000b5f40: 203a 2074 696d 6520 7068 6920 3d20 696e : time phi = in │ │ │ │ 000b5f50: 7665 7273 654d 6170 2074 6f4d 6170 286d verseMap toMap(m │ │ │ │ 000b5f60: 696e 6f72 7328 322c 6d61 7472 6978 7b7b inors(2,matrix{{ │ │ │ │ 000b5f70: 785f 302c 785f 312c 785f 332c 785f 342c x_0,x_1,x_3,x_4, │ │ │ │ 000b5f80: 785f 357d 2c7b 785f 312c 7c0a 7c20 2d2d x_5},{x_1,|.| -- │ │ │ │ -000b5f90: 2075 7365 6420 302e 3233 3731 3034 7320 used 0.237104s │ │ │ │ -000b5fa0: 2863 7075 293b 2030 2e31 3235 3632 3573 (cpu); 0.125625s │ │ │ │ -000b5fb0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -000b5fc0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +000b5f90: 2075 7365 6420 302e 3036 3631 3431 3973 used 0.0661419s │ │ │ │ +000b5fa0: 2028 6370 7529 3b20 302e 3036 3631 3433 (cpu); 0.066143 │ │ │ │ +000b5fb0: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ +000b5fc0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000b5fd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b5fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b5ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6020: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b6030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -46784,18 +46784,18 @@ │ │ │ │ 000b6bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b6c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 ----------+.|i11 │ │ │ │ 000b6c10: 203a 2074 696d 6520 5365 6772 6543 6c61 : time SegreCla │ │ │ │ 000b6c20: 7373 2070 6869 2020 2020 2020 2020 2020 ss phi │ │ │ │ 000b6c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6c50: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000b6c60: 2075 7365 6420 302e 3338 3531 3435 7320 used 0.385145s │ │ │ │ -000b6c70: 2863 7075 293b 2030 2e32 3530 3736 3673 (cpu); 0.250766s │ │ │ │ -000b6c80: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -000b6c90: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ +000b6c60: 2075 7365 6420 302e 3337 3436 3835 7320 used 0.374685s │ │ │ │ +000b6c70: 2863 7075 293b 2030 2e32 3436 3373 2028 (cpu); 0.2463s ( │ │ │ │ +000b6c80: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000b6c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6ca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b6cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b6cf0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b6d00: 2020 2020 2020 3920 2020 2020 2038 2020 9 8 │ │ │ │ @@ -46944,16 +46944,16 @@ │ │ │ │ 000b75f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7600: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b7610: 2020 2074 696d 6520 5365 6772 6543 6c61 time SegreCla │ │ │ │ 000b7620: 7373 2042 2020 2020 2020 2020 2020 2020 ss B │ │ │ │ 000b7630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7650: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000b7660: 2075 7365 6420 302e 3430 3539 3737 7320 used 0.405977s │ │ │ │ -000b7670: 2863 7075 293b 2030 2e32 3637 3335 3373 (cpu); 0.267353s │ │ │ │ +000b7660: 2075 7365 6420 302e 3339 3134 3232 7320 used 0.391422s │ │ │ │ +000b7670: 2863 7075 293b 2030 2e32 3637 3239 3873 (cpu); 0.267298s │ │ │ │ 000b7680: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000b7690: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000b76a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b76b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b76c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b76d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b76e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -47004,18 +47004,18 @@ │ │ │ │ 000b79b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b79c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b79d0: 2020 2074 696d 6520 5365 6772 6543 6c61 time SegreCla │ │ │ │ 000b79e0: 7373 206c 6966 7428 422c 616d 6269 656e ss lift(B,ambien │ │ │ │ 000b79f0: 7420 7269 6e67 2042 2920 2020 2020 2020 t ring B) │ │ │ │ 000b7a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7a10: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000b7a20: 2075 7365 6420 312e 3633 3131 3973 2028 used 1.63119s ( │ │ │ │ -000b7a30: 6370 7529 3b20 302e 3935 3132 3634 7320 cpu); 0.951264s │ │ │ │ -000b7a40: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -000b7a50: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +000b7a20: 2075 7365 6420 312e 3430 3432 3273 2028 used 1.40422s ( │ │ │ │ +000b7a30: 6370 7529 3b20 302e 3833 3039 3573 2028 cpu); 0.83095s ( │ │ │ │ +000b7a40: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000b7a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7a60: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b7a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b7ab0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000b7ac0: 2020 2020 2020 2020 3920 2020 2020 2020 9 │ │ │ │ @@ -47254,18 +47254,18 @@ │ │ │ │ 000b8950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000b8960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 000b8970: 203a 2074 696d 6520 6170 706c 7928 312e : time apply(1. │ │ │ │ 000b8980: 2e31 322c 6920 2d3e 2064 6573 6372 6962 .12,i -> describ │ │ │ │ 000b8990: 6520 7370 6563 6961 6c43 7265 6d6f 6e61 e specialCremona │ │ │ │ 000b89a0: 5472 616e 7366 6f72 6d61 7469 6f6e 2869 Transformation(i │ │ │ │ 000b89b0: 2c5a 5a2f 3333 3331 2929 207c 0a7c 202d ,ZZ/3331)) |.| - │ │ │ │ -000b89c0: 2d20 7573 6564 2031 2e33 3733 3673 2028 - used 1.3736s ( │ │ │ │ -000b89d0: 6370 7529 3b20 312e 3133 3731 3473 2028 cpu); 1.13714s ( │ │ │ │ -000b89e0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -000b89f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000b89c0: 2d20 7573 6564 2031 2e30 3031 3933 7320 - used 1.00193s │ │ │ │ +000b89d0: 2863 7075 293b 2030 2e39 3131 3132 3173 (cpu); 0.911121s │ │ │ │ +000b89e0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +000b89f0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000b8a00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000b8a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b8a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b8a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b8a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000b8a50: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 000b8a60: 203d 2028 7261 7469 6f6e 616c 206d 6170 = (rational map │ │ │ │ @@ -48077,914 +48077,914 @@ │ │ │ │ 000bbcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000bbcd0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7469 ------+.|i1 : ti │ │ │ │ 000bbce0: 6d65 2073 7065 6369 616c 4375 6269 6354 me specialCubicT │ │ │ │ 000bbcf0: 7261 6e73 666f 726d 6174 696f 6e20 3920 ransformation 9 │ │ │ │ 000bbd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd20: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -000bbd30: 6420 302e 3130 3233 3031 7320 2863 7075 d 0.102301s (cpu │ │ │ │ -000bbd40: 293b 2030 2e31 3032 3330 3273 2028 7468 ); 0.102302s (th │ │ │ │ -000bbd50: 7265 6164 293b 2030 7320 2867 6320 2020 read); 0s (gc │ │ │ │ +000bbd30: 6420 302e 3038 3135 3434 7320 2863 7075 d 0.081544s (cpu │ │ │ │ +000bbd40: 293b 2030 2e30 3831 3534 3438 7320 2874 ); 0.0815448s (t │ │ │ │ +000bbd50: 6872 6561 6420 2020 2020 2020 2020 2020 hread │ │ │ │ 000bbd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd70: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000bbd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbdc0: 2020 2020 2020 7c0a 7c6f 3120 3d20 2d2d |.|o1 = -- │ │ │ │ 000bbdd0: 2072 6174 696f 6e61 6c20 6d61 7020 2d2d rational map -- │ │ │ │ 000bbde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbe00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbe10: 2020 2020 2020 7c0a 7c20 2020 2020 736f |.| so │ │ │ │ 000bbe20: 7572 6365 3a20 5072 6f6a 2851 515b 7820 urce: Proj(QQ[x │ │ │ │ 000bbe30: 2c20 7820 2c20 7820 2c20 7820 2c20 7820 , x , x , x , x │ │ │ │ -000bbe40: 2c20 7820 2c20 7820 5d29 2020 2020 2020 , x , x ]) │ │ │ │ +000bbe40: 2c20 7820 2c20 2020 2020 2020 2020 2020 , x , │ │ │ │ 000bbe50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbe60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000bbe70: 2020 2020 2020 2020 2020 2020 2020 2030 0 │ │ │ │ 000bbe80: 2020 2031 2020 2032 2020 2033 2020 2034 1 2 3 4 │ │ │ │ -000bbe90: 2020 2035 2020 2036 2020 2020 2020 2020 5 6 │ │ │ │ +000bbe90: 2020 2035 2020 2020 2020 2020 2020 2020 5 │ │ │ │ 000bbea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbeb0: 2020 2020 2020 7c0a 7c20 2020 2020 7461 |.| ta │ │ │ │ 000bbec0: 7267 6574 3a20 7375 6276 6172 6965 7479 rget: subvariety │ │ │ │ 000bbed0: 206f 6620 5072 6f6a 2851 515b 7420 2c20 of Proj(QQ[t , │ │ │ │ -000bbee0: 7420 2c20 7420 2c20 7420 2c20 7420 2020 t , t , t , t │ │ │ │ +000bbee0: 7420 2c20 7420 2020 2020 2020 2020 2020 t , t │ │ │ │ 000bbef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbf00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000bbf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbf20: 2020 2020 2020 2020 2020 2020 2030 2020 0 │ │ │ │ -000bbf30: 2031 2020 2032 2020 2033 2020 2020 2020 1 2 3 │ │ │ │ +000bbf30: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 000bbf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbf50: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000bbf60: 2020 2020 2020 7b20 2020 2020 2020 2020 { │ │ │ │ 000bbf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbfa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000bbfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bbff0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000bc000: 2020 2020 2020 2033 7420 7420 202b 2031 3t t + 1 │ │ │ │ 000bc010: 3074 2074 2020 2b20 3130 7420 7420 202d 0t t + 10t t - │ │ │ │ -000bc020: 2033 7420 7420 202d 2031 3074 2020 2020 3t t - 10t │ │ │ │ +000bc020: 2033 7420 7420 2020 2020 2020 2020 2020 3t t │ │ │ │ 000bc030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bc040: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000bc050: 2020 2020 2020 2020 2032 2034 2020 2020 2 4 │ │ │ │ 000bc060: 2020 3320 3420 2020 2020 2030 2035 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+000bf240: 2020 2020 2020 7c0a 7c20 7820 7820 7820 |.| x x x │ │ │ │ +000bf250: 202d 2034 3778 2078 2078 2020 2b20 3131 - 47x x x + 11 │ │ │ │ +000bf260: 7820 7820 202d 2032 7820 7820 202d 2036 x x - 2x x - 6 │ │ │ │ +000bf270: 3678 2078 2020 2b20 3232 7820 7820 202b 6x x + 22x x + │ │ │ │ +000bf280: 2032 3078 2020 2d20 3278 2078 2020 2b20 20x - 2x x + │ │ │ │ +000bf290: 3138 7820 2020 7c0a 7c20 2030 2032 2033 18x |.| 0 2 3 │ │ │ │ +000bf2a0: 2020 2020 2020 3120 3220 3320 2020 2020 1 2 3 │ │ │ │ +000bf2b0: 2032 2033 2020 2020 2030 2033 2020 2020 2 3 0 3 │ │ │ │ +000bf2c0: 2020 3120 3320 2020 2020 2032 2033 2020 1 3 2 3 │ │ │ │ +000bf2d0: 2020 2020 3320 2020 2020 3020 3420 2020 3 0 4 │ │ │ │ +000bf2e0: 2020 2030 2020 7c0a 7c20 2020 2020 2020 0 |.| │ │ │ │ 000bf2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bf300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bf310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bf320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bf330: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -000bf340: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -000bf350: 2032 2020 2020 2020 2020 3220 2020 2020 2 2 │ │ │ │ -000bf360: 2020 2032 2020 2020 2033 2020 2020 2020 2 3 │ │ │ │ -000bf370: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -000bf380: 2020 2020 2020 7c0a 7c20 3236 7820 7820 |.| 26x x │ │ │ │ -000bf390: 7820 202d 2036 7820 7820 202b 2034 7820 x - 6x x + 4x │ │ │ │ -000bf3a0: 7820 202b 2032 3878 2078 2020 2d20 3132 x + 28x x - 12 │ │ │ │ -000bf3b0: 7820 7820 202d 2038 7820 202d 2031 3078 x x - 8x - 10x │ │ │ │ -000bf3c0: 2078 2078 2020 2d20 3278 2078 2020 2b20 x x - 2x x + │ │ │ │ -000bf3d0: 3130 7820 2020 7c0a 7c20 2020 2031 2032 10x |.| 1 2 │ │ │ │ -000bf3e0: 2033 2020 2020 2032 2033 2020 2020 2030 3 2 3 0 │ │ │ │ -000bf3f0: 2033 2020 2020 2020 3120 3320 2020 2020 3 1 3 │ │ │ │ -000bf400: 2032 2033 2020 2020 2033 2020 2020 2020 2 3 3 │ │ │ │ -000bf410: 3020 3120 3420 2020 2020 3120 3420 2020 0 1 4 1 4 │ │ │ │ +000bf340: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +000bf350: 2020 3220 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ +000bf360: 2020 2020 3220 2020 2020 3320 2020 2020 2 3 │ │ │ │ +000bf370: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +000bf380: 2020 2020 2020 7c0a 7c2b 2032 3678 2078 |.|+ 26x x │ │ │ │ +000bf390: 2078 2020 2d20 3678 2078 2020 2b20 3478 x - 6x x + 4x │ │ │ │ +000bf3a0: 2078 2020 2b20 3238 7820 7820 202d 2031 x + 28x x - 1 │ │ │ │ +000bf3b0: 3278 2078 2020 2d20 3878 2020 2d20 3130 2x x - 8x - 10 │ │ │ │ +000bf3c0: 7820 7820 7820 202d 2032 7820 7820 202b x x x - 2x x + │ │ │ │ +000bf3d0: 2031 3078 2020 7c0a 7c20 2020 2020 3120 10x |.| 1 │ │ │ │ +000bf3e0: 3220 3320 2020 2020 3220 3320 2020 2020 2 3 2 3 │ │ │ │ +000bf3f0: 3020 3320 2020 2020 2031 2033 2020 2020 0 3 1 3 │ │ │ │ +000bf400: 2020 3220 3320 2020 2020 3320 2020 2020 2 3 3 │ │ │ │ +000bf410: 2030 2031 2034 2020 2020 2031 2034 2020 0 1 4 1 4 │ │ │ │ 000bf420: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000bf430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bf440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bf450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000bf460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000bf470: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000bf470: 2020 2020 2020 7c0a 7c32 2020 2020 2020 |.|2 │ │ │ │ 000bf480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000bf490: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -000bf4a0: 2020 3220 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ -000bf4b0: 2020 2020 3220 2020 2020 2033 2020 2020 2 3 │ │ │ │ -000bf4c0: 2032 2020 2020 7c0a 7c78 2020 2d20 3231 2 |.|x - 21 │ │ │ │ -000bf4d0: 7820 7820 7820 202b 2034 3778 2078 2078 x x x + 47x x x │ │ │ │ -000bf4e0: 2020 2d20 3133 7820 7820 202d 2031 3478 - 13x x - 14x │ │ │ │ -000bf4f0: 2078 2020 2b20 3636 7820 7820 202d 2032 x + 66x x - 2 │ │ │ │ -000bf500: 3278 2078 2020 2d20 3230 7820 202b 2032 2x x - 20x + 2 │ │ │ │ -000bf510: 7820 7820 2020 7c0a 7c20 3320 2020 2020 x x |.| 3 │ │ │ │ -000bf520: 2030 2032 2033 2020 2020 2020 3120 3220 0 2 3 1 2 │ │ │ │ -000bf530: 3320 2020 2020 2032 2033 2020 2020 2020 3 2 3 │ │ │ │ -000bf540: 3020 3320 2020 2020 2031 2033 2020 2020 0 3 1 3 │ │ │ │ -000bf550: 2020 3220 3320 2020 2020 2033 2020 2020 2 3 3 │ │ │ │ -000bf560: 2030 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 0 |.|------- │ │ │ │ +000bf490: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +000bf4a0: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +000bf4b0: 2020 2020 2032 2020 2020 2020 3320 2020 2 3 │ │ │ │ +000bf4c0: 2020 3220 2020 7c0a 7c20 7820 202d 2032 2 |.| x - 2 │ │ │ │ +000bf4d0: 3178 2078 2078 2020 2b20 3437 7820 7820 1x x x + 47x x │ │ │ │ +000bf4e0: 7820 202d 2031 3378 2078 2020 2d20 3134 x - 13x x - 14 │ │ │ │ +000bf4f0: 7820 7820 202b 2036 3678 2078 2020 2d20 x x + 66x x - │ │ │ │ +000bf500: 3232 7820 7820 202d 2032 3078 2020 2b20 22x x - 20x + │ │ │ │ +000bf510: 3278 2078 2020 7c0a 7c31 2033 2020 2020 2x x |.|1 3 │ │ │ │ +000bf520: 2020 3020 3220 3320 2020 2020 2031 2032 0 2 3 1 2 │ │ │ │ +000bf530: 2033 2020 2020 2020 3220 3320 2020 2020 3 2 3 │ │ │ │ +000bf540: 2030 2033 2020 2020 2020 3120 3320 2020 0 3 1 3 │ │ │ │ +000bf550: 2020 2032 2033 2020 2020 2020 3320 2020 2 3 3 │ │ │ │ +000bf560: 2020 3020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 0 |.|------- │ │ │ │ 000bf570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000bf580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000bf590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000bf5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000bf5b0: 2d2d 2d2d 2d2d 7c0a 7c20 7420 202d 2031 ------|.| t - 1 │ │ │ │ 000bf5c0: 3374 2074 2020 2b20 3133 7420 7420 202b 3t t + 13t t + │ │ │ │ 000bf5d0: 2031 3874 2074 2020 2d20 3474 2074 2020 18t t - 4t t │ │ │ │ @@ -50527,17 +50527,17 @@ │ │ │ │ 000c55e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000c55f0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 7469 ------+.|i2 : ti │ │ │ │ 000c5600: 6d65 2064 6573 6372 6962 6520 6f6f 2020 me describe oo │ │ │ │ 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 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -000c5650: 6420 302e 3032 3435 3432 3873 2028 6370 d 0.0245428s (cp │ │ │ │ -000c5660: 7529 3b20 302e 3032 3435 3434 3273 2028 u); 0.0245442s ( │ │ │ │ -000c5670: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000c5650: 6420 302e 3031 3733 3731 3873 2028 6370 d 0.0173718s (cp │ │ │ │ +000c5660: 7529 3b20 302e 3031 3733 3733 7320 2874 u); 0.017373s (t │ │ │ │ +000c5670: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 000c5680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c5690: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000c56a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c56b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c56c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c56d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c56e0: 2020 2020 2020 7c0a 7c6f 3220 3d20 7261 |.|o2 = ra │ │ │ │ @@ -50741,16 +50741,16 @@ │ │ │ │ 000c6340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000c6350: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ 000c6360: 7469 6d65 2073 7065 6369 616c 5175 6164 time specialQuad │ │ │ │ 000c6370: 7261 7469 6354 7261 6e73 666f 726d 6174 raticTransformat │ │ │ │ 000c6380: 696f 6e20 3420 2020 2020 2020 2020 2020 ion 4 │ │ │ │ 000c6390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c63a0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000c63b0: 7365 6420 302e 3038 3739 3434 3473 2028 sed 0.0879444s ( │ │ │ │ -000c63c0: 6370 7529 3b20 302e 3038 3739 3435 3773 cpu); 0.0879457s │ │ │ │ +000c63b0: 7365 6420 302e 3037 3335 3530 3773 2028 sed 0.0735507s ( │ │ │ │ +000c63c0: 6370 7529 3b20 302e 3037 3335 3530 3373 cpu); 0.0735503s │ │ │ │ 000c63d0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000c63e0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000c63f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000c6400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c6410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c6420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c6430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -51296,17 +51296,17 @@ │ │ │ │ 000c85f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000c8600: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ 000c8610: 7469 6d65 2064 6573 6372 6962 6520 6f6f time describe oo │ │ │ │ 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 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000c8660: 7365 6420 302e 3030 3834 3734 3235 7320 sed 0.00847425s │ │ │ │ -000c8670: 2863 7075 293b 2030 2e30 3038 3437 3639 (cpu); 0.0084769 │ │ │ │ -000c8680: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +000c8660: 7365 6420 302e 3030 3639 3937 3433 7320 sed 0.00699743s │ │ │ │ +000c8670: 2863 7075 293b 2030 2e30 3036 3939 3639 (cpu); 0.0069969 │ │ │ │ +000c8680: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ 000c8690: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000c86a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000c86b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c86c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c86d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c86e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000c86f0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ @@ -52407,17 +52407,17 @@ │ │ │ │ 000ccb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000ccb70: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 696d -----+.|i4 : tim │ │ │ │ 000ccb80: 6520 7068 6927 203d 2076 616c 7565 2073 e phi' = value s │ │ │ │ 000ccb90: 7472 3b20 2020 2020 2020 2020 2020 2020 tr; │ │ │ │ 000ccba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccbc0: 207c 0a7c 202d 2d20 7573 6564 2030 2e30 |.| -- used 0.0 │ │ │ │ -000ccbd0: 3232 3739 3937 7320 2863 7075 293b 2030 227997s (cpu); 0 │ │ │ │ -000ccbe0: 2e30 3232 3739 3873 2028 7468 7265 6164 .022798s (thread │ │ │ │ -000ccbf0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +000ccbd0: 3233 3837 3635 7320 2863 7075 293b 2030 238765s (cpu); 0 │ │ │ │ +000ccbe0: 2e30 3233 3837 3633 7320 2874 6872 6561 .0238763s (threa │ │ │ │ +000ccbf0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 000ccc00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000ccc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccc50: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ 000ccc60: 2052 6174 696f 6e61 6c4d 6170 2028 6375 RationalMap (cu │ │ │ │ @@ -52430,18 +52430,18 @@ │ │ │ │ 000cccd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000ccce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000cccf0: 2d2b 0a7c 6935 203a 2074 696d 6520 6465 -+.|i5 : time de │ │ │ │ 000ccd00: 7363 7269 6265 2070 6869 2720 2020 2020 scribe phi' │ │ │ │ 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 207c 0a7c |.| │ │ │ │ -000ccd40: 202d 2d20 7573 6564 2030 2e30 3035 3534 -- used 0.00554 │ │ │ │ -000ccd50: 3135 3373 2028 6370 7529 3b20 302e 3030 153s (cpu); 0.00 │ │ │ │ -000ccd60: 3535 3431 3937 7320 2874 6872 6561 6429 554197s (thread) │ │ │ │ -000ccd70: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +000ccd40: 202d 2d20 7573 6564 2030 2e30 3036 3037 -- used 0.00607 │ │ │ │ +000ccd50: 3837 3773 2028 6370 7529 3b20 302e 3030 877s (cpu); 0.00 │ │ │ │ +000ccd60: 3630 3833 3473 2028 7468 7265 6164 293b 60834s (thread); │ │ │ │ +000ccd70: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 000ccd80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000ccd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000ccdd0: 2020 2020 207c 0a7c 6f35 203d 2072 6174 |.|o5 = rat │ │ │ │ 000ccde0: 696f 6e61 6c20 6d61 7020 6465 6669 6e65 ional map define │ │ │ │ @@ -52497,18 +52497,18 @@ │ │ │ │ 000cd100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000cd110: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ 000cd120: 2074 696d 6520 6465 7363 7269 6265 2069 time describe i │ │ │ │ 000cd130: 6e76 6572 7365 2070 6869 2720 2020 2020 nverse phi' │ │ │ │ 000cd140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd160: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000cd170: 2030 2e30 3034 3535 3435 3673 2028 6370 0.00455456s (cp │ │ │ │ -000cd180: 7529 3b20 302e 3030 3435 3535 3735 7320 u); 0.00455575s │ │ │ │ -000cd190: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -000cd1a0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +000cd170: 2030 2e30 3035 3132 3339 3373 2028 6370 0.00512393s (cp │ │ │ │ +000cd180: 7529 3b20 302e 3030 3531 3238 3673 2028 u); 0.0051286s ( │ │ │ │ +000cd190: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000cd1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd1b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000cd1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000cd1f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000cd200: 6f36 203d 2072 6174 696f 6e61 6c20 6d61 o6 = rational ma │ │ │ │ 000cd210: 7020 6465 6669 6e65 6420 6279 2066 6f72 p defined by for │ │ ├── ./usr/share/info/DGAlgebras.info.gz │ │ │ ├── DGAlgebras.info │ │ │ │ @@ -3626,16 +3626,16 @@ │ │ │ │ 0000e290: 2042 2020 2020 2020 2020 2020 2020 2020 B │ │ │ │ 0000e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e2d0: 207c 0a7c 4669 6e64 696e 6720 6561 7379 |.|Finding easy │ │ │ │ 0000e2e0: 2072 656c 6174 696f 6e73 2020 2020 2020 relations │ │ │ │ 0000e2f0: 2020 2020 203a 2020 2d2d 2075 7365 6420 : -- used │ │ │ │ -0000e300: 302e 3031 3938 3038 3973 2028 6370 7529 0.0198089s (cpu) │ │ │ │ -0000e310: 3b20 302e 3031 3835 3637 3773 2020 2020 ; 0.0185677s │ │ │ │ +0000e300: 302e 3230 3333 3135 7320 2863 7075 293b 0.203315s (cpu); │ │ │ │ +0000e310: 2030 2e30 3432 3931 3173 2020 2020 2020 0.042911s │ │ │ │ 0000e320: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e370: 207c 0a7c 6f35 203d 2048 4220 2020 2020 |.|o5 = HB │ │ │ │ 0000e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3889,17 +3889,17 @@ │ │ │ │ 0000f300: 2068 6f6d 6f6c 6f67 7941 6c67 6562 7261 homologyAlgebra │ │ │ │ 0000f310: 2843 2c47 656e 4465 6772 6565 4c69 6d69 (C,GenDegreeLimi │ │ │ │ 0000f320: 743d 3e34 2c52 656c 4465 6772 6565 4c69 t=>4,RelDegreeLi │ │ │ │ 0000f330: 6d69 743d 3e34 2920 2020 2020 2020 2020 mit=>4) │ │ │ │ 0000f340: 2020 2020 2020 2020 7c0a 7c46 696e 6469 |.|Findi │ │ │ │ 0000f350: 6e67 2065 6173 7920 7265 6c61 7469 6f6e ng easy relation │ │ │ │ 0000f360: 7320 2020 2020 2020 2020 2020 3a20 202d s : - │ │ │ │ -0000f370: 2d20 7573 6564 2030 2e30 3139 3939 3673 - used 0.019996s │ │ │ │ -0000f380: 2028 6370 7529 3b20 302e 3031 3834 3330 (cpu); 0.018430 │ │ │ │ -0000f390: 3673 2020 2020 2020 7c0a 7c20 2020 2020 6s |.| │ │ │ │ +0000f370: 2d20 7573 6564 2030 2e30 3330 3830 3534 - used 0.0308054 │ │ │ │ +0000f380: 7320 2863 7075 293b 2030 2e30 3139 3235 s (cpu); 0.01925 │ │ │ │ +0000f390: 3635 7320 2020 2020 7c0a 7c20 2020 2020 65s |.| │ │ │ │ 0000f3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0000f3f0: 2020 5a5a 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ 0000f400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5293,17 +5293,17 @@ │ │ │ │ 00014ac0: 2048 4b52 203d 2048 4820 4b52 2020 2020 HKR = HH KR │ │ │ │ 00014ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b00: 2020 2020 2020 2020 207c 0a7c 4669 6e64 |.|Find │ │ │ │ 00014b10: 696e 6720 6561 7379 2072 656c 6174 696f ing easy relatio │ │ │ │ 00014b20: 6e73 2020 2020 2020 2020 2020 203a 2020 ns : │ │ │ │ -00014b30: 2d2d 2075 7365 6420 302e 3032 3432 3738 -- used 0.024278 │ │ │ │ -00014b40: 3773 2028 6370 7529 3b20 302e 3032 3331 7s (cpu); 0.0231 │ │ │ │ -00014b50: 3736 3373 2020 2020 207c 0a7c 2020 2020 763s |.| │ │ │ │ +00014b30: 2d2d 2075 7365 6420 302e 3137 3431 3734 -- used 0.174174 │ │ │ │ +00014b40: 7320 2863 7075 293b 2030 2e30 3334 3439 s (cpu); 0.03449 │ │ │ │ +00014b50: 3538 7320 2020 2020 207c 0a7c 2020 2020 58s |.| │ │ │ │ 00014b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ba0: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ 00014bb0: 2048 4b52 2020 2020 2020 2020 2020 2020 HKR │ │ │ │ 00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5408,17 +5408,17 @@ │ │ │ │ 000151f0: 3a20 484b 5227 203d 2048 4820 6b6f 737a : HKR' = HH kosz │ │ │ │ 00015200: 756c 436f 6d70 6c65 7844 4741 2052 2720 ulComplexDGA R' │ │ │ │ 00015210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015230: 2020 2020 2020 2020 207c 0a7c 4669 6e64 |.|Find │ │ │ │ 00015240: 696e 6720 6561 7379 2072 656c 6174 696f ing easy relatio │ │ │ │ 00015250: 6e73 2020 2020 2020 2020 2020 203a 2020 ns : │ │ │ │ -00015260: 2d2d 2075 7365 6420 302e 3639 3133 3931 -- used 0.691391 │ │ │ │ -00015270: 7320 2863 7075 293b 2030 2e35 3932 3030 s (cpu); 0.59200 │ │ │ │ -00015280: 3773 2020 2020 2020 207c 0a7c 2020 2020 7s |.| │ │ │ │ +00015260: 2d2d 2075 7365 6420 302e 3637 3333 3532 -- used 0.673352 │ │ │ │ +00015270: 7320 2863 7075 293b 2030 2e36 3539 3430 s (cpu); 0.65940 │ │ │ │ +00015280: 3373 2020 2020 2020 207c 0a7c 2020 2020 3s |.| │ │ │ │ 00015290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152d0: 2020 2020 2020 2020 207c 0a7c 6f31 3020 |.|o10 │ │ │ │ 000152e0: 3d20 484b 5227 2020 2020 2020 2020 2020 = HKR' │ │ │ │ 000152f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7277,17 +7277,17 @@ │ │ │ │ 0001c6c0: 3137 203a 2048 4867 203d 2048 4820 6720 17 : HHg = HH g │ │ │ │ 0001c6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c700: 2020 2020 2020 2020 2020 2020 7c0a 7c46 |.|F │ │ │ │ 0001c710: 696e 6469 6e67 2065 6173 7920 7265 6c61 inding easy rela │ │ │ │ 0001c720: 7469 6f6e 7320 2020 2020 2020 2020 2020 tions │ │ │ │ -0001c730: 3a20 202d 2d20 7573 6564 2030 2e30 3136 : -- used 0.016 │ │ │ │ -0001c740: 3439 3039 7320 2863 7075 293b 2030 2e30 4909s (cpu); 0.0 │ │ │ │ -0001c750: 3135 3531 3131 7320 2020 2020 7c0a 7c20 155111s |.| │ │ │ │ +0001c730: 3a20 202d 2d20 7573 6564 2030 2e30 3233 : -- used 0.023 │ │ │ │ +0001c740: 3933 3138 7320 2863 7075 293b 2030 2e30 9318s (cpu); 0.0 │ │ │ │ +0001c750: 3134 3137 3338 7320 2020 2020 7c0a 7c20 141738s |.| │ │ │ │ 0001c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001c7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c7c0: 2020 2020 2020 2020 205a 5a20 2020 2020 ZZ │ │ │ │ @@ -31584,17 +31584,17 @@ │ │ │ │ 0007b5f0: 6935 203a 2048 4d20 3d20 686f 6d6f 6c6f i5 : HM = homolo │ │ │ │ 0007b600: 6779 204d 2020 2020 2020 2020 2020 2020 gy M │ │ │ │ 0007b610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b630: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0007b640: 4669 6e64 696e 6720 6561 7379 2072 656c Finding easy rel │ │ │ │ 0007b650: 6174 696f 6e73 2020 2020 2020 2020 2020 ations │ │ │ │ -0007b660: 203a 2020 2d2d 2075 7365 6420 302e 3134 : -- used 0.14 │ │ │ │ -0007b670: 3130 3373 2028 6370 7529 3b20 302e 3036 103s (cpu); 0.06 │ │ │ │ -0007b680: 3533 3334 3473 2020 2020 2020 207c 0a7c 53344s |.| │ │ │ │ +0007b660: 203a 2020 2d2d 2075 7365 6420 302e 3233 : -- used 0.23 │ │ │ │ +0007b670: 3530 3035 7320 2863 7075 293b 2030 2e30 5005s (cpu); 0.0 │ │ │ │ +0007b680: 3532 3735 3133 7320 2020 2020 207c 0a7c 527513s |.| │ │ │ │ 0007b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0007b6d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0007b6e0: 6f35 203d 2063 6f6b 6572 6e65 6c20 7b30 o5 = cokernel {0 │ │ │ │ 0007b6f0: 2c20 307d 207c 2058 5f32 2058 5f31 2030 , 0} | X_2 X_1 0 │ │ │ │ @@ -68938,16 +68938,16 @@ │ │ │ │ 0010d490: 3d20 686f 6d6f 6c6f 6779 416c 6765 6272 = homologyAlgebr │ │ │ │ 0010d4a0: 6120 4120 2020 2020 2020 2020 2020 2020 a A │ │ │ │ 0010d4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d4d0: 2020 2020 207c 0a7c 4669 6e64 696e 6720 |.|Finding │ │ │ │ 0010d4e0: 6561 7379 2072 656c 6174 696f 6e73 2020 easy relations │ │ │ │ 0010d4f0: 2020 2020 2020 2020 203a 2020 2d2d 2075 : -- u │ │ │ │ -0010d500: 7365 6420 302e 3032 3130 3137 3573 2028 sed 0.0210175s ( │ │ │ │ -0010d510: 6370 7529 3b20 302e 3031 3837 3735 3973 cpu); 0.0187759s │ │ │ │ +0010d500: 7365 6420 302e 3134 3636 3239 7320 2863 sed 0.146629s (c │ │ │ │ +0010d510: 7075 293b 2030 2e30 3236 3536 3938 7320 pu); 0.0265698s │ │ │ │ 0010d520: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0010d530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0010d570: 2020 2020 207c 0a7c 6f34 203d 2048 4120 |.|o4 = HA │ │ │ │ 0010d580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -79760,17 +79760,17 @@ │ │ │ │ 001378f0: 203a 2048 4120 3d20 4848 204b 523b 2020 : HA = HH KR; │ │ │ │ 00137900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00137910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00137920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00137930: 2020 2020 2020 2020 2020 207c 0a7c 4669 |.|Fi │ │ │ │ 00137940: 6e64 696e 6720 6561 7379 2072 656c 6174 nding easy relat │ │ │ │ 00137950: 696f 6e73 2020 2020 2020 2020 2020 203a ions : │ │ │ │ -00137960: 2020 2d2d 2075 7365 6420 302e 3031 3535 -- used 0.0155 │ │ │ │ -00137970: 3139 3473 2028 6370 7529 3b20 302e 3031 194s (cpu); 0.01 │ │ │ │ -00137980: 3434 3173 2020 2020 2020 207c 0a7c 2d2d 441s |.|-- │ │ │ │ +00137960: 2020 2d2d 2075 7365 6420 302e 3137 3930 -- used 0.1790 │ │ │ │ +00137970: 3638 7320 2863 7075 293b 2030 2e30 3430 68s (cpu); 0.040 │ │ │ │ +00137980: 3831 3331 7320 2020 2020 207c 0a7c 2d2d 8131s |.|-- │ │ │ │ 00137990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001379a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001379b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001379c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001379d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2874 -----------|.|(t │ │ │ │ 001379e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 001379f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -81142,17 +81142,17 @@ │ │ │ │ 0013cf50: 3320 3a20 4841 203d 2048 4820 4120 2020 3 : HA = HH A │ │ │ │ 0013cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013cf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013cf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013cf90: 2020 2020 2020 2020 2020 2020 7c0a 7c46 |.|F │ │ │ │ 0013cfa0: 696e 6469 6e67 2065 6173 7920 7265 6c61 inding easy rela │ │ │ │ 0013cfb0: 7469 6f6e 7320 2020 2020 2020 2020 2020 tions │ │ │ │ -0013cfc0: 3a20 202d 2d20 7573 6564 2030 2e30 3534 : -- used 0.054 │ │ │ │ -0013cfd0: 3138 3139 7320 2863 7075 293b 2030 2e30 1819s (cpu); 0.0 │ │ │ │ -0013cfe0: 3531 3433 3736 7320 2020 2020 7c0a 7c20 514376s |.| │ │ │ │ +0013cfc0: 3a20 202d 2d20 7573 6564 2030 2e33 3531 : -- used 0.351 │ │ │ │ +0013cfd0: 3439 3673 2028 6370 7529 3b20 302e 3036 496s (cpu); 0.06 │ │ │ │ +0013cfe0: 3537 3431 3173 2020 2020 2020 7c0a 7c20 57411s |.| │ │ │ │ 0013cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013d030: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0013d040: 3320 3d20 4841 2020 2020 2020 2020 2020 3 = HA │ │ │ │ 0013d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -81518,17 +81518,17 @@ │ │ │ │ 0013e6d0: 3520 3a20 4848 6720 3d20 4848 2067 2020 5 : HHg = HH g │ │ │ │ 0013e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e710: 2020 2020 2020 2020 2020 2020 7c0a 7c46 |.|F │ │ │ │ 0013e720: 696e 6469 6e67 2065 6173 7920 7265 6c61 inding easy rela │ │ │ │ 0013e730: 7469 6f6e 7320 2020 2020 2020 2020 2020 tions │ │ │ │ -0013e740: 3a20 202d 2d20 7573 6564 2030 2e30 3136 : -- used 0.016 │ │ │ │ -0013e750: 3538 3238 7320 2863 7075 293b 2030 2e30 5828s (cpu); 0.0 │ │ │ │ -0013e760: 3135 3438 3173 2020 2020 2020 7c0a 7c20 15481s |.| │ │ │ │ +0013e740: 3a20 202d 2d20 7573 6564 2030 2e30 3337 : -- used 0.037 │ │ │ │ +0013e750: 3030 3237 7320 2863 7075 293b 2030 2e30 0027s (cpu); 0.0 │ │ │ │ +0013e760: 3136 3633 3135 7320 2020 2020 7c0a 7c20 166315s |.| │ │ │ │ 0013e770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0013e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0013e7d0: 2020 2020 2020 2020 5a5a 2020 2020 2020 ZZ │ │ │ │ @@ -81989,17 +81989,17 @@ │ │ │ │ 00140440: 6935 203a 2068 203d 2068 6f6d 6f6c 6f67 i5 : h = homolog │ │ │ │ 00140450: 7920 6964 4d20 2020 2020 2020 2020 2020 y idM │ │ │ │ 00140460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00140470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00140480: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00140490: 4669 6e64 696e 6720 6561 7379 2072 656c Finding easy rel │ │ │ │ 001404a0: 6174 696f 6e73 2020 2020 2020 2020 2020 ations │ │ │ │ -001404b0: 203a 2020 2d2d 2075 7365 6420 302e 3137 : -- used 0.17 │ │ │ │ -001404c0: 3230 3237 7320 2863 7075 293b 2030 2e30 2027s (cpu); 0.0 │ │ │ │ -001404d0: 3637 3636 3436 7320 2020 2020 207c 0a7c 676646s |.| │ │ │ │ +001404b0: 203a 2020 2d2d 2075 7365 6420 302e 3330 : -- used 0.30 │ │ │ │ +001404c0: 3439 3532 7320 2863 7075 293b 2030 2e30 4952s (cpu); 0.0 │ │ │ │ +001404d0: 3631 3832 3336 7320 2020 2020 207c 0a7c 618236s |.| │ │ │ │ 001404e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001404f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00140500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00140510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00140520: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00140530: 6f35 203d 207b 302c 2030 7d20 7c20 3120 o5 = {0, 0} | 1 │ │ │ │ 00140540: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ @@ -82433,17 +82433,17 @@ │ │ │ │ 00142000: 3420 3a20 4841 203d 2068 6f6d 6f6c 6f67 4 : HA = homolog │ │ │ │ 00142010: 7941 6c67 6562 7261 2841 2920 2020 2020 yAlgebra(A) │ │ │ │ 00142020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142040: 2020 2020 2020 2020 2020 2020 7c0a 7c46 |.|F │ │ │ │ 00142050: 696e 6469 6e67 2065 6173 7920 7265 6c61 inding easy rela │ │ │ │ 00142060: 7469 6f6e 7320 2020 2020 2020 2020 2020 tions │ │ │ │ -00142070: 3a20 202d 2d20 7573 6564 2030 2e30 3232 : -- used 0.022 │ │ │ │ -00142080: 3337 3435 7320 2863 7075 293b 2030 2e30 3745s (cpu); 0.0 │ │ │ │ -00142090: 3231 3234 3239 7320 2020 2020 7c0a 7c20 212429s |.| │ │ │ │ +00142070: 3a20 202d 2d20 7573 6564 2030 2e30 3330 : -- used 0.030 │ │ │ │ +00142080: 3331 3534 7320 2863 7075 293b 2030 2e30 3154s (cpu); 0.0 │ │ │ │ +00142090: 3139 3432 3538 7320 2020 2020 7c0a 7c20 194258s |.| │ │ │ │ 001420a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001420b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001420c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001420d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001420e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 001420f0: 3420 3d20 4841 2020 2020 2020 2020 2020 4 = HA │ │ │ │ 00142100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -82596,16 +82596,16 @@ │ │ │ │ 00142a30: 6f6d 6f6c 6f67 7941 6c67 6562 7261 2841 omologyAlgebra(A │ │ │ │ 00142a40: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00142a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142a70: 2020 7c0a 7c46 696e 6469 6e67 2065 6173 |.|Finding eas │ │ │ │ 00142a80: 7920 7265 6c61 7469 6f6e 7320 2020 2020 y relations │ │ │ │ 00142a90: 2020 2020 2020 3a20 202d 2d20 7573 6564 : -- used │ │ │ │ -00142aa0: 2030 2e30 3931 3932 3636 7320 2863 7075 0.0919266s (cpu │ │ │ │ -00142ab0: 293b 2030 2e30 3839 3031 7320 2020 2020 ); 0.08901s │ │ │ │ +00142aa0: 2030 2e30 3937 3532 3837 7320 2863 7075 0.0975287s (cpu │ │ │ │ +00142ab0: 293b 2030 2e30 3837 3231 3735 7320 2020 ); 0.0872175s │ │ │ │ 00142ac0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00142ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00142b10: 2020 7c0a 7c6f 3820 3d20 4841 2020 2020 |.|o8 = HA │ │ │ │ 00142b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -83001,16 +83001,16 @@ │ │ │ │ 00144380: 686f 6d6f 6c6f 6779 416c 6765 6272 6128 homologyAlgebra( │ │ │ │ 00144390: 4129 2020 2020 2020 2020 2020 2020 2020 A) │ │ │ │ 001443a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001443b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001443c0: 2020 7c0a 7c46 696e 6469 6e67 2065 6173 |.|Finding eas │ │ │ │ 001443d0: 7920 7265 6c61 7469 6f6e 7320 2020 2020 y relations │ │ │ │ 001443e0: 2020 2020 2020 3a20 202d 2d20 7573 6564 : -- used │ │ │ │ -001443f0: 2030 2e30 3631 3231 3539 7320 2863 7075 0.0612159s (cpu │ │ │ │ -00144400: 293b 2030 2e30 3539 3132 3836 7320 2020 ); 0.0591286s │ │ │ │ +001443f0: 2030 2e30 3631 3537 7320 2863 7075 293b 0.06157s (cpu); │ │ │ │ +00144400: 2030 2e30 3531 3236 3331 7320 2020 2020 0.0512631s │ │ │ │ 00144410: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00144420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00144460: 2020 7c0a 7c6f 3136 203d 2048 4120 2020 |.|o16 = HA │ │ │ │ 00144470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -83212,17 +83212,17 @@ │ │ │ │ 001450b0: 7c69 3231 203a 2048 4220 3d20 686f 6d6f |i21 : HB = homo │ │ │ │ 001450c0: 6c6f 6779 416c 6765 6272 6128 422c 4765 logyAlgebra(B,Ge │ │ │ │ 001450d0: 6e44 6567 7265 654c 696d 6974 3d3e 372c nDegreeLimit=>7, │ │ │ │ 001450e0: 5265 6c44 6567 7265 654c 696d 6974 3d3e RelDegreeLimit=> │ │ │ │ 001450f0: 3134 2920 2020 2020 2020 2020 2020 7c0a 14) |. │ │ │ │ 00145100: 7c46 696e 6469 6e67 2065 6173 7920 7265 |Finding easy re │ │ │ │ 00145110: 6c61 7469 6f6e 7320 2020 2020 2020 2020 lations │ │ │ │ -00145120: 2020 3a20 202d 2d20 7573 6564 2030 2e30 : -- used 0.0 │ │ │ │ -00145130: 3230 3137 3973 2028 6370 7529 3b20 302e 20179s (cpu); 0. │ │ │ │ -00145140: 3031 3835 3136 3973 2020 2020 2020 7c0a 0185169s |. │ │ │ │ +00145120: 2020 3a20 202d 2d20 7573 6564 2030 2e31 : -- used 0.1 │ │ │ │ +00145130: 3931 3436 3973 2028 6370 7529 3b20 302e 91469s (cpu); 0. │ │ │ │ +00145140: 3034 3231 3536 3673 2020 2020 2020 7c0a 0421566s |. │ │ │ │ 00145150: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00145160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00145170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00145180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00145190: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 001451a0: 7c6f 3231 203d 2048 4220 2020 2020 2020 |o21 = HB │ │ │ │ 001451b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -84058,17 +84058,17 @@ │ │ │ │ 00148590: 2048 203d 2048 4828 4b52 2920 2020 2020 H = HH(KR) │ │ │ │ 001485a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001485b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001485c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001485d0: 2020 2020 2020 2020 207c 0a7c 4669 6e64 |.|Find │ │ │ │ 001485e0: 696e 6720 6561 7379 2072 656c 6174 696f ing easy relatio │ │ │ │ 001485f0: 6e73 2020 2020 2020 2020 2020 203a 2020 ns : │ │ │ │ -00148600: 2d2d 2075 7365 6420 302e 3031 3631 3632 -- used 0.016162 │ │ │ │ -00148610: 3373 2028 6370 7529 3b20 302e 3031 3531 3s (cpu); 0.0151 │ │ │ │ -00148620: 3332 3873 2020 2020 207c 0a7c 2020 2020 328s |.| │ │ │ │ +00148600: 2d2d 2075 7365 6420 302e 3033 3436 3535 -- used 0.034655 │ │ │ │ +00148610: 3273 2028 6370 7529 3b20 302e 3031 3635 2s (cpu); 0.0165 │ │ │ │ +00148620: 3133 3273 2020 2020 207c 0a7c 2020 2020 132s |.| │ │ │ │ 00148630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00148640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00148650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00148660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00148670: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ 00148680: 2048 2020 2020 2020 2020 2020 2020 2020 H │ │ │ │ 00148690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -84433,17 +84433,17 @@ │ │ │ │ 00149d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00149d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00149d20: 2d2d 2d2d 2d2b 0a7c 6936 203a 2048 4b52 -----+.|i6 : HKR │ │ │ │ 00149d30: 203d 2048 4828 4b52 2920 2020 2020 2020 = HH(KR) │ │ │ │ 00149d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00149d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00149d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00149d70: 7c0a 7c20 2d2d 2075 7365 6420 302e 3131 |.| -- used 0.11 │ │ │ │ -00149d80: 3433 3732 7320 2863 7075 293b 2030 2e31 4372s (cpu); 0.1 │ │ │ │ -00149d90: 3038 3239 3173 2028 7468 7265 6164 293b 08291s (thread); │ │ │ │ +00149d70: 7c0a 7c20 2d2d 2075 7365 6420 302e 3132 |.| -- used 0.12 │ │ │ │ +00149d80: 3231 3938 7320 2863 7075 293b 2030 2e31 2198s (cpu); 0.1 │ │ │ │ +00149d90: 3031 3530 3873 2028 7468 7265 6164 293b 01508s (thread); │ │ │ │ 00149da0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00149db0: 2020 2020 2020 2020 2020 207c 0a7c 4669 |.|Fi │ │ │ │ 00149dc0: 6e64 696e 6720 6561 7379 2072 656c 6174 nding easy relat │ │ │ │ 00149dd0: 696f 6e73 2020 2020 2020 2020 2020 203a ions : │ │ │ │ 00149de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00149df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00149e00: 2020 2020 2020 7c0a 7c6f 3620 3d20 484b |.|o6 = HK │ │ │ │ @@ -91727,16 +91727,16 @@ │ │ │ │ 001664e0: 3233 203d 206d 6173 7365 7954 7269 706c 23 = masseyTripl │ │ │ │ 001664f0: 6550 726f 6475 6374 284b 522c 7a31 2c7a eProduct(KR,z1,z │ │ │ │ 00166500: 322c 7a33 2920 2020 2020 2020 2020 2020 2,z3) │ │ │ │ 00166510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00166520: 2020 2020 207c 0a7c 4669 6e64 696e 6720 |.|Finding │ │ │ │ 00166530: 6561 7379 2072 656c 6174 696f 6e73 2020 easy relations │ │ │ │ 00166540: 2020 2020 2020 2020 203a 2020 2d2d 2075 : -- u │ │ │ │ -00166550: 7365 6420 302e 3539 3630 3634 7320 2863 sed 0.596064s (c │ │ │ │ -00166560: 7075 293b 2030 2e35 3038 3830 3573 2020 pu); 0.508805s │ │ │ │ +00166550: 7365 6420 302e 3838 3738 3432 7320 2863 sed 0.887842s (c │ │ │ │ +00166560: 7075 293b 2030 2e36 3039 3230 3673 2020 pu); 0.609206s │ │ │ │ 00166570: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00166580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00166590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001665a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001665b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001665c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 001665d0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ @@ -92231,16 +92231,16 @@ │ │ │ │ 00168460: 203d 2048 4828 4b52 2920 2020 2020 2020 = HH(KR) │ │ │ │ 00168470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001684a0: 2020 2020 2020 207c 0a7c 4669 6e64 696e |.|Findin │ │ │ │ 001684b0: 6720 6561 7379 2072 656c 6174 696f 6e73 g easy relations │ │ │ │ 001684c0: 2020 2020 2020 2020 2020 203a 2020 2d2d : -- │ │ │ │ -001684d0: 2075 7365 6420 302e 3137 3433 3433 7320 used 0.174343s │ │ │ │ -001684e0: 2863 7075 293b 2030 2e31 3730 3737 3273 (cpu); 0.170772s │ │ │ │ +001684d0: 2075 7365 6420 302e 3138 3333 3131 7320 used 0.183311s │ │ │ │ +001684e0: 2863 7075 293b 2030 2e31 3732 3631 3673 (cpu); 0.172616s │ │ │ │ 001684f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00168500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00168540: 2020 2020 2020 207c 0a7c 6f35 203d 2048 |.|o5 = H │ │ │ │ 00168550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -98928,16 +98928,16 @@ │ │ │ │ 001826f0: 203d 2074 6f72 416c 6765 6272 6128 522c = torAlgebra(R, │ │ │ │ 00182700: 532c 4765 6e44 6567 7265 654c 696d 6974 S,GenDegreeLimit │ │ │ │ 00182710: 3d3e 342c 5265 6c44 6567 7265 654c 696d =>4,RelDegreeLim │ │ │ │ 00182720: 6974 3d3e 3829 2020 2020 2020 2020 2020 it=>8) │ │ │ │ 00182730: 2020 2020 2020 7c0a 7c46 696e 6469 6e67 |.|Finding │ │ │ │ 00182740: 2065 6173 7920 7265 6c61 7469 6f6e 7320 easy relations │ │ │ │ 00182750: 2020 2020 2020 2020 2020 3a20 202d 2d20 : -- │ │ │ │ -00182760: 7573 6564 2030 2e35 3332 3536 3673 2028 used 0.532566s ( │ │ │ │ -00182770: 6370 7529 3b20 302e 3431 3937 3437 7320 cpu); 0.419747s │ │ │ │ +00182760: 7573 6564 2030 2e36 3434 3438 3673 2028 used 0.644486s ( │ │ │ │ +00182770: 6370 7529 3b20 302e 3533 3935 3334 7320 cpu); 0.539534s │ │ │ │ 00182780: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00182790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001827a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001827b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001827c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001827d0: 2020 2020 2020 7c0a 7c6f 3420 3d20 4842 |.|o4 = HB │ │ │ │ 001827e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/DirectSummands.info.gz │ │ │ ├── DirectSummands.info │ │ │ │ @@ -271,15 +271,15 @@ │ │ │ │ 000010e0: 2077 6974 6820 4746 2034 3920 2020 2020 with GF 49 │ │ │ │ 000010f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001100: 2020 2020 2020 207c 0a7c 202d 2d20 7472 |.| -- tr │ │ │ │ 00001110: 7920 7573 696e 6720 6368 616e 6765 4261 y using changeBa │ │ │ │ 00001120: 7365 4669 656c 6420 7769 7468 2047 4620 seField with GF │ │ │ │ 00001130: 3439 2020 2020 2020 2020 2020 2020 2020 49 │ │ │ │ 00001140: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00001150: 7c20 2d2d 2033 2e33 3533 3936 7320 656c | -- 3.35396s el │ │ │ │ +00001150: 7c20 2d2d 2032 2e35 3630 3236 7320 656c | -- 2.56026s el │ │ │ │ 00001160: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00001170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001190: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000011a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000011b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000011c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -386,16 +386,16 @@ │ │ │ │ 00001810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00001840: 6935 203a 2065 6c61 7073 6564 5469 6d65 i5 : elapsedTime │ │ │ │ 00001850: 2074 616c 6c79 5375 6d6d 616e 6473 2073 tallySummands s │ │ │ │ 00001860: 756d 6d61 6e64 7320 6368 616e 6765 4261 ummands changeBa │ │ │ │ 00001870: 7365 4669 656c 6428 4746 2837 2c20 3229 seField(GF(7, 2) │ │ │ │ -00001880: 2c20 4629 7c0a 7c20 2d2d 2031 312e 3535 , F)|.| -- 11.55 │ │ │ │ -00001890: 3938 7320 656c 6170 7365 6420 2020 2020 98s elapsed │ │ │ │ +00001880: 2c20 4629 7c0a 7c20 2d2d 2038 2e33 3636 , F)|.| -- 8.366 │ │ │ │ +00001890: 3531 7320 656c 6170 7365 6420 2020 2020 51s elapsed │ │ │ │ 000018a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000018b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000018c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000018d0: 2020 2020 2020 2020 2020 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 │ │ │ │ @@ -4317,16 +4317,16 @@ │ │ │ │ 00010dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010de0: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2061 ------+.|i10 : a │ │ │ │ 00010df0: 7373 6572 7420 656c 6170 7365 6454 696d ssert elapsedTim │ │ │ │ 00010e00: 6520 6973 496e 6465 636f 6d70 6f73 6162 e isIndecomposab │ │ │ │ 00010e10: 6c65 2046 484d 2020 2020 2020 2020 2020 le FHM │ │ │ │ 00010e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010e30: 2020 2020 2020 7c0a 7c20 2d2d 2031 2e30 |.| -- 1.0 │ │ │ │ -00010e40: 3032 3233 7320 656c 6170 7365 6420 2020 0223s elapsed │ │ │ │ +00010e30: 2020 2020 2020 7c0a 7c20 2d2d 202e 3838 |.| -- .88 │ │ │ │ +00010e40: 3936 3138 7320 656c 6170 7365 6420 2020 9618s elapsed │ │ │ │ 00010e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010e80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00010e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/EdgeIdeals.info.gz │ │ │ ├── EdgeIdeals.info │ │ │ │ @@ -3838,17 +3838,17 @@ │ │ │ │ 0000efd0: 6c65 2020 2020 2020 2020 2020 2020 2020 le │ │ │ │ 0000efe0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f030: 207c 0a7c 6f33 203d 2047 7261 7068 7b22 |.|o3 = Graph{" │ │ │ │ -0000f040: 6564 6765 7322 203d 3e20 7b7b 612c 2064 edges" => {{a, d │ │ │ │ -0000f050: 7d2c 207b 612c 2063 7d2c 207b 622c 2065 }, {a, c}, {b, e │ │ │ │ -0000f060: 7d2c 207b 622c 2064 7d2c 207b 632c 2065 }, {b, d}, {c, e │ │ │ │ +0000f040: 6564 6765 7322 203d 3e20 7b7b 612c 2063 edges" => {{a, c │ │ │ │ +0000f050: 7d2c 207b 622c 2065 7d2c 207b 622c 2064 }, {b, e}, {b, d │ │ │ │ +0000f060: 7d2c 207b 632c 2065 7d2c 207b 612c 2064 }, {c, e}, {a, d │ │ │ │ 0000f070: 7d7d 7d20 2020 2020 2020 2020 2020 2020 }}} │ │ │ │ 0000f080: 207c 0a7c 2020 2020 2020 2020 2020 2022 |.| " │ │ │ │ 0000f090: 7269 6e67 2220 3d3e 2052 2020 2020 2020 ring" => R │ │ │ │ 0000f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0d0: 207c 0a7c 2020 2020 2020 2020 2020 2022 |.| " │ │ │ │ @@ -7842,16 +7842,16 @@ │ │ │ │ 0001ea10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ea30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea60: 2020 7c0a 7c6f 3420 3d20 4879 7065 7247 |.|o4 = HyperG │ │ │ │ 0001ea70: 7261 7068 7b22 6564 6765 7322 203d 3e20 raph{"edges" => │ │ │ │ -0001ea80: 7b7b 622c 2063 7d2c 207b 612c 2065 7d2c {{b, c}, {a, e}, │ │ │ │ -0001ea90: 207b 632c 2064 7d7d 7d20 2020 2020 2020 {c, d}}} │ │ │ │ +0001ea80: 7b7b 632c 2064 7d2c 207b 622c 2063 7d2c {{c, d}, {b, c}, │ │ │ │ +0001ea90: 207b 612c 2065 7d7d 7d20 2020 2020 2020 {a, e}}} │ │ │ │ 0001eaa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0001eab0: 2020 2020 2020 2022 7269 6e67 2220 3d3e "ring" => │ │ │ │ 0001eac0: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 0001ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eae0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001eaf0: 2020 2020 2020 2020 2022 7665 7274 6963 "vertic │ │ │ │ 0001eb00: 6573 2220 3d3e 207b 612c 2062 2c20 632c es" => {a, b, c, │ │ │ │ @@ -21417,17 +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 3220 2020 3420 2020 3520 2020 2 4 5 │ │ │ │ -00053b00: 2020 3120 2020 3520 2020 2020 3120 2020 1 5 1 │ │ │ │ -00053b10: 3220 2020 3320 2020 3420 2020 2020 2020 2 3 4 │ │ │ │ +00053af0: 2020 2020 3120 2020 3220 2020 3320 2020 1 2 3 │ │ │ │ +00053b00: 2020 3320 2020 3420 2020 2020 3120 2020 3 4 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 3120 2020 3220 2020 3420 2020 1 2 4 │ │ │ │ -00053e20: 2020 3120 2020 3520 2020 2020 3220 2020 1 5 2 │ │ │ │ -00053e30: 3320 2020 3420 2020 3520 2020 2020 2020 3 4 5 │ │ │ │ +00053e10: 2020 2020 3220 2020 3320 2020 3420 2020 2 3 4 │ │ │ │ +00053e20: 2020 3320 2020 3520 2020 2020 3120 2020 3 5 1 │ │ │ │ +00053e30: 3220 2020 3420 2020 3520 2020 2020 2020 2 4 5 │ │ │ │ 00053e40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053e50: 2020 2020 2020 2272 696e 6722 203d 3e20 "ring" => │ │ │ │ 00053e60: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00053e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053e90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053ea0: 2020 2020 2020 2276 6572 7469 6365 7322 "vertices" │ │ │ │ @@ -22868,17 +22868,17 @@ │ │ │ │ 00059530: 2020 207c 0a7c 6f33 203d 2047 7261 7068 |.|o3 = Graph │ │ │ │ 00059540: 7b22 6564 6765 7322 203d 3e20 7b7b 7820 {"edges" => {{x │ │ │ │ 00059550: 2c20 7820 7d2c 207b 7820 2c20 7820 7d2c , x }, {x , x }, │ │ │ │ 00059560: 207b 7820 2c20 7820 7d2c 207b 7820 2c20 {x , x }, {x , │ │ │ │ 00059570: 7820 7d2c 207b 7820 2c20 7820 7d7d 7d20 x }, {x , x }}} │ │ │ │ 00059580: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059590: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -000595a0: 2020 2032 2020 2020 2032 2020 2033 2020 2 2 3 │ │ │ │ -000595b0: 2020 2033 2020 2034 2020 2020 2034 2020 3 4 4 │ │ │ │ -000595c0: 2035 2020 2020 2035 2020 2036 2020 2020 5 5 6 │ │ │ │ +000595a0: 2020 2032 2020 2020 2033 2020 2034 2020 2 3 4 │ │ │ │ +000595b0: 2020 2034 2020 2035 2020 2020 2031 2020 4 5 1 │ │ │ │ +000595c0: 2036 2020 2020 2035 2020 2036 2020 2020 6 5 6 │ │ │ │ 000595d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000595e0: 2022 7269 6e67 2220 3d3e 2052 2020 2020 "ring" => R │ │ │ │ 000595f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059620: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059630: 2022 7665 7274 6963 6573 2220 3d3e 207b "vertices" => { │ │ │ │ @@ -22978,16 +22978,16 @@ │ │ │ │ 00059c10: 2020 207c 0a7c 6f37 203d 2047 7261 7068 |.|o7 = Graph │ │ │ │ 00059c20: 7b22 6564 6765 7322 203d 3e20 7b7b 7820 {"edges" => {{x │ │ │ │ 00059c30: 2c20 7820 7d2c 207b 7820 2c20 7820 7d2c , x }, {x , x }, │ │ │ │ 00059c40: 207b 7820 2c20 7820 7d2c 207b 7820 2c20 {x , x }, {x , │ │ │ │ 00059c50: 7820 7d7d 7d20 2020 2020 2020 2020 2020 x }}} │ │ │ │ 00059c60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059c70: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -00059c80: 2020 2032 2020 2020 2031 2020 2033 2020 2 1 3 │ │ │ │ -00059c90: 2020 2034 2020 2035 2020 2020 2034 2020 4 5 4 │ │ │ │ +00059c80: 2020 2033 2020 2020 2032 2020 2033 2020 3 2 3 │ │ │ │ +00059c90: 2020 2034 2020 2036 2020 2020 2035 2020 4 6 5 │ │ │ │ 00059ca0: 2036 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ 00059cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059cc0: 2022 7269 6e67 2220 3d3e 2052 2020 2020 "ring" => R │ │ │ │ 00059cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059d00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ ├── ./usr/share/info/EigenSolver.info.gz │ │ │ ├── EigenSolver.info │ │ │ │ @@ -171,15 +171,15 @@ │ │ │ │ 00000aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000ac0: 2b0a 7c69 3320 3a20 656c 6170 7365 6454 +.|i3 : elapsedT │ │ │ │ 00000ad0: 696d 6520 736f 6c73 203d 207a 6572 6f44 ime sols = zeroD │ │ │ │ 00000ae0: 696d 536f 6c76 6520 493b 2020 2020 2020 imSolve I; │ │ │ │ 00000af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00000b10: 7c0a 7c20 2d2d 202e 3231 3730 3938 7320 |.| -- .217098s │ │ │ │ +00000b10: 7c0a 7c20 2d2d 202e 3230 3833 3434 7320 |.| -- .208344s │ │ │ │ 00000b20: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00000b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000b60: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00000b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/Elimination.info.gz │ │ │ ├── Elimination.info │ │ │ │ @@ -336,18 +336,18 @@ │ │ │ │ 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 3538 3532 3173 used 0.00258521s │ │ │ │ -00001570: 2028 6370 7529 3b20 302e 3030 3235 3832 (cpu); 0.002582 │ │ │ │ -00001580: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ -00001590: 2867 6329 207c 0a7c 2020 2020 2020 2020 (gc) |.| │ │ │ │ +00001560: 7573 6564 2030 2e30 3032 3938 3033 3273 used 0.00298032s │ │ │ │ +00001570: 2028 6370 7529 3b20 302e 3030 3239 3931 (cpu); 0.002991 │ │ │ │ +00001580: 3131 7320 2874 6872 6561 6429 3b20 3073 11s (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 │ │ │ │ 00001600: 2020 2020 2020 2020 2032 2020 207c 0a7c 2 |.| │ │ │ │ @@ -366,18 +366,18 @@ │ │ │ │ 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 3531 3338 3673 used 0.00151386s │ │ │ │ -00001750: 2028 6370 7529 3b20 302e 3030 3135 3134 (cpu); 0.001514 │ │ │ │ -00001760: 3837 7320 2874 6872 6561 6429 3b20 3073 87s (thread); 0s │ │ │ │ -00001770: 2028 6763 297c 0a7c 2020 2020 2020 2020 (gc)|.| │ │ │ │ +00001740: 7573 6564 2030 2e30 3031 3639 3831 3773 used 0.00169817s │ │ │ │ +00001750: 2028 6370 7529 3b20 302e 3030 3136 3937 (cpu); 0.001697 │ │ │ │ +00001760: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ +00001770: 2867 6329 207c 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 │ │ │ │ 000017e0: 2020 2020 2020 2020 2020 2032 207c 0a7c 2 |.| │ │ │ │ @@ -618,17 +618,17 @@ │ │ │ │ 00002690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000026a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000026b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 000026c0: 3420 3a20 7469 6d65 2065 6c69 6d69 6e61 4 : time elimina │ │ │ │ 000026d0: 7465 2878 2c69 6465 616c 2866 2c67 2929 te(x,ideal(f,g)) │ │ │ │ 000026e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000026f0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00002700: 7365 6420 302e 3030 3237 3137 3632 7320 sed 0.00271762s │ │ │ │ -00002710: 2863 7075 293b 2030 2e30 3032 3731 3434 (cpu); 0.0027144 │ │ │ │ -00002720: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ +00002700: 7365 6420 302e 3030 3236 3536 3633 7320 sed 0.00265663s │ │ │ │ +00002710: 2863 7075 293b 2030 2e30 3032 3635 3238 (cpu); 0.0026528 │ │ │ │ +00002720: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ 00002730: 2867 6329 7c0a 7c20 2020 2020 2020 2020 (gc)|.| │ │ │ │ 00002740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002770: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00002780: 2020 2020 2020 2020 2032 2020 2020 3220 2 2 │ │ │ │ 00002790: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ @@ -648,18 +648,18 @@ │ │ │ │ 00002870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 000028a0: 3520 3a20 7469 6d65 2069 6465 616c 2072 5 : time ideal r │ │ │ │ 000028b0: 6573 756c 7461 6e74 2866 2c67 2c78 2920 esultant(f,g,x) │ │ │ │ 000028c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000028d0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000028e0: 7365 6420 302e 3030 3136 3037 3938 7320 sed 0.00160798s │ │ │ │ -000028f0: 2863 7075 293b 2030 2e30 3031 3630 3930 (cpu); 0.0016090 │ │ │ │ -00002900: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ -00002910: 2867 6329 7c0a 7c20 2020 2020 2020 2020 (gc)|.| │ │ │ │ +000028e0: 7365 6420 302e 3030 3134 3935 3133 7320 sed 0.00149513s │ │ │ │ +000028f0: 2863 7075 293b 2030 2e30 3031 3439 3536 (cpu); 0.0014956 │ │ │ │ +00002900: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00002910: 6763 2920 7c0a 7c20 2020 2020 2020 2020 gc) |.| │ │ │ │ 00002920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002950: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00002960: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ 00002970: 3220 2020 2020 2020 2020 2020 2020 3220 2 2 │ │ │ │ 00002980: 2020 2020 2020 2020 2020 3220 7c0a 7c6f 2 |.|o │ │ │ │ @@ -990,17 +990,17 @@ │ │ │ │ 00003dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ 00003df0: 3a20 7469 6d65 2065 6c69 6d69 6e61 7465 : time eliminate │ │ │ │ 00003e00: 2869 6465 616c 2866 2c67 292c 7829 2020 (ideal(f,g),x) │ │ │ │ 00003e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003e30: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00003e40: 2075 7365 6420 312e 3731 3136 3773 2028 used 1.71167s ( │ │ │ │ -00003e50: 6370 7529 3b20 312e 3339 3634 3773 2028 cpu); 1.39647s ( │ │ │ │ -00003e60: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00003e40: 2075 7365 6420 312e 3335 3138 3573 2028 used 1.35185s ( │ │ │ │ +00003e50: 6370 7529 3b20 312e 3133 3635 7320 2874 cpu); 1.1365s (t │ │ │ │ +00003e60: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00003e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003e80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00003e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 │ │ │ │ 00003ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ @@ -1270,17 +1270,17 @@ │ │ │ │ 00004f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ 00004f70: 3a20 7469 6d65 2069 6465 616c 2072 6573 : time ideal res │ │ │ │ 00004f80: 756c 7461 6e74 2866 2c67 2c78 2920 2020 ultant(f,g,x) │ │ │ │ 00004f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004fb0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00004fc0: 2075 7365 6420 302e 3031 3638 3235 3673 used 0.0168256s │ │ │ │ -00004fd0: 2028 6370 7529 3b20 302e 3031 3638 3239 (cpu); 0.016829 │ │ │ │ -00004fe0: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ +00004fc0: 2075 7365 6420 302e 3031 3433 3930 3173 used 0.0143901s │ │ │ │ +00004fd0: 2028 6370 7529 3b20 302e 3031 3433 3930 (cpu); 0.014390 │ │ │ │ +00004fe0: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ 00004ff0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00005000: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00005010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005050: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ @@ -1913,16 +1913,16 @@ │ │ │ │ 00007780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007790: 2d2d 2b0a 7c69 3520 3a20 7469 6d65 2065 --+.|i5 : time e │ │ │ │ 000077a0: 6c69 6d69 6e61 7465 2869 6465 616c 2866 liminate(ideal(f │ │ │ │ 000077b0: 2c67 292c 7829 2020 2020 2020 2020 2020 ,g),x) │ │ │ │ 000077c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000077d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000077e0: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e |.| -- used 1. │ │ │ │ -000077f0: 3834 3935 3773 2028 6370 7529 3b20 312e 84957s (cpu); 1. │ │ │ │ -00007800: 3437 3034 3973 2028 7468 7265 6164 293b 47049s (thread); │ │ │ │ +000077f0: 3335 3533 3573 2028 6370 7529 3b20 312e 35535s (cpu); 1. │ │ │ │ +00007800: 3134 3138 3973 2028 7468 7265 6164 293b 14189s (thread); │ │ │ │ 00007810: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00007820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007830: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00007840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2193,17 +2193,17 @@ │ │ │ │ 00008900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008910: 2d2d 2b0a 7c69 3620 3a20 7469 6d65 2069 --+.|i6 : time i │ │ │ │ 00008920: 6465 616c 2072 6573 756c 7461 6e74 2866 deal resultant(f │ │ │ │ 00008930: 2c67 2c78 2920 2020 2020 2020 2020 2020 ,g,x) │ │ │ │ 00008940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008960: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00008970: 3031 3735 3736 3873 2028 6370 7529 3b20 0175768s (cpu); │ │ │ │ -00008980: 302e 3031 3735 3834 7320 2874 6872 6561 0.017584s (threa │ │ │ │ -00008990: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +00008970: 3031 3237 3637 3473 2028 6370 7529 3b20 0127674s (cpu); │ │ │ │ +00008980: 302e 3031 3237 3636 3773 2028 7468 7265 0.0127667s (thre │ │ │ │ +00008990: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 000089a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000089c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ ├── ./usr/share/info/EnumerationCurves.info.gz │ │ │ ├── EnumerationCurves.info │ │ │ │ @@ -256,17 +256,17 @@ │ │ │ │ 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 3033 3133 3939 7320 2863 7075 d 0.031399s (cpu │ │ │ │ -00001070: 293b 2030 2e30 3331 3339 3734 7320 2874 ); 0.0313974s (t │ │ │ │ -00001080: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00001060: 6420 302e 3032 3239 3231 3173 2028 6370 d 0.0229211s (cp │ │ │ │ +00001070: 7529 3b20 302e 3032 3239 3231 3573 2028 u); 0.0229215s ( │ │ │ │ +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 │ │ │ │ 000010f0: 2020 2020 2020 7c0a 7c6f 3120 3d20 7b31 |.|o1 = {1 │ │ │ │ @@ -674,16 +674,16 @@ │ │ │ │ 00002a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002a30: 2d2d 2b0a 7c69 3720 3a20 7469 6d65 2066 --+.|i7 : time f │ │ │ │ 00002a40: 6f72 2044 2069 6e20 5420 6c69 7374 2072 or D in T list r │ │ │ │ 00002a50: 6174 696f 6e61 6c43 7572 7665 2832 2c44 ationalCurve(2,D │ │ │ │ 00002a60: 2920 2d20 7261 7469 6f6e 616c 4375 7276 ) - rationalCurv │ │ │ │ 00002a70: 6528 312c 4429 2f38 7c0a 7c20 2d2d 2075 e(1,D)/8|.| -- u │ │ │ │ -00002a80: 7365 6420 302e 3335 3332 3436 7320 2863 sed 0.353246s (c │ │ │ │ -00002a90: 7075 293b 2030 2e32 3834 3130 3573 2028 pu); 0.284105s ( │ │ │ │ +00002a80: 7365 6420 302e 3330 3930 3637 7320 2863 sed 0.309067s (c │ │ │ │ +00002a90: 7075 293b 2030 2e32 3530 3137 3273 2028 pu); 0.250172s ( │ │ │ │ 00002aa0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00002ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00002ac0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00002ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002b00: 2020 2020 7c0a 7c6f 3720 3d20 7b36 3039 |.|o7 = {609 │ │ │ │ @@ -710,16 +710,16 @@ │ │ │ │ 00002c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00002c80: 3820 3a20 7469 6d65 2072 6174 696f 6e61 8 : time rationa │ │ │ │ 00002c90: 6c43 7572 7665 2833 2920 2020 2020 2020 lCurve(3) │ │ │ │ 00002ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002cb0: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00002cc0: 6564 2030 2e32 3330 3234 7320 2863 7075 ed 0.23024s (cpu │ │ │ │ -00002cd0: 293b 2030 2e31 3633 3331 3673 2028 7468 ); 0.163316s (th │ │ │ │ +00002cc0: 6564 2030 2e31 3131 3733 7320 2863 7075 ed 0.11173s (cpu │ │ │ │ +00002cd0: 293b 2030 2e31 3131 3733 3773 2028 7468 ); 0.111737s (th │ │ │ │ 00002ce0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00002cf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00002d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002d20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00002d30: 2020 2020 2038 3536 3435 3735 3030 3020 8564575000 │ │ │ │ 00002d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -743,17 +743,17 @@ │ │ │ │ 00002e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00002e90: 0a7c 6939 203a 2074 696d 6520 666f 7220 .|i9 : time for │ │ │ │ 00002ea0: 4420 696e 2054 206c 6973 7420 7261 7469 D in T list rati │ │ │ │ 00002eb0: 6f6e 616c 4375 7276 6528 332c 4429 2020 onalCurve(3,D) │ │ │ │ 00002ec0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00002ed0: 2075 7365 6420 362e 3437 3337 7320 2863 used 6.4737s (c │ │ │ │ -00002ee0: 7075 293b 2035 2e35 3037 3733 7320 2874 pu); 5.50773s (t │ │ │ │ -00002ef0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00002ed0: 2075 7365 6420 342e 3332 3434 3673 2028 used 4.32446s ( │ │ │ │ +00002ee0: 6370 7529 3b20 332e 3734 3138 3173 2028 cpu); 3.74181s ( │ │ │ │ +00002ef0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00002f00: 2020 2020 207c 0a7c 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 2020 2020 │ │ │ │ 00002f40: 7c0a 7c20 2020 2020 2038 3536 3435 3735 |.| 8564575 │ │ │ │ 00002f50: 3030 3020 2034 3232 3639 3038 3136 2020 000 422690816 │ │ │ │ 00002f60: 2020 2020 2020 2020 2034 3833 3435 3932 4834592 │ │ │ │ @@ -786,16 +786,16 @@ │ │ │ │ 00003110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003140: 2d2d 2d2b 0a7c 6931 3020 3a20 7469 6d65 ---+.|i10 : time │ │ │ │ 00003150: 2072 6174 696f 6e61 6c43 7572 7665 2833 rationalCurve(3 │ │ │ │ 00003160: 2920 2d20 7261 7469 6f6e 616c 4375 7276 ) - rationalCurv │ │ │ │ 00003170: 6528 3129 2f32 3720 2020 207c 0a7c 202d e(1)/27 |.| - │ │ │ │ -00003180: 2d20 7573 6564 2030 2e32 3332 3739 3573 - used 0.232795s │ │ │ │ -00003190: 2028 6370 7529 3b20 302e 3137 3535 3635 (cpu); 0.175565 │ │ │ │ +00003180: 2d20 7573 6564 2030 2e31 3935 3832 3673 - used 0.195826s │ │ │ │ +00003190: 2028 6370 7529 3b20 302e 3133 3736 3733 (cpu); 0.137673 │ │ │ │ 000031a0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 000031b0: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ 000031c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000031d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000031e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 000031f0: 3020 3d20 3331 3732 3036 3337 3520 2020 0 = 317206375 │ │ │ │ 00003200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -824,16 +824,16 @@ │ │ │ │ 00003370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003390: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 7469 -----+.|i11 : ti │ │ │ │ 000033a0: 6d65 2066 6f72 2044 2069 6e20 5420 6c69 me for D in T li │ │ │ │ 000033b0: 7374 2072 6174 696f 6e61 6c43 7572 7665 st rationalCurve │ │ │ │ 000033c0: 2833 2c44 2920 2d20 7261 7469 6f6e 616c (3,D) - rational │ │ │ │ 000033d0: 4375 7276 6528 312c 4429 2f32 377c 0a7c Curve(1,D)/27|.| │ │ │ │ -000033e0: 202d 2d20 7573 6564 2036 2e30 3835 3632 -- used 6.08562 │ │ │ │ -000033f0: 7320 2863 7075 293b 2035 2e31 3934 3234 s (cpu); 5.19424 │ │ │ │ +000033e0: 202d 2d20 7573 6564 2034 2e34 3731 3431 -- used 4.47141 │ │ │ │ +000033f0: 7320 2863 7075 293b 2033 2e38 3934 3134 s (cpu); 3.89414 │ │ │ │ 00003400: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00003410: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00003420: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00003430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003460: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -860,18 +860,18 @@ │ │ │ │ 000035b0: 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ....+----------- │ │ │ │ 000035c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000035d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000035e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a --------+.|i12 : │ │ │ │ 000035f0: 2074 696d 6520 7261 7469 6f6e 616c 4375 time rationalCu │ │ │ │ 00003600: 7276 6528 3429 2020 2020 2020 2020 2020 rve(4) │ │ │ │ 00003610: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00003620: 7c20 2d2d 2075 7365 6420 312e 3835 3530 | -- used 1.8550 │ │ │ │ -00003630: 3173 2028 6370 7529 3b20 312e 3439 3932 1s (cpu); 1.4992 │ │ │ │ -00003640: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ -00003650: 2867 6329 7c0a 7c20 2020 2020 2020 2020 (gc)|.| │ │ │ │ +00003620: 7c20 2d2d 2075 7365 6420 312e 3430 3832 | -- used 1.4082 │ │ │ │ +00003630: 3273 2028 6370 7529 3b20 312e 3231 3135 2s (cpu); 1.2115 │ │ │ │ +00003640: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00003650: 6763 2920 7c0a 7c20 2020 2020 2020 2020 gc) |.| │ │ │ │ 00003660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003680: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00003690: 2020 2031 3535 3137 3932 3637 3936 3837 1551792679687 │ │ │ │ 000036a0: 3520 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ 000036b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000036c0: 7c0a 7c6f 3132 203d 202d 2d2d 2d2d 2d2d |.|o12 = ------- │ │ │ │ @@ -891,17 +891,17 @@ │ │ │ │ 000037a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000037b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000037c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000037d0: 7c69 3133 203a 2074 696d 6520 7261 7469 |i13 : time rati │ │ │ │ 000037e0: 6f6e 616c 4375 7276 6528 342c 7b34 2c32 onalCurve(4,{4,2 │ │ │ │ 000037f0: 7d29 2020 2020 2020 2020 2020 2020 2020 }) │ │ │ │ 00003800: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -00003810: 382e 3733 3435 7320 2863 7075 293b 2036 8.7345s (cpu); 6 │ │ │ │ -00003820: 2e33 3535 3537 7320 2874 6872 6561 6429 .35557s (thread) │ │ │ │ -00003830: 3b20 3073 2028 6763 2920 7c0a 7c20 2020 ; 0s (gc) |.| │ │ │ │ +00003810: 362e 3136 3937 3373 2028 6370 7529 3b20 6.16973s (cpu); │ │ │ │ +00003820: 342e 3839 3836 3673 2028 7468 7265 6164 4.89866s (thread │ │ │ │ +00003830: 293b 2030 7320 2867 6329 7c0a 7c20 2020 ); 0s (gc)|.| │ │ │ │ 00003840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003870: 7c0a 7c6f 3133 203d 2033 3838 3339 3134 |.|o13 = 3883914 │ │ │ │ 00003880: 3038 3420 2020 2020 2020 2020 2020 2020 084 │ │ │ │ 00003890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000038a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ @@ -924,16 +924,16 @@ │ │ │ │ 000039b0: 0a0a 0a2b 2d2d 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 2d2b 0a7c 6931 3420 3a20 -------+.|i14 : │ │ │ │ 000039f0: 7469 6d65 2072 6174 696f 6e61 6c43 7572 time rationalCur │ │ │ │ 00003a00: 7665 2834 2920 2d20 7261 7469 6f6e 616c ve(4) - rational │ │ │ │ 00003a10: 4375 7276 6528 3229 2f38 2020 207c 0a7c Curve(2)/8 |.| │ │ │ │ -00003a20: 202d 2d20 7573 6564 2032 2e32 3730 3036 -- used 2.27006 │ │ │ │ -00003a30: 7320 2863 7075 293b 2031 2e39 3733 3534 s (cpu); 1.97354 │ │ │ │ +00003a20: 202d 2d20 7573 6564 2031 2e34 3132 3732 -- used 1.41272 │ │ │ │ +00003a30: 7320 2863 7075 293b 2031 2e32 3134 3231 s (cpu); 1.21421 │ │ │ │ 00003a40: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00003a50: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ 00003a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003a80: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ 00003a90: 3d20 3234 3234 3637 3533 3030 3030 2020 = 242467530000 │ │ │ │ 00003aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -962,17 +962,17 @@ │ │ │ │ 00003c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003c30: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 7469 -----+.|i15 : ti │ │ │ │ 00003c40: 6d65 2072 6174 696f 6e61 6c43 7572 7665 me rationalCurve │ │ │ │ 00003c50: 2834 2c7b 342c 327d 2920 2d20 7261 7469 (4,{4,2}) - rati │ │ │ │ 00003c60: 6f6e 616c 4375 7276 6528 322c 7b34 2c32 onalCurve(2,{4,2 │ │ │ │ 00003c70: 7d29 2f38 7c0a 7c20 2d2d 2075 7365 6420 })/8|.| -- used │ │ │ │ -00003c80: 382e 3437 3730 3373 2028 6370 7529 3b20 8.47703s (cpu); │ │ │ │ -00003c90: 362e 3334 3533 7320 2874 6872 6561 6429 6.3453s (thread) │ │ │ │ -00003ca0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +00003c80: 362e 3136 3637 3673 2028 6370 7529 3b20 6.16676s (cpu); │ │ │ │ +00003c90: 342e 3838 3537 3873 2028 7468 7265 6164 4.88578s (thread │ │ │ │ +00003ca0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00003cb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00003cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003cf0: 2020 7c0a 7c6f 3135 203d 2033 3838 3339 |.|o15 = 38839 │ │ │ │ 00003d00: 3032 3532 3820 2020 2020 2020 2020 2020 02528 │ │ │ │ 00003d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -989,16 +989,16 @@ │ │ │ │ 00003dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00003df0: 7c69 3136 203a 2074 696d 6520 7261 7469 |i16 : time rati │ │ │ │ 00003e00: 6f6e 616c 4375 7276 6528 342c 7b33 2c33 onalCurve(4,{3,3 │ │ │ │ 00003e10: 7d29 202d 2072 6174 696f 6e61 6c43 7572 }) - rationalCur │ │ │ │ 00003e20: 7665 2832 2c7b 332c 337d 292f 387c 0a7c ve(2,{3,3})/8|.| │ │ │ │ -00003e30: 202d 2d20 7573 6564 2039 2e32 3230 3935 -- used 9.22095 │ │ │ │ -00003e40: 7320 2863 7075 293b 2036 2e37 3730 3036 s (cpu); 6.77006 │ │ │ │ +00003e30: 202d 2d20 7573 6564 2036 2e32 3930 3439 -- used 6.29049 │ │ │ │ +00003e40: 7320 2863 7075 293b 2035 2e30 3539 3632 s (cpu); 5.05962 │ │ │ │ 00003e50: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00003e60: 6763 2920 2020 2020 2020 2020 7c0a 7c20 gc) |.| │ │ │ │ 00003e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003ea0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 00003eb0: 3620 3d20 3131 3339 3434 3833 3834 2020 6 = 1139448384 │ │ ├── ./usr/share/info/EquivariantGB.info.gz │ │ │ ├── EquivariantGB.info │ │ │ │ @@ -1917,21 +1917,21 @@ │ │ │ │ 000077c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000077d0: 2020 2020 2020 2020 2020 7c0a 7c33 2020 |.|3 │ │ │ │ 000077e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000077f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 |.| │ │ │ │ -00007830: 2020 2d2d 2075 7365 6420 2e30 3032 3234 -- used .00224 │ │ │ │ -00007840: 3435 2073 6563 6f6e 6473 2020 2020 2020 45 seconds │ │ │ │ +00007830: 2020 2d2d 2075 7365 6420 2e30 3031 3832 -- used .00182 │ │ │ │ +00007840: 3320 7365 636f 6e64 7320 2020 2020 2020 3 seconds │ │ │ │ 00007850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007870: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007880: 2020 2d2d 2075 7365 6420 2e30 3030 3638 -- used .00068 │ │ │ │ -00007890: 3834 2073 6563 6f6e 6473 2020 2020 2020 84 seconds │ │ │ │ +00007880: 2020 2d2d 2075 7365 6420 2e30 3030 3539 -- used .00059 │ │ │ │ +00007890: 3535 3232 2073 6563 6f6e 6473 2020 2020 5522 seconds │ │ │ │ 000078a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000078b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000078c0: 2020 2020 2020 2020 2020 7c0a 7c28 392c |.|(9, │ │ │ │ 000078d0: 2039 2920 2020 2020 2020 2020 2020 2020 9) │ │ │ │ 000078e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000078f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1942,21 +1942,21 @@ │ │ │ │ 00007950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007960: 2020 2020 2020 2020 2020 7c0a 7c34 2020 |.|4 │ │ │ │ 00007970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000079a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000079b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000079c0: 2020 2d2d 2075 7365 6420 2e30 3033 3730 -- used .00370 │ │ │ │ -000079d0: 3138 3720 7365 636f 6e64 7320 2020 2020 187 seconds │ │ │ │ +000079c0: 2020 2d2d 2075 7365 6420 2e30 3033 3431 -- used .00341 │ │ │ │ +000079d0: 3032 3820 7365 636f 6e64 7320 2020 2020 028 seconds │ │ │ │ 000079e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000079f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007a10: 2020 2d2d 2075 7365 6420 2e30 3034 3934 -- used .00494 │ │ │ │ -00007a20: 3934 3820 7365 636f 6e64 7320 2020 2020 948 seconds │ │ │ │ +00007a10: 2020 2d2d 2075 7365 6420 2e30 3034 3530 -- used .00450 │ │ │ │ +00007a20: 3139 2073 6563 6f6e 6473 2020 2020 2020 19 seconds │ │ │ │ 00007a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a50: 2020 2020 2020 2020 2020 7c0a 7c28 3136 |.|(16 │ │ │ │ 00007a60: 2c20 3236 2920 2020 2020 2020 2020 2020 , 26) │ │ │ │ 00007a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1967,61 +1967,61 @@ │ │ │ │ 00007ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007af0: 2020 2020 2020 2020 2020 7c0a 7c35 2020 |.|5 │ │ │ │ 00007b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007b50: 2020 2d2d 2075 7365 6420 2e30 3037 3639 -- used .00769 │ │ │ │ -00007b60: 3738 3320 7365 636f 6e64 7320 2020 2020 783 seconds │ │ │ │ +00007b50: 2020 2d2d 2075 7365 6420 2e30 3038 3031 -- used .00801 │ │ │ │ +00007b60: 3131 3320 7365 636f 6e64 7320 2020 2020 113 seconds │ │ │ │ 00007b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007b90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007ba0: 2020 2d2d 2075 7365 6420 2e31 3138 3631 -- used .11861 │ │ │ │ -00007bb0: 3620 7365 636f 6e64 7320 2020 2020 2020 6 seconds │ │ │ │ +00007ba0: 2020 2d2d 2075 7365 6420 2e31 3137 3236 -- used .11726 │ │ │ │ +00007bb0: 3220 7365 636f 6e64 7320 2020 2020 2020 2 seconds │ │ │ │ 00007bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007be0: 2020 2020 2020 2020 2020 7c0a 7c28 3235 |.|(25 │ │ │ │ 00007bf0: 2c20 3630 2920 2020 2020 2020 2020 2020 , 60) │ │ │ │ 00007c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c30: 2020 2020 2020 2020 2020 7c0a 7c36 2020 |.|6 │ │ │ │ 00007c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007c80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007c90: 2020 2d2d 2075 7365 6420 2e30 3138 3030 -- used .01800 │ │ │ │ -00007ca0: 3036 2073 6563 6f6e 6473 2020 2020 2020 06 seconds │ │ │ │ +00007c90: 2020 2d2d 2075 7365 6420 2e30 3137 3734 -- used .01774 │ │ │ │ +00007ca0: 3639 2073 6563 6f6e 6473 2020 2020 2020 69 seconds │ │ │ │ 00007cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007cd0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007ce0: 2020 2d2d 2075 7365 6420 2e33 3139 3337 -- used .31937 │ │ │ │ -00007cf0: 3820 7365 636f 6e64 7320 2020 2020 2020 8 seconds │ │ │ │ +00007ce0: 2020 2d2d 2075 7365 6420 2e32 3839 3930 -- used .28990 │ │ │ │ +00007cf0: 3320 7365 636f 6e64 7320 2020 2020 2020 3 seconds │ │ │ │ 00007d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d20: 2020 2020 2020 2020 2020 7c0a 7c28 3336 |.|(36 │ │ │ │ 00007d30: 2c20 3132 3029 2020 2020 2020 2020 2020 , 120) │ │ │ │ 00007d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d70: 2020 2020 2020 2020 2020 7c0a 7c37 2020 |.|7 │ │ │ │ 00007d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007dc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007dd0: 2020 2d2d 2075 7365 6420 2e30 3339 3333 -- used .03933 │ │ │ │ -00007de0: 3820 7365 636f 6e64 7320 2020 2020 2020 8 seconds │ │ │ │ +00007dd0: 2020 2d2d 2075 7365 6420 2e30 3337 3036 -- used .03706 │ │ │ │ +00007de0: 3535 2073 6563 6f6e 6473 2020 2020 2020 55 seconds │ │ │ │ 00007df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00007e20: 2020 2d2d 2075 7365 6420 312e 3132 3233 -- used 1.1223 │ │ │ │ -00007e30: 3920 7365 636f 6e64 7320 2020 2020 2020 9 seconds │ │ │ │ +00007e20: 2020 2d2d 2075 7365 6420 2e39 3933 3032 -- used .99302 │ │ │ │ +00007e30: 3520 7365 636f 6e64 7320 2020 2020 2020 5 seconds │ │ │ │ 00007e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e60: 2020 2020 2020 2020 2020 7c0a 7c28 3439 |.|(49 │ │ │ │ 00007e70: 2c20 3231 3729 2020 2020 2020 2020 2020 , 217) │ │ │ │ 00007e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/FastMinors.info.gz │ │ │ ├── FastMinors.info │ │ │ │ @@ -4169,17 +4169,17 @@ │ │ │ │ 00010480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000104a0: 6932 3820 3a20 7469 6d65 2064 696d 2028 i28 : time dim ( │ │ │ │ 000104b0: 4a20 2b20 6368 6f6f 7365 476f 6f64 4d69 J + chooseGoodMi │ │ │ │ 000104c0: 6e6f 7273 2838 2c20 362c 204d 2c20 4a2c nors(8, 6, M, J, │ │ │ │ 000104d0: 2053 7472 6174 6567 793d 3e52 616e 646f Strategy=>Rando │ │ │ │ 000104e0: 6d29 2920 2020 2020 2020 2020 207c 0a7c m)) |.| │ │ │ │ -000104f0: 202d 2d20 7573 6564 2030 2e30 3339 3937 -- used 0.03997 │ │ │ │ -00010500: 3734 7320 2863 7075 293b 2030 2e30 3433 74s (cpu); 0.043 │ │ │ │ -00010510: 3434 3636 7320 2874 6872 6561 6429 3b20 4466s (thread); │ │ │ │ +000104f0: 202d 2d20 7573 6564 2030 2e30 3532 3032 -- used 0.05202 │ │ │ │ +00010500: 3633 7320 2863 7075 293b 2030 2e30 3438 63s (cpu); 0.048 │ │ │ │ +00010510: 3035 3632 7320 2874 6872 6561 6429 3b20 0562s (thread); │ │ │ │ 00010520: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00010530: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00010540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010580: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -4194,18 +4194,18 @@ │ │ │ │ 00010610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00010630: 6932 3920 3a20 7469 6d65 2064 696d 2028 i29 : time dim ( │ │ │ │ 00010640: 4a20 2b20 6368 6f6f 7365 476f 6f64 4d69 J + chooseGoodMi │ │ │ │ 00010650: 6e6f 7273 2838 2c20 362c 204d 2c20 4a2c nors(8, 6, M, J, │ │ │ │ 00010660: 2053 7472 6174 6567 793d 3e4c 6578 536d Strategy=>LexSm │ │ │ │ 00010670: 616c 6c65 7374 2929 2020 2020 207c 0a7c allest)) |.| │ │ │ │ -00010680: 202d 2d20 7573 6564 2030 2e32 3231 3832 -- used 0.22182 │ │ │ │ -00010690: 3373 2028 6370 7529 3b20 302e 3137 3435 3s (cpu); 0.1745 │ │ │ │ -000106a0: 3535 7320 2874 6872 6561 6429 3b20 3073 55s (thread); 0s │ │ │ │ -000106b0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00010680: 202d 2d20 7573 6564 2030 2e32 3333 3530 -- used 0.23350 │ │ │ │ +00010690: 3773 2028 6370 7529 3b20 302e 3136 3331 7s (cpu); 0.1631 │ │ │ │ +000106a0: 3573 2028 7468 7265 6164 293b 2030 7320 5s (thread); 0s │ │ │ │ +000106b0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000106c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000106d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000106e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000106f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010710: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00010720: 6f32 3920 3d20 3320 2020 2020 2020 2020 o29 = 3 │ │ │ │ @@ -4219,18 +4219,18 @@ │ │ │ │ 000107a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000107b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000107c0: 6933 3020 3a20 7469 6d65 2064 696d 2028 i30 : time dim ( │ │ │ │ 000107d0: 4a20 2b20 6368 6f6f 7365 476f 6f64 4d69 J + chooseGoodMi │ │ │ │ 000107e0: 6e6f 7273 2838 2c20 362c 204d 2c20 4a2c nors(8, 6, M, J, │ │ │ │ 000107f0: 2053 7472 6174 6567 793d 3e4c 6578 536d Strategy=>LexSm │ │ │ │ 00010800: 616c 6c65 7374 5465 726d 2929 207c 0a7c allestTerm)) |.| │ │ │ │ -00010810: 202d 2d20 7573 6564 2030 2e33 3538 3133 -- used 0.35813 │ │ │ │ -00010820: 3473 2028 6370 7529 3b20 302e 3330 3234 4s (cpu); 0.3024 │ │ │ │ -00010830: 3873 2028 7468 7265 6164 293b 2030 7320 8s (thread); 0s │ │ │ │ -00010840: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00010810: 202d 2d20 7573 6564 2030 2e33 3430 3630 -- used 0.34060 │ │ │ │ +00010820: 3673 2028 6370 7529 3b20 302e 3237 3033 6s (cpu); 0.2703 │ │ │ │ +00010830: 3136 7320 2874 6872 6561 6429 3b20 3073 16s (thread); 0s │ │ │ │ +00010840: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00010850: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00010860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000108a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000108b0: 6f33 3020 3d20 3120 2020 2020 2020 2020 o30 = 1 │ │ │ │ @@ -4244,17 +4244,17 @@ │ │ │ │ 00010930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00010950: 6933 3120 3a20 7469 6d65 2064 696d 2028 i31 : time dim ( │ │ │ │ 00010960: 4a20 2b20 6368 6f6f 7365 476f 6f64 4d69 J + chooseGoodMi │ │ │ │ 00010970: 6e6f 7273 2838 2c20 362c 204d 2c20 4a2c nors(8, 6, M, J, │ │ │ │ 00010980: 2053 7472 6174 6567 793d 3e4c 6578 4c61 Strategy=>LexLa │ │ │ │ 00010990: 7267 6573 7429 2920 2020 2020 207c 0a7c rgest)) |.| │ │ │ │ -000109a0: 202d 2d20 7573 6564 2030 2e31 3639 3835 -- used 0.16985 │ │ │ │ -000109b0: 3573 2028 6370 7529 3b20 302e 3137 3230 5s (cpu); 0.1720 │ │ │ │ -000109c0: 3734 7320 2874 6872 6561 6429 3b20 3073 74s (thread); 0s │ │ │ │ +000109a0: 202d 2d20 7573 6564 2030 2e31 3635 3232 -- used 0.16522 │ │ │ │ +000109b0: 3573 2028 6370 7529 3b20 302e 3136 3631 5s (cpu); 0.1661 │ │ │ │ +000109c0: 3233 7320 2874 6872 6561 6429 3b20 3073 23s (thread); 0s │ │ │ │ 000109d0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 000109e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000109f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010a30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -4269,18 +4269,18 @@ │ │ │ │ 00010ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00010ae0: 6933 3220 3a20 7469 6d65 2064 696d 2028 i32 : time dim ( │ │ │ │ 00010af0: 4a20 2b20 6368 6f6f 7365 476f 6f64 4d69 J + chooseGoodMi │ │ │ │ 00010b00: 6e6f 7273 2838 2c20 362c 204d 2c20 4a2c nors(8, 6, M, J, │ │ │ │ 00010b10: 2053 7472 6174 6567 793d 3e47 5265 764c Strategy=>GRevL │ │ │ │ 00010b20: 6578 536d 616c 6c65 7374 2929 207c 0a7c exSmallest)) |.| │ │ │ │ -00010b30: 202d 2d20 7573 6564 2030 2e32 3339 3973 -- used 0.2399s │ │ │ │ -00010b40: 2028 6370 7529 3b20 302e 3137 3237 3835 (cpu); 0.172785 │ │ │ │ -00010b50: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -00010b60: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +00010b30: 202d 2d20 7573 6564 2030 2e32 3534 3734 -- used 0.25474 │ │ │ │ +00010b40: 3573 2028 6370 7529 3b20 302e 3136 3732 5s (cpu); 0.1672 │ │ │ │ +00010b50: 3631 7320 2874 6872 6561 6429 3b20 3073 61s (thread); 0s │ │ │ │ +00010b60: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00010b70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00010b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00010bd0: 6f33 3220 3d20 3320 2020 2020 2020 2020 o32 = 3 │ │ │ │ @@ -4294,17 +4294,17 @@ │ │ │ │ 00010c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00010c70: 6933 3320 3a20 7469 6d65 2064 696d 2028 i33 : time dim ( │ │ │ │ 00010c80: 4a20 2b20 6368 6f6f 7365 476f 6f64 4d69 J + chooseGoodMi │ │ │ │ 00010c90: 6e6f 7273 2838 2c20 362c 204d 2c20 4a2c nors(8, 6, M, J, │ │ │ │ 00010ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010cb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00010cc0: 202d 2d20 7573 6564 2030 2e32 3836 3631 -- used 0.28661 │ │ │ │ -00010cd0: 3773 2028 6370 7529 3b20 302e 3231 3431 7s (cpu); 0.2141 │ │ │ │ -00010ce0: 3132 7320 2874 6872 6561 6429 3b20 3073 12s (thread); 0s │ │ │ │ +00010cc0: 202d 2d20 7573 6564 2030 2e32 3635 3838 -- used 0.26588 │ │ │ │ +00010cd0: 3373 2028 6370 7529 3b20 302e 3139 3830 3s (cpu); 0.1980 │ │ │ │ +00010ce0: 3538 7320 2874 6872 6561 6429 3b20 3073 58s (thread); 0s │ │ │ │ 00010cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00010d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -4334,18 +4334,18 @@ │ │ │ │ 00010ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00010ef0: 6933 3420 3a20 7469 6d65 2064 696d 2028 i34 : time dim ( │ │ │ │ 00010f00: 4a20 2b20 6368 6f6f 7365 476f 6f64 4d69 J + chooseGoodMi │ │ │ │ 00010f10: 6e6f 7273 2838 2c20 362c 204d 2c20 4a2c nors(8, 6, M, J, │ │ │ │ 00010f20: 2053 7472 6174 6567 793d 3e47 5265 764c Strategy=>GRevL │ │ │ │ 00010f30: 6578 4c61 7267 6573 7429 2920 207c 0a7c exLargest)) |.| │ │ │ │ -00010f40: 202d 2d20 7573 6564 2030 2e31 3335 3734 -- used 0.13574 │ │ │ │ -00010f50: 3773 2028 6370 7529 3b20 302e 3133 3632 7s (cpu); 0.1362 │ │ │ │ -00010f60: 3973 2028 7468 7265 6164 293b 2030 7320 9s (thread); 0s │ │ │ │ -00010f70: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00010f40: 202d 2d20 7573 6564 2030 2e31 3137 3437 -- used 0.11747 │ │ │ │ +00010f50: 3373 2028 6370 7529 3b20 302e 3132 3032 3s (cpu); 0.1202 │ │ │ │ +00010f60: 3631 7320 2874 6872 6561 6429 3b20 3073 61s (thread); 0s │ │ │ │ +00010f70: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00010f80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00010f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 207c 0a7c |.| │ │ │ │ 00010fe0: 6f33 3420 3d20 3320 2020 2020 2020 2020 o34 = 3 │ │ │ │ @@ -4359,16 +4359,16 @@ │ │ │ │ 00011060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00011080: 6933 3520 3a20 7469 6d65 2064 696d 2028 i35 : time dim ( │ │ │ │ 00011090: 4a20 2b20 6368 6f6f 7365 476f 6f64 4d69 J + chooseGoodMi │ │ │ │ 000110a0: 6e6f 7273 2838 2c20 362c 204d 2c20 4a2c nors(8, 6, M, J, │ │ │ │ 000110b0: 2053 7472 6174 6567 793d 3e50 6f69 6e74 Strategy=>Point │ │ │ │ 000110c0: 7329 2920 2020 2020 2020 2020 207c 0a7c s)) |.| │ │ │ │ -000110d0: 202d 2d20 7573 6564 2031 322e 3136 3534 -- used 12.1654 │ │ │ │ -000110e0: 7320 2863 7075 293b 2039 2e37 3037 3139 s (cpu); 9.70719 │ │ │ │ +000110d0: 202d 2d20 7573 6564 2031 312e 3732 3335 -- used 11.7235 │ │ │ │ +000110e0: 7320 2863 7075 293b 2039 2e30 3631 3338 s (cpu); 9.06138 │ │ │ │ 000110f0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00011100: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00011110: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00011120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4593,17 +4593,17 @@ │ │ │ │ 00011f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00011f20: 0a7c 202d 2d20 696e 7465 726e 616c 4368 .| -- internalCh │ │ │ │ 00011f30: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ 00011f40: 696e 6720 5261 6e64 6f6d 2020 2020 2020 ing Random │ │ │ │ 00011f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011f70: 0a7c 202d 2d20 7573 6564 2030 2e34 3135 .| -- used 0.415 │ │ │ │ -00011f80: 3631 3773 2028 6370 7529 3b20 302e 3335 617s (cpu); 0.35 │ │ │ │ -00011f90: 3431 3539 7320 2874 6872 6561 6429 3b20 4159s (thread); │ │ │ │ +00011f70: 0a7c 202d 2d20 7573 6564 2030 2e33 3937 .| -- used 0.397 │ │ │ │ +00011f80: 3935 3673 2028 6370 7529 3b20 302e 3333 956s (cpu); 0.33 │ │ │ │ +00011f90: 3035 3733 7320 2874 6872 6561 6429 3b20 0573s (thread); │ │ │ │ 00011fa0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00011fb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00011fc0: 0a7c 6368 6f6f 7365 476f 6f64 4d69 6e6f .|chooseGoodMino │ │ │ │ 00011fd0: 7273 3a20 666f 756e 6420 3d32 302c 2061 rs: found =20, a │ │ │ │ 00011fe0: 7474 656d 7074 6564 203d 2032 3220 2020 ttempted = 22 │ │ │ │ 00011ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012000: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -4894,16 +4894,16 @@ │ │ │ │ 000131d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000131e0: 2d2d 2d2d 2d2d 2b0a 7c69 3433 203a 2074 ------+.|i43 : t │ │ │ │ 000131f0: 696d 6520 6469 6d20 284a 202b 2063 686f ime dim (J + cho │ │ │ │ 00013200: 6f73 6547 6f6f 644d 696e 6f72 7328 312c oseGoodMinors(1, │ │ │ │ 00013210: 2036 2c20 4d2c 204a 2c20 5374 7261 7465 6, M, J, Strate │ │ │ │ 00013220: 6779 3d3e 506f 696e 7473 2c20 2020 2020 gy=>Points, │ │ │ │ 00013230: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00013240: 6420 302e 3733 3132 3739 7320 2863 7075 d 0.731279s (cpu │ │ │ │ -00013250: 293b 2030 2e35 3834 3839 3573 2028 7468 ); 0.584895s (th │ │ │ │ +00013240: 6420 302e 3732 3339 3333 7320 2863 7075 d 0.723933s (cpu │ │ │ │ +00013250: 293b 2030 2e35 3737 3138 3473 2028 7468 ); 0.577184s (th │ │ │ │ 00013260: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00013270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013280: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00013290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000132a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000132b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000132c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5029,16 +5029,16 @@ │ │ │ │ 00013a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013a50: 2d2d 2d2d 2d2d 2b0a 7c69 3436 203a 2074 ------+.|i46 : t │ │ │ │ 00013a60: 696d 6520 6469 6d20 284a 202b 2063 686f ime dim (J + cho │ │ │ │ 00013a70: 6f73 6547 6f6f 644d 696e 6f72 7328 312c oseGoodMinors(1, │ │ │ │ 00013a80: 2036 2c20 4d2c 204a 2c20 5374 7261 7465 6, M, J, Strate │ │ │ │ 00013a90: 6779 3d3e 506f 696e 7473 2c20 2020 2020 gy=>Points, │ │ │ │ 00013aa0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00013ab0: 6420 302e 3632 3831 3332 7320 2863 7075 d 0.628132s (cpu │ │ │ │ -00013ac0: 293b 2030 2e34 3733 3831 3373 2028 7468 ); 0.473813s (th │ │ │ │ +00013ab0: 6420 302e 3637 3232 3031 7320 2863 7075 d 0.672201s (cpu │ │ │ │ +00013ac0: 293b 2030 2e35 3032 3332 3673 2028 7468 ); 0.502326s (th │ │ │ │ 00013ad0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 00013ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00013b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5121,16 +5121,16 @@ │ │ │ │ 00014000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00014020: 3437 203a 2074 696d 6520 7265 6775 6c61 47 : time regula │ │ │ │ 00014030: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00014040: 2c20 532f 4a2c 204d 6178 4d69 6e6f 7273 , S/J, MaxMinors │ │ │ │ 00014050: 203d 3e20 3130 302c 2020 2020 2020 2020 => 100, │ │ │ │ 00014060: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00014070: 2d2d 2075 7365 6420 322e 3136 3732 3573 -- used 2.16725s │ │ │ │ -00014080: 2028 6370 7529 3b20 312e 3839 3839 3473 (cpu); 1.89894s │ │ │ │ +00014070: 2d2d 2075 7365 6420 322e 3232 3137 3973 -- used 2.22179s │ │ │ │ +00014080: 2028 6370 7529 3b20 312e 3931 3435 3273 (cpu); 1.91452s │ │ │ │ 00014090: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000140a0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000140b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 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 2020 │ │ │ │ @@ -5156,18 +5156,18 @@ │ │ │ │ 00014230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00014250: 3438 203a 2074 696d 6520 7265 6775 6c61 48 : time regula │ │ │ │ 00014260: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00014270: 2c20 532f 4a2c 204d 6178 4d69 6e6f 7273 , S/J, MaxMinors │ │ │ │ 00014280: 203d 3e20 3130 302c 2020 2020 2020 2020 => 100, │ │ │ │ 00014290: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000142a0: 2d2d 2075 7365 6420 312e 3238 3431 7320 -- used 1.2841s │ │ │ │ -000142b0: 2863 7075 293b 2031 2e30 3730 3637 7320 (cpu); 1.07067s │ │ │ │ -000142c0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -000142d0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +000142a0: 2d2d 2075 7365 6420 312e 3233 3339 3673 -- used 1.23396s │ │ │ │ +000142b0: 2028 6370 7529 3b20 312e 3031 3533 3673 (cpu); 1.01536s │ │ │ │ +000142c0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +000142d0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000142e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 000142f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014330: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 00014340: 3438 203d 2074 7275 6520 2020 2020 2020 48 = true │ │ │ │ @@ -5191,31 +5191,31 @@ │ │ │ │ 00014460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00014480: 3439 203a 2074 696d 6520 7265 6775 6c61 49 : time regula │ │ │ │ 00014490: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 000144a0: 2c20 532f 4a2c 204d 6178 4d69 6e6f 7273 , S/J, MaxMinors │ │ │ │ 000144b0: 203d 3e20 3130 302c 2053 7472 6174 6567 => 100, Strateg │ │ │ │ 000144c0: 793d 3e52 616e 646f 6d29 2020 7c0a 7c20 y=>Random) |.| │ │ │ │ -000144d0: 2d2d 2075 7365 6420 322e 3832 3636 3173 -- used 2.82661s │ │ │ │ -000144e0: 2028 6370 7529 3b20 322e 3533 3335 7320 (cpu); 2.5335s │ │ │ │ -000144f0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00014500: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +000144d0: 2d2d 2075 7365 6420 322e 3834 3739 3873 -- used 2.84798s │ │ │ │ +000144e0: 2028 6370 7529 3b20 322e 3636 3130 3273 (cpu); 2.66102s │ │ │ │ +000144f0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +00014500: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00014510: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00014520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00014570: 3530 203a 2074 696d 6520 7265 6775 6c61 50 : time regula │ │ │ │ 00014580: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00014590: 2c20 532f 4a2c 204d 6178 4d69 6e6f 7273 , S/J, MaxMinors │ │ │ │ 000145a0: 203d 3e20 3130 302c 2020 2020 2020 2020 => 100, │ │ │ │ 000145b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000145c0: 2d2d 2075 7365 6420 322e 3832 3138 3573 -- used 2.82185s │ │ │ │ -000145d0: 2028 6370 7529 3b20 322e 3132 3830 3673 (cpu); 2.12806s │ │ │ │ +000145c0: 2d2d 2075 7365 6420 322e 3534 3137 3673 -- used 2.54176s │ │ │ │ +000145d0: 2028 6370 7529 3b20 312e 3937 3336 3773 (cpu); 1.97367s │ │ │ │ 000145e0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000145f0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00014600: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 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 2d2d ---------------- │ │ │ │ @@ -5231,17 +5231,17 @@ │ │ │ │ 000146e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00014700: 3531 203a 2074 696d 6520 7265 6775 6c61 51 : time regula │ │ │ │ 00014710: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00014720: 2c20 532f 4a2c 204d 6178 4d69 6e6f 7273 , S/J, MaxMinors │ │ │ │ 00014730: 203d 3e20 3130 302c 2020 2020 2020 2020 => 100, │ │ │ │ 00014740: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00014750: 2d2d 2075 7365 6420 302e 3238 3236 3739 -- used 0.282679 │ │ │ │ -00014760: 7320 2863 7075 293b 2030 2e32 3832 3939 s (cpu); 0.28299 │ │ │ │ -00014770: 3573 2028 7468 7265 6164 293b 2030 7320 5s (thread); 0s │ │ │ │ +00014750: 2d2d 2075 7365 6420 302e 3238 3531 3339 -- used 0.285139 │ │ │ │ +00014760: 7320 2863 7075 293b 2030 2e32 3838 3732 s (cpu); 0.28872 │ │ │ │ +00014770: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ 00014780: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00014790: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 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 2020 │ │ │ │ 000147e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -5266,16 +5266,16 @@ │ │ │ │ 00014910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00014930: 3532 203a 2074 696d 6520 7265 6775 6c61 52 : time regula │ │ │ │ 00014940: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00014950: 2c20 532f 4a2c 204d 6178 4d69 6e6f 7273 , S/J, MaxMinors │ │ │ │ 00014960: 203d 3e20 3130 302c 2020 2020 2020 2020 => 100, │ │ │ │ 00014970: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00014980: 2d2d 2075 7365 6420 322e 3936 3136 3573 -- used 2.96165s │ │ │ │ -00014990: 2028 6370 7529 3b20 322e 3331 3533 3973 (cpu); 2.31539s │ │ │ │ +00014980: 2d2d 2075 7365 6420 322e 3933 3239 3273 -- used 2.93292s │ │ │ │ +00014990: 2028 6370 7529 3b20 322e 3231 3537 3873 (cpu); 2.21578s │ │ │ │ 000149a0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000149b0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000149c0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 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 2d2d ---------------- │ │ │ │ @@ -5291,16 +5291,16 @@ │ │ │ │ 00014aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00014ac0: 3533 203a 2074 696d 6520 7265 6775 6c61 53 : time regula │ │ │ │ 00014ad0: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00014ae0: 2c20 532f 4a2c 204d 6178 4d69 6e6f 7273 , S/J, MaxMinors │ │ │ │ 00014af0: 203d 3e20 3130 302c 2020 2020 2020 2020 => 100, │ │ │ │ 00014b00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00014b10: 2d2d 2075 7365 6420 332e 3436 3638 3273 -- used 3.46682s │ │ │ │ -00014b20: 2028 6370 7529 3b20 322e 3832 3831 3373 (cpu); 2.82813s │ │ │ │ +00014b10: 2d2d 2075 7365 6420 332e 3237 3033 3273 -- used 3.27032s │ │ │ │ +00014b20: 2028 6370 7529 3b20 322e 3636 3432 3173 (cpu); 2.66421s │ │ │ │ 00014b30: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 00014b40: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00014b50: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00014b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5316,16 +5316,16 @@ │ │ │ │ 00014c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00014c50: 3534 203a 2074 696d 6520 7265 6775 6c61 54 : time regula │ │ │ │ 00014c60: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00014c70: 2c20 532f 4a2c 204d 6178 4d69 6e6f 7273 , S/J, MaxMinors │ │ │ │ 00014c80: 203d 3e20 3130 302c 2053 7472 6174 6567 => 100, Strateg │ │ │ │ 00014c90: 793d 3e50 6f69 6e74 7329 2020 7c0a 7c20 y=>Points) |.| │ │ │ │ -00014ca0: 2d2d 2075 7365 6420 3633 2e38 3234 3873 -- used 63.8248s │ │ │ │ -00014cb0: 2028 6370 7529 3b20 3531 2e33 3034 3773 (cpu); 51.3047s │ │ │ │ +00014ca0: 2d2d 2075 7365 6420 3531 2e36 3333 3973 -- used 51.6339s │ │ │ │ +00014cb0: 2028 6370 7529 3b20 3433 2e39 3339 3773 (cpu); 43.9397s │ │ │ │ 00014cc0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 00014cd0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00014ce0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00014cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5341,16 +5341,16 @@ │ │ │ │ 00014dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00014de0: 3535 203a 2074 696d 6520 7265 6775 6c61 55 : time regula │ │ │ │ 00014df0: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 00014e00: 2c20 532f 4a2c 204d 6178 4d69 6e6f 7273 , S/J, MaxMinors │ │ │ │ 00014e10: 203d 3e20 3130 302c 2020 2020 2020 2020 => 100, │ │ │ │ 00014e20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00014e30: 2d2d 2075 7365 6420 322e 3938 3836 3573 -- used 2.98865s │ │ │ │ -00014e40: 2028 6370 7529 3b20 322e 3335 3032 3573 (cpu); 2.35025s │ │ │ │ +00014e30: 2d2d 2075 7365 6420 322e 3839 3037 3973 -- used 2.89079s │ │ │ │ +00014e40: 2028 6370 7529 3b20 322e 3232 3038 3873 (cpu); 2.22088s │ │ │ │ 00014e50: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 00014e60: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00014e70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00014e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6006,16 +6006,16 @@ │ │ │ │ 00017750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017760: 2d2d 2d2d 2b0a 7c69 3720 3a20 7469 6d65 ----+.|i7 : time │ │ │ │ 00017770: 2069 7343 6f64 696d 4174 4c65 6173 7428 isCodimAtLeast( │ │ │ │ 00017780: 332c 204a 2920 2020 2020 2020 2020 2020 3, J) │ │ │ │ 00017790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000177a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000177b0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -000177c0: 302e 3030 3039 3437 3036 3673 2028 6370 0.000947066s (cp │ │ │ │ -000177d0: 7529 3b20 302e 3030 3236 3339 3535 7320 u); 0.00263955s │ │ │ │ +000177c0: 302e 3030 3032 3832 3032 3173 2028 6370 0.000282021s (cp │ │ │ │ +000177d0: 7529 3b20 302e 3030 3235 3634 3933 7320 u); 0.00256493s │ │ │ │ 000177e0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 000177f0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00017800: 2020 2020 7c0a 7c20 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 2020 │ │ │ │ @@ -6216,18 +6216,18 @@ │ │ │ │ 00018470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018480: 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 696d -----+.|i9 : tim │ │ │ │ 00018490: 6520 6973 436f 6469 6d41 744c 6561 7374 e isCodimAtLeast │ │ │ │ 000184a0: 2835 2c20 492c 2050 6169 724c 696d 6974 (5, I, PairLimit │ │ │ │ 000184b0: 203d 3e20 352c 2056 6572 626f 7365 3d3e => 5, Verbose=> │ │ │ │ 000184c0: 7472 7565 2920 2020 2020 2020 2020 2020 true) │ │ │ │ 000184d0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000184e0: 2030 2e30 3034 3030 3037 3473 2028 6370 0.00400074s (cp │ │ │ │ -000184f0: 7529 3b20 302e 3030 3235 3134 3437 7320 u); 0.00251447s │ │ │ │ -00018500: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00018510: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +000184e0: 2030 2e30 3030 3736 3935 3673 2028 6370 0.00076956s (cp │ │ │ │ +000184f0: 7529 3b20 302e 3030 3234 3539 3373 2028 u); 0.0024593s ( │ │ │ │ +00018500: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00018510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018520: 2020 2020 207c 0a7c 6973 436f 6469 6d41 |.|isCodimA │ │ │ │ 00018530: 744c 6561 7374 3a20 436f 6d70 7574 696e tLeast: Computin │ │ │ │ 00018540: 6720 636f 6469 6d20 6f66 206d 6f6e 6f6d g codim of monom │ │ │ │ 00018550: 6961 6c73 2062 6173 6564 206f 6e20 6964 ials based on id │ │ │ │ 00018560: 6561 6c20 6765 6e65 7261 746f 7273 2e20 eal generators. │ │ │ │ 00018570: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00018580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6246,18 +6246,18 @@ │ │ │ │ 00018650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018660: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 7469 -----+.|i10 : ti │ │ │ │ 00018670: 6d65 2069 7343 6f64 696d 4174 4c65 6173 me isCodimAtLeas │ │ │ │ 00018680: 7428 352c 2049 2c20 5061 6972 4c69 6d69 t(5, I, PairLimi │ │ │ │ 00018690: 7420 3d3e 2032 3030 2c20 5665 7262 6f73 t => 200, Verbos │ │ │ │ 000186a0: 653d 3e66 616c 7365 2920 2020 2020 2020 e=>false) │ │ │ │ 000186b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000186c0: 2036 2e35 3337 3365 2d30 3573 2028 6370 6.5373e-05s (cp │ │ │ │ -000186d0: 7529 3b20 302e 3030 3232 3936 3373 2028 u); 0.0022963s ( │ │ │ │ -000186e0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -000186f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000186c0: 2030 2e30 3030 3833 3939 3738 7320 2863 0.000839978s (c │ │ │ │ +000186d0: 7075 293b 2030 2e30 3032 3338 3832 3773 pu); 0.00238827s │ │ │ │ +000186e0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +000186f0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00018700: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00018710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018750: 2020 2020 207c 0a7c 6f31 3020 3d20 7472 |.|o10 = tr │ │ │ │ 00018760: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ @@ -7483,17 +7483,17 @@ │ │ │ │ 0001d3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ 0001d3c0: 3a20 7469 6d65 2070 726f 6a44 696d 286d : time projDim(m │ │ │ │ 0001d3d0: 6f64 756c 6520 492c 2053 7472 6174 6567 odule I, Strateg │ │ │ │ 0001d3e0: 793d 3e53 7472 6174 6567 7952 616e 646f y=>StrategyRando │ │ │ │ 0001d3f0: 6d29 2020 2020 2020 2020 2020 2020 2020 m) │ │ │ │ 0001d400: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001d410: 2030 2e32 3936 3937 3173 2028 6370 7529 0.296971s (cpu) │ │ │ │ -0001d420: 3b20 302e 3135 3939 3333 7320 2874 6872 ; 0.159933s (thr │ │ │ │ -0001d430: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +0001d410: 2030 2e33 3132 3732 3473 2028 6370 7529 0.312724s (cpu) │ │ │ │ +0001d420: 3b20 302e 3137 3635 7320 2874 6872 6561 ; 0.1765s (threa │ │ │ │ +0001d430: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0001d440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d450: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d490: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ 0001d4a0: 203d 2031 2020 2020 2020 2020 2020 2020 = 1 │ │ │ │ @@ -7506,17 +7506,17 @@ │ │ │ │ 0001d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d530: 2d2b 0a7c 6935 203a 2074 696d 6520 7072 -+.|i5 : time pr │ │ │ │ 0001d540: 6f6a 4469 6d28 6d6f 6475 6c65 2049 2c20 ojDim(module I, │ │ │ │ 0001d550: 5374 7261 7465 6779 3d3e 5374 7261 7465 Strategy=>Strate │ │ │ │ 0001d560: 6779 5261 6e64 6f6d 2c20 4d69 6e44 696d gyRandom, MinDim │ │ │ │ 0001d570: 656e 7369 6f6e 203d 3e20 3129 7c0a 7c20 ension => 1)|.| │ │ │ │ -0001d580: 2d2d 2075 7365 6420 302e 3030 3938 3833 -- used 0.009883 │ │ │ │ -0001d590: 3839 7320 2863 7075 293b 2030 2e30 3132 89s (cpu); 0.012 │ │ │ │ -0001d5a0: 3030 3937 7320 2874 6872 6561 6429 3b20 0097s (thread); │ │ │ │ +0001d580: 2d2d 2075 7365 6420 302e 3030 3838 3436 -- used 0.008846 │ │ │ │ +0001d590: 3939 7320 2863 7075 293b 2030 2e30 3131 99s (cpu); 0.011 │ │ │ │ +0001d5a0: 3730 3934 7320 2874 6872 6561 6429 3b20 7094s (thread); │ │ │ │ 0001d5b0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001d5c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001d5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d610: 2020 7c0a 7c6f 3520 3d20 3120 2020 2020 |.|o5 = 1 │ │ │ │ @@ -7724,17 +7724,17 @@ │ │ │ │ 0001e2b0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0001e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e2e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 0001e2f0: 7469 6d65 2049 3220 3d20 7265 6375 7273 time I2 = recurs │ │ │ │ 0001e300: 6976 654d 696e 6f72 7328 342c 204d 2c20 iveMinors(4, M, │ │ │ │ 0001e310: 5468 7265 6164 733d 3e30 293b 2020 2020 Threads=>0); │ │ │ │ -0001e320: 7c0a 7c20 2d2d 2075 7365 6420 302e 3535 |.| -- used 0.55 │ │ │ │ -0001e330: 3536 3633 7320 2863 7075 293b 2030 2e34 5663s (cpu); 0.4 │ │ │ │ -0001e340: 3831 3236 3473 2028 7468 7265 6164 293b 81264s (thread); │ │ │ │ +0001e320: 7c0a 7c20 2d2d 2075 7365 6420 302e 3435 |.| -- used 0.45 │ │ │ │ +0001e330: 3932 3331 7320 2863 7075 293b 2030 2e34 9231s (cpu); 0.4 │ │ │ │ +0001e340: 3037 3530 3373 2028 7468 7265 6164 293b 07503s (thread); │ │ │ │ 0001e350: 2030 7320 2867 6329 7c0a 7c20 2020 2020 0s (gc)|.| │ │ │ │ 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: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66 |.|o3 : Ideal of │ │ │ │ 0001e3a0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0001e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7742,16 +7742,16 @@ │ │ │ │ 0001e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e400: 2b0a 7c69 3420 3a20 7469 6d65 2049 3120 +.|i4 : time I1 │ │ │ │ 0001e410: 3d20 6d69 6e6f 7273 2834 2c20 4d2c 2053 = minors(4, M, S │ │ │ │ 0001e420: 7472 6174 6567 793d 3e43 6f66 6163 746f trategy=>Cofacto │ │ │ │ 0001e430: 7229 3b20 2020 2020 7c0a 7c20 2d2d 2075 r); |.| -- u │ │ │ │ -0001e440: 7365 6420 312e 3537 3734 3873 2028 6370 sed 1.57748s (cp │ │ │ │ -0001e450: 7529 3b20 312e 3337 3837 3973 2028 7468 u); 1.37879s (th │ │ │ │ +0001e440: 7365 6420 312e 3135 3631 3373 2028 6370 sed 1.15613s (cp │ │ │ │ +0001e450: 7529 3b20 312e 3034 3331 3673 2028 7468 u); 1.04316s (th │ │ │ │ 0001e460: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 0001e470: 7c0a 7c20 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: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ 0001e4b0: 4964 6561 6c20 6f66 2052 2020 2020 2020 Ideal of R │ │ │ │ 0001e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8188,17 +8188,17 @@ │ │ │ │ 0001ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ffc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0001ffd0: 3820 3a20 7469 6d65 2072 6567 756c 6172 8 : time regular │ │ │ │ 0001ffe0: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 0001fff0: 2053 2920 2020 2020 2020 2020 2020 2020 S) │ │ │ │ 00020000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00020020: 2d2d 2075 7365 6420 302e 3932 3139 3134 -- used 0.921914 │ │ │ │ -00020030: 7320 2863 7075 293b 2030 2e36 3137 3931 s (cpu); 0.61791 │ │ │ │ -00020040: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ +00020020: 2d2d 2075 7365 6420 302e 3939 3436 3434 -- used 0.994644 │ │ │ │ +00020030: 7320 2863 7075 293b 2030 2e35 3833 3436 s (cpu); 0.58346 │ │ │ │ +00020040: 3573 2028 7468 7265 6164 293b 2030 7320 5s (thread); 0s │ │ │ │ 00020050: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00020060: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00020070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000200a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000200b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ @@ -8213,16 +8213,16 @@ │ │ │ │ 00020140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00020160: 3920 3a20 7469 6d65 2072 6567 756c 6172 9 : time regular │ │ │ │ 00020170: 496e 436f 6469 6d65 6e73 696f 6e28 322c InCodimension(2, │ │ │ │ 00020180: 2053 2920 2020 2020 2020 2020 2020 2020 S) │ │ │ │ 00020190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000201a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000201b0: 2d2d 2075 7365 6420 3130 2e31 3034 3473 -- used 10.1044s │ │ │ │ -000201c0: 2028 6370 7529 3b20 362e 3735 3739 3373 (cpu); 6.75793s │ │ │ │ +000201b0: 2d2d 2075 7365 6420 382e 3036 3533 3773 -- used 8.06537s │ │ │ │ +000201c0: 2028 6370 7529 3b20 352e 3033 3639 3373 (cpu); 5.03693s │ │ │ │ 000201d0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 000201e0: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 000201f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00020200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -8343,16 +8343,16 @@ │ │ │ │ 00020960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020970: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 7469 -----+.|i12 : ti │ │ │ │ 00020980: 6d65 2028 6469 6d20 7369 6e67 756c 6172 me (dim singular │ │ │ │ 00020990: 4c6f 6375 7320 2852 2929 2020 2020 2020 Locus (R)) │ │ │ │ 000209a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000209b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000209c0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000209d0: 2030 2e30 3237 3935 3931 7320 2863 7075 0.0279591s (cpu │ │ │ │ -000209e0: 293b 2030 2e30 3236 3136 3237 7320 2874 ); 0.0261627s (t │ │ │ │ +000209d0: 2030 2e30 3136 3031 3738 7320 2863 7075 0.0160178s (cpu │ │ │ │ +000209e0: 293b 2030 2e30 3136 3537 3836 7320 2874 ); 0.0165786s (t │ │ │ │ 000209f0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00020a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020a10: 2020 2020 207c 0a7c 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 2020 2020 2020 │ │ │ │ 00020a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8368,17 +8368,17 @@ │ │ │ │ 00020af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020b00: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 7469 -----+.|i13 : ti │ │ │ │ 00020b10: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00020b20: 6d65 6e73 696f 6e28 322c 2052 2920 2020 mension(2, R) │ │ │ │ 00020b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b50: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00020b60: 2030 2e35 3539 3033 3873 2028 6370 7529 0.559038s (cpu) │ │ │ │ -00020b70: 3b20 302e 3331 3636 3432 7320 2874 6872 ; 0.316642s (thr │ │ │ │ -00020b80: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00020b60: 2030 2e34 3837 3330 3173 2028 6370 7529 0.487301s (cpu) │ │ │ │ +00020b70: 3b20 302e 3236 3533 3473 2028 7468 7265 ; 0.26534s (thre │ │ │ │ +00020b80: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ 00020b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ba0: 2020 2020 207c 0a7c 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 2020 2020 2020 │ │ │ │ 00020be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020bf0: 2020 2020 207c 0a7c 6f31 3320 3d20 7472 |.|o13 = tr │ │ │ │ @@ -8393,16 +8393,16 @@ │ │ │ │ 00020c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020c90: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 7469 -----+.|i14 : ti │ │ │ │ 00020ca0: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00020cb0: 6d65 6e73 696f 6e28 322c 2052 2920 2020 mension(2, R) │ │ │ │ 00020cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ce0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00020cf0: 2030 2e34 3431 3334 3373 2028 6370 7529 0.441343s (cpu) │ │ │ │ -00020d00: 3b20 302e 3330 3331 3031 7320 2874 6872 ; 0.303101s (thr │ │ │ │ +00020cf0: 2030 2e33 3038 3931 3573 2028 6370 7529 0.308915s (cpu) │ │ │ │ +00020d00: 3b20 302e 3230 3037 3332 7320 2874 6872 ; 0.200732s (thr │ │ │ │ 00020d10: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00020d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d30: 2020 2020 207c 0a7c 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 2020 2020 2020 │ │ │ │ 00020d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8418,16 +8418,16 @@ │ │ │ │ 00020e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020e20: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 7469 -----+.|i15 : ti │ │ │ │ 00020e30: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 00020e40: 6d65 6e73 696f 6e28 322c 2052 2920 2020 mension(2, R) │ │ │ │ 00020e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020e70: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00020e80: 2030 2e34 3234 3738 3773 2028 6370 7529 0.424787s (cpu) │ │ │ │ -00020e90: 3b20 302e 3231 3739 3731 7320 2874 6872 ; 0.217971s (thr │ │ │ │ +00020e80: 2030 2e33 3632 3030 3973 2028 6370 7529 0.362009s (cpu) │ │ │ │ +00020e90: 3b20 302e 3138 3838 3534 7320 2874 6872 ; 0.188854s (thr │ │ │ │ 00020ea0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00020eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00020ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -10104,18 +10104,18 @@ │ │ │ │ 00027770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027780: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00027790: 2d2d 2069 6e74 6572 6e61 6c43 686f 6f73 -- internalChoos │ │ │ │ 000277a0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ 000277b0: 2047 5265 764c 6578 536d 616c 6c65 7374 GRevLexSmallest │ │ │ │ 000277c0: 5465 726d 2020 2020 2020 2020 2020 2020 Term │ │ │ │ 000277d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000277e0: 2d2d 2075 7365 6420 392e 3534 3936 3273 -- used 9.54962s │ │ │ │ -000277f0: 2028 6370 7529 3b20 362e 3236 3938 7320 (cpu); 6.2698s │ │ │ │ -00027800: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00027810: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +000277e0: 2d2d 2075 7365 6420 382e 3430 3735 3373 -- used 8.40753s │ │ │ │ +000277f0: 2028 6370 7529 3b20 352e 3239 3936 3973 (cpu); 5.29969s │ │ │ │ +00027800: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +00027810: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00027820: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ 00027830: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ 00027840: 696f 6e3a 2072 696e 6720 6469 6d65 6e73 ion: ring dimens │ │ │ │ 00027850: 696f 6e20 3d33 2c20 7468 6572 6520 6172 ion =3, there ar │ │ │ │ 00027860: 6520 3137 3332 3520 706f 7373 6962 6c65 e 17325 possible │ │ │ │ 00027870: 2034 2062 7920 3420 6d69 6e6f 7c0a 7c72 4 by 4 mino|.|r │ │ │ │ 00027880: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ @@ -10793,18 +10793,18 @@ │ │ │ │ 0002a280: 7374 5465 726d 2020 2020 2020 2020 2020 stTerm │ │ │ │ 0002a290: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0002a2a0: 7c20 2d2d 2069 6e74 6572 6e61 6c43 686f | -- internalCho │ │ │ │ 0002a2b0: 6f73 654d 696e 6f72 3a20 4368 6f6f 7369 oseMinor: Choosi │ │ │ │ 0002a2c0: 6e67 2052 616e 646f 6d4e 6f6e 5a65 726f ng RandomNonZero │ │ │ │ 0002a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a2e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a2f0: 7c20 2d2d 2075 7365 6420 312e 3639 3934 | -- used 1.6994 │ │ │ │ -0002a300: 3273 2028 6370 7529 3b20 312e 3133 3034 2s (cpu); 1.1304 │ │ │ │ -0002a310: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ -0002a320: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0002a2f0: 7c20 2d2d 2075 7365 6420 312e 3535 3636 | -- used 1.5566 │ │ │ │ +0002a300: 7320 2863 7075 293b 2031 2e30 3532 3473 s (cpu); 1.0524s │ │ │ │ +0002a310: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +0002a320: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 0002a330: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0002a340: 7c72 6567 756c 6172 496e 436f 6469 6d65 |regularInCodime │ │ │ │ 0002a350: 6e73 696f 6e3a 2072 696e 6720 6469 6d65 nsion: ring dime │ │ │ │ 0002a360: 6e73 696f 6e20 3d33 2c20 7468 6572 6520 nsion =3, there │ │ │ │ 0002a370: 6172 6520 3137 3332 3520 706f 7373 6962 are 17325 possib │ │ │ │ 0002a380: 6c65 2034 2062 7920 3420 6d69 6e6f 7c0a le 4 by 4 mino|. │ │ │ │ 0002a390: 7c72 6567 756c 6172 496e 436f 6469 6d65 |regularInCodime │ │ │ │ @@ -11063,16 +11063,16 @@ │ │ │ │ 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 2b0a 7c69 ------------+.|i │ │ │ │ 0002b390: 3138 203a 2074 696d 6520 7265 6775 6c61 18 : time regula │ │ │ │ 0002b3a0: 7249 6e43 6f64 696d 656e 7369 6f6e 2832 rInCodimension(2 │ │ │ │ 0002b3b0: 2c20 532c 2056 6572 6966 794e 6f6e 5265 , S, VerifyNonRe │ │ │ │ 0002b3c0: 6775 6c61 723d 3e74 7275 6529 7c0a 7c20 gular=>true)|.| │ │ │ │ -0002b3d0: 2d2d 2075 7365 6420 312e 3633 3239 3673 -- used 1.63296s │ │ │ │ -0002b3e0: 2028 6370 7529 3b20 302e 3937 3236 3435 (cpu); 0.972645 │ │ │ │ +0002b3d0: 2d2d 2075 7365 6420 312e 3533 3933 3173 -- used 1.53931s │ │ │ │ +0002b3e0: 2028 6370 7529 3b20 302e 3930 3937 3031 (cpu); 0.909701 │ │ │ │ 0002b3f0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 0002b400: 6763 2920 2020 2020 2020 2020 7c0a 7c20 gc) |.| │ │ │ │ 0002b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b440: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0002b450: 3138 203d 2066 616c 7365 2020 2020 2020 18 = false │ │ │ │ @@ -11166,17 +11166,17 @@ │ │ │ │ 0002b9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0002ba00: 6932 3220 3a20 7469 6d65 2072 6567 756c i22 : time regul │ │ │ │ 0002ba10: 6172 496e 436f 6469 6d65 6e73 696f 6e28 arInCodimension( │ │ │ │ 0002ba20: 322c 2052 2c20 5374 7261 7465 6779 3d3e 2, R, Strategy=> │ │ │ │ 0002ba30: 5374 7261 7465 6779 4375 7272 656e 7429 StrategyCurrent) │ │ │ │ -0002ba40: 7c0a 7c20 2d2d 2075 7365 6420 302e 3539 |.| -- used 0.59 │ │ │ │ -0002ba50: 3733 3633 7320 2863 7075 293b 2030 2e33 7363s (cpu); 0.3 │ │ │ │ -0002ba60: 3232 3536 3873 2028 7468 7265 6164 293b 22568s (thread); │ │ │ │ +0002ba40: 7c0a 7c20 2d2d 2075 7365 6420 302e 3534 |.| -- used 0.54 │ │ │ │ +0002ba50: 3732 3434 7320 2863 7075 293b 2030 2e33 7244s (cpu); 0.3 │ │ │ │ +0002ba60: 3031 3733 3673 2028 7468 7265 6164 293b 01736s (thread); │ │ │ │ 0002ba70: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0002ba80: 2020 207c 0a7c 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 2020 7c0a 7c6f 3232 203d 2074 |.|o22 = t │ │ │ │ 0002bad0: 7275 6520 2020 2020 2020 2020 2020 2020 rue │ │ │ │ @@ -11187,18 +11187,18 @@ │ │ │ │ 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 2b0a 7c69 ------------+.|i │ │ │ │ 0002bb50: 3233 203a 2074 696d 6520 7265 6775 6c61 23 : time regula │ │ │ │ 0002bb60: 7249 6e43 6f64 696d 656e 7369 6f6e 2832 rInCodimension(2 │ │ │ │ 0002bb70: 2c20 522c 2053 7472 6174 6567 793d 3e53 , R, Strategy=>S │ │ │ │ 0002bb80: 7472 6174 6567 7943 7572 7265 6e74 297c trategyCurrent)| │ │ │ │ -0002bb90: 0a7c 202d 2d20 7573 6564 2030 2e37 3036 .| -- used 0.706 │ │ │ │ -0002bba0: 3036 3873 2028 6370 7529 3b20 302e 3336 068s (cpu); 0.36 │ │ │ │ -0002bbb0: 3636 3235 7320 2874 6872 6561 6429 3b20 6625s (thread); │ │ │ │ -0002bbc0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +0002bb90: 0a7c 202d 2d20 7573 6564 2030 2e36 3232 .| -- used 0.622 │ │ │ │ +0002bba0: 3836 7320 2863 7075 293b 2030 2e32 3937 86s (cpu); 0.297 │ │ │ │ +0002bbb0: 3934 7320 2874 6872 6561 6429 3b20 3073 94s (thread); 0s │ │ │ │ +0002bbc0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0002bbd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc10: 2020 2020 207c 0a7c 6f32 3320 3d20 7472 |.|o23 = tr │ │ │ │ 0002bc20: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ 0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -11208,18 +11208,18 @@ │ │ │ │ 0002bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 0002bca0: 3420 3a20 7469 6d65 2072 6567 756c 6172 4 : time regular │ │ │ │ 0002bcb0: 496e 436f 6469 6d65 6e73 696f 6e28 312c InCodimension(1, │ │ │ │ 0002bcc0: 2053 2c20 5374 7261 7465 6779 3d3e 5374 S, Strategy=>St │ │ │ │ 0002bcd0: 7261 7465 6779 4375 7272 656e 7429 7c0a rategyCurrent)|. │ │ │ │ -0002bce0: 7c20 2d2d 2075 7365 6420 302e 3439 3832 | -- used 0.4982 │ │ │ │ -0002bcf0: 3673 2028 6370 7529 3b20 302e 3335 3738 6s (cpu); 0.3578 │ │ │ │ -0002bd00: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ -0002bd10: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0002bce0: 7c20 2d2d 2075 7365 6420 302e 3338 3533 | -- used 0.3853 │ │ │ │ +0002bcf0: 3536 7320 2863 7075 293b 2030 2e32 3535 56s (cpu); 0.255 │ │ │ │ +0002bd00: 3930 3373 2028 7468 7265 6164 293b 2030 903s (thread); 0 │ │ │ │ +0002bd10: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0002bd20: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002bd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd60: 2020 2020 7c0a 7c6f 3234 203d 2074 7275 |.|o24 = tru │ │ │ │ 0002bd70: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ 0002bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -11229,17 +11229,17 @@ │ │ │ │ 0002bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3235 ----------+.|i25 │ │ │ │ 0002bdf0: 203a 2074 696d 6520 7265 6775 6c61 7249 : time regularI │ │ │ │ 0002be00: 6e43 6f64 696d 656e 7369 6f6e 2831 2c20 nCodimension(1, │ │ │ │ 0002be10: 532c 2053 7472 6174 6567 793d 3e53 7472 S, Strategy=>Str │ │ │ │ 0002be20: 6174 6567 7943 7572 7265 6e74 297c 0a7c ategyCurrent)|.| │ │ │ │ -0002be30: 202d 2d20 7573 6564 2030 2e35 3332 3737 -- used 0.53277 │ │ │ │ -0002be40: 3173 2028 6370 7529 3b20 302e 3331 3734 1s (cpu); 0.3174 │ │ │ │ -0002be50: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ +0002be30: 202d 2d20 7573 6564 2030 2e34 3437 3238 -- used 0.44728 │ │ │ │ +0002be40: 7320 2863 7075 293b 2030 2e32 3532 3138 s (cpu); 0.25218 │ │ │ │ +0002be50: 3573 2028 7468 7265 6164 293b 2030 7320 5s (thread); 0s │ │ │ │ 0002be60: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0002be70: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002beb0: 2020 207c 0a7c 6f32 3520 3d20 7472 7565 |.|o25 = true │ │ │ │ 0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -11267,43 +11267,43 @@ │ │ │ │ 0002c020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c040: 2d2d 2d2d 2d2b 0a7c 6932 3820 3a20 7469 -----+.|i28 : ti │ │ │ │ 0002c050: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 0002c060: 6d65 6e73 696f 6e28 322c 2052 2c20 5374 mension(2, R, St │ │ │ │ 0002c070: 7261 7465 6779 3d3e 5374 7261 7465 6779 rategy=>Strategy │ │ │ │ 0002c080: 4375 7272 656e 7429 7c0a 7c20 2d2d 2075 Current)|.| -- u │ │ │ │ -0002c090: 7365 6420 332e 3538 3838 3173 2028 6370 sed 3.58881s (cp │ │ │ │ -0002c0a0: 7529 3b20 322e 3133 3939 3173 2028 7468 u); 2.13991s (th │ │ │ │ +0002c090: 7365 6420 332e 3036 3138 3773 2028 6370 sed 3.06187s (cp │ │ │ │ +0002c0a0: 7529 3b20 312e 3732 3537 3173 2028 7468 u); 1.72571s (th │ │ │ │ 0002c0b0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 0002c0c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 0002c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0002c110: 7c69 3239 203a 2074 696d 6520 7265 6775 |i29 : time regu │ │ │ │ 0002c120: 6c61 7249 6e43 6f64 696d 656e 7369 6f6e larInCodimension │ │ │ │ 0002c130: 2832 2c20 522c 2053 7472 6174 6567 793d (2, R, Strategy= │ │ │ │ 0002c140: 3e53 7472 6174 6567 7943 7572 7265 6e74 >StrategyCurrent │ │ │ │ -0002c150: 297c 0a7c 202d 2d20 7573 6564 2033 2e32 )|.| -- used 3.2 │ │ │ │ -0002c160: 3135 3732 7320 2863 7075 293b 2031 2e38 1572s (cpu); 1.8 │ │ │ │ -0002c170: 3631 3336 7320 2874 6872 6561 6429 3b20 6136s (thread); │ │ │ │ +0002c150: 297c 0a7c 202d 2d20 7573 6564 2032 2e39 )|.| -- used 2.9 │ │ │ │ +0002c160: 3630 3333 7320 2863 7075 293b 2031 2e36 6033s (cpu); 1.6 │ │ │ │ +0002c170: 3734 3339 7320 2874 6872 6561 6429 3b20 7439s (thread); │ │ │ │ 0002c180: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 0002c190: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0002c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c1d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 3020 3a20 -------+.|i30 : │ │ │ │ 0002c1e0: 7469 6d65 2072 6567 756c 6172 496e 436f time regularInCo │ │ │ │ 0002c1f0: 6469 6d65 6e73 696f 6e28 312c 2053 2c20 dimension(1, S, │ │ │ │ 0002c200: 5374 7261 7465 6779 3d3e 5374 7261 7465 Strategy=>Strate │ │ │ │ 0002c210: 6779 4375 7272 656e 7429 7c0a 7c20 2d2d gyCurrent)|.| -- │ │ │ │ -0002c220: 2075 7365 6420 302e 3332 3833 3138 7320 used 0.328318s │ │ │ │ -0002c230: 2863 7075 293b 2030 2e32 3235 3738 3373 (cpu); 0.225783s │ │ │ │ -0002c240: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -0002c250: 6329 2020 2020 2020 2020 2020 207c 0a7c c) |.| │ │ │ │ +0002c220: 2075 7365 6420 302e 3331 3436 3439 7320 used 0.314649s │ │ │ │ +0002c230: 2863 7075 293b 2030 2e31 3934 3833 7320 (cpu); 0.19483s │ │ │ │ +0002c240: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +0002c250: 2920 2020 2020 2020 2020 2020 207c 0a7c ) |.| │ │ │ │ 0002c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c2a0: 7c0a 7c6f 3330 203d 2074 7275 6520 2020 |.|o30 = true │ │ │ │ 0002c2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -11313,18 +11313,18 @@ │ │ │ │ 0002c300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c320: 2d2d 2d2d 2d2d 2b0a 7c69 3331 203a 2074 ------+.|i31 : t │ │ │ │ 0002c330: 696d 6520 7265 6775 6c61 7249 6e43 6f64 ime regularInCod │ │ │ │ 0002c340: 696d 656e 7369 6f6e 2831 2c20 532c 2053 imension(1, S, S │ │ │ │ 0002c350: 7472 6174 6567 793d 3e53 7472 6174 6567 trategy=>Strateg │ │ │ │ 0002c360: 7943 7572 7265 6e74 297c 0a7c 202d 2d20 yCurrent)|.| -- │ │ │ │ -0002c370: 7573 6564 2030 2e35 3636 3539 3473 2028 used 0.566594s ( │ │ │ │ -0002c380: 6370 7529 3b20 302e 3337 3131 3831 7320 cpu); 0.371181s │ │ │ │ -0002c390: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0002c3a0: 2920 2020 2020 2020 2020 2020 7c0a 7c20 ) |.| │ │ │ │ +0002c370: 7573 6564 2030 2e34 3738 3437 3673 2028 used 0.478476s ( │ │ │ │ +0002c380: 6370 7529 3b20 302e 3331 3236 3473 2028 cpu); 0.31264s ( │ │ │ │ +0002c390: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +0002c3a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0002c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c3e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0002c3f0: 0a7c 6f33 3120 3d20 7472 7565 2020 2020 .|o31 = true │ │ │ │ 0002c400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -11334,17 +11334,17 @@ │ │ │ │ 0002c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c470: 2d2d 2d2d 2d2b 0a7c 6933 3220 3a20 7469 -----+.|i32 : ti │ │ │ │ 0002c480: 6d65 2072 6567 756c 6172 496e 436f 6469 me regularInCodi │ │ │ │ 0002c490: 6d65 6e73 696f 6e28 312c 2053 2c20 5374 mension(1, S, St │ │ │ │ 0002c4a0: 7261 7465 6779 3d3e 5374 7261 7465 6779 rategy=>Strategy │ │ │ │ 0002c4b0: 5261 6e64 6f6d 2920 7c0a 7c20 2d2d 2075 Random) |.| -- u │ │ │ │ -0002c4c0: 7365 6420 322e 3234 3237 7320 2863 7075 sed 2.2427s (cpu │ │ │ │ -0002c4d0: 293b 2031 2e36 3235 3937 7320 2874 6872 ); 1.62597s (thr │ │ │ │ -0002c4e0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +0002c4c0: 7365 6420 312e 3737 3731 3673 2028 6370 sed 1.77716s (cp │ │ │ │ +0002c4d0: 7529 3b20 312e 3233 3835 3673 2028 7468 u); 1.23856s (th │ │ │ │ +0002c4e0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 0002c4f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002c500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c530: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0002c540: 7c6f 3332 203d 2074 7275 6520 2020 2020 |o32 = true │ │ │ │ 0002c550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -11355,16 +11355,16 @@ │ │ │ │ 0002c5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c5c0: 2d2d 2d2d 2b0a 7c69 3333 203a 2074 696d ----+.|i33 : tim │ │ │ │ 0002c5d0: 6520 7265 6775 6c61 7249 6e43 6f64 696d e regularInCodim │ │ │ │ 0002c5e0: 656e 7369 6f6e 2831 2c20 532c 2053 7472 ension(1, S, Str │ │ │ │ 0002c5f0: 6174 6567 793d 3e53 7472 6174 6567 7952 ategy=>StrategyR │ │ │ │ 0002c600: 616e 646f 6d29 207c 0a7c 202d 2d20 7573 andom) |.| -- us │ │ │ │ -0002c610: 6564 2031 2e37 3838 3035 7320 2863 7075 ed 1.78805s (cpu │ │ │ │ -0002c620: 293b 2031 2e32 3230 3638 7320 2874 6872 ); 1.22068s (thr │ │ │ │ +0002c610: 6564 2031 2e35 3135 3436 7320 2863 7075 ed 1.51546s (cpu │ │ │ │ +0002c620: 293b 2031 2e30 3437 3437 7320 2874 6872 ); 1.04747s (thr │ │ │ │ 0002c630: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 0002c640: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c680: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0002c690: 6f33 3320 3d20 7472 7565 2020 2020 2020 o33 = true │ │ │ │ @@ -11626,34 +11626,34 @@ │ │ │ │ 0002d690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d6c0: 2d2d 2b0a 7c69 3420 3a20 7469 6d65 2072 --+.|i4 : time r │ │ │ │ 0002d6d0: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ 0002d6e0: 696f 6e28 312c 2053 2f4a 2920 2020 2020 ion(1, S/J) │ │ │ │ 0002d6f0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -0002d700: 7365 6420 332e 3934 3932 3473 2028 6370 sed 3.94924s (cp │ │ │ │ -0002d710: 7529 3b20 322e 3532 3432 7320 2874 6872 u); 2.5242s (thr │ │ │ │ -0002d720: 6561 6429 3b20 3073 2028 6763 2920 7c0a ead); 0s (gc) |. │ │ │ │ +0002d700: 7365 6420 332e 3130 3530 3673 2028 6370 sed 3.10506s (cp │ │ │ │ +0002d710: 7529 3b20 312e 3832 3233 3573 2028 7468 u); 1.82235s (th │ │ │ │ +0002d720: 7265 6164 293b 2030 7320 2867 6329 7c0a read); 0s (gc)|. │ │ │ │ 0002d730: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002d740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d760: 2020 2020 7c0a 7c6f 3420 3d20 7472 7565 |.|o4 = true │ │ │ │ 0002d770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d790: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0002d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d7d0: 2b0a 7c69 3520 3a20 7469 6d65 2072 6567 +.|i5 : time reg │ │ │ │ 0002d7e0: 756c 6172 496e 436f 6469 6d65 6e73 696f ularInCodimensio │ │ │ │ 0002d7f0: 6e28 322c 2053 2f4a 2920 2020 2020 2020 n(2, S/J) │ │ │ │ 0002d800: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -0002d810: 6420 3134 2e34 3633 3273 2028 6370 7529 d 14.4632s (cpu) │ │ │ │ -0002d820: 3b20 392e 3036 3131 3273 2028 7468 7265 ; 9.06112s (thre │ │ │ │ -0002d830: 6164 293b 2030 7320 2867 6329 7c0a 7c20 ad); 0s (gc)|.| │ │ │ │ +0002d810: 6420 3132 2e38 3635 7320 2863 7075 293b d 12.865s (cpu); │ │ │ │ +0002d820: 2037 2e38 3430 3736 7320 2874 6872 6561 7.84076s (threa │ │ │ │ +0002d830: 6429 3b20 3073 2028 6763 2920 7c0a 7c20 d); 0s (gc) |.| │ │ │ │ 0002d840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d870: 2020 7c0a 7c6f 3520 3d20 7472 7565 2020 |.|o5 = true │ │ │ │ 0002d880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d8a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ @@ -12262,17 +12262,17 @@ │ │ │ │ 0002fe50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fe60: 2020 2020 7c0a 7c20 2d2d 2069 6e74 6572 |.| -- inter │ │ │ │ 0002fe70: 6e61 6c43 686f 6f73 654d 696e 6f72 3a20 nalChooseMinor: │ │ │ │ 0002fe80: 4368 6f6f 7369 6e67 204c 6578 536d 616c Choosing LexSmal │ │ │ │ 0002fe90: 6c65 7374 2020 2020 2020 2020 2020 2020 lest │ │ │ │ 0002fea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002feb0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ -0002fec0: 332e 3636 3633 3473 2028 6370 7529 3b20 3.66634s (cpu); │ │ │ │ -0002fed0: 322e 3333 3036 7320 2874 6872 6561 6429 2.3306s (thread) │ │ │ │ -0002fee0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0002fec0: 332e 3538 3235 3373 2028 6370 7529 3b20 3.58253s (cpu); │ │ │ │ +0002fed0: 322e 3036 3530 3573 2028 7468 7265 6164 2.06505s (thread │ │ │ │ +0002fee0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 0002fef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ff00: 2020 2020 7c0a 7c72 6567 756c 6172 496e |.|regularIn │ │ │ │ 0002ff10: 436f 6469 6d65 6e73 696f 6e3a 2072 696e Codimension: rin │ │ │ │ 0002ff20: 6720 6469 6d65 6e73 696f 6e20 3d34 2c20 g dimension =4, │ │ │ │ 0002ff30: 7468 6572 6520 6172 6520 3134 3635 3132 there are 146512 │ │ │ │ 0002ff40: 3820 706f 7373 6962 6c65 2035 2062 7920 8 possible 5 by │ │ │ │ 0002ff50: 3520 6d69 7c0a 7c72 6567 756c 6172 496e 5 mi|.|regularIn │ │ │ │ @@ -12686,17 +12686,17 @@ │ │ │ │ 000318d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000318e0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2069 |.| -- i │ │ │ │ 000318f0: 6e74 6572 6e61 6c43 686f 6f73 654d 696e nternalChooseMin │ │ │ │ 00031900: 6f72 3a20 4368 6f6f 7369 6e67 2047 5265 or: Choosing GRe │ │ │ │ 00031910: 764c 6578 536d 616c 6c65 7374 5465 726d vLexSmallestTerm │ │ │ │ 00031920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031930: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -00031940: 7365 6420 302e 3331 3432 3173 2028 6370 sed 0.31421s (cp │ │ │ │ -00031950: 7529 3b20 302e 3139 3438 3934 7320 2874 u); 0.194894s (t │ │ │ │ -00031960: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00031940: 7365 6420 302e 3236 3638 3134 7320 2863 sed 0.266814s (c │ │ │ │ +00031950: 7075 293b 2030 2e31 3433 3530 3173 2028 pu); 0.143501s ( │ │ │ │ +00031960: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00031970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031980: 2020 2020 2020 2020 7c0a 7c72 6567 756c |.|regul │ │ │ │ 00031990: 6172 496e 436f 6469 6d65 6e73 696f 6e3a arInCodimension: │ │ │ │ 000319a0: 2072 696e 6720 6469 6d65 6e73 696f 6e20 ring dimension │ │ │ │ 000319b0: 3d34 2c20 7468 6572 6520 6172 6520 3134 =4, there are 14 │ │ │ │ 000319c0: 3635 3132 3820 706f 7373 6962 6c65 2035 65128 possible 5 │ │ │ │ 000319d0: 2062 7920 3520 6d69 7c0a 7c72 6567 756c by 5 mi|.|regul │ │ │ │ @@ -12874,16 +12874,16 @@ │ │ │ │ 00032490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000324a0: 2020 2020 207c 0a7c 202d 2d20 696e 7465 |.| -- inte │ │ │ │ 000324b0: 726e 616c 4368 6f6f 7365 4d69 6e6f 723a rnalChooseMinor: │ │ │ │ 000324c0: 2043 686f 6f73 696e 6720 5261 6e64 6f6d Choosing Random │ │ │ │ 000324d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000324e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000324f0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00032500: 2030 2e32 3635 3936 3973 2028 6370 7529 0.265969s (cpu) │ │ │ │ -00032510: 3b20 302e 3137 3038 3233 7320 2874 6872 ; 0.170823s (thr │ │ │ │ +00032500: 2030 2e32 3637 3639 3973 2028 6370 7529 0.267699s (cpu) │ │ │ │ +00032510: 3b20 302e 3135 3037 3132 7320 2874 6872 ; 0.150712s (thr │ │ │ │ 00032520: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00032530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032540: 2020 2020 207c 0a7c 7265 6775 6c61 7249 |.|regularI │ │ │ │ 00032550: 6e43 6f64 696d 656e 7369 6f6e 3a20 7269 nCodimension: ri │ │ │ │ 00032560: 6e67 2064 696d 656e 7369 6f6e 203d 342c ng dimension =4, │ │ │ │ 00032570: 2074 6865 7265 2061 7265 2031 3436 3531 there are 14651 │ │ │ │ 00032580: 3238 2070 6f73 7369 626c 6520 3520 6279 28 possible 5 by │ │ │ │ @@ -13150,17 +13150,17 @@ │ │ │ │ 000335d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000335e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 000335f0: 0a7c 202d 2d20 696e 7465 726e 616c 4368 .| -- internalCh │ │ │ │ 00033600: 6f6f 7365 4d69 6e6f 723a 2043 686f 6f73 ooseMinor: Choos │ │ │ │ 00033610: 696e 6720 5261 6e64 6f6d 2020 2020 2020 ing Random │ │ │ │ 00033620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033630: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00033640: 0a7c 202d 2d20 7573 6564 2030 2e34 3336 .| -- used 0.436 │ │ │ │ -00033650: 3238 3873 2028 6370 7529 3b20 302e 3238 288s (cpu); 0.28 │ │ │ │ -00033660: 3736 3339 7320 2874 6872 6561 6429 3b20 7639s (thread); │ │ │ │ +00033640: 0a7c 202d 2d20 7573 6564 2030 2e34 3431 .| -- used 0.441 │ │ │ │ +00033650: 3838 3873 2028 6370 7529 3b20 302e 3234 888s (cpu); 0.24 │ │ │ │ +00033660: 3532 3033 7320 2874 6872 6561 6429 3b20 5203s (thread); │ │ │ │ 00033670: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00033680: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00033690: 0a7c 7265 6775 6c61 7249 6e43 6f64 696d .|regularInCodim │ │ │ │ 000336a0: 656e 7369 6f6e 3a20 7269 6e67 2064 696d ension: ring dim │ │ │ │ 000336b0: 656e 7369 6f6e 203d 342c 2074 6865 7265 ension =4, there │ │ │ │ 000336c0: 2061 7265 2031 3436 3531 3238 2070 6f73 are 1465128 pos │ │ │ │ 000336d0: 7369 626c 6520 3520 6279 2035 206d 697c sible 5 by 5 mi| │ │ │ │ @@ -13564,16 +13564,16 @@ │ │ │ │ 00034fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00034fd0: 2d2d 2069 6e74 6572 6e61 6c43 686f 6f73 -- internalChoos │ │ │ │ 00034fe0: 654d 696e 6f72 3a20 4368 6f6f 7369 6e67 eMinor: Choosing │ │ │ │ 00034ff0: 204c 6578 536d 616c 6c65 7374 2020 2020 LexSmallest │ │ │ │ 00035000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00035020: 2d2d 2075 7365 6420 312e 3232 3233 3173 -- used 1.22231s │ │ │ │ -00035030: 2028 6370 7529 3b20 302e 3830 3933 3738 (cpu); 0.809378 │ │ │ │ +00035020: 2d2d 2075 7365 6420 312e 3337 3335 3273 -- used 1.37352s │ │ │ │ +00035030: 2028 6370 7529 3b20 302e 3833 3333 3632 (cpu); 0.833362 │ │ │ │ 00035040: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00035050: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 00035060: 2020 2020 2020 2020 2020 2020 7c0a 7c72 |.|r │ │ │ │ 00035070: 6567 756c 6172 496e 436f 6469 6d65 6e73 egularInCodimens │ │ │ │ 00035080: 696f 6e3a 2072 696e 6720 6469 6d65 6e73 ion: ring dimens │ │ │ │ 00035090: 696f 6e20 3d34 2c20 7468 6572 6520 6172 ion =4, there ar │ │ │ │ 000350a0: 6520 3134 3635 3132 3820 706f 7373 6962 e 1465128 possib │ │ │ │ @@ -14108,18 +14108,18 @@ │ │ │ │ 000371b0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 000371c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000371d0: 202d 2d20 696e 7465 726e 616c 4368 6f6f -- internalChoo │ │ │ │ 000371e0: 7365 4d69 6e6f 723a 2043 686f 6f73 696e seMinor: Choosin │ │ │ │ 000371f0: 6720 5261 6e64 6f6d 4e6f 6e5a 6572 6f20 g RandomNonZero │ │ │ │ 00037200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037210: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00037220: 202d 2d20 7573 6564 2031 2e30 3732 3833 -- used 1.07283 │ │ │ │ -00037230: 7320 2863 7075 293b 2030 2e36 3838 3631 s (cpu); 0.68861 │ │ │ │ -00037240: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ -00037250: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00037220: 202d 2d20 7573 6564 2030 2e39 3730 3930 -- used 0.97090 │ │ │ │ +00037230: 3273 2028 6370 7529 3b20 302e 3536 3239 2s (cpu); 0.5629 │ │ │ │ +00037240: 3536 7320 2874 6872 6561 6429 3b20 3073 56s (thread); 0s │ │ │ │ +00037250: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00037260: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00037270: 7265 6775 6c61 7249 6e43 6f64 696d 656e regularInCodimen │ │ │ │ 00037280: 7369 6f6e 3a20 7269 6e67 2064 696d 656e sion: ring dimen │ │ │ │ 00037290: 7369 6f6e 203d 342c 2074 6865 7265 2061 sion =4, there a │ │ │ │ 000372a0: 7265 2031 3436 3531 3238 2070 6f73 7369 re 1465128 possi │ │ │ │ 000372b0: 626c 6520 3520 6279 2035 206d 697c 0a7c ble 5 by 5 mi|.| │ │ │ │ 000372c0: 7265 6775 6c61 7249 6e43 6f64 696d 656e regularInCodimen │ │ │ │ @@ -14960,16 +14960,16 @@ │ │ │ │ 0003a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003a710: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 656c ------+.|i2 : el │ │ │ │ 0003a720: 6170 7365 6454 696d 6520 7265 6775 6c61 apsedTime regula │ │ │ │ 0003a730: 7249 6e43 6f64 696d 656e 7369 6f6e 2831 rInCodimension(1 │ │ │ │ 0003a740: 2c20 542c 2053 7472 6174 6567 793d 3e53 , T, Strategy=>S │ │ │ │ 0003a750: 7472 6174 6567 7944 6566 6175 6c74 2920 trategyDefault) │ │ │ │ -0003a760: 2020 2020 2020 7c0a 7c20 2d2d 2034 2e31 |.| -- 4.1 │ │ │ │ -0003a770: 3635 3173 2065 6c61 7073 6564 2020 2020 651s elapsed │ │ │ │ +0003a760: 2020 2020 2020 7c0a 7c20 2d2d 2032 2e38 |.| -- 2.8 │ │ │ │ +0003a770: 3736 3434 7320 656c 6170 7365 6420 2020 7644s elapsed │ │ │ │ 0003a780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a7b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0003a7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003a7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -15315,15 +15315,15 @@ │ │ │ │ 0003bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003bd40: 2d2d 2d2b 0a7c 6934 203a 2065 6c61 7073 ---+.|i4 : elaps │ │ │ │ 0003bd50: 6564 5469 6d65 2072 6567 756c 6172 496e edTime regularIn │ │ │ │ 0003bd60: 436f 6469 6d65 6e73 696f 6e28 312c 2054 Codimension(1, T │ │ │ │ 0003bd70: 2c20 5374 7261 7465 6779 3d3e 4c65 7853 , Strategy=>LexS │ │ │ │ 0003bd80: 6d61 6c6c 6573 7454 6572 6d29 7c0a 7c20 mallestTerm)|.| │ │ │ │ -0003bd90: 2d2d 2031 2e36 3734 3537 7320 656c 6170 -- 1.67457s elap │ │ │ │ +0003bd90: 2d2d 2031 2e31 3735 3736 7320 656c 6170 -- 1.17576s elap │ │ │ │ 0003bda0: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0003bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bdd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0003bde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/FiniteFittingIdeals.info.gz │ │ │ ├── FiniteFittingIdeals.info │ │ │ │ @@ -1044,18 +1044,18 @@ │ │ │ │ 00004130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004150: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520 --+.|i15 : time │ │ │ │ 00004160: 493d 636f 3146 6974 7469 6e67 284b 3329 I=co1Fitting(K3) │ │ │ │ 00004170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004190: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000041a0: 202d 2d20 7573 6564 2030 2e30 3032 3533 -- used 0.00253 │ │ │ │ -000041b0: 3838 3973 2028 6370 7529 3b20 302e 3030 889s (cpu); 0.00 │ │ │ │ -000041c0: 3235 3336 3136 7320 2874 6872 6561 6429 253616s (thread) │ │ │ │ -000041d0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +000041a0: 202d 2d20 7573 6564 2030 2e30 3032 3330 -- used 0.00230 │ │ │ │ +000041b0: 3633 7320 2863 7075 293b 2030 2e30 3032 63s (cpu); 0.002 │ │ │ │ +000041c0: 3330 3434 7320 2874 6872 6561 6429 3b20 3044s (thread); │ │ │ │ +000041d0: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 000041e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000041f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004230: 2020 207c 0a7c 6f31 3520 3d20 6964 6561 |.|o15 = idea │ │ │ │ 00004240: 6c20 2861 2061 2020 202b 2061 2020 2d20 l (a a + a - │ │ │ │ @@ -1082,16 +1082,16 @@ │ │ │ │ 00004390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000043a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3136 ----------+.|i16 │ │ │ │ 000043b0: 203a 2074 696d 6520 4a3d 6669 7474 696e : time J=fittin │ │ │ │ 000043c0: 6749 6465 616c 2832 2d31 2c63 6f6b 6572 gIdeal(2-1,coker │ │ │ │ 000043d0: 204b 3329 3b20 2020 2020 2020 2020 2020 K3); │ │ │ │ 000043e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000043f0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00004400: 2030 2e30 3036 3632 3035 3673 2028 6370 0.00662056s (cp │ │ │ │ -00004410: 7529 3b20 302e 3030 3636 3237 3934 7320 u); 0.00662794s │ │ │ │ +00004400: 2030 2e30 3035 3431 3236 3573 2028 6370 0.00541265s (cp │ │ │ │ +00004410: 7529 3b20 302e 3030 3534 3133 3235 7320 u); 0.00541325s │ │ │ │ 00004420: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 00004430: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00004440: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00004450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004480: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ ├── ./usr/share/info/ForeignFunctions.info.gz │ │ │ ├── ForeignFunctions.info │ │ │ │ @@ -3505,15 +3505,15 @@ │ │ │ │ 0000db00: 6d61 6c6c 6f63 2038 2020 2020 2020 2020 malloc 8 │ │ │ │ 0000db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000db20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000db30: 0a7c 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: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0000db70: 3720 3d20 3078 3766 3035 3538 3037 3066 7 = 0x7f0558070f │ │ │ │ +0000db70: 3720 3d20 3078 3766 6230 3938 3037 3066 7 = 0x7fb098070f │ │ │ │ 0000db80: 3630 2020 2020 2020 2020 2020 2020 2020 60 │ │ │ │ 0000db90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dba0: 2020 2020 2020 207c 0a7c 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 2020 2020 2020 2020 │ │ │ │ 0000dbe0: 2020 207c 0a7c 6f31 3720 3a20 466f 7265 |.|o17 : Fore │ │ │ │ @@ -3639,15 +3639,15 @@ │ │ │ │ 0000e360: 2d2d 2d2b 0a7c 6932 203a 2070 6565 6b20 ---+.|i2 : peek │ │ │ │ 0000e370: 6d70 6672 2020 2020 2020 2020 2020 2020 mpfr │ │ │ │ 0000e380: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0000e390: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0000e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3b0: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ 0000e3c0: 2053 6861 7265 644c 6962 7261 7279 7b30 SharedLibrary{0 │ │ │ │ -0000e3d0: 7837 6634 3066 3962 3433 6163 302c 206d x7f40f9b43ac0, m │ │ │ │ +0000e3d0: 7837 6638 3538 3838 3539 6163 302c 206d x7f8588859ac0, m │ │ │ │ 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2020 2020 2020 207c | │ │ │ │ -00015180: 0a7c 6f31 203d 2030 7837 6634 3066 3236 .|o1 = 0x7f40f26 │ │ │ │ -00015190: 6133 6461 3020 2020 2020 2020 2020 2020 a3da0 │ │ │ │ +00015180: 0a7c 6f31 203d 2030 7837 6638 3537 3366 .|o1 = 0x7f8573f │ │ │ │ +00015190: 6530 3938 3020 2020 2020 2020 2020 2020 e0980 │ │ │ │ 000151a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000151b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151c0: 2020 2020 207c 0a7c 6f31 203a 2050 6f69 |.|o1 : Poi │ │ │ │ 000151d0: 6e74 6572 2020 2020 2020 2020 2020 2020 nter │ │ │ │ 000151e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 000151f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 00015210: 203a 2076 6f69 6473 7461 7220 7074 7220 : voidstar ptr │ │ │ │ 00015220: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00015230: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00015240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015250: 207c 0a7c 6f32 203d 2030 7837 6634 3066 |.|o2 = 0x7f40f │ │ │ │ -00015260: 3236 6133 6461 3020 2020 2020 2020 2020 26a3da0 │ │ │ │ +00015250: 207c 0a7c 6f32 203d 2030 7837 6638 3537 |.|o2 = 0x7f857 │ │ │ │ +00015260: 3366 6530 3938 3020 2020 2020 2020 2020 3fe0980 │ │ │ │ 00015270: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00015280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015290: 2020 2020 2020 207c 0a7c 6f32 203a 2046 |.|o2 : F │ │ │ │ 000152a0: 6f72 6569 676e 4f62 6a65 6374 206f 6620 oreignObject of │ │ │ │ 000152b0: 7479 7065 2076 6f69 642a 7c0a 2b2d 2d2d type void*|.+--- │ │ │ │ 000152c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000152d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ @@ -5497,16 +5497,16 @@ │ │ │ │ 00015780: 696e 7433 327c 0a2b 2d2d 2d2d 2d2d 2d2d int32|.+-------- │ │ │ │ 00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000157a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ 000157b0: 7074 7220 3d20 6164 6472 6573 7320 7820 ptr = address x │ │ │ │ 000157c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000157d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000157e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000157f0: 7c6f 3220 3d20 3078 3766 3430 6632 3164 |o2 = 0x7f40f21d │ │ │ │ -00015800: 3932 3230 2020 2020 2020 2020 2020 2020 9220 │ │ │ │ +000157f0: 7c6f 3220 3d20 3078 3766 3835 3733 6230 |o2 = 0x7f8573b0 │ │ │ │ +00015800: 6637 3530 2020 2020 2020 2020 2020 2020 f750 │ │ │ │ 00015810: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015830: 2020 2020 7c0a 7c6f 3220 3a20 506f 696e |.|o2 : Poin │ │ │ │ 00015840: 7465 7220 2020 2020 2020 2020 2020 2020 ter │ │ │ │ 00015850: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00015860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015870: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ @@ -5647,16 +5647,16 @@ │ │ │ │ 000160e0: 7273 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d rs...+---------- │ │ │ │ 000160f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016100: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2070 -------+.|i1 : p │ │ │ │ 00016110: 7472 203d 2076 6f69 6473 7461 7220 6164 tr = voidstar ad │ │ │ │ 00016120: 6472 6573 7320 696e 7420 357c 0a7c 2020 dress int 5|.| │ │ │ │ 00016130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016140: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016150: 0a7c 6f31 203d 2030 7837 6634 3066 3231 .|o1 = 0x7f40f21 │ │ │ │ -00016160: 6439 3166 3020 2020 2020 2020 2020 2020 d91f0 │ │ │ │ +00016150: 0a7c 6f31 203d 2030 7837 6638 3537 3362 .|o1 = 0x7f8573b │ │ │ │ +00016160: 3066 6164 3020 2020 2020 2020 2020 2020 0fad0 │ │ │ │ 00016170: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00016180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016190: 2020 2020 2020 207c 0a7c 6f31 203a 2046 |.|o1 : F │ │ │ │ 000161a0: 6f72 6569 676e 4f62 6a65 6374 206f 6620 oreignObject of │ │ │ │ 000161b0: 7479 7065 2076 6f69 642a 207c 0a2b 2d2d type void* |.+-- │ │ │ │ 000161c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000161d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ @@ -5745,15 +5745,15 @@ │ │ │ │ 00016700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016720: 2b0a 7c69 3120 3a20 7074 7220 3d20 6765 +.|i1 : ptr = ge │ │ │ │ 00016730: 744d 656d 6f72 7920 3820 2020 2020 2020 tMemory 8 │ │ │ │ 00016740: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00016750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016760: 2020 2020 2020 7c0a 7c6f 3120 3d20 3078 |.|o1 = 0x │ │ │ │ -00016770: 3766 3430 6635 3063 6133 3630 2020 2020 7f40f50ca360 │ │ │ │ +00016770: 3766 3835 3734 3062 3638 3830 2020 2020 7f85740b6880 │ │ │ │ 00016780: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00016790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000167a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 000167b0: 3120 3a20 466f 7265 6967 6e4f 626a 6563 1 : ForeignObjec │ │ │ │ 000167c0: 7420 6f66 2074 7970 6520 766f 6964 2a7c t of type void*| │ │ │ │ 000167d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000167e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5768,16 +5768,16 @@ │ │ │ │ 00016870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016880: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ 00016890: 3a20 7074 7220 3d20 6765 744d 656d 6f72 : ptr = getMemor │ │ │ │ 000168a0: 7928 382c 2041 746f 6d69 6320 3d3e 2074 y(8, Atomic => t │ │ │ │ 000168b0: 7275 6529 7c0a 7c20 2020 2020 2020 2020 rue)|.| │ │ │ │ 000168c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000168d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000168e0: 7c6f 3220 3d20 3078 3766 3430 6632 3164 |o2 = 0x7f40f21d │ │ │ │ -000168f0: 3961 6530 2020 2020 2020 2020 2020 2020 9ae0 │ │ │ │ +000168e0: 7c6f 3220 3d20 3078 3766 3835 3733 6230 |o2 = 0x7f8573b0 │ │ │ │ +000168f0: 6661 6330 2020 2020 2020 2020 2020 2020 fac0 │ │ │ │ 00016900: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00016910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016930: 2020 7c0a 7c6f 3220 3a20 466f 7265 6967 |.|o2 : Foreig │ │ │ │ 00016940: 6e4f 626a 6563 7420 6f66 2074 7970 6520 nObject of type │ │ │ │ 00016950: 766f 6964 2a20 2020 2020 2020 7c0a 2b2d void* |.+- │ │ │ │ 00016960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5796,15 +5796,15 @@ │ │ │ │ 00016a30: 792e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d y...+----------- │ │ │ │ 00016a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016a50: 2d2d 2d2d 2d2b 0a7c 6933 203a 2070 7472 -----+.|i3 : ptr │ │ │ │ 00016a60: 203d 2067 6574 4d65 6d6f 7279 2069 6e74 = getMemory int │ │ │ │ 00016a70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00016a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016a90: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -00016aa0: 203d 2030 7837 6634 3066 3231 6439 6130 = 0x7f40f21d9a0 │ │ │ │ +00016aa0: 203d 2030 7837 6638 3537 3362 3066 3964 = 0x7f8573b0f9d │ │ │ │ 00016ab0: 3020 2020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ 00016ac0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00016ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ae0: 207c 0a7c 6f33 203a 2046 6f72 6569 676e |.|o3 : Foreign │ │ │ │ 00016af0: 4f62 6a65 6374 206f 6620 7479 7065 2076 Object of type v │ │ │ │ 00016b00: 6f69 642a 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d oid*|.+--------- │ │ │ │ 00016b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5966,18 +5966,18 @@ │ │ │ │ 000174d0: 7420 312c 2061 6464 7265 7373 2069 6e74 t 1, address int │ │ │ │ 000174e0: 2032 7d20 2020 2020 2020 2020 2020 7c0a 2} |. │ │ │ │ 000174f0: 7c20 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 2020 │ │ │ │ 00017530: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -00017540: 203d 207b 3078 3766 3430 6632 3231 3765 = {0x7f40f2217e │ │ │ │ -00017550: 3930 2c20 3078 3766 3430 6632 3231 3765 90, 0x7f40f2217e │ │ │ │ -00017560: 3830 2c20 3078 3766 3430 6632 3231 3765 80, 0x7f40f2217e │ │ │ │ -00017570: 3730 7d20 2020 2020 2020 2020 2020 2020 70} │ │ │ │ +00017540: 203d 207b 3078 3766 3835 3733 6232 6162 = {0x7f8573b2ab │ │ │ │ +00017550: 3130 2c20 3078 3766 3835 3733 6232 6162 10, 0x7f8573b2ab │ │ │ │ +00017560: 3030 2c20 3078 3766 3835 3733 6232 6161 00, 0x7f8573b2aa │ │ │ │ +00017570: 6630 7d20 2020 2020 2020 2020 2020 2020 f0} │ │ │ │ 00017580: 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 │ │ │ │ 000175d0: 2020 2020 207c 0a7c 6f33 203a 2046 6f72 |.|o3 : For │ │ │ │ 000175e0: 6569 676e 4f62 6a65 6374 206f 6620 7479 eignObject of ty │ │ │ │ @@ -6432,17 +6432,17 @@ │ │ │ │ 000191f0: 6e74 2030 2c20 6164 6472 6573 7320 696e nt 0, address in │ │ │ │ 00019200: 7420 312c 2061 6464 7265 7373 2069 6e74 t 1, address int │ │ │ │ 00019210: 2032 7d7c 0a7c 2020 2020 2020 2020 2020 2}|.| │ │ │ │ 00019220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019250: 2020 2020 207c 0a7c 6f32 203d 207b 3078 |.|o2 = {0x │ │ │ │ -00019260: 3766 3430 6632 3231 3764 3030 2c20 3078 7f40f2217d00, 0x │ │ │ │ -00019270: 3766 3430 6632 3231 3763 6630 2c20 3078 7f40f2217cf0, 0x │ │ │ │ -00019280: 3766 3430 6632 3231 3763 6530 7d20 2020 7f40f2217ce0} │ │ │ │ +00019260: 3766 3835 3733 6232 6139 3630 2c20 3078 7f8573b2a960, 0x │ │ │ │ +00019270: 3766 3835 3733 6232 6139 3530 2c20 3078 7f8573b2a950, 0x │ │ │ │ +00019280: 3766 3835 3733 6232 6139 3430 7d20 2020 7f8573b2a940} │ │ │ │ 00019290: 2020 2020 2020 207c 0a7c 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 2020 2020 │ │ │ │ 000192d0: 2020 2020 2020 2020 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ 000192e0: 2046 6f72 6569 676e 4f62 6a65 6374 206f ForeignObject o │ │ │ │ 000192f0: 6620 7479 7065 2076 6f69 642a 2a20 2020 f type void** │ │ │ │ @@ -7910,15 +7910,15 @@ │ │ │ │ 0001ee50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 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 2020 │ │ │ │ 0001eea0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ 0001eeb0: 3d20 4861 7368 5461 626c 657b 2262 6172 = HashTable{"bar │ │ │ │ -0001eec0: 2220 3d3e 2036 2e39 3132 3631 652d 3331 " => 6.91261e-31 │ │ │ │ +0001eec0: 2220 3d3e 2036 2e39 3237 3236 652d 3331 " => 6.92726e-31 │ │ │ │ 0001eed0: 307d 2020 2020 2020 2020 2020 2020 2020 0} │ │ │ │ 0001eee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0001ef00: 2020 2020 2020 2020 2020 2266 6f6f 2220 "foo" │ │ │ │ 0001ef10: 3d3e 2032 3720 2020 2020 2020 2020 2020 => 27 │ │ │ │ 0001ef20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8160,27 +8160,27 @@ │ │ │ │ 0001fdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fe00: 2d2d 2b0a 7c69 3220 3a20 7065 656b 2078 --+.|i2 : peek x │ │ │ │ 0001fe10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0001fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe50: 2020 7c0a 7c6f 3220 3d20 696e 7433 327b |.|o2 = int32{ │ │ │ │ -0001fe60: 4164 6472 6573 7320 3d3e 2030 7837 6634 Address => 0x7f4 │ │ │ │ -0001fe70: 3066 3231 6439 3035 307d 7c0a 2b2d 2d2d 0f21d9050}|.+--- │ │ │ │ +0001fe60: 4164 6472 6573 7320 3d3e 2030 7837 6638 Address => 0x7f8 │ │ │ │ +0001fe70: 3537 3361 6464 3531 307d 7c0a 2b2d 2d2d 573add510}|.+--- │ │ │ │ 0001fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fea0: 2d2d 2b0a 0a54 6f20 6765 7420 7468 6973 --+..To get this │ │ │ │ 0001feb0: 2c20 7573 6520 2a6e 6f74 6520 6164 6472 , use *note addr │ │ │ │ 0001fec0: 6573 733a 2061 6464 7265 7373 2c2e 0a0a ess: address,... │ │ │ │ 0001fed0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0001fee0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6164 6472 ----+.|i3 : addr │ │ │ │ 0001fef0: 6573 7320 7820 2020 2020 7c0a 7c20 2020 ess x |.| │ │ │ │ 0001ff00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ff10: 7c0a 7c6f 3320 3d20 3078 3766 3430 6632 |.|o3 = 0x7f40f2 │ │ │ │ -0001ff20: 3164 3930 3530 7c0a 7c20 2020 2020 2020 1d9050|.| │ │ │ │ +0001ff10: 7c0a 7c6f 3320 3d20 3078 3766 3835 3733 |.|o3 = 0x7f8573 │ │ │ │ +0001ff20: 6164 6435 3130 7c0a 7c20 2020 2020 2020 add510|.| │ │ │ │ 0001ff30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 0001ff40: 3320 3a20 506f 696e 7465 7220 2020 2020 3 : Pointer │ │ │ │ 0001ff50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0001ff60: 2d2d 2d2d 2d2d 2d2d 2b0a 0a55 7365 202a --------+..Use * │ │ │ │ 0001ff70: 6e6f 7465 2063 6c61 7373 3a20 284d 6163 note class: (Mac │ │ │ │ 0001ff80: 6175 6c61 7932 446f 6329 636c 6173 732c aulay2Doc)class, │ │ │ │ 0001ff90: 2074 6f20 6465 7465 726d 696e 6520 7468 to determine th │ │ │ │ @@ -8882,29 +8882,29 @@ │ │ │ │ 00022b10: 7473 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ts...+---------- │ │ │ │ 00022b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022b30: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7820 ------+.|i5 : x │ │ │ │ 00022b40: 3d20 766f 6964 7374 6172 2061 6464 7265 = voidstar addre │ │ │ │ 00022b50: 7373 2069 6e74 2035 207c 0a7c 2020 2020 ss int 5 |.| │ │ │ │ 00022b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00022b80: 3520 3d20 3078 3766 3430 6632 3166 6439 5 = 0x7f40f21fd9 │ │ │ │ -00022b90: 3930 2020 2020 2020 2020 2020 2020 207c 90 | │ │ │ │ +00022b80: 3520 3d20 3078 3766 3835 3733 6230 6632 5 = 0x7f8573b0f2 │ │ │ │ +00022b90: 3030 2020 2020 2020 2020 2020 2020 207c 00 | │ │ │ │ 00022ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00022bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022bc0: 2020 7c0a 7c6f 3520 3a20 466f 7265 6967 |.|o5 : Foreig │ │ │ │ 00022bd0: 6e4f 626a 6563 7420 6f66 2074 7970 6520 nObject of type │ │ │ │ 00022be0: 766f 6964 2a7c 0a2b 2d2d 2d2d 2d2d 2d2d void*|.+-------- │ │ │ │ 00022bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022c00: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 00022c10: 7661 6c75 6520 7820 2020 2020 2020 2020 value x │ │ │ │ 00022c20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00022c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00022c50: 7c6f 3620 3d20 3078 3766 3430 6632 3166 |o6 = 0x7f40f21f │ │ │ │ -00022c60: 6439 3930 2020 2020 2020 2020 2020 2020 d990 │ │ │ │ +00022c50: 7c6f 3620 3d20 3078 3766 3835 3733 6230 |o6 = 0x7f8573b0 │ │ │ │ +00022c60: 6632 3030 2020 2020 2020 2020 2020 2020 f200 │ │ │ │ 00022c70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c90: 2020 2020 7c0a 7c6f 3620 3a20 506f 696e |.|o6 : Poin │ │ │ │ 00022ca0: 7465 7220 2020 2020 2020 2020 2020 2020 ter │ │ │ │ 00022cb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00022cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 6f72 ----------+..For │ │ │ │ @@ -9431,50 +9431,50 @@ │ │ │ │ 00024d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024d70: 2d2d 2d2b 0a7c 6935 203a 2063 6f6c 6c65 ---+.|i5 : colle │ │ │ │ 00024d80: 6374 4761 7262 6167 6528 2920 2020 2020 ctGarbage() │ │ │ │ 00024d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024db0: 2020 2020 2020 2020 207c 0a7c 6672 6565 |.|free │ │ │ │ 00024dc0: 696e 6720 6d65 6d6f 7279 2061 7420 3078 ing memory at 0x │ │ │ │ -00024dd0: 3766 3430 6463 3038 3438 6230 2020 2020 7f40dc0848b0 │ │ │ │ +00024dd0: 3766 3835 3563 3038 3431 6630 2020 2020 7f855c0841f0 │ │ │ │ 00024de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024df0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00024e00: 0a7c 6672 6565 696e 6720 6d65 6d6f 7279 .|freeing memory │ │ │ │ -00024e10: 2061 7420 3078 3766 3430 6463 3038 3438 at 0x7f40dc0848 │ │ │ │ -00024e20: 3730 2020 2020 2020 2020 2020 2020 2020 70 │ │ │ │ +00024e10: 2061 7420 3078 3766 3835 3563 3038 3438 at 0x7f855c0848 │ │ │ │ +00024e20: 3130 2020 2020 2020 2020 2020 2020 2020 10 │ │ │ │ 00024e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024e40: 2020 2020 207c 0a7c 6672 6565 696e 6720 |.|freeing │ │ │ │ -00024e50: 6d65 6d6f 7279 2061 7420 3078 3766 3430 memory at 0x7f40 │ │ │ │ -00024e60: 6463 3038 3438 3330 2020 2020 2020 2020 dc084830 │ │ │ │ +00024e50: 6d65 6d6f 7279 2061 7420 3078 3766 3835 memory at 0x7f85 │ │ │ │ +00024e60: 3563 3038 3438 3330 2020 2020 2020 2020 5c084830 │ │ │ │ 00024e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024e80: 2020 2020 2020 2020 2020 207c 0a7c 6672 |.|fr │ │ │ │ 00024e90: 6565 696e 6720 6d65 6d6f 7279 2061 7420 eeing memory at │ │ │ │ -00024ea0: 3078 3766 3430 6463 3038 3431 6430 2020 0x7f40dc0841d0 │ │ │ │ +00024ea0: 3078 3766 3835 3563 3038 3438 3530 2020 0x7f855c084850 │ │ │ │ 00024eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ed0: 207c 0a7c 6672 6565 696e 6720 6d65 6d6f |.|freeing memo │ │ │ │ -00024ee0: 7279 2061 7420 3078 3766 3430 6463 3038 ry at 0x7f40dc08 │ │ │ │ -00024ef0: 3431 6630 2020 2020 2020 2020 2020 2020 41f0 │ │ │ │ +00024ee0: 7279 2061 7420 3078 3766 3835 3563 3038 ry at 0x7f855c08 │ │ │ │ +00024ef0: 3438 6230 2020 2020 2020 2020 2020 2020 48b0 │ │ │ │ 00024f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f10: 2020 2020 2020 207c 0a7c 6672 6565 696e |.|freein │ │ │ │ 00024f20: 6720 6d65 6d6f 7279 2061 7420 3078 3766 g memory at 0x7f │ │ │ │ -00024f30: 3430 6463 3038 3438 6430 2020 2020 2020 40dc0848d0 │ │ │ │ +00024f30: 3835 3563 3038 3438 3930 2020 2020 2020 855c084890 │ │ │ │ 00024f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00024f60: 6672 6565 696e 6720 6d65 6d6f 7279 2061 freeing memory a │ │ │ │ -00024f70: 7420 3078 3766 3430 6463 3038 3438 3130 t 0x7f40dc084810 │ │ │ │ +00024f70: 7420 3078 3766 3835 3563 3038 3438 6430 t 0x7f855c0848d0 │ │ │ │ 00024f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024fa0: 2020 207c 0a7c 6672 6565 696e 6720 6d65 |.|freeing me │ │ │ │ -00024fb0: 6d6f 7279 2061 7420 3078 3766 3430 6463 mory at 0x7f40dc │ │ │ │ -00024fc0: 3038 3438 3530 2020 2020 2020 2020 2020 084850 │ │ │ │ +00024fb0: 6d6f 7279 2061 7420 3078 3766 3835 3563 mory at 0x7f855c │ │ │ │ +00024fc0: 3038 3438 3730 2020 2020 2020 2020 2020 084870 │ │ │ │ 00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024fe0: 2020 2020 2020 2020 207c 0a7c 6672 6565 |.|free │ │ │ │ 00024ff0: 696e 6720 6d65 6d6f 7279 2061 7420 3078 ing memory at 0x │ │ │ │ -00025000: 3766 3430 6463 3038 3438 3930 2020 2020 7f40dc084890 │ │ │ │ +00025000: 3766 3835 3563 3038 3431 6430 2020 2020 7f855c0841d0 │ │ │ │ 00025010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025020: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00025030: 0a2b 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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025070: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ @@ -9547,49 +9547,49 @@ │ │ │ │ 000254a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 000254b0: 0a7c 6932 203a 2070 6565 6b20 7820 2020 .|i2 : peek x │ │ │ │ 000254c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 000254e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00025500: 0a7c 6f32 203d 2069 6e74 3332 7b41 6464 .|o2 = int32{Add │ │ │ │ -00025510: 7265 7373 203d 3e20 3078 3766 3430 6632 ress => 0x7f40f2 │ │ │ │ -00025520: 3164 3939 3630 7d7c 0a2b 2d2d 2d2d 2d2d 1d9960}|.+------ │ │ │ │ +00025510: 7265 7373 203d 3e20 3078 3766 3835 3733 ress => 0x7f8573 │ │ │ │ +00025520: 6230 6663 3430 7d7c 0a2b 2d2d 2d2d 2d2d b0fc40}|.+------ │ │ │ │ 00025530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00025550: 0a0a 5468 6573 6520 706f 696e 7465 7273 ..These pointers │ │ │ │ 00025560: 2063 616e 2062 6520 6163 6365 7373 6564 can be accessed │ │ │ │ 00025570: 2075 7369 6e67 202a 6e6f 7465 2061 6464 using *note add │ │ │ │ 00025580: 7265 7373 3a20 6164 6472 6573 732c 2e0a ress: address,.. │ │ │ │ 00025590: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000255a0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7074 ------+.|i3 : pt │ │ │ │ 000255b0: 7220 3d20 6164 6472 6573 7320 787c 0a7c r = address x|.| │ │ │ │ 000255c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000255d0: 2020 2020 7c0a 7c6f 3320 3d20 3078 3766 |.|o3 = 0x7f │ │ │ │ -000255e0: 3430 6632 3164 3939 3630 207c 0a7c 2020 40f21d9960 |.| │ │ │ │ +000255e0: 3835 3733 6230 6663 3430 207c 0a7c 2020 8573b0fc40 |.| │ │ │ │ 000255f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025600: 2020 7c0a 7c6f 3320 3a20 506f 696e 7465 |.|o3 : Pointe │ │ │ │ 00025610: 7220 2020 2020 2020 207c 0a2b 2d2d 2d2d r |.+---- │ │ │ │ 00025620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025630: 2b0a 0a53 696d 706c 6520 6172 6974 686d +..Simple arithm │ │ │ │ 00025640: 6574 6963 2063 616e 2062 6520 7065 7266 etic can be perf │ │ │ │ 00025650: 6f72 6d65 6420 6f6e 2070 6f69 6e74 6572 ormed on pointer │ │ │ │ 00025660: 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s...+----------- │ │ │ │ 00025670: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ 00025680: 7074 7220 2b20 3520 2020 2020 2020 7c0a ptr + 5 |. │ │ │ │ 00025690: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000256a0: 2020 2020 7c0a 7c6f 3420 3d20 3078 3766 |.|o4 = 0x7f │ │ │ │ -000256b0: 3430 6632 3164 3939 3635 7c0a 7c20 2020 40f21d9965|.| │ │ │ │ +000256b0: 3835 3733 6230 6663 3435 7c0a 7c20 2020 8573b0fc45|.| │ │ │ │ 000256c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000256d0: 7c0a 7c6f 3420 3a20 506f 696e 7465 7220 |.|o4 : Pointer │ │ │ │ 000256e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000256f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00025700: 3520 3a20 7074 7220 2d20 3320 2020 2020 5 : ptr - 3 │ │ │ │ 00025710: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00025720: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -00025730: 3078 3766 3430 6632 3164 3939 3564 7c0a 0x7f40f21d995d|. │ │ │ │ +00025730: 3078 3766 3835 3733 6230 6663 3364 7c0a 0x7f8573b0fc3d|. │ │ │ │ 00025740: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00025750: 2020 2020 7c0a 7c6f 3520 3a20 506f 696e |.|o5 : Poin │ │ │ │ 00025760: 7465 7220 2020 2020 2020 7c0a 2b2d 2d2d ter |.+--- │ │ │ │ 00025770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025780: 2b0a 2a20 4d65 6e75 3a0a 0a2a 206e 756c +.* Menu:..* nul │ │ │ │ 00025790: 6c50 6f69 6e74 6572 3a3a 2020 2020 2020 lPointer:: │ │ │ │ 000257a0: 2020 2020 2020 2020 2020 2074 6865 206e the n │ │ │ │ @@ -9758,16 +9758,16 @@ │ │ │ │ 000261d0: 695f 7479 7065 2073 7472 7563 740a 7573 i_type struct.us │ │ │ │ 000261e0: 6564 2062 7920 6c69 6266 6669 2074 6f20 ed by libffi to │ │ │ │ 000261f0: 6964 656e 7469 6679 2074 6865 2074 7970 identify the typ │ │ │ │ 00026200: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d e...+----------- │ │ │ │ 00026210: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ 00026220: 6164 6472 6573 7320 696e 7420 2020 7c0a address int |. │ │ │ │ 00026230: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00026240: 2020 2020 7c0a 7c6f 3120 3d20 3078 3536 |.|o1 = 0x56 │ │ │ │ -00026250: 3266 6264 3036 3330 3830 7c0a 7c20 2020 2fbd063080|.| │ │ │ │ +00026240: 2020 2020 7c0a 7c6f 3120 3d20 3078 3535 |.|o1 = 0x55 │ │ │ │ +00026250: 6336 3738 6533 3230 3830 7c0a 7c20 2020 c678e32080|.| │ │ │ │ 00026260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026270: 7c0a 7c6f 3120 3a20 506f 696e 7465 7220 |.|o1 : Pointer │ │ │ │ 00026280: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00026290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 ------------+..I │ │ │ │ 000262a0: 6620 7820 6973 2061 2066 6f72 6569 676e f x is a foreign │ │ │ │ 000262b0: 206f 626a 6563 742c 2074 6865 6e20 7468 object, then th │ │ │ │ 000262c0: 6973 2072 6574 7572 6e73 2074 6865 2061 is returns the a │ │ │ │ @@ -9776,15 +9776,15 @@ │ │ │ │ 000262f0: 7320 6c69 6b65 2074 6865 2026 2022 6164 s like the & "ad │ │ │ │ 00026300: 6472 6573 732d 6f66 2220 6f70 6572 6174 dress-of" operat │ │ │ │ 00026310: 6f72 2069 6e20 432e 0a0a 2b2d 2d2d 2d2d or in C...+----- │ │ │ │ 00026320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00026330: 7c69 3220 3a20 6164 6472 6573 7320 696e |i2 : address in │ │ │ │ 00026340: 7420 3520 7c0a 7c20 2020 2020 2020 2020 t 5 |.| │ │ │ │ 00026350: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -00026360: 3d20 3078 3766 3430 6632 3164 3935 6630 = 0x7f40f21d95f0 │ │ │ │ +00026360: 3d20 3078 3766 3835 3733 6230 6664 6530 = 0x7f8573b0fde0 │ │ │ │ 00026370: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00026380: 2020 2020 2020 7c0a 7c6f 3220 3a20 506f |.|o2 : Po │ │ │ │ 00026390: 696e 7465 7220 2020 2020 2020 7c0a 2b2d inter |.+- │ │ │ │ 000263a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000263b0: 2d2d 2b0a 0a57 6179 7320 746f 2075 7365 --+..Ways to use │ │ │ │ 000263c0: 2061 6464 7265 7373 3a0a 3d3d 3d3d 3d3d address:.====== │ │ │ │ 000263d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ @@ -9867,16 +9867,16 @@ │ │ │ │ 000268a0: 696e 7433 327c 0a2b 2d2d 2d2d 2d2d 2d2d int32|.+-------- │ │ │ │ 000268b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000268c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ 000268d0: 7074 7220 3d20 6164 6472 6573 7320 7820 ptr = address x │ │ │ │ 000268e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000268f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026900: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00026910: 7c6f 3220 3d20 3078 3766 3430 6632 3164 |o2 = 0x7f40f21d │ │ │ │ -00026920: 3937 6630 2020 2020 2020 2020 2020 2020 97f0 │ │ │ │ +00026910: 7c6f 3220 3d20 3078 3766 3835 3733 6230 |o2 = 0x7f8573b0 │ │ │ │ +00026920: 6666 3030 2020 2020 2020 2020 2020 2020 ff00 │ │ │ │ 00026930: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00026940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026950: 2020 2020 7c0a 7c6f 3220 3a20 506f 696e |.|o2 : Poin │ │ │ │ 00026960: 7465 7220 2020 2020 2020 2020 2020 2020 ter │ │ │ │ 00026970: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00026980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026990: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ ├── ./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: 3432 332d 302f 3020 2020 2020 2020 2020 423-0/0 │ │ │ │ +000034c0: 7c6f 3220 3d20 2f74 6d70 2f4d 322d 3139 |o2 = /tmp/M2-19 │ │ │ │ +000034d0: 3330 332d 302f 3020 2020 2020 2020 2020 303-0/0 │ │ │ │ 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 3432 332d 302f 3020 2020 2020 2-15423-0/0 │ │ │ │ +00003560: 322d 3139 3330 332d 302f 3020 2020 2020 2-19303-0/0 │ │ │ │ 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 3432 332d = /tmp/M2-15423- │ │ │ │ +00003680: 3d20 2f74 6d70 2f4d 322d 3139 3330 332d = /tmp/M2-19303- │ │ │ │ 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 2031 2e35 3639 3237 7320 - used 1.56927s │ │ │ │ -0000a8b0: 2863 7075 293b 2031 2e31 3233 3232 7320 (cpu); 1.12322s │ │ │ │ +0000a8a0: 2d20 7573 6564 2031 2e36 3431 3435 7320 - used 1.64145s │ │ │ │ +0000a8b0: 2863 7075 293b 2031 2e31 3336 3534 7320 (cpu); 1.13654s │ │ │ │ 0000a8c0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0000a8d0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0000a8e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a920: 2020 2020 2020 207c 0a7c 6f32 3720 3d20 |.|o27 = │ │ │ │ @@ -2723,17 +2723,17 @@ │ │ │ │ 0000aa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000aa40: 0a7c 6932 3820 3a20 7469 6d65 2066 7074 .|i28 : time fpt │ │ │ │ 0000aa50: 2866 2c20 4465 7074 684f 6653 6561 7263 (f, DepthOfSearc │ │ │ │ 0000aa60: 6820 3d3e 2033 2c20 4174 7465 6d70 7473 h => 3, Attempts │ │ │ │ 0000aa70: 203d 3e20 3729 2020 2020 2020 2020 2020 => 7) │ │ │ │ 0000aa80: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0000aa90: 2031 2e30 3236 3273 2028 6370 7529 3b20 1.0262s (cpu); │ │ │ │ -0000aaa0: 302e 3735 3332 3338 7320 2874 6872 6561 0.753238s (threa │ │ │ │ -0000aab0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0000aa90: 2030 2e38 3938 3335 3773 2028 6370 7529 0.898357s (cpu) │ │ │ │ +0000aaa0: 3b20 302e 3639 3735 3231 7320 2874 6872 ; 0.697521s (thr │ │ │ │ +0000aab0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 0000aac0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0000aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab10: 207c 0a7c 2020 2020 2020 3120 2020 2020 |.| 1 │ │ │ │ 0000ab20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2762,16 +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 2030 2e37 3430 3336 3973 - used 0.740369s │ │ │ │ -0000ad10: 2028 6370 7529 3b20 302e 3533 3338 3534 (cpu); 0.533854 │ │ │ │ +0000ad00: 2d20 7573 6564 2030 2e37 3738 3534 3773 - used 0.778547s │ │ │ │ +0000ad10: 2028 6370 7529 3b20 302e 3537 3234 3735 (cpu); 0.572475 │ │ │ │ 0000ad20: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 0000ad30: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 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 3433 | -- used 0.0043 │ │ │ │ -0000e5d0: 3338 3135 7320 2863 7075 293b 2030 2e30 3815s (cpu); 0.0 │ │ │ │ -0000e5e0: 3034 3333 3431 3673 2028 7468 7265 6164 0433416s (thread │ │ │ │ +0000e5c0: 7c20 2d2d 2075 7365 6420 302e 3030 3432 | -- used 0.0042 │ │ │ │ +0000e5d0: 3639 3736 7320 2863 7075 293b 2030 2e30 6976s (cpu); 0.0 │ │ │ │ +0000e5e0: 3034 3236 3635 3973 2028 7468 7265 6164 0426659s (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 3239 3336 3934 7320 2863 7075 293b 0.293694s (cpu); │ │ │ │ -0000e710: 2030 2e32 3239 3434 3573 2028 7468 7265 0.229445s (thre │ │ │ │ -0000e720: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +0000e700: 302e 3332 3131 3836 7320 2863 7075 293b 0.321186s (cpu); │ │ │ │ +0000e710: 2030 2e32 3439 3338 7320 2874 6872 6561 0.24938s (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 │ │ │ │ @@ -3805,17 +3805,17 @@ │ │ │ │ 0000edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ede0: 2b0a 7c69 3139 203a 2074 696d 6520 6672 +.|i19 : time fr │ │ │ │ 0000edf0: 6f62 656e 6975 734e 7528 342c 2066 2920 obeniusNu(4, f) │ │ │ │ 0000ee00: 2d2d 2043 6f6e 7461 696e 6d65 6e74 5465 -- ContainmentTe │ │ │ │ 0000ee10: 7374 2069 7320 7365 7420 746f 2046 726f st is set to Fro │ │ │ │ 0000ee20: 6265 6e69 7573 526f 6f74 2c20 6279 2020 beniusRoot, by │ │ │ │ -0000ee30: 7c0a 7c20 2d2d 2075 7365 6420 302e 3235 |.| -- used 0.25 │ │ │ │ -0000ee40: 3237 3238 7320 2863 7075 293b 2030 2e31 2728s (cpu); 0.1 │ │ │ │ -0000ee50: 3736 3534 3473 2028 7468 7265 6164 293b 76544s (thread); │ │ │ │ +0000ee30: 7c0a 7c20 2d2d 2075 7365 6420 302e 3237 |.| -- used 0.27 │ │ │ │ +0000ee40: 3433 3833 7320 2863 7075 293b 2030 2e31 4383s (cpu); 0.1 │ │ │ │ +0000ee50: 3932 3832 3973 2028 7468 7265 6164 293b 92829s (thread); │ │ │ │ 0000ee60: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0000ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ee80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3840,17 +3840,17 @@ │ │ │ │ 0000eff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f010: 2b0a 7c69 3230 203a 2074 696d 6520 6672 +.|i20 : time fr │ │ │ │ 0000f020: 6f62 656e 6975 734e 7528 342c 2066 2c20 obeniusNu(4, f, │ │ │ │ 0000f030: 436f 6e74 6169 6e6d 656e 7454 6573 7420 ContainmentTest │ │ │ │ 0000f040: 3d3e 2053 7461 6e64 6172 6450 6f77 6572 => StandardPower │ │ │ │ 0000f050: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0000f060: 7c0a 7c20 2d2d 2075 7365 6420 312e 3831 |.| -- used 1.81 │ │ │ │ -0000f070: 3034 3573 2028 6370 7529 3b20 312e 3335 045s (cpu); 1.35 │ │ │ │ -0000f080: 3134 3173 2028 7468 7265 6164 293b 2030 141s (thread); 0 │ │ │ │ +0000f060: 7c0a 7c20 2d2d 2075 7365 6420 312e 3331 |.| -- used 1.31 │ │ │ │ +0000f070: 3534 3973 2028 6370 7529 3b20 312e 3032 549s (cpu); 1.02 │ │ │ │ +0000f080: 3536 3973 2028 7468 7265 6164 293b 2030 569s (thread); 0 │ │ │ │ 0000f090: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4015,18 +4015,18 @@ │ │ │ │ 0000fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000fb00: 0a7c 6932 3720 3a20 7469 6d65 2066 726f .|i27 : time fro │ │ │ │ 0000fb10: 6265 6e69 7573 4e75 2835 2c20 6629 202d beniusNu(5, f) - │ │ │ │ 0000fb20: 2d20 7573 6573 2062 696e 6172 7920 7365 - uses binary se │ │ │ │ 0000fb30: 6172 6368 2028 6465 6661 756c 7429 2020 arch (default) │ │ │ │ 0000fb40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000fb50: 0a7c 202d 2d20 7573 6564 2030 2e37 3738 .| -- used 0.778 │ │ │ │ -0000fb60: 3930 3773 2028 6370 7529 3b20 302e 3633 907s (cpu); 0.63 │ │ │ │ -0000fb70: 3530 3039 7320 2874 6872 6561 6429 3b20 5009s (thread); │ │ │ │ -0000fb80: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +0000fb50: 0a7c 202d 2d20 7573 6564 2030 2e36 3931 .| -- used 0.691 │ │ │ │ +0000fb60: 3736 7320 2863 7075 293b 2030 2e35 3537 76s (cpu); 0.557 │ │ │ │ +0000fb70: 3536 3373 2028 7468 7265 6164 293b 2030 563s (thread); 0 │ │ │ │ +0000fb80: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000fb90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000fba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fbe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000fbf0: 0a7c 6f32 3720 3d20 3131 3234 2020 2020 .|o27 = 1124 │ │ │ │ @@ -4040,17 +4040,17 @@ │ │ │ │ 0000fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000fc90: 0a7c 6932 3820 3a20 7469 6d65 2066 726f .|i28 : time fro │ │ │ │ 0000fca0: 6265 6e69 7573 4e75 2835 2c20 662c 2053 beniusNu(5, f, S │ │ │ │ 0000fcb0: 6561 7263 6820 3d3e 204c 696e 6561 7229 earch => Linear) │ │ │ │ 0000fcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fcd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000fce0: 0a7c 202d 2d20 7573 6564 2031 2e30 3832 .| -- used 1.082 │ │ │ │ -0000fcf0: 3332 7320 2863 7075 293b 2030 2e38 3237 32s (cpu); 0.827 │ │ │ │ -0000fd00: 3139 3773 2028 7468 7265 6164 293b 2030 197s (thread); 0 │ │ │ │ +0000fce0: 0a7c 202d 2d20 7573 6564 2031 2e30 3932 .| -- used 1.092 │ │ │ │ +0000fcf0: 3031 7320 2863 7075 293b 2030 2e38 3233 01s (cpu); 0.823 │ │ │ │ +0000fd00: 3937 3673 2028 7468 7265 6164 293b 2030 976s (thread); 0 │ │ │ │ 0000fd10: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000fd20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000fd30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fd70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -4085,18 +4085,18 @@ │ │ │ │ 0000ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000ff60: 0a7c 6933 3020 3a20 7469 6d65 2066 726f .|i30 : time fro │ │ │ │ 0000ff70: 6265 6e69 7573 4e75 2832 2c20 4d2c 204d beniusNu(2, M, M │ │ │ │ 0000ff80: 5e32 2920 2d2d 2075 7365 7320 6269 6e61 ^2) -- uses bina │ │ │ │ 0000ff90: 7279 2073 6561 7263 6820 2864 6566 6175 ry search (defau │ │ │ │ 0000ffa0: 6c74 2920 2020 2020 2020 2020 2020 207c lt) | │ │ │ │ -0000ffb0: 0a7c 202d 2d20 7573 6564 2032 2e31 3630 .| -- used 2.160 │ │ │ │ -0000ffc0: 3535 7320 2863 7075 293b 2031 2e36 3038 55s (cpu); 1.608 │ │ │ │ -0000ffd0: 3535 7320 2874 6872 6561 6429 3b20 3073 55s (thread); 0s │ │ │ │ -0000ffe0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0000ffb0: 0a7c 202d 2d20 7573 6564 2031 2e34 3635 .| -- used 1.465 │ │ │ │ +0000ffc0: 3039 7320 2863 7075 293b 2031 2e31 3734 09s (cpu); 1.174 │ │ │ │ +0000ffd0: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ +0000ffe0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0000fff0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00010000: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00010010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010040: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00010050: 0a7c 6f33 3020 3d20 3937 2020 2020 2020 .|o30 = 97 │ │ │ │ @@ -4110,17 +4110,17 @@ │ │ │ │ 000100d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000100e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 000100f0: 0a7c 6933 3120 3a20 7469 6d65 2066 726f .|i31 : time fro │ │ │ │ 00010100: 6265 6e69 7573 4e75 2832 2c20 4d2c 204d beniusNu(2, M, M │ │ │ │ 00010110: 5e32 2c20 5365 6172 6368 203d 3e20 4c69 ^2, Search => Li │ │ │ │ 00010120: 6e65 6172 2920 2d2d 2062 7574 206c 696e near) -- but lin │ │ │ │ 00010130: 6561 7220 7365 6172 6368 2067 6574 737c ear search gets| │ │ │ │ -00010140: 0a7c 202d 2d20 7573 6564 2030 2e36 3336 .| -- used 0.636 │ │ │ │ -00010150: 3735 3773 2028 6370 7529 3b20 302e 3439 757s (cpu); 0.49 │ │ │ │ -00010160: 3934 3533 7320 2874 6872 6561 6429 3b20 9453s (thread); │ │ │ │ +00010140: 0a7c 202d 2d20 7573 6564 2030 2e34 3932 .| -- used 0.492 │ │ │ │ +00010150: 3939 3173 2028 6370 7529 3b20 302e 3432 991s (cpu); 0.42 │ │ │ │ +00010160: 3233 3239 7320 2874 6872 6561 6429 3b20 2329s (thread); │ │ │ │ 00010170: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 00010180: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00010190: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000101a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000101d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ ├── ./usr/share/info/GKMVarieties.info.gz │ │ │ ├── GKMVarieties.info │ │ │ │ @@ -17563,17 +17563,17 @@ │ │ │ │ 000449a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000449b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000449c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3720 ---------+.|i27 │ │ │ │ 000449d0: 3a20 7469 6d65 2043 203d 206f 7262 6974 : time C = orbit │ │ │ │ 000449e0: 436c 6f73 7572 6528 582c 4d61 7429 2020 Closure(X,Mat) │ │ │ │ 000449f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a00: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00044a10: 7573 6564 2030 2e36 3835 3031 7320 2863 used 0.68501s (c │ │ │ │ -00044a20: 7075 293b 2030 2e34 3236 3834 3973 2028 pu); 0.426849s ( │ │ │ │ -00044a30: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +00044a10: 7573 6564 2036 2e36 3835 3133 7320 2863 used 6.68513s (c │ │ │ │ +00044a20: 7075 293b 2031 2e31 3235 3032 7320 2874 pu); 1.12502s (t │ │ │ │ +00044a30: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00044a40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00044a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a80: 2020 2020 2020 2020 207c 0a7c 6f32 3720 |.|o27 │ │ │ │ 00044a90: 3d20 616e 2022 6571 7569 7661 7269 616e = an "equivarian │ │ │ │ 00044aa0: 7420 4b2d 636c 6173 7322 206f 6e20 6120 t K-class" on a │ │ │ │ @@ -17591,17 +17591,17 @@ │ │ │ │ 00044b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00044b80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3820 ---------+.|i28 │ │ │ │ 00044b90: 3a20 7469 6d65 2043 203d 206f 7262 6974 : time C = orbit │ │ │ │ 00044ba0: 436c 6f73 7572 6528 582c 4d61 742c 2052 Closure(X,Mat, R │ │ │ │ 00044bb0: 5245 464d 6574 686f 6420 3d3e 2074 7275 REFMethod => tru │ │ │ │ 00044bc0: 6529 2020 2020 2020 207c 0a7c 202d 2d20 e) |.| -- │ │ │ │ -00044bd0: 7573 6564 2032 2e30 3539 3973 2028 6370 used 2.0599s (cp │ │ │ │ -00044be0: 7529 3b20 312e 3035 3033 3673 2028 7468 u); 1.05036s (th │ │ │ │ -00044bf0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +00044bd0: 7573 6564 2033 2e30 3831 3133 7320 2863 used 3.08113s (c │ │ │ │ +00044be0: 7075 293b 2030 2e39 3932 3635 3373 2028 pu); 0.992653s ( │ │ │ │ +00044bf0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 00044c00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00044c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044c40: 2020 2020 2020 2020 207c 0a7c 6f32 3820 |.|o28 │ │ │ │ 00044c50: 3d20 616e 2022 6571 7569 7661 7269 616e = an "equivarian │ │ │ │ 00044c60: 7420 4b2d 636c 6173 7322 206f 6e20 6120 t K-class" on a │ │ ├── ./usr/share/info/Graphs.info.gz │ │ │ ├── Graphs.info │ │ │ │ @@ -18078,16 +18078,16 @@ │ │ │ │ 000469d0: 7973 2048 2020 2020 2020 2020 2020 2020 ys H │ │ │ │ 000469e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000469f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00046a00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00046a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a30: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00046a40: 7b64 6967 7261 7068 2c20 6d61 702c 206e {digraph, map, n │ │ │ │ -00046a50: 6577 4469 6772 6170 687d 2020 2020 2020 ewDigraph} │ │ │ │ +00046a40: 7b6d 6170 2c20 6e65 7744 6967 7261 7068 {map, newDigraph │ │ │ │ +00046a50: 2c20 6469 6772 6170 687d 2020 2020 2020 , digraph} │ │ │ │ 00046a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00046a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046aa0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ 00046ab0: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ 00046ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/GroebnerStrata.info.gz │ │ │ ├── GroebnerStrata.info │ │ │ │ @@ -2491,25 +2491,25 @@ │ │ │ │ 00009ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009bb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00009bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009c00: 2020 2020 207c 0a7c 6f31 3220 3d20 7c20 |.|o12 = | │ │ │ │ -00009c10: 2d32 3620 2d32 3420 3620 3339 2031 3720 -26 -24 6 39 17 │ │ │ │ -00009c20: 3238 202d 3436 202d 3132 2031 3120 2d32 28 -46 -12 11 -2 │ │ │ │ -00009c30: 3920 2d31 3020 2d34 3820 2d33 3620 2d33 9 -10 -48 -36 -3 │ │ │ │ -00009c40: 3020 3339 202d 3239 202d 3820 2d32 3220 0 39 -29 -8 -22 │ │ │ │ -00009c50: 2d31 3020 207c 0a7c 2020 2020 2020 2d2d -10 |.| -- │ │ │ │ +00009c10: 3337 202d 3220 2d31 3920 2d34 202d 3133 37 -2 -19 -4 -13 │ │ │ │ +00009c20: 2032 3020 3530 202d 3334 202d 3132 202d 20 50 -34 -12 - │ │ │ │ +00009c30: 3820 3233 2033 3820 2d33 3620 2d31 3020 8 23 38 -36 -10 │ │ │ │ +00009c40: 3332 2031 3920 2d32 3220 2d32 3920 3139 32 19 -22 -29 19 │ │ │ │ +00009c50: 2031 3120 207c 0a7c 2020 2020 2020 2d2d 11 |.| -- │ │ │ │ 00009c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009ca0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3131 -----|.| 11 │ │ │ │ -00009cb0: 2031 3920 3139 2032 3420 2d32 3920 7c20 19 19 24 -29 | │ │ │ │ +00009ca0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2d33 -----|.| -3 │ │ │ │ +00009cb0: 3020 2d32 3920 3234 202d 3239 207c 2020 0 -29 24 -29 | │ │ │ │ 00009cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00009d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2536,25 +2536,25 @@ │ │ │ │ 00009e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009e80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00009e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ed0: 2020 2020 207c 0a7c 6f31 3320 3d20 7c20 |.|o13 = | │ │ │ │ -00009ee0: 2d33 3020 2d34 3120 2d33 3920 3330 202d -30 -41 -39 30 - │ │ │ │ -00009ef0: 3133 2033 3220 3137 202d 3132 2032 3920 13 32 17 -12 29 │ │ │ │ -00009f00: 3235 202d 3439 2031 3120 3433 2032 3120 25 -49 11 43 21 │ │ │ │ -00009f10: 2d34 3220 2d34 2031 3920 2d34 3720 3339 -42 -4 19 -47 39 │ │ │ │ -00009f20: 202d 3338 207c 0a7c 2020 2020 2020 2d2d -38 |.| -- │ │ │ │ +00009ee0: 2d34 202d 3620 2d32 3920 3433 2031 3220 -4 -6 -29 43 12 │ │ │ │ +00009ef0: 2d38 202d 3437 2033 3120 2d31 3920 2d33 -8 -47 31 -19 -3 │ │ │ │ +00009f00: 202d 3137 202d 3220 2d34 3920 3231 2031 -17 -2 -49 21 1 │ │ │ │ +00009f10: 3120 3320 2d34 3720 3139 202d 3136 202d 1 3 -47 19 -16 - │ │ │ │ +00009f20: 3338 2033 397c 0a7c 2020 2020 2020 2d2d 38 39|.| -- │ │ │ │ 00009f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00009f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00009f70: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2d31 -----|.| -1 │ │ │ │ -00009f80: 3620 3020 2d32 3420 3334 207c 2020 2020 6 0 -24 34 | │ │ │ │ +00009f70: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3020 -----|.| 0 │ │ │ │ +00009f80: 2d32 3420 3334 207c 2020 2020 2020 2020 -24 34 | │ │ │ │ 00009f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00009fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00009ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2582,52 +2582,52 @@ │ │ │ │ 0000a150: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a1b0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0000a1c0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +0000a1c0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a1e0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0000a1e0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a1f0: 2020 2020 207c 0a7c 6f31 3420 3d20 6964 |.|o14 = id │ │ │ │ -0000a200: 6561 6c20 2861 2020 2b20 3131 622a 6320 eal (a + 11b*c │ │ │ │ -0000a210: 2b20 3339 6320 202d 2032 3961 2a64 202b + 39c - 29a*d + │ │ │ │ -0000a220: 2031 3762 2a64 202d 2032 3463 2a64 202d 17b*d - 24c*d - │ │ │ │ -0000a230: 2032 3664 202c 2061 2a62 202b 2033 3962 26d , a*b + 39b │ │ │ │ -0000a240: 2a63 202d 207c 0a7c 2020 2020 2020 2d2d *c - |.| -- │ │ │ │ +0000a200: 6561 6c20 2861 2020 2d20 3132 622a 6320 eal (a - 12b*c │ │ │ │ +0000a210: 2d20 3463 2020 2d20 3861 2a64 202d 2031 - 4c - 8a*d - 1 │ │ │ │ +0000a220: 3362 2a64 202d 2032 632a 6420 2b20 3337 3b*d - 2c*d + 37 │ │ │ │ +0000a230: 6420 2c20 612a 6220 2b20 3332 622a 6320 d , a*b + 32b*c │ │ │ │ +0000a240: 2b20 2020 207c 0a7c 2020 2020 2020 2d2d + |.| -- │ │ │ │ 0000a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a290: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ 0000a2a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a2b0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -0000a2c0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a2d0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a2e0: 2020 2020 207c 0a7c 2020 2020 2020 3130 |.| 10 │ │ │ │ -0000a2f0: 6320 202d 2032 3961 2a64 202d 2034 3862 c - 29a*d - 48b │ │ │ │ -0000a300: 2a64 202b 2032 3863 2a64 202b 2036 6420 *d + 28c*d + 6d │ │ │ │ -0000a310: 2c20 6220 202b 2031 3962 2a63 202d 2038 , b + 19b*c - 8 │ │ │ │ -0000a320: 6320 202b 2031 3961 2a64 202d 2032 3262 c + 19a*d - 22b │ │ │ │ -0000a330: 2a64 202d 207c 0a7c 2020 2020 2020 2d2d *d - |.| -- │ │ │ │ +0000a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000a2c0: 3220 2020 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ +0000a2d0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0000a2e0: 2020 2020 207c 0a7c 2020 2020 2020 3233 |.| 23 │ │ │ │ +0000a2f0: 6320 202b 2031 3961 2a64 202b 2033 3862 c + 19a*d + 38b │ │ │ │ +0000a300: 2a64 202b 2032 3063 2a64 202d 2031 3964 *d + 20c*d - 19d │ │ │ │ +0000a310: 202c 2062 2020 2d20 3330 622a 6320 2d20 , b - 30b*c - │ │ │ │ +0000a320: 3232 6320 202d 2032 3961 2a64 202d 2032 22c - 29a*d - 2 │ │ │ │ +0000a330: 3962 2a64 207c 0a7c 2020 2020 2020 2d2d 9b*d |.| -- │ │ │ │ 0000a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a380: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0000a390: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -0000a3a0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +0000a390: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0000a3a0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a3c0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -0000a3d0: 2020 2020 207c 0a7c 2020 2020 2020 3336 |.| 36 │ │ │ │ -0000a3e0: 632a 6420 2d20 3436 6420 2c20 612a 6320 c*d - 46d , a*c │ │ │ │ -0000a3f0: 2b20 3234 622a 6320 2d20 3130 6320 202d + 24b*c - 10c - │ │ │ │ -0000a400: 2032 3961 2a64 202b 2031 3162 2a64 202d 29a*d + 11b*d - │ │ │ │ -0000a410: 2033 3063 2a64 202d 2031 3264 2029 2020 30c*d - 12d ) │ │ │ │ +0000a3c0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +0000a3d0: 2020 2020 207c 0a7c 2020 2020 2020 2d20 |.| - │ │ │ │ +0000a3e0: 3336 632a 6420 2b20 3530 6420 2c20 612a 36c*d + 50d , a* │ │ │ │ +0000a3f0: 6320 2b20 3234 622a 6320 2b20 3139 6320 c + 24b*c + 19c │ │ │ │ +0000a400: 202d 2032 3961 2a64 202b 2031 3162 2a64 - 29a*d + 11b*d │ │ │ │ +0000a410: 202d 2031 3063 2a64 202d 2033 3464 2029 - 10c*d - 34d ) │ │ │ │ 0000a420: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a470: 2020 2020 207c 0a7c 6f31 3420 3a20 4964 |.|o14 : Id │ │ │ │ 0000a480: 6561 6c20 6f66 2053 2020 2020 2020 2020 eal of S │ │ │ │ @@ -2649,50 +2649,50 @@ │ │ │ │ 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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a5c0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ 0000a5d0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000a5f0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0000a5f0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a600: 2020 2020 207c 0a7c 6f31 3520 3d20 6964 |.|o15 = id │ │ │ │ -0000a610: 6561 6c20 2861 2020 2b20 3239 622a 6320 eal (a + 29b*c │ │ │ │ -0000a620: 2b20 3330 6320 202b 2032 3561 2a64 202d + 30c + 25a*d - │ │ │ │ -0000a630: 2031 3362 2a64 202d 2034 3163 2a64 202d 13b*d - 41c*d - │ │ │ │ -0000a640: 2033 3064 202c 2061 2a62 202d 2034 3262 30d , a*b - 42b │ │ │ │ -0000a650: 2a63 202d 207c 0a7c 2020 2020 2020 2d2d *c - |.| -- │ │ │ │ +0000a610: 6561 6c20 2861 2020 2d20 3139 622a 6320 eal (a - 19b*c │ │ │ │ +0000a620: 2b20 3433 6320 202d 2033 612a 6420 2b20 + 43c - 3a*d + │ │ │ │ +0000a630: 3132 622a 6420 2d20 3663 2a64 202d 2034 12b*d - 6c*d - 4 │ │ │ │ +0000a640: 6420 2c20 612a 6220 2b20 3131 622a 6320 d , a*b + 11b*c │ │ │ │ +0000a650: 2d20 2020 207c 0a7c 2020 2020 2020 2d2d - |.| -- │ │ │ │ 0000a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a6a0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ 0000a6b0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a6c0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -0000a6d0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a6e0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a6f0: 2020 2020 207c 0a7c 2020 2020 2020 3439 |.| 49 │ │ │ │ -0000a700: 6320 202d 2034 612a 6420 2b20 3131 622a c - 4a*d + 11b* │ │ │ │ -0000a710: 6420 2b20 3332 632a 6420 2d20 3339 6420 d + 32c*d - 39d │ │ │ │ -0000a720: 2c20 6220 202d 2031 3662 2a63 202b 2031 , b - 16b*c + 1 │ │ │ │ -0000a730: 3963 2020 2d20 3437 622a 6420 2b20 3433 9c - 47b*d + 43 │ │ │ │ -0000a740: 632a 6420 2b7c 0a7c 2020 2020 2020 2d2d c*d +|.| -- │ │ │ │ +0000a6c0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +0000a6d0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0000a6e0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0000a6f0: 2020 2020 207c 0a7c 2020 2020 2020 3137 |.| 17 │ │ │ │ +0000a700: 6320 202b 2033 612a 6420 2d20 3262 2a64 c + 3a*d - 2b*d │ │ │ │ +0000a710: 202d 2038 632a 6420 2d20 3239 6420 2c20 - 8c*d - 29d , │ │ │ │ +0000a720: 6220 202b 2033 3962 2a63 202d 2034 3763 b + 39b*c - 47c │ │ │ │ +0000a730: 2020 2b20 3139 622a 6420 2d20 3439 632a + 19b*d - 49c* │ │ │ │ +0000a740: 6420 2d20 207c 0a7c 2020 2020 2020 2d2d d - |.| -- │ │ │ │ 0000a750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a790: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ 0000a7a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a7b0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 0000a7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a7d0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0000a7e0: 2020 2020 207c 0a7c 2020 2020 2020 3137 |.| 17 │ │ │ │ +0000a7e0: 2020 2020 207c 0a7c 2020 2020 2020 3437 |.| 47 │ │ │ │ 0000a7f0: 6420 2c20 612a 6320 2d20 3234 622a 6320 d , a*c - 24b*c │ │ │ │ -0000a800: 2b20 3339 6320 202b 2033 3461 2a64 202d + 39c + 34a*d - │ │ │ │ -0000a810: 2033 3862 2a64 202b 2032 3163 2a64 202d 38b*d + 21c*d - │ │ │ │ -0000a820: 2031 3264 2029 2020 2020 2020 2020 2020 12d ) │ │ │ │ +0000a800: 2d20 3136 6320 202b 2033 3461 2a64 202d - 16c + 34a*d - │ │ │ │ +0000a810: 2033 3862 2a64 202b 2032 3163 2a64 202b 38b*d + 21c*d + │ │ │ │ +0000a820: 2033 3164 2029 2020 2020 2020 2020 2020 31d ) │ │ │ │ 0000a830: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a880: 2020 2020 207c 0a7c 6f31 3520 3a20 4964 |.|o15 : Id │ │ │ │ 0000a890: 6561 6c20 6f66 2053 2020 2020 2020 2020 eal of S │ │ │ │ @@ -2713,29 +2713,29 @@ │ │ │ │ 0000a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a9c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000a9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a9e0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -0000a9f0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0000a9f0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ 0000aa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000aa10: 2032 2020 207c 0a7c 6f31 3620 3d20 7b69 2 |.|o16 = {i │ │ │ │ -0000aa20: 6465 616c 2028 6120 2b20 3234 6220 2d20 deal (a + 24b - │ │ │ │ -0000aa30: 3130 6320 2d20 3137 642c 2062 2020 2b20 10c - 17d, b + │ │ │ │ -0000aa40: 3139 622a 6320 2d20 3863 2020 2b20 3237 19b*c - 8c + 27 │ │ │ │ -0000aa50: 622a 6420 2d20 3438 632a 6420 2d20 3236 b*d - 48c*d - 26 │ │ │ │ -0000aa60: 6420 292c 207c 0a7c 2020 2020 2020 2d2d d ), |.| -- │ │ │ │ +0000aa10: 2020 3220 207c 0a7c 6f31 3620 3d20 7b69 2 |.|o16 = {i │ │ │ │ +0000aa20: 6465 616c 2028 6120 2b20 3234 6220 2b20 deal (a + 24b + │ │ │ │ +0000aa30: 3139 6320 2b20 3336 642c 2062 2020 2d20 19c + 36d, b - │ │ │ │ +0000aa40: 3330 622a 6320 2d20 3232 6320 202d 2034 30b*c - 22c - 4 │ │ │ │ +0000aa50: 3062 2a64 202b 2031 3063 2a64 202d 2031 0b*d + 10c*d - 1 │ │ │ │ +0000aa60: 3764 2029 2c7c 0a7c 2020 2020 2020 2d2d 7d ),|.| -- │ │ │ │ 0000aa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000aab0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 6964 -----|.| id │ │ │ │ 0000aac0: 6561 6c20 2863 202d 2032 3964 2c20 6220 eal (c - 29d, b │ │ │ │ -0000aad0: 2b20 3233 642c 2061 202d 2031 3464 297d + 23d, a - 14d)} │ │ │ │ +0000aad0: 2b20 3330 642c 2061 202d 2033 3864 297d + 30d, a - 38d)} │ │ │ │ 0000aae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000ab10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ab40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2755,9986 +2755,10066 @@ │ │ │ │ 0000ac20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0000ac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ac80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ac90: 2020 2020 207c 0a7c 6f31 3720 3d20 7b69 |.|o17 = {i │ │ │ │ -0000aca0: 6465 616c 2028 6220 2d20 3432 6320 2d20 deal (b - 42c - │ │ │ │ -0000acb0: 3138 642c 2061 202b 2034 3163 202b 2033 18d, a + 41c + 3 │ │ │ │ -0000acc0: 3364 292c 2069 6465 616c 2028 6220 2b20 3d), ideal (b + │ │ │ │ -0000acd0: 3236 6320 2d20 3239 642c 2061 202d 2034 26c - 29d, a - 4 │ │ │ │ -0000ace0: 3463 202d 207c 0a7c 2020 2020 2020 2d2d 4c - |.| -- │ │ │ │ -0000acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ad20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ad30: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3964 -----|.| 9d │ │ │ │ -0000ad40: 297d 2020 2020 2020 2020 2020 2020 2020 )} │ │ │ │ -0000ad50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ad80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0000ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ac90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0000aca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000acb0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0000acc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000acd0: 2020 2020 2020 3220 2020 3220 2020 2020 2 2 │ │ │ │ +0000ace0: 2020 2020 207c 0a7c 6f31 3720 3d20 7b69 |.|o17 = {i │ │ │ │ +0000acf0: 6465 616c 2028 612a 6320 2d20 3234 622a deal (a*c - 24b* │ │ │ │ +0000ad00: 6320 2d20 3136 6320 202b 2033 3461 2a64 c - 16c + 34a*d │ │ │ │ +0000ad10: 202d 2033 3862 2a64 202b 2032 3163 2a64 - 38b*d + 21c*d │ │ │ │ +0000ad20: 202b 2033 3164 202c 2062 2020 2b20 3339 + 31d , b + 39 │ │ │ │ +0000ad30: 622a 6320 2d7c 0a7c 2020 2020 2020 2d2d b*c -|.| -- │ │ │ │ +0000ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ad60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ad70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ad80: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +0000ad90: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0000ada0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0000adb0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ 0000adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000add0: 2020 2020 207c 0a7c 6f31 3720 3a20 4c69 |.|o17 : Li │ │ │ │ -0000ade0: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ -0000adf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ae10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ae20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0000add0: 2020 2020 207c 0a7c 2020 2020 2020 3437 |.| 47 │ │ │ │ +0000ade0: 6320 202b 2031 3962 2a64 202d 2034 3963 c + 19b*d - 49c │ │ │ │ +0000adf0: 2a64 202d 2034 3764 202c 2061 2a62 202b *d - 47d , a*b + │ │ │ │ +0000ae00: 2031 3162 2a63 202d 2031 3763 2020 2b20 11b*c - 17c + │ │ │ │ +0000ae10: 3361 2a64 202d 2032 622a 6420 2d20 3863 3a*d - 2b*d - 8c │ │ │ │ +0000ae20: 2a64 202d 207c 0a7c 2020 2020 2020 2d2d *d - |.| -- │ │ │ │ 0000ae30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ae40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ae70: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2c20 7468 -----+..Note, th │ │ │ │ -0000ae80: 6520 6765 6e65 7261 6c20 656c 656d 656e e general elemen │ │ │ │ -0000ae90: 7420 6f66 206f 6e65 2063 6f6d 706f 6e65 t of one compone │ │ │ │ -0000aea0: 6e74 2069 7320 6120 706c 616e 6520 636f nt is a plane co │ │ │ │ -0000aeb0: 6e69 6320 756e 696f 6e20 6120 706f 696e nic union a poin │ │ │ │ -0000aec0: 742e 2028 5468 650a 6469 6d65 6e73 696f t. (The.dimensio │ │ │ │ -0000aed0: 6e20 6f66 2074 6865 206c 6f63 7573 206f n of the locus o │ │ │ │ -0000aee0: 6620 616c 6c20 7375 6368 2069 733a 2028 f all such is: ( │ │ │ │ -0000aef0: 6368 6f69 6365 206f 6620 706c 616e 6529 choice of plane) │ │ │ │ -0000af00: 202b 2028 6368 6f69 6365 206f 6620 6465 + (choice of de │ │ │ │ -0000af10: 6772 6565 2032 0a69 6e20 706c 616e 6529 gree 2.in plane) │ │ │ │ -0000af20: 202b 2063 686f 6963 6520 6f66 2070 6f69 + choice of poi │ │ │ │ -0000af30: 6e74 2e20 5468 6973 2069 7320 3320 2b20 nt. This is 3 + │ │ │ │ -0000af40: 3520 2b20 3320 3d20 3131 2e0a 0a54 6865 5 + 3 = 11...The │ │ │ │ -0000af50: 206f 7468 6572 2063 6f6d 706f 6e65 6e74 other component │ │ │ │ -0000af60: 2063 6f6e 7369 7374 7320 6f66 2074 776f consists of two │ │ │ │ -0000af70: 2073 6b65 7720 6c69 6e65 732e 2020 5468 skew lines. Th │ │ │ │ -0000af80: 6973 2068 6173 2064 696d 656e 7369 6f6e is has dimension │ │ │ │ -0000af90: 2028 6368 6f69 6365 206f 660a 6c69 6e65 (choice of.line │ │ │ │ -0000afa0: 2920 2b20 2863 686f 6963 6520 6f66 206c ) + (choice of l │ │ │ │ -0000afb0: 696e 6529 2e20 5468 6973 2069 7320 3420 ine). This is 4 │ │ │ │ -0000afc0: 2b20 3420 3d20 382e 2020 416c 736f 206e + 4 = 8. Also n │ │ │ │ -0000afd0: 6f74 6963 6520 7468 6174 2074 6865 2032 otice that the 2 │ │ │ │ -0000afe0: 2073 6b65 7720 6c69 6e65 730a 646f 206e skew lines.do n │ │ │ │ -0000aff0: 6f74 2068 6176 6520 746f 2062 6520 6465 ot have to be de │ │ │ │ -0000b000: 6669 6e65 6420 6f76 6572 2074 6865 2062 fined over the b │ │ │ │ -0000b010: 6173 6520 6669 656c 642c 2061 7320 696e ase field, as in │ │ │ │ -0000b020: 2074 6869 7320 6361 7365 2e0a 0a0a 0a41 this case.....A │ │ │ │ -0000b030: 7574 686f 7273 0a3d 3d3d 3d3d 3d3d 0a0a uthors.=======.. │ │ │ │ -0000b040: 2020 2a20 4d69 6b65 2053 7469 6c6c 6d61 * Mike Stillma │ │ │ │ -0000b050: 6e20 2873 6565 2068 7474 703a 2f2f 7777 n (see http://ww │ │ │ │ -0000b060: 772e 6d61 7468 2e63 6f72 6e65 6c6c 2e65 w.math.cornell.e │ │ │ │ -0000b070: 6475 2f7e 6d69 6b65 2f20 290a 2020 2020 du/~mike/ ). │ │ │ │ -0000b080: 3c6d 696b 6540 6d61 7468 2e63 6f72 6e65 . * Kris │ │ │ │ -0000b0a0: 7469 6e65 204a 6f6e 6573 203c 6b65 6a6f tine Jones │ │ │ │ -0000b0c0: 0a0a 5665 7273 696f 6e0a 3d3d 3d3d 3d3d ..Version.====== │ │ │ │ -0000b0d0: 3d0a 0a54 6869 7320 646f 6375 6d65 6e74 =..This document │ │ │ │ -0000b0e0: 6174 696f 6e20 6465 7363 7269 6265 7320 ation describes │ │ │ │ -0000b0f0: 7665 7273 696f 6e20 302e 3920 6f66 2047 version 0.9 of G │ │ │ │ -0000b100: 726f 6562 6e65 7253 7472 6174 612c 2072 roebnerStrata, r │ │ │ │ -0000b110: 656c 6561 7365 6420 3131 204e 6f76 0a32 eleased 11 Nov.2 │ │ │ │ -0000b120: 3032 312e 0a0a 4369 7461 7469 6f6e 0a3d 021...Citation.= │ │ │ │ -0000b130: 3d3d 3d3d 3d3d 3d0a 0a49 6620 796f 7520 =======..If you │ │ │ │ -0000b140: 6861 7665 2075 7365 6420 7468 6973 2070 have used this p │ │ │ │ -0000b150: 6163 6b61 6765 2069 6e20 796f 7572 2072 ackage in your r │ │ │ │ -0000b160: 6573 6561 7263 682c 2070 6c65 6173 6520 esearch, please │ │ │ │ -0000b170: 6369 7465 2069 7420 6173 2066 6f6c 6c6f cite it as follo │ │ │ │ -0000b180: 7773 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ws:..+---------- │ │ │ │ -0000b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b1d0: 2d2d 2d2b 0a7c 406d 6973 637b 4772 6f65 ---+.|@misc{Groe │ │ │ │ -0000b1e0: 626e 6572 5374 7261 7461 536f 7572 6365 bnerStrataSource │ │ │ │ -0000b1f0: 2c20 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 207c 0a7c 2020 7469 746c 6520 3d20 |.| title = │ │ │ │ -0000b230: 7b7b 4772 6f65 626e 6572 5374 7261 7461 {{GroebnerStrata │ │ │ │ -0000b240: 3a20 636f 6d70 7574 696e 6720 4772 6f65 : computing Groe │ │ │ │ -0000b250: 626e 6572 206c 6f63 6920 696e 2020 2020 bner loci in │ │ │ │ -0000b260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b270: 2020 207c 0a7c 2020 6175 7468 6f72 203d |.| author = │ │ │ │ -0000b280: 207b 4d69 6b65 2053 7469 6c6c 6d61 6e20 {Mike Stillman │ │ │ │ -0000b290: 616e 6420 4b72 6973 7469 6e65 204a 6f6e and Kristine Jon │ │ │ │ -0000b2a0: 6573 7d2c 2020 2020 2020 2020 2020 2020 es}, │ │ │ │ -0000b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b2c0: 2020 207c 0a7c 2020 686f 7770 7562 6c69 |.| howpubli │ │ │ │ -0000b2d0: 7368 6564 203d 207b 4120 5c65 6d70 687b shed = {A \emph{ │ │ │ │ -0000b2e0: 4d61 6361 756c 6179 327d 2070 6163 6b61 Macaulay2} packa │ │ │ │ -0000b2f0: 6765 2061 7661 696c 6162 6c65 2020 2020 ge available │ │ │ │ -0000b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b310: 2020 207c 0a7c 2020 2020 5c75 726c 7b68 |.| \url{h │ │ │ │ -0000b320: 7474 7073 3a2f 2f67 6974 6875 622e 636f ttps://github.co │ │ │ │ -0000b330: 6d2f 4d61 6361 756c 6179 322f 4d32 2f74 m/Macaulay2/M2/t │ │ │ │ -0000b340: 7265 652f 7374 6162 6c65 2f4d 3220 2020 ree/stable/M2 │ │ │ │ -0000b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b360: 2020 207c 0a7c 7d20 2020 2020 2020 2020 |.|} │ │ │ │ -0000b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ae70: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +0000ae80: 2032 2020 2032 2020 2020 2020 2020 2020 2 2 │ │ │ │ +0000ae90: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0000aea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000aeb0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0000aec0: 2020 2020 207c 0a7c 2020 2020 2020 3239 |.| 29 │ │ │ │ +0000aed0: 6420 2c20 6120 202d 2031 3962 2a63 202b d , a - 19b*c + │ │ │ │ +0000aee0: 2034 3363 2020 2d20 3361 2a64 202b 2031 43c - 3a*d + 1 │ │ │ │ +0000aef0: 3262 2a64 202d 2036 632a 6420 2d20 3464 2b*d - 6c*d - 4d │ │ │ │ +0000af00: 2029 7d20 2020 2020 2020 2020 2020 2020 )} │ │ │ │ +0000af10: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0000af20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000af30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000af40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000af60: 2020 2020 207c 0a7c 6f31 3720 3a20 4c69 |.|o17 : Li │ │ │ │ +0000af70: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ +0000af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000afb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0000afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000b000: 2d2d 2d2d 2d2b 0a0a 4e6f 7465 2c20 7468 -----+..Note, th │ │ │ │ +0000b010: 6520 6765 6e65 7261 6c20 656c 656d 656e e general elemen │ │ │ │ +0000b020: 7420 6f66 206f 6e65 2063 6f6d 706f 6e65 t of one compone │ │ │ │ +0000b030: 6e74 2069 7320 6120 706c 616e 6520 636f nt is a plane co │ │ │ │ +0000b040: 6e69 6320 756e 696f 6e20 6120 706f 696e nic union a poin │ │ │ │ +0000b050: 742e 2028 5468 650a 6469 6d65 6e73 696f t. (The.dimensio │ │ │ │ +0000b060: 6e20 6f66 2074 6865 206c 6f63 7573 206f n of the locus o │ │ │ │ +0000b070: 6620 616c 6c20 7375 6368 2069 733a 2028 f all such is: ( │ │ │ │ +0000b080: 6368 6f69 6365 206f 6620 706c 616e 6529 choice of plane) │ │ │ │ +0000b090: 202b 2028 6368 6f69 6365 206f 6620 6465 + (choice of de │ │ │ │ +0000b0a0: 6772 6565 2032 0a69 6e20 706c 616e 6529 gree 2.in plane) │ │ │ │ +0000b0b0: 202b 2063 686f 6963 6520 6f66 2070 6f69 + choice of poi │ │ │ │ +0000b0c0: 6e74 2e20 5468 6973 2069 7320 3320 2b20 nt. This is 3 + │ │ │ │ +0000b0d0: 3520 2b20 3320 3d20 3131 2e0a 0a54 6865 5 + 3 = 11...The │ │ │ │ +0000b0e0: 206f 7468 6572 2063 6f6d 706f 6e65 6e74 other component │ │ │ │ +0000b0f0: 2063 6f6e 7369 7374 7320 6f66 2074 776f consists of two │ │ │ │ +0000b100: 2073 6b65 7720 6c69 6e65 732e 2020 5468 skew lines. Th │ │ │ │ +0000b110: 6973 2068 6173 2064 696d 656e 7369 6f6e is has dimension │ │ │ │ +0000b120: 2028 6368 6f69 6365 206f 660a 6c69 6e65 (choice of.line │ │ │ │ +0000b130: 2920 2b20 2863 686f 6963 6520 6f66 206c ) + (choice of l │ │ │ │ +0000b140: 696e 6529 2e20 5468 6973 2069 7320 3420 ine). This is 4 │ │ │ │ +0000b150: 2b20 3420 3d20 382e 2020 416c 736f 206e + 4 = 8. Also n │ │ │ │ +0000b160: 6f74 6963 6520 7468 6174 2074 6865 2032 otice that the 2 │ │ │ │ +0000b170: 2073 6b65 7720 6c69 6e65 730a 646f 206e skew lines.do n │ │ │ │ +0000b180: 6f74 2068 6176 6520 746f 2062 6520 6465 ot have to be de │ │ │ │ +0000b190: 6669 6e65 6420 6f76 6572 2074 6865 2062 fined over the b │ │ │ │ +0000b1a0: 6173 6520 6669 656c 642c 2061 7320 696e ase field, as in │ │ │ │ +0000b1b0: 2074 6869 7320 6361 7365 2e0a 0a0a 0a41 this case.....A │ │ │ │ +0000b1c0: 7574 686f 7273 0a3d 3d3d 3d3d 3d3d 0a0a uthors.=======.. │ │ │ │ +0000b1d0: 2020 2a20 4d69 6b65 2053 7469 6c6c 6d61 * Mike Stillma │ │ │ │ +0000b1e0: 6e20 2873 6565 2068 7474 703a 2f2f 7777 n (see http://ww │ │ │ │ +0000b1f0: 772e 6d61 7468 2e63 6f72 6e65 6c6c 2e65 w.math.cornell.e │ │ │ │ +0000b200: 6475 2f7e 6d69 6b65 2f20 290a 2020 2020 du/~mike/ ). │ │ │ │ +0000b210: 3c6d 696b 6540 6d61 7468 2e63 6f72 6e65 . * Kris │ │ │ │ +0000b230: 7469 6e65 204a 6f6e 6573 203c 6b65 6a6f tine Jones │ │ │ │ +0000b250: 0a0a 5665 7273 696f 6e0a 3d3d 3d3d 3d3d ..Version.====== │ │ │ │ +0000b260: 3d0a 0a54 6869 7320 646f 6375 6d65 6e74 =..This document │ │ │ │ +0000b270: 6174 696f 6e20 6465 7363 7269 6265 7320 ation describes │ │ │ │ +0000b280: 7665 7273 696f 6e20 302e 3920 6f66 2047 version 0.9 of G │ │ │ │ +0000b290: 726f 6562 6e65 7253 7472 6174 612c 2072 roebnerStrata, r │ │ │ │ +0000b2a0: 656c 6561 7365 6420 3131 204e 6f76 0a32 eleased 11 Nov.2 │ │ │ │ +0000b2b0: 3032 312e 0a0a 4369 7461 7469 6f6e 0a3d 021...Citation.= │ │ │ │ +0000b2c0: 3d3d 3d3d 3d3d 3d0a 0a49 6620 796f 7520 =======..If you │ │ │ │ +0000b2d0: 6861 7665 2075 7365 6420 7468 6973 2070 have used this p │ │ │ │ +0000b2e0: 6163 6b61 6765 2069 6e20 796f 7572 2072 ackage in your r │ │ │ │ +0000b2f0: 6573 6561 7263 682c 2070 6c65 6173 6520 esearch, please │ │ │ │ +0000b300: 6369 7465 2069 7420 6173 2066 6f6c 6c6f cite it as follo │ │ │ │ +0000b310: 7773 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ws:..+---------- │ │ │ │ +0000b320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000b330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000b360: 2d2d 2d2b 0a7c 406d 6973 637b 4772 6f65 ---+.|@misc{Groe │ │ │ │ +0000b370: 626e 6572 5374 7261 7461 536f 7572 6365 bnerStrataSource │ │ │ │ +0000b380: 2c20 2020 2020 2020 2020 2020 2020 2020 , │ │ │ │ 0000b390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b3b0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ -0000b3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b400: 2d2d 2d7c 0a7c 4869 6c62 6572 7420 7363 ---|.|Hilbert sc │ │ │ │ -0000b410: 6865 6d65 732e 2056 6572 7369 6f6e 7e30 hemes. Version~0 │ │ │ │ -0000b420: 2e39 7d7d 2c20 2020 2020 2020 2020 2020 .9}}, │ │ │ │ -0000b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b3b0: 2020 207c 0a7c 2020 7469 746c 6520 3d20 |.| title = │ │ │ │ +0000b3c0: 7b7b 4772 6f65 626e 6572 5374 7261 7461 {{GroebnerStrata │ │ │ │ +0000b3d0: 3a20 636f 6d70 7574 696e 6720 4772 6f65 : computing Groe │ │ │ │ +0000b3e0: 626e 6572 206c 6f63 6920 696e 2020 2020 bner loci in │ │ │ │ +0000b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b400: 2020 207c 0a7c 2020 6175 7468 6f72 203d |.| author = │ │ │ │ +0000b410: 207b 4d69 6b65 2053 7469 6c6c 6d61 6e20 {Mike Stillman │ │ │ │ +0000b420: 616e 6420 4b72 6973 7469 6e65 204a 6f6e and Kristine Jon │ │ │ │ +0000b430: 6573 7d2c 2020 2020 2020 2020 2020 2020 es}, │ │ │ │ 0000b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b450: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0000b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b450: 2020 207c 0a7c 2020 686f 7770 7562 6c69 |.| howpubli │ │ │ │ +0000b460: 7368 6564 203d 207b 4120 5c65 6d70 687b shed = {A \emph{ │ │ │ │ +0000b470: 4d61 6361 756c 6179 327d 2070 6163 6b61 Macaulay2} packa │ │ │ │ +0000b480: 6765 2061 7661 696c 6162 6c65 2020 2020 ge available │ │ │ │ 0000b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b4a0: 2020 207c 0a7c 6174 2020 2020 2020 2020 |.|at │ │ │ │ -0000b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b4a0: 2020 207c 0a7c 2020 2020 5c75 726c 7b68 |.| \url{h │ │ │ │ +0000b4b0: 7474 7073 3a2f 2f67 6974 6875 622e 636f ttps://github.co │ │ │ │ +0000b4c0: 6d2f 4d61 6361 756c 6179 322f 4d32 2f74 m/Macaulay2/M2/t │ │ │ │ +0000b4d0: 7265 652f 7374 6162 6c65 2f4d 3220 2020 ree/stable/M2 │ │ │ │ 0000b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b4f0: 2020 207c 0a7c 2f4d 6163 6175 6c61 7932 |.|/Macaulay2 │ │ │ │ -0000b500: 2f70 6163 6b61 6765 737d 7d20 2020 2020 /packages}} │ │ │ │ +0000b4f0: 2020 207c 0a7c 7d20 2020 2020 2020 2020 |.|} │ │ │ │ +0000b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000b540: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0000b540: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 0000b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000b590: 2d2d 2d2b 0a0a 4578 706f 7274 730a 3d3d ---+..Exports.== │ │ │ │ -0000b5a0: 3d3d 3d3d 3d0a 0a20 202a 2046 756e 6374 =====.. * Funct │ │ │ │ -0000b5b0: 696f 6e73 2061 6e64 2063 6f6d 6d61 6e64 ions and command │ │ │ │ -0000b5c0: 730a 2020 2020 2020 2a20 2a6e 6f74 6520 s. * *note │ │ │ │ -0000b5d0: 6669 6e64 5765 6967 6874 436f 6e73 7472 findWeightConstr │ │ │ │ -0000b5e0: 6169 6e74 733a 2066 696e 6457 6569 6768 aints: findWeigh │ │ │ │ -0000b5f0: 7443 6f6e 7374 7261 696e 7473 2c20 2d2d tConstraints, -- │ │ │ │ -0000b600: 2072 6574 7572 6e73 2061 206d 6174 7269 returns a matri │ │ │ │ -0000b610: 780a 2020 2020 2020 2020 6f66 2077 6569 x. of wei │ │ │ │ -0000b620: 6768 7420 636f 6e73 7472 6169 6e74 730a ght constraints. │ │ │ │ -0000b630: 2020 2020 2020 2a20 2a6e 6f74 6520 6669 * *note fi │ │ │ │ -0000b640: 6e64 5765 6967 6874 5665 6374 6f72 3a20 ndWeightVector: │ │ │ │ -0000b650: 6669 6e64 5765 6967 6874 5665 6374 6f72 findWeightVector │ │ │ │ -0000b660: 2c20 2d2d 2072 6574 7572 6e73 2061 2077 , -- returns a w │ │ │ │ -0000b670: 6569 6768 7420 7665 6374 6f72 0a20 2020 eight vector. │ │ │ │ -0000b680: 2020 202a 202a 6e6f 7465 2067 726f 6562 * *note groeb │ │ │ │ -0000b690: 6e65 7246 616d 696c 793a 2067 726f 6562 nerFamily: groeb │ │ │ │ -0000b6a0: 6e65 7246 616d 696c 792c 202d 2d20 636f nerFamily, -- co │ │ │ │ -0000b6b0: 6d70 7574 6573 2066 616d 696c 6965 7320 mputes families │ │ │ │ -0000b6c0: 6f66 2069 6465 616c 730a 2020 2020 2020 of ideals. │ │ │ │ -0000b6d0: 2020 7769 7468 2061 2073 7065 6369 6669 with a specifi │ │ │ │ -0000b6e0: 6564 2069 6e69 7469 616c 2069 6465 616c ed initial ideal │ │ │ │ -0000b6f0: 0a20 2020 2020 202a 202a 6e6f 7465 2067 . * *note g │ │ │ │ -0000b700: 726f 6562 6e65 7253 7472 6174 756d 3a20 roebnerStratum: │ │ │ │ -0000b710: 6772 6f65 626e 6572 5374 7261 7475 6d2c groebnerStratum, │ │ │ │ -0000b720: 202d 2d20 636f 6d70 7574 6520 7468 6520 -- compute the │ │ │ │ -0000b730: 6964 6561 6c20 7768 6572 6520 610a 2020 ideal where a. │ │ │ │ -0000b740: 2020 2020 2020 6769 7665 6e20 6973 2061 given is a │ │ │ │ -0000b750: 2047 726f 6562 6e65 7220 6261 7369 730a Groebner basis. │ │ │ │ -0000b760: 2020 2020 2020 2a20 2a6e 6f74 6520 6c69 * *note li │ │ │ │ -0000b770: 6e65 6172 5061 7274 3a20 6c69 6e65 6172 nearPart: linear │ │ │ │ -0000b780: 5061 7274 2c20 2d2d 2072 6574 7572 6e73 Part, -- returns │ │ │ │ -0000b790: 206c 696e 6561 7220 7061 7274 206f 6620 linear part of │ │ │ │ -0000b7a0: 6120 706f 6c79 6e6f 6d69 616c 0a20 2020 a polynomial. │ │ │ │ -0000b7b0: 2020 202a 202a 6e6f 7465 206e 6f6e 6d69 * *note nonmi │ │ │ │ -0000b7c0: 6e69 6d61 6c4d 6170 733a 206e 6f6e 6d69 nimalMaps: nonmi │ │ │ │ -0000b7d0: 6e69 6d61 6c4d 6170 732c 202d 2d20 6669 nimalMaps, -- fi │ │ │ │ -0000b7e0: 6e64 2074 6865 2064 6567 7265 6520 7a65 nd the degree ze │ │ │ │ -0000b7f0: 726f 206d 6170 7320 696e 0a20 2020 2020 ro maps in. │ │ │ │ -0000b800: 2020 2074 6865 2053 6368 7265 7965 7220 the Schreyer │ │ │ │ -0000b810: 7265 736f 6c75 7469 6f6e 206f 6620 616e resolution of an │ │ │ │ -0000b820: 2069 6465 616c 0a20 2020 2020 202a 2022 ideal. * " │ │ │ │ -0000b830: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ -0000b840: 696f 6e61 6c56 6172 6965 7479 2220 2d2d ionalVariety" -- │ │ │ │ -0000b850: 2073 6565 202a 6e6f 7465 0a20 2020 2020 see *note. │ │ │ │ -0000b860: 2020 2072 616e 646f 6d50 6f69 6e74 4f6e randomPointOn │ │ │ │ -0000b870: 5261 7469 6f6e 616c 5661 7269 6574 7928 RationalVariety( │ │ │ │ -0000b880: 4964 6561 6c29 3a0a 2020 2020 2020 2020 Ideal):. │ │ │ │ -0000b890: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ -0000b8a0: 696f 6e61 6c56 6172 6965 7479 5f6c 7049 ionalVariety_lpI │ │ │ │ -0000b8b0: 6465 616c 5f72 702c 202d 2d20 6669 6e64 deal_rp, -- find │ │ │ │ -0000b8c0: 2061 2072 616e 646f 6d20 706f 696e 7420 a random point │ │ │ │ -0000b8d0: 6f6e 2061 0a20 2020 2020 2020 2076 6172 on a. var │ │ │ │ -0000b8e0: 6965 7479 2074 6861 7420 6361 6e20 6265 iety that can be │ │ │ │ -0000b8f0: 2064 6574 6563 7465 6420 746f 2062 6520 detected to be │ │ │ │ -0000b900: 7261 7469 6f6e 616c 0a20 2020 2020 202a rational. * │ │ │ │ -0000b910: 2022 7261 6e64 6f6d 506f 696e 7473 4f6e "randomPointsOn │ │ │ │ -0000b920: 5261 7469 6f6e 616c 5661 7269 6574 7922 RationalVariety" │ │ │ │ -0000b930: 202d 2d20 7365 6520 2a6e 6f74 650a 2020 -- see *note. │ │ │ │ -0000b940: 2020 2020 2020 7261 6e64 6f6d 506f 696e randomPoin │ │ │ │ -0000b950: 7473 4f6e 5261 7469 6f6e 616c 5661 7269 tsOnRationalVari │ │ │ │ -0000b960: 6574 7928 4964 6561 6c2c 5a5a 293a 0a20 ety(Ideal,ZZ):. │ │ │ │ -0000b970: 2020 2020 2020 2072 616e 646f 6d50 6f69 randomPoi │ │ │ │ -0000b980: 6e74 734f 6e52 6174 696f 6e61 6c56 6172 ntsOnRationalVar │ │ │ │ -0000b990: 6965 7479 5f6c 7049 6465 616c 5f63 6d5a iety_lpIdeal_cmZ │ │ │ │ -0000b9a0: 5a5f 7270 2c20 2d2d 2066 696e 6420 7261 Z_rp, -- find ra │ │ │ │ -0000b9b0: 6e64 6f6d 2070 6f69 6e74 7320 6f6e 0a20 ndom points on. │ │ │ │ -0000b9c0: 2020 2020 2020 2061 2076 6172 6965 7479 a variety │ │ │ │ -0000b9d0: 2074 6861 7420 6361 6e20 6265 2064 6574 that can be det │ │ │ │ -0000b9e0: 6563 7465 6420 746f 2062 6520 7261 7469 ected to be rati │ │ │ │ -0000b9f0: 6f6e 616c 0a20 2020 2020 202a 202a 6e6f onal. * *no │ │ │ │ -0000ba00: 7465 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 te smallerMonomi │ │ │ │ -0000ba10: 616c 733a 2073 6d61 6c6c 6572 4d6f 6e6f als: smallerMono │ │ │ │ -0000ba20: 6d69 616c 732c 202d 2d20 7265 7475 726e mials, -- return │ │ │ │ -0000ba30: 7320 7468 6520 7374 616e 6461 7264 0a20 s the standard. │ │ │ │ -0000ba40: 2020 2020 2020 206d 6f6e 6f6d 6961 6c73 monomials │ │ │ │ -0000ba50: 2073 6d61 6c6c 6572 2062 7574 206f 6620 smaller but of │ │ │ │ -0000ba60: 7468 6520 7361 6d65 2064 6567 7265 6520 the same degree │ │ │ │ -0000ba70: 6173 2067 6976 656e 206d 6f6e 6f6d 6961 as given monomia │ │ │ │ -0000ba80: 6c28 7329 0a20 2020 2020 202a 202a 6e6f l(s). * *no │ │ │ │ -0000ba90: 7465 2073 7461 6e64 6172 644d 6f6e 6f6d te standardMonom │ │ │ │ -0000baa0: 6961 6c73 3a20 7374 616e 6461 7264 4d6f ials: standardMo │ │ │ │ -0000bab0: 6e6f 6d69 616c 732c 202d 2d20 636f 6d70 nomials, -- comp │ │ │ │ -0000bac0: 7574 6573 2073 7461 6e64 6172 640a 2020 utes standard. │ │ │ │ -0000bad0: 2020 2020 2020 6d6f 6e6f 6d69 616c 730a monomials. │ │ │ │ -0000bae0: 2020 2020 2020 2a20 2a6e 6f74 6520 7461 * *note ta │ │ │ │ -0000baf0: 696c 4d6f 6e6f 6d69 616c 733a 2074 6169 ilMonomials: tai │ │ │ │ -0000bb00: 6c4d 6f6e 6f6d 6961 6c73 2c20 2d2d 2066 lMonomials, -- f │ │ │ │ -0000bb10: 696e 6420 7461 696c 206d 6f6e 6f6d 6961 ind tail monomia │ │ │ │ -0000bb20: 6c73 0a20 202a 204d 6574 686f 6473 0a20 ls. * Methods. │ │ │ │ -0000bb30: 2020 2020 202a 2022 6669 6e64 5765 6967 * "findWeig │ │ │ │ -0000bb40: 6874 436f 6e73 7472 6169 6e74 7328 4964 htConstraints(Id │ │ │ │ -0000bb50: 6561 6c2c 4c69 7374 2922 202d 2d20 7365 eal,List)" -- se │ │ │ │ -0000bb60: 6520 2a6e 6f74 6520 6669 6e64 5765 6967 e *note findWeig │ │ │ │ -0000bb70: 6874 436f 6e73 7472 6169 6e74 733a 0a20 htConstraints:. │ │ │ │ -0000bb80: 2020 2020 2020 2066 696e 6457 6569 6768 findWeigh │ │ │ │ -0000bb90: 7443 6f6e 7374 7261 696e 7473 2c20 2d2d tConstraints, -- │ │ │ │ -0000bba0: 2072 6574 7572 6e73 2061 206d 6174 7269 returns a matri │ │ │ │ -0000bbb0: 7820 6f66 2077 6569 6768 7420 636f 6e73 x of weight cons │ │ │ │ -0000bbc0: 7472 6169 6e74 730a 2020 2020 2020 2a20 traints. * │ │ │ │ -0000bbd0: 2266 696e 6457 6569 6768 7456 6563 746f "findWeightVecto │ │ │ │ -0000bbe0: 7228 4964 6561 6c2c 4c69 7374 2922 202d r(Ideal,List)" - │ │ │ │ -0000bbf0: 2d20 7365 6520 2a6e 6f74 6520 6669 6e64 - see *note find │ │ │ │ -0000bc00: 5765 6967 6874 5665 6374 6f72 3a0a 2020 WeightVector:. │ │ │ │ -0000bc10: 2020 2020 2020 6669 6e64 5765 6967 6874 findWeight │ │ │ │ -0000bc20: 5665 6374 6f72 2c20 2d2d 2072 6574 7572 Vector, -- retur │ │ │ │ -0000bc30: 6e73 2061 2077 6569 6768 7420 7665 6374 ns a weight vect │ │ │ │ -0000bc40: 6f72 0a20 2020 2020 202a 2022 6772 6f65 or. * "groe │ │ │ │ -0000bc50: 626e 6572 4661 6d69 6c79 2849 6465 616c bnerFamily(Ideal │ │ │ │ -0000bc60: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -0000bc70: 6772 6f65 626e 6572 4661 6d69 6c79 3a20 groebnerFamily: │ │ │ │ -0000bc80: 6772 6f65 626e 6572 4661 6d69 6c79 2c20 groebnerFamily, │ │ │ │ -0000bc90: 2d2d 0a20 2020 2020 2020 2063 6f6d 7075 --. compu │ │ │ │ -0000bca0: 7465 7320 6661 6d69 6c69 6573 206f 6620 tes families of │ │ │ │ -0000bcb0: 6964 6561 6c73 2077 6974 6820 6120 7370 ideals with a sp │ │ │ │ -0000bcc0: 6563 6966 6965 6420 696e 6974 6961 6c20 ecified initial │ │ │ │ -0000bcd0: 6964 6561 6c0a 2020 2020 2020 2a20 2267 ideal. * "g │ │ │ │ -0000bce0: 726f 6562 6e65 7246 616d 696c 7928 4964 roebnerFamily(Id │ │ │ │ -0000bcf0: 6561 6c2c 4c69 7374 2922 202d 2d20 7365 eal,List)" -- se │ │ │ │ -0000bd00: 6520 2a6e 6f74 6520 6772 6f65 626e 6572 e *note groebner │ │ │ │ -0000bd10: 4661 6d69 6c79 3a0a 2020 2020 2020 2020 Family:. │ │ │ │ -0000bd20: 6772 6f65 626e 6572 4661 6d69 6c79 2c20 groebnerFamily, │ │ │ │ -0000bd30: 2d2d 2063 6f6d 7075 7465 7320 6661 6d69 -- computes fami │ │ │ │ -0000bd40: 6c69 6573 206f 6620 6964 6561 6c73 2077 lies of ideals w │ │ │ │ -0000bd50: 6974 6820 6120 7370 6563 6966 6965 6420 ith a specified │ │ │ │ -0000bd60: 696e 6974 6961 6c0a 2020 2020 2020 2020 initial. │ │ │ │ -0000bd70: 6964 6561 6c0a 2020 2020 2020 2a20 2267 ideal. * "g │ │ │ │ -0000bd80: 726f 6562 6e65 7253 7472 6174 756d 2849 roebnerStratum(I │ │ │ │ -0000bd90: 6465 616c 2922 202d 2d20 7365 6520 2a6e deal)" -- see *n │ │ │ │ -0000bda0: 6f74 6520 6772 6f65 626e 6572 5374 7261 ote groebnerStra │ │ │ │ -0000bdb0: 7475 6d3a 2067 726f 6562 6e65 7253 7472 tum: groebnerStr │ │ │ │ -0000bdc0: 6174 756d 2c0a 2020 2020 2020 2020 2d2d atum,. -- │ │ │ │ -0000bdd0: 2063 6f6d 7075 7465 2074 6865 2069 6465 compute the ide │ │ │ │ -0000bde0: 616c 2077 6865 7265 2061 2067 6976 656e al where a given │ │ │ │ -0000bdf0: 2069 7320 6120 4772 6f65 626e 6572 2062 is a Groebner b │ │ │ │ -0000be00: 6173 6973 0a20 2020 2020 202a 2022 6e6f asis. * "no │ │ │ │ -0000be10: 6e6d 696e 696d 616c 4d61 7073 2849 6465 nminimalMaps(Ide │ │ │ │ -0000be20: 616c 2922 202d 2d20 7365 6520 2a6e 6f74 al)" -- see *not │ │ │ │ -0000be30: 6520 6e6f 6e6d 696e 696d 616c 4d61 7073 e nonminimalMaps │ │ │ │ -0000be40: 3a20 6e6f 6e6d 696e 696d 616c 4d61 7073 : nonminimalMaps │ │ │ │ -0000be50: 2c20 2d2d 0a20 2020 2020 2020 2066 696e , --. fin │ │ │ │ -0000be60: 6420 7468 6520 6465 6772 6565 207a 6572 d the degree zer │ │ │ │ -0000be70: 6f20 6d61 7073 2069 6e20 7468 6520 5363 o maps in the Sc │ │ │ │ -0000be80: 6872 6579 6572 2072 6573 6f6c 7574 696f hreyer resolutio │ │ │ │ -0000be90: 6e20 6f66 2061 6e20 6964 6561 6c0a 2020 n of an ideal. │ │ │ │ -0000bea0: 2020 2020 2a20 2a6e 6f74 6520 7261 6e64 * *note rand │ │ │ │ -0000beb0: 6f6d 506f 696e 744f 6e52 6174 696f 6e61 omPointOnRationa │ │ │ │ -0000bec0: 6c56 6172 6965 7479 2849 6465 616c 293a lVariety(Ideal): │ │ │ │ -0000bed0: 0a20 2020 2020 2020 2072 616e 646f 6d50 . randomP │ │ │ │ -0000bee0: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ -0000bef0: 7269 6574 795f 6c70 4964 6561 6c5f 7270 riety_lpIdeal_rp │ │ │ │ -0000bf00: 2c20 2d2d 2066 696e 6420 6120 7261 6e64 , -- find a rand │ │ │ │ -0000bf10: 6f6d 2070 6f69 6e74 206f 6e20 610a 2020 om point on a. │ │ │ │ -0000bf20: 2020 2020 2020 7661 7269 6574 7920 7468 variety th │ │ │ │ -0000bf30: 6174 2063 616e 2062 6520 6465 7465 6374 at can be detect │ │ │ │ -0000bf40: 6564 2074 6f20 6265 2072 6174 696f 6e61 ed to be rationa │ │ │ │ -0000bf50: 6c0a 2020 2020 2020 2a20 2a6e 6f74 6520 l. * *note │ │ │ │ -0000bf60: 7261 6e64 6f6d 506f 696e 7473 4f6e 5261 randomPointsOnRa │ │ │ │ -0000bf70: 7469 6f6e 616c 5661 7269 6574 7928 4964 tionalVariety(Id │ │ │ │ -0000bf80: 6561 6c2c 5a5a 293a 0a20 2020 2020 2020 eal,ZZ):. │ │ │ │ -0000bf90: 2072 616e 646f 6d50 6f69 6e74 734f 6e52 randomPointsOnR │ │ │ │ -0000bfa0: 6174 696f 6e61 6c56 6172 6965 7479 5f6c ationalVariety_l │ │ │ │ -0000bfb0: 7049 6465 616c 5f63 6d5a 5a5f 7270 2c20 pIdeal_cmZZ_rp, │ │ │ │ -0000bfc0: 2d2d 2066 696e 6420 7261 6e64 6f6d 2070 -- find random p │ │ │ │ -0000bfd0: 6f69 6e74 7320 6f6e 0a20 2020 2020 2020 oints on. │ │ │ │ -0000bfe0: 2061 2076 6172 6965 7479 2074 6861 7420 a variety that │ │ │ │ -0000bff0: 6361 6e20 6265 2064 6574 6563 7465 6420 can be detected │ │ │ │ -0000c000: 746f 2062 6520 7261 7469 6f6e 616c 0a20 to be rational. │ │ │ │ -0000c010: 2020 2020 202a 2022 736d 616c 6c65 724d * "smallerM │ │ │ │ -0000c020: 6f6e 6f6d 6961 6c73 2849 6465 616c 2922 onomials(Ideal)" │ │ │ │ -0000c030: 202d 2d20 7365 6520 2a6e 6f74 6520 736d -- see *note sm │ │ │ │ -0000c040: 616c 6c65 724d 6f6e 6f6d 6961 6c73 3a0a allerMonomials:. │ │ │ │ -0000c050: 2020 2020 2020 2020 736d 616c 6c65 724d smallerM │ │ │ │ -0000c060: 6f6e 6f6d 6961 6c73 2c20 2d2d 2072 6574 onomials, -- ret │ │ │ │ -0000c070: 7572 6e73 2074 6865 2073 7461 6e64 6172 urns the standar │ │ │ │ -0000c080: 6420 6d6f 6e6f 6d69 616c 7320 736d 616c d monomials smal │ │ │ │ -0000c090: 6c65 7220 6275 7420 6f66 2074 6865 0a20 ler but of the. │ │ │ │ -0000c0a0: 2020 2020 2020 2073 616d 6520 6465 6772 same degr │ │ │ │ -0000c0b0: 6565 2061 7320 6769 7665 6e20 6d6f 6e6f ee as given mono │ │ │ │ -0000c0c0: 6d69 616c 2873 290a 2020 2020 2020 2a20 mial(s). * │ │ │ │ -0000c0d0: 2273 6d61 6c6c 6572 4d6f 6e6f 6d69 616c "smallerMonomial │ │ │ │ -0000c0e0: 7328 4964 6561 6c2c 5269 6e67 456c 656d s(Ideal,RingElem │ │ │ │ -0000c0f0: 656e 7429 2220 2d2d 2073 6565 202a 6e6f ent)" -- see *no │ │ │ │ -0000c100: 7465 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 te smallerMonomi │ │ │ │ -0000c110: 616c 733a 0a20 2020 2020 2020 2073 6d61 als:. sma │ │ │ │ -0000c120: 6c6c 6572 4d6f 6e6f 6d69 616c 732c 202d llerMonomials, - │ │ │ │ -0000c130: 2d20 7265 7475 726e 7320 7468 6520 7374 - returns the st │ │ │ │ -0000c140: 616e 6461 7264 206d 6f6e 6f6d 6961 6c73 andard monomials │ │ │ │ -0000c150: 2073 6d61 6c6c 6572 2062 7574 206f 6620 smaller but of │ │ │ │ -0000c160: 7468 650a 2020 2020 2020 2020 7361 6d65 the. same │ │ │ │ -0000c170: 2064 6567 7265 6520 6173 2067 6976 656e degree as given │ │ │ │ -0000c180: 206d 6f6e 6f6d 6961 6c28 7329 0a20 2020 monomial(s). │ │ │ │ -0000c190: 2020 202a 2022 7374 616e 6461 7264 4d6f * "standardMo │ │ │ │ -0000c1a0: 6e6f 6d69 616c 7328 4964 6561 6c29 2220 nomials(Ideal)" │ │ │ │ -0000c1b0: 2d2d 2073 6565 202a 6e6f 7465 2073 7461 -- see *note sta │ │ │ │ -0000c1c0: 6e64 6172 644d 6f6e 6f6d 6961 6c73 3a0a ndardMonomials:. │ │ │ │ -0000c1d0: 2020 2020 2020 2020 7374 616e 6461 7264 standard │ │ │ │ -0000c1e0: 4d6f 6e6f 6d69 616c 732c 202d 2d20 636f Monomials, -- co │ │ │ │ -0000c1f0: 6d70 7574 6573 2073 7461 6e64 6172 6420 mputes standard │ │ │ │ -0000c200: 6d6f 6e6f 6d69 616c 730a 2020 2020 2020 monomials. │ │ │ │ -0000c210: 2a20 2273 7461 6e64 6172 644d 6f6e 6f6d * "standardMonom │ │ │ │ -0000c220: 6961 6c73 284c 6973 742c 4964 6561 6c29 ials(List,Ideal) │ │ │ │ -0000c230: 2220 2d2d 2073 6565 202a 6e6f 7465 2073 " -- see *note s │ │ │ │ -0000c240: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ -0000c250: 3a0a 2020 2020 2020 2020 7374 616e 6461 :. standa │ │ │ │ -0000c260: 7264 4d6f 6e6f 6d69 616c 732c 202d 2d20 rdMonomials, -- │ │ │ │ -0000c270: 636f 6d70 7574 6573 2073 7461 6e64 6172 computes standar │ │ │ │ -0000c280: 6420 6d6f 6e6f 6d69 616c 730a 2020 2020 d monomials. │ │ │ │ -0000c290: 2020 2a20 2273 7461 6e64 6172 644d 6f6e * "standardMon │ │ │ │ -0000c2a0: 6f6d 6961 6c73 285a 5a2c 4964 6561 6c29 omials(ZZ,Ideal) │ │ │ │ -0000c2b0: 2220 2d2d 2073 6565 202a 6e6f 7465 2073 " -- see *note s │ │ │ │ -0000c2c0: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ -0000c2d0: 3a0a 2020 2020 2020 2020 7374 616e 6461 :. standa │ │ │ │ -0000c2e0: 7264 4d6f 6e6f 6d69 616c 732c 202d 2d20 rdMonomials, -- │ │ │ │ -0000c2f0: 636f 6d70 7574 6573 2073 7461 6e64 6172 computes standar │ │ │ │ -0000c300: 6420 6d6f 6e6f 6d69 616c 730a 2020 2020 d monomials. │ │ │ │ -0000c310: 2020 2a20 2274 6169 6c4d 6f6e 6f6d 6961 * "tailMonomia │ │ │ │ -0000c320: 6c73 2849 6465 616c 2922 202d 2d20 7365 ls(Ideal)" -- se │ │ │ │ -0000c330: 6520 2a6e 6f74 6520 7461 696c 4d6f 6e6f e *note tailMono │ │ │ │ -0000c340: 6d69 616c 733a 2074 6169 6c4d 6f6e 6f6d mials: tailMonom │ │ │ │ -0000c350: 6961 6c73 2c20 2d2d 0a20 2020 2020 2020 ials, --. │ │ │ │ -0000c360: 2066 696e 6420 7461 696c 206d 6f6e 6f6d find tail monom │ │ │ │ -0000c370: 6961 6c73 0a20 2020 2020 202a 2022 7461 ials. * "ta │ │ │ │ -0000c380: 696c 4d6f 6e6f 6d69 616c 7328 4964 6561 ilMonomials(Idea │ │ │ │ -0000c390: 6c2c 5269 6e67 456c 656d 656e 7429 2220 l,RingElement)" │ │ │ │ -0000c3a0: 2d2d 2073 6565 202a 6e6f 7465 2074 6169 -- see *note tai │ │ │ │ -0000c3b0: 6c4d 6f6e 6f6d 6961 6c73 3a0a 2020 2020 lMonomials:. │ │ │ │ -0000c3c0: 2020 2020 7461 696c 4d6f 6e6f 6d69 616c tailMonomial │ │ │ │ -0000c3d0: 732c 202d 2d20 6669 6e64 2074 6169 6c20 s, -- find tail │ │ │ │ -0000c3e0: 6d6f 6e6f 6d69 616c 730a 2020 2a20 5379 monomials. * Sy │ │ │ │ -0000c3f0: 6d62 6f6c 730a 2020 2020 2020 2a20 2a6e mbols. * *n │ │ │ │ -0000c400: 6f74 6520 416c 6c53 7461 6e64 6172 643a ote AllStandard: │ │ │ │ -0000c410: 2041 6c6c 5374 616e 6461 7264 2c20 2d2d AllStandard, -- │ │ │ │ -0000c420: 2062 6f6f 6c65 616e 206f 7074 696f 6e20 boolean option │ │ │ │ -0000c430: 666f 7220 6465 7465 726d 696e 696e 6720 for determining │ │ │ │ -0000c440: 7468 650a 2020 2020 2020 2020 7573 6520 the. use │ │ │ │ -0000c450: 6f66 2061 6c6c 2073 7461 6e64 6172 6420 of all standard │ │ │ │ -0000c460: 6f72 2073 6d61 6c6c 6572 2073 7461 6e64 or smaller stand │ │ │ │ -0000c470: 6172 6420 6d6f 6e6f 6d69 616c 730a 2020 ard monomials. │ │ │ │ -0000c480: 2020 2020 2a20 2a6e 6f74 6520 4d69 6e69 * *note Mini │ │ │ │ -0000c490: 6d61 6c69 7a65 3a20 4d69 6e69 6d61 6c69 malize: Minimali │ │ │ │ -0000c4a0: 7a65 2c20 2d2d 2062 6f6f 6c65 616e 206f ze, -- boolean o │ │ │ │ -0000c4b0: 7074 696f 6e20 6f66 2067 726f 6562 6e65 ption of groebne │ │ │ │ -0000c4c0: 7253 7472 6174 756d 0a20 2020 2020 2020 rStratum. │ │ │ │ -0000c4d0: 2028 6375 7272 656e 746c 7920 6120 6e6f (currently a no │ │ │ │ -0000c4e0: 2d6f 7029 0a0a 466f 7220 7468 6520 7072 -op)..For the pr │ │ │ │ -0000c4f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -0000c500: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -0000c510: 206f 626a 6563 7420 2a6e 6f74 6520 4772 object *note Gr │ │ │ │ -0000c520: 6f65 626e 6572 5374 7261 7461 3a20 546f oebnerStrata: To │ │ │ │ -0000c530: 702c 2069 7320 6120 2a6e 6f74 6520 7061 p, is a *note pa │ │ │ │ -0000c540: 636b 6167 653a 0a28 4d61 6361 756c 6179 ckage:.(Macaulay │ │ │ │ -0000c550: 3244 6f63 2950 6163 6b61 6765 2c2c 2064 2Doc)Package,, d │ │ │ │ -0000c560: 6566 696e 6564 2069 6e20 4772 6f65 626e efined in Groebn │ │ │ │ -0000c570: 6572 5374 7261 7461 2e6d 322e 0a0a 2d2d erStrata.m2...-- │ │ │ │ -0000c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000c5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0000c5d0: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0000c5e0: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0000c5f0: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0000c600: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0000c610: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ -0000c620: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0000c630: 6167 6573 2f0a 4772 6f65 626e 6572 5374 ages/.GroebnerSt │ │ │ │ -0000c640: 7261 7461 2e6d 323a 3435 393a 302e 0a1f rata.m2:459:0... │ │ │ │ -0000c650: 0a46 696c 653a 2047 726f 6562 6e65 7253 .File: GroebnerS │ │ │ │ -0000c660: 7472 6174 612e 696e 666f 2c20 4e6f 6465 trata.info, Node │ │ │ │ -0000c670: 3a20 416c 6c53 7461 6e64 6172 642c 204e : AllStandard, N │ │ │ │ -0000c680: 6578 743a 2066 696e 6457 6569 6768 7443 ext: findWeightC │ │ │ │ -0000c690: 6f6e 7374 7261 696e 7473 2c20 5072 6576 onstraints, Prev │ │ │ │ -0000c6a0: 3a20 546f 702c 2055 703a 2054 6f70 0a0a : Top, Up: Top.. │ │ │ │ -0000c6b0: 416c 6c53 7461 6e64 6172 6420 2d2d 2062 AllStandard -- b │ │ │ │ -0000c6c0: 6f6f 6c65 616e 206f 7074 696f 6e20 666f oolean option fo │ │ │ │ -0000c6d0: 7220 6465 7465 726d 696e 696e 6720 7468 r determining th │ │ │ │ -0000c6e0: 6520 7573 6520 6f66 2061 6c6c 2073 7461 e use of all sta │ │ │ │ -0000c6f0: 6e64 6172 6420 6f72 2073 6d61 6c6c 6572 ndard or smaller │ │ │ │ -0000c700: 2073 7461 6e64 6172 6420 6d6f 6e6f 6d69 standard monomi │ │ │ │ -0000c710: 616c 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a als.************ │ │ │ │ -0000c720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000c730: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000c740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000c750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000c760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000c770: 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 6970 *******..Descrip │ │ │ │ -0000c780: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -0000c790: 0a0a 5468 6973 2069 7320 616e 206f 7074 ..This is an opt │ │ │ │ -0000c7a0: 696f 6e61 6c20 696e 7075 7420 666f 7220 ional input for │ │ │ │ -0000c7b0: 7468 6520 2a6e 6f74 6520 6772 6f65 626e the *note groebn │ │ │ │ -0000c7c0: 6572 5374 7261 7475 6d3a 2067 726f 6562 erStratum: groeb │ │ │ │ -0000c7d0: 6e65 7253 7472 6174 756d 2c20 616e 640a nerStratum, and. │ │ │ │ -0000c7e0: 2a6e 6f74 6520 6772 6f65 626e 6572 4661 *note groebnerFa │ │ │ │ -0000c7f0: 6d69 6c79 3a20 6772 6f65 626e 6572 4661 mily: groebnerFa │ │ │ │ -0000c800: 6d69 6c79 2c20 6675 6e63 7469 6f6e 732e mily, functions. │ │ │ │ -0000c810: 2020 4974 2074 616b 6573 2076 616c 7565 It takes value │ │ │ │ -0000c820: 7320 2474 7275 6524 2061 6e64 0a24 6661 s $true$ and.$fa │ │ │ │ -0000c830: 6c73 6524 2e20 2057 6865 6e20 6173 7369 lse$. When assi │ │ │ │ -0000c840: 676e 6564 2074 6865 2076 616c 7565 2024 gned the value $ │ │ │ │ -0000c850: 7472 7565 242c 2074 6865 2066 756e 6374 true$, the funct │ │ │ │ -0000c860: 696f 6e73 2061 7265 2063 6f6d 7075 7465 ions are compute │ │ │ │ -0000c870: 6420 7769 7468 0a72 6573 7065 6374 2074 d with.respect t │ │ │ │ -0000c880: 6f20 616c 6c20 7374 616e 6461 7264 206d o all standard m │ │ │ │ -0000c890: 6f6e 6f6d 6961 6c73 2066 6f72 2074 6865 onomials for the │ │ │ │ -0000c8a0: 2073 7065 6369 6669 6564 2067 656e 6572 specified gener │ │ │ │ -0000c8b0: 6174 6f72 732e 2020 5768 656e 2061 7373 ators. When ass │ │ │ │ -0000c8c0: 6967 6e65 640a 7468 6520 7661 6c75 6520 igned.the value │ │ │ │ -0000c8d0: 2466 616c 7365 242c 2074 6865 2066 756e $false$, the fun │ │ │ │ -0000c8e0: 6374 696f 6e73 2061 7265 2063 6f6d 7075 ctions are compu │ │ │ │ -0000c8f0: 7465 6420 7769 7468 2072 6573 7065 6374 ted with respect │ │ │ │ -0000c900: 2074 6f20 736d 616c 6c65 7220 6d6f 6e6f to smaller mono │ │ │ │ -0000c910: 6d69 616c 730a 6f66 2074 6865 2073 616d mials.of the sam │ │ │ │ -0000c920: 6520 6465 6772 6565 2061 7320 7468 6520 e degree as the │ │ │ │ -0000c930: 7370 6563 6966 6965 6420 6765 6e65 7261 specified genera │ │ │ │ -0000c940: 746f 7273 2062 7574 2077 6869 6368 2064 tors but which d │ │ │ │ -0000c950: 6f20 6e6f 7420 6c69 6520 696e 2074 6865 o not lie in the │ │ │ │ -0000c960: 0a69 6465 616c 2e0a 0a53 6565 2061 6c73 .ideal...See als │ │ │ │ -0000c970: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ -0000c980: 2a6e 6f74 6520 6772 6f65 626e 6572 5374 *note groebnerSt │ │ │ │ -0000c990: 7261 7475 6d3a 2067 726f 6562 6e65 7253 ratum: groebnerS │ │ │ │ -0000c9a0: 7472 6174 756d 2c20 2d2d 2063 6f6d 7075 tratum, -- compu │ │ │ │ -0000c9b0: 7465 2074 6865 2069 6465 616c 2077 6865 te the ideal whe │ │ │ │ -0000c9c0: 7265 2061 2067 6976 656e 0a20 2020 2069 re a given. i │ │ │ │ -0000c9d0: 7320 6120 4772 6f65 626e 6572 2062 6173 s a Groebner bas │ │ │ │ -0000c9e0: 6973 0a20 202a 202a 6e6f 7465 2067 726f is. * *note gro │ │ │ │ -0000c9f0: 6562 6e65 7246 616d 696c 793a 2067 726f ebnerFamily: gro │ │ │ │ -0000ca00: 6562 6e65 7246 616d 696c 792c 202d 2d20 ebnerFamily, -- │ │ │ │ -0000ca10: 636f 6d70 7574 6573 2066 616d 696c 6965 computes familie │ │ │ │ -0000ca20: 7320 6f66 2069 6465 616c 7320 7769 7468 s of ideals with │ │ │ │ -0000ca30: 2061 0a20 2020 2073 7065 6369 6669 6564 a. specified │ │ │ │ -0000ca40: 2069 6e69 7469 616c 2069 6465 616c 0a20 initial ideal. │ │ │ │ -0000ca50: 202a 202a 6e6f 7465 2073 6d61 6c6c 6572 * *note smaller │ │ │ │ -0000ca60: 4d6f 6e6f 6d69 616c 733a 2073 6d61 6c6c Monomials: small │ │ │ │ -0000ca70: 6572 4d6f 6e6f 6d69 616c 732c 202d 2d20 erMonomials, -- │ │ │ │ -0000ca80: 7265 7475 726e 7320 7468 6520 7374 616e returns the stan │ │ │ │ -0000ca90: 6461 7264 206d 6f6e 6f6d 6961 6c73 0a20 dard monomials. │ │ │ │ -0000caa0: 2020 2073 6d61 6c6c 6572 2062 7574 206f smaller but o │ │ │ │ -0000cab0: 6620 7468 6520 7361 6d65 2064 6567 7265 f the same degre │ │ │ │ -0000cac0: 6520 6173 2067 6976 656e 206d 6f6e 6f6d e as given monom │ │ │ │ -0000cad0: 6961 6c28 7329 0a20 202a 202a 6e6f 7465 ial(s). * *note │ │ │ │ -0000cae0: 2073 7461 6e64 6172 644d 6f6e 6f6d 6961 standardMonomia │ │ │ │ -0000caf0: 6c73 3a20 7374 616e 6461 7264 4d6f 6e6f ls: standardMono │ │ │ │ -0000cb00: 6d69 616c 732c 202d 2d20 636f 6d70 7574 mials, -- comput │ │ │ │ -0000cb10: 6573 2073 7461 6e64 6172 6420 6d6f 6e6f es standard mono │ │ │ │ -0000cb20: 6d69 616c 730a 0a46 756e 6374 696f 6e73 mials..Functions │ │ │ │ -0000cb30: 2077 6974 6820 6f70 7469 6f6e 616c 2061 with optional a │ │ │ │ -0000cb40: 7267 756d 656e 7420 6e61 6d65 6420 416c rgument named Al │ │ │ │ -0000cb50: 6c53 7461 6e64 6172 643a 0a3d 3d3d 3d3d lStandard:.===== │ │ │ │ -0000cb60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0000cb70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0000cb80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -0000cb90: 2020 2a20 2267 726f 6562 6e65 7246 616d * "groebnerFam │ │ │ │ -0000cba0: 696c 7928 2e2e 2e2c 416c 6c53 7461 6e64 ily(...,AllStand │ │ │ │ -0000cbb0: 6172 643d 3e2e 2e2e 2922 202d 2d20 7365 ard=>...)" -- se │ │ │ │ -0000cbc0: 6520 2a6e 6f74 6520 6772 6f65 626e 6572 e *note groebner │ │ │ │ -0000cbd0: 4661 6d69 6c79 3a0a 2020 2020 6772 6f65 Family:. groe │ │ │ │ -0000cbe0: 626e 6572 4661 6d69 6c79 2c20 2d2d 2063 bnerFamily, -- c │ │ │ │ -0000cbf0: 6f6d 7075 7465 7320 6661 6d69 6c69 6573 omputes families │ │ │ │ -0000cc00: 206f 6620 6964 6561 6c73 2077 6974 6820 of ideals with │ │ │ │ -0000cc10: 6120 7370 6563 6966 6965 6420 696e 6974 a specified init │ │ │ │ -0000cc20: 6961 6c0a 2020 2020 6964 6561 6c0a 2020 ial. ideal. │ │ │ │ -0000cc30: 2a20 2274 6169 6c4d 6f6e 6f6d 6961 6c73 * "tailMonomials │ │ │ │ -0000cc40: 282e 2e2e 2c41 6c6c 5374 616e 6461 7264 (...,AllStandard │ │ │ │ -0000cc50: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ -0000cc60: 6e6f 7465 2074 6169 6c4d 6f6e 6f6d 6961 note tailMonomia │ │ │ │ -0000cc70: 6c73 3a0a 2020 2020 7461 696c 4d6f 6e6f ls:. tailMono │ │ │ │ -0000cc80: 6d69 616c 732c 202d 2d20 6669 6e64 2074 mials, -- find t │ │ │ │ -0000cc90: 6169 6c20 6d6f 6e6f 6d69 616c 730a 0a46 ail monomials..F │ │ │ │ -0000cca0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -0000ccb0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -0000ccc0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -0000ccd0: 202a 6e6f 7465 2041 6c6c 5374 616e 6461 *note AllStanda │ │ │ │ -0000cce0: 7264 3a20 416c 6c53 7461 6e64 6172 642c rd: AllStandard, │ │ │ │ -0000ccf0: 2069 7320 6120 2a6e 6f74 6520 7379 6d62 is a *note symb │ │ │ │ -0000cd00: 6f6c 3a0a 284d 6163 6175 6c61 7932 446f ol:.(Macaulay2Do │ │ │ │ -0000cd10: 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d c)Symbol,...---- │ │ │ │ -0000cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000cd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000cd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0000cd70: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0000cd80: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0000cd90: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0000cda0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0000cdb0: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ -0000cdc0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0000cdd0: 6573 2f0a 4772 6f65 626e 6572 5374 7261 es/.GroebnerStra │ │ │ │ -0000cde0: 7461 2e6d 323a 3130 3633 3a30 2e0a 1f0a ta.m2:1063:0.... │ │ │ │ -0000cdf0: 4669 6c65 3a20 4772 6f65 626e 6572 5374 File: GroebnerSt │ │ │ │ -0000ce00: 7261 7461 2e69 6e66 6f2c 204e 6f64 653a rata.info, Node: │ │ │ │ -0000ce10: 2066 696e 6457 6569 6768 7443 6f6e 7374 findWeightConst │ │ │ │ -0000ce20: 7261 696e 7473 2c20 4e65 7874 3a20 6669 raints, Next: fi │ │ │ │ -0000ce30: 6e64 5765 6967 6874 5665 6374 6f72 2c20 ndWeightVector, │ │ │ │ -0000ce40: 5072 6576 3a20 416c 6c53 7461 6e64 6172 Prev: AllStandar │ │ │ │ -0000ce50: 642c 2055 703a 2054 6f70 0a0a 6669 6e64 d, Up: Top..find │ │ │ │ -0000ce60: 5765 6967 6874 436f 6e73 7472 6169 6e74 WeightConstraint │ │ │ │ -0000ce70: 7320 2d2d 2072 6574 7572 6e73 2061 206d s -- returns a m │ │ │ │ -0000ce80: 6174 7269 7820 6f66 2077 6569 6768 7420 atrix of weight │ │ │ │ -0000ce90: 636f 6e73 7472 6169 6e74 730a 2a2a 2a2a constraints.**** │ │ │ │ -0000cea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000ceb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000cec0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000ced0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0000cee0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0000cef0: 2063 6f6e 7374 7261 696e 7473 203d 2066 constraints = f │ │ │ │ -0000cf00: 696e 6457 6569 6768 7443 6f6e 7374 7261 indWeightConstra │ │ │ │ -0000cf10: 696e 7473 284d 2c20 4c29 0a20 202a 2049 ints(M, L). * I │ │ │ │ -0000cf20: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ -0000cf30: 2c20 616e 202a 6e6f 7465 2069 6465 616c , an *note ideal │ │ │ │ -0000cf40: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0000cf50: 4964 6561 6c2c 2c20 4d20 7368 6f75 6c64 Ideal,, M should │ │ │ │ -0000cf60: 2062 6520 6120 6d6f 6e6f 6d69 616c 2069 be a monomial i │ │ │ │ -0000cf70: 6465 616c 0a20 2020 2020 202a 204c 2c20 deal. * L, │ │ │ │ -0000cf80: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ -0000cf90: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ -0000cfa0: 2c2c 2061 206c 6973 7420 6f66 206c 6973 ,, a list of lis │ │ │ │ -0000cfb0: 7473 206f 6620 7461 696c 206d 6f6e 6f6d ts of tail monom │ │ │ │ -0000cfc0: 6961 6c73 0a20 202a 204f 7574 7075 7473 ials. * Outputs │ │ │ │ -0000cfd0: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ -0000cfe0: 6520 6d61 7472 6978 3a20 284d 6163 6175 e matrix: (Macau │ │ │ │ -0000cff0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c lay2Doc)Matrix,, │ │ │ │ -0000d000: 2077 6974 6820 7468 6520 636f 6e73 7472 with the constr │ │ │ │ -0000d010: 6169 6e74 7320 6f6e 2061 2077 6569 6768 aints on a weigh │ │ │ │ -0000d020: 740a 2020 2020 2020 2020 7665 6374 6f72 t. vector │ │ │ │ -0000d030: 2066 6f72 2024 4d24 2074 6f20 6265 2074 for $M$ to be t │ │ │ │ -0000d040: 6865 206c 6561 6420 6d6f 6e6f 6d69 616c he lead monomial │ │ │ │ -0000d050: 2063 6f6e 7369 6465 7269 6e67 2074 6865 considering the │ │ │ │ -0000d060: 2067 6976 656e 2074 6169 6c0a 2020 2020 given tail. │ │ │ │ -0000d070: 2020 2020 6d6f 6e6f 6d69 616c 7320 696e monomials in │ │ │ │ -0000d080: 2024 4c24 2e20 5468 6520 6e75 6d62 6572 $L$. The number │ │ │ │ -0000d090: 206f 6620 726f 7773 206f 6620 7468 6520 of rows of the │ │ │ │ -0000d0a0: 6d61 7472 6978 2069 7320 7468 6520 6e75 matrix is the nu │ │ │ │ -0000d0b0: 6d62 6572 206f 660a 2020 2020 2020 2020 mber of. │ │ │ │ -0000d0c0: 7661 7269 6162 6c65 7320 696e 2074 6865 variables in the │ │ │ │ -0000d0d0: 2072 696e 6720 6f66 2024 4d24 2c20 5468 ring of $M$, Th │ │ │ │ -0000d0e0: 6520 6e75 6d62 6572 206f 6620 636f 6c75 e number of colu │ │ │ │ -0000d0f0: 6d6e 7320 6973 2074 6865 2074 6f74 616c mns is the total │ │ │ │ -0000d100: 2073 697a 650a 2020 2020 2020 2020 6f66 size. of │ │ │ │ -0000d110: 2024 4c24 0a0a 4465 7363 7269 7074 696f $L$..Descriptio │ │ │ │ -0000d120: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b n.===========..+ │ │ │ │ -0000d130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000d180: 6931 203a 2052 203d 205a 5a2f 3332 3030 i1 : R = ZZ/3200 │ │ │ │ -0000d190: 335b 612c 622c 632c 2064 5d3b 2020 2020 3[a,b,c, d]; │ │ │ │ -0000d1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d1c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000d1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000d220: 6932 203a 204d 203d 2069 6465 616c 2028 i2 : M = ideal ( │ │ │ │ -0000d230: 615e 322c 2061 2a62 2c20 625e 3229 3b20 a^2, a*b, b^2); │ │ │ │ -0000d240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d260: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000d270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d2b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000d2c0: 6f32 203a 2049 6465 616c 206f 6620 5220 o2 : Ideal of R │ │ │ │ -0000d2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d300: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000d310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000d360: 6933 203a 204c 203d 2073 6d61 6c6c 6572 i3 : L = smaller │ │ │ │ -0000d370: 4d6f 6e6f 6d69 616c 7320 4d3b 2020 2020 Monomials M; │ │ │ │ -0000d380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d3a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000d3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000d400: 6934 203a 206d 6174 203d 2066 696e 6457 i4 : mat = findW │ │ │ │ -0000d410: 6569 6768 7443 6f6e 7374 7261 696e 7473 eightConstraints │ │ │ │ -0000d420: 284d 2c4c 2920 2020 2020 2020 2020 2020 (M,L) │ │ │ │ +0000b590: 2d2d 2d7c 0a7c 4869 6c62 6572 7420 7363 ---|.|Hilbert sc │ │ │ │ +0000b5a0: 6865 6d65 732e 2056 6572 7369 6f6e 7e30 hemes. Version~0 │ │ │ │ +0000b5b0: 2e39 7d7d 2c20 2020 2020 2020 2020 2020 .9}}, │ │ │ │ +0000b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b5e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0000b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b630: 2020 207c 0a7c 6174 2020 2020 2020 2020 |.|at │ │ │ │ +0000b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b680: 2020 207c 0a7c 2f4d 6163 6175 6c61 7932 |.|/Macaulay2 │ │ │ │ +0000b690: 2f70 6163 6b61 6765 737d 7d20 2020 2020 /packages}} │ │ │ │ +0000b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000b6d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0000b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000b700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000b710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000b720: 2d2d 2d2b 0a0a 4578 706f 7274 730a 3d3d ---+..Exports.== │ │ │ │ +0000b730: 3d3d 3d3d 3d0a 0a20 202a 2046 756e 6374 =====.. * Funct │ │ │ │ +0000b740: 696f 6e73 2061 6e64 2063 6f6d 6d61 6e64 ions and command │ │ │ │ +0000b750: 730a 2020 2020 2020 2a20 2a6e 6f74 6520 s. * *note │ │ │ │ +0000b760: 6669 6e64 5765 6967 6874 436f 6e73 7472 findWeightConstr │ │ │ │ +0000b770: 6169 6e74 733a 2066 696e 6457 6569 6768 aints: findWeigh │ │ │ │ +0000b780: 7443 6f6e 7374 7261 696e 7473 2c20 2d2d tConstraints, -- │ │ │ │ +0000b790: 2072 6574 7572 6e73 2061 206d 6174 7269 returns a matri │ │ │ │ +0000b7a0: 780a 2020 2020 2020 2020 6f66 2077 6569 x. of wei │ │ │ │ +0000b7b0: 6768 7420 636f 6e73 7472 6169 6e74 730a ght constraints. │ │ │ │ +0000b7c0: 2020 2020 2020 2a20 2a6e 6f74 6520 6669 * *note fi │ │ │ │ +0000b7d0: 6e64 5765 6967 6874 5665 6374 6f72 3a20 ndWeightVector: │ │ │ │ +0000b7e0: 6669 6e64 5765 6967 6874 5665 6374 6f72 findWeightVector │ │ │ │ +0000b7f0: 2c20 2d2d 2072 6574 7572 6e73 2061 2077 , -- returns a w │ │ │ │ +0000b800: 6569 6768 7420 7665 6374 6f72 0a20 2020 eight vector. │ │ │ │ +0000b810: 2020 202a 202a 6e6f 7465 2067 726f 6562 * *note groeb │ │ │ │ +0000b820: 6e65 7246 616d 696c 793a 2067 726f 6562 nerFamily: groeb │ │ │ │ +0000b830: 6e65 7246 616d 696c 792c 202d 2d20 636f nerFamily, -- co │ │ │ │ +0000b840: 6d70 7574 6573 2066 616d 696c 6965 7320 mputes families │ │ │ │ +0000b850: 6f66 2069 6465 616c 730a 2020 2020 2020 of ideals. │ │ │ │ +0000b860: 2020 7769 7468 2061 2073 7065 6369 6669 with a specifi │ │ │ │ +0000b870: 6564 2069 6e69 7469 616c 2069 6465 616c ed initial ideal │ │ │ │ +0000b880: 0a20 2020 2020 202a 202a 6e6f 7465 2067 . * *note g │ │ │ │ +0000b890: 726f 6562 6e65 7253 7472 6174 756d 3a20 roebnerStratum: │ │ │ │ +0000b8a0: 6772 6f65 626e 6572 5374 7261 7475 6d2c groebnerStratum, │ │ │ │ +0000b8b0: 202d 2d20 636f 6d70 7574 6520 7468 6520 -- compute the │ │ │ │ +0000b8c0: 6964 6561 6c20 7768 6572 6520 610a 2020 ideal where a. │ │ │ │ +0000b8d0: 2020 2020 2020 6769 7665 6e20 6973 2061 given is a │ │ │ │ +0000b8e0: 2047 726f 6562 6e65 7220 6261 7369 730a Groebner basis. │ │ │ │ +0000b8f0: 2020 2020 2020 2a20 2a6e 6f74 6520 6c69 * *note li │ │ │ │ +0000b900: 6e65 6172 5061 7274 3a20 6c69 6e65 6172 nearPart: linear │ │ │ │ +0000b910: 5061 7274 2c20 2d2d 2072 6574 7572 6e73 Part, -- returns │ │ │ │ +0000b920: 206c 696e 6561 7220 7061 7274 206f 6620 linear part of │ │ │ │ +0000b930: 6120 706f 6c79 6e6f 6d69 616c 0a20 2020 a polynomial. │ │ │ │ +0000b940: 2020 202a 202a 6e6f 7465 206e 6f6e 6d69 * *note nonmi │ │ │ │ +0000b950: 6e69 6d61 6c4d 6170 733a 206e 6f6e 6d69 nimalMaps: nonmi │ │ │ │ +0000b960: 6e69 6d61 6c4d 6170 732c 202d 2d20 6669 nimalMaps, -- fi │ │ │ │ +0000b970: 6e64 2074 6865 2064 6567 7265 6520 7a65 nd the degree ze │ │ │ │ +0000b980: 726f 206d 6170 7320 696e 0a20 2020 2020 ro maps in. │ │ │ │ +0000b990: 2020 2074 6865 2053 6368 7265 7965 7220 the Schreyer │ │ │ │ +0000b9a0: 7265 736f 6c75 7469 6f6e 206f 6620 616e resolution of an │ │ │ │ +0000b9b0: 2069 6465 616c 0a20 2020 2020 202a 2022 ideal. * " │ │ │ │ +0000b9c0: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ +0000b9d0: 696f 6e61 6c56 6172 6965 7479 2220 2d2d ionalVariety" -- │ │ │ │ +0000b9e0: 2073 6565 202a 6e6f 7465 0a20 2020 2020 see *note. │ │ │ │ +0000b9f0: 2020 2072 616e 646f 6d50 6f69 6e74 4f6e randomPointOn │ │ │ │ +0000ba00: 5261 7469 6f6e 616c 5661 7269 6574 7928 RationalVariety( │ │ │ │ +0000ba10: 4964 6561 6c29 3a0a 2020 2020 2020 2020 Ideal):. │ │ │ │ +0000ba20: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ +0000ba30: 696f 6e61 6c56 6172 6965 7479 5f6c 7049 ionalVariety_lpI │ │ │ │ +0000ba40: 6465 616c 5f72 702c 202d 2d20 6669 6e64 deal_rp, -- find │ │ │ │ +0000ba50: 2061 2072 616e 646f 6d20 706f 696e 7420 a random point │ │ │ │ +0000ba60: 6f6e 2061 0a20 2020 2020 2020 2076 6172 on a. var │ │ │ │ +0000ba70: 6965 7479 2074 6861 7420 6361 6e20 6265 iety that can be │ │ │ │ +0000ba80: 2064 6574 6563 7465 6420 746f 2062 6520 detected to be │ │ │ │ +0000ba90: 7261 7469 6f6e 616c 0a20 2020 2020 202a rational. * │ │ │ │ +0000baa0: 2022 7261 6e64 6f6d 506f 696e 7473 4f6e "randomPointsOn │ │ │ │ +0000bab0: 5261 7469 6f6e 616c 5661 7269 6574 7922 RationalVariety" │ │ │ │ +0000bac0: 202d 2d20 7365 6520 2a6e 6f74 650a 2020 -- see *note. │ │ │ │ +0000bad0: 2020 2020 2020 7261 6e64 6f6d 506f 696e randomPoin │ │ │ │ +0000bae0: 7473 4f6e 5261 7469 6f6e 616c 5661 7269 tsOnRationalVari │ │ │ │ +0000baf0: 6574 7928 4964 6561 6c2c 5a5a 293a 0a20 ety(Ideal,ZZ):. │ │ │ │ +0000bb00: 2020 2020 2020 2072 616e 646f 6d50 6f69 randomPoi │ │ │ │ +0000bb10: 6e74 734f 6e52 6174 696f 6e61 6c56 6172 ntsOnRationalVar │ │ │ │ +0000bb20: 6965 7479 5f6c 7049 6465 616c 5f63 6d5a iety_lpIdeal_cmZ │ │ │ │ +0000bb30: 5a5f 7270 2c20 2d2d 2066 696e 6420 7261 Z_rp, -- find ra │ │ │ │ +0000bb40: 6e64 6f6d 2070 6f69 6e74 7320 6f6e 0a20 ndom points on. │ │ │ │ +0000bb50: 2020 2020 2020 2061 2076 6172 6965 7479 a variety │ │ │ │ +0000bb60: 2074 6861 7420 6361 6e20 6265 2064 6574 that can be det │ │ │ │ +0000bb70: 6563 7465 6420 746f 2062 6520 7261 7469 ected to be rati │ │ │ │ +0000bb80: 6f6e 616c 0a20 2020 2020 202a 202a 6e6f onal. * *no │ │ │ │ +0000bb90: 7465 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 te smallerMonomi │ │ │ │ +0000bba0: 616c 733a 2073 6d61 6c6c 6572 4d6f 6e6f als: smallerMono │ │ │ │ +0000bbb0: 6d69 616c 732c 202d 2d20 7265 7475 726e mials, -- return │ │ │ │ +0000bbc0: 7320 7468 6520 7374 616e 6461 7264 0a20 s the standard. │ │ │ │ +0000bbd0: 2020 2020 2020 206d 6f6e 6f6d 6961 6c73 monomials │ │ │ │ +0000bbe0: 2073 6d61 6c6c 6572 2062 7574 206f 6620 smaller but of │ │ │ │ +0000bbf0: 7468 6520 7361 6d65 2064 6567 7265 6520 the same degree │ │ │ │ +0000bc00: 6173 2067 6976 656e 206d 6f6e 6f6d 6961 as given monomia │ │ │ │ +0000bc10: 6c28 7329 0a20 2020 2020 202a 202a 6e6f l(s). * *no │ │ │ │ +0000bc20: 7465 2073 7461 6e64 6172 644d 6f6e 6f6d te standardMonom │ │ │ │ +0000bc30: 6961 6c73 3a20 7374 616e 6461 7264 4d6f ials: standardMo │ │ │ │ +0000bc40: 6e6f 6d69 616c 732c 202d 2d20 636f 6d70 nomials, -- comp │ │ │ │ +0000bc50: 7574 6573 2073 7461 6e64 6172 640a 2020 utes standard. │ │ │ │ +0000bc60: 2020 2020 2020 6d6f 6e6f 6d69 616c 730a monomials. │ │ │ │ +0000bc70: 2020 2020 2020 2a20 2a6e 6f74 6520 7461 * *note ta │ │ │ │ +0000bc80: 696c 4d6f 6e6f 6d69 616c 733a 2074 6169 ilMonomials: tai │ │ │ │ +0000bc90: 6c4d 6f6e 6f6d 6961 6c73 2c20 2d2d 2066 lMonomials, -- f │ │ │ │ +0000bca0: 696e 6420 7461 696c 206d 6f6e 6f6d 6961 ind tail monomia │ │ │ │ +0000bcb0: 6c73 0a20 202a 204d 6574 686f 6473 0a20 ls. * Methods. │ │ │ │ +0000bcc0: 2020 2020 202a 2022 6669 6e64 5765 6967 * "findWeig │ │ │ │ +0000bcd0: 6874 436f 6e73 7472 6169 6e74 7328 4964 htConstraints(Id │ │ │ │ +0000bce0: 6561 6c2c 4c69 7374 2922 202d 2d20 7365 eal,List)" -- se │ │ │ │ +0000bcf0: 6520 2a6e 6f74 6520 6669 6e64 5765 6967 e *note findWeig │ │ │ │ +0000bd00: 6874 436f 6e73 7472 6169 6e74 733a 0a20 htConstraints:. │ │ │ │ +0000bd10: 2020 2020 2020 2066 696e 6457 6569 6768 findWeigh │ │ │ │ +0000bd20: 7443 6f6e 7374 7261 696e 7473 2c20 2d2d tConstraints, -- │ │ │ │ +0000bd30: 2072 6574 7572 6e73 2061 206d 6174 7269 returns a matri │ │ │ │ +0000bd40: 7820 6f66 2077 6569 6768 7420 636f 6e73 x of weight cons │ │ │ │ +0000bd50: 7472 6169 6e74 730a 2020 2020 2020 2a20 traints. * │ │ │ │ +0000bd60: 2266 696e 6457 6569 6768 7456 6563 746f "findWeightVecto │ │ │ │ +0000bd70: 7228 4964 6561 6c2c 4c69 7374 2922 202d r(Ideal,List)" - │ │ │ │ +0000bd80: 2d20 7365 6520 2a6e 6f74 6520 6669 6e64 - see *note find │ │ │ │ +0000bd90: 5765 6967 6874 5665 6374 6f72 3a0a 2020 WeightVector:. │ │ │ │ +0000bda0: 2020 2020 2020 6669 6e64 5765 6967 6874 findWeight │ │ │ │ +0000bdb0: 5665 6374 6f72 2c20 2d2d 2072 6574 7572 Vector, -- retur │ │ │ │ +0000bdc0: 6e73 2061 2077 6569 6768 7420 7665 6374 ns a weight vect │ │ │ │ +0000bdd0: 6f72 0a20 2020 2020 202a 2022 6772 6f65 or. * "groe │ │ │ │ +0000bde0: 626e 6572 4661 6d69 6c79 2849 6465 616c bnerFamily(Ideal │ │ │ │ +0000bdf0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ +0000be00: 6772 6f65 626e 6572 4661 6d69 6c79 3a20 groebnerFamily: │ │ │ │ +0000be10: 6772 6f65 626e 6572 4661 6d69 6c79 2c20 groebnerFamily, │ │ │ │ +0000be20: 2d2d 0a20 2020 2020 2020 2063 6f6d 7075 --. compu │ │ │ │ +0000be30: 7465 7320 6661 6d69 6c69 6573 206f 6620 tes families of │ │ │ │ +0000be40: 6964 6561 6c73 2077 6974 6820 6120 7370 ideals with a sp │ │ │ │ +0000be50: 6563 6966 6965 6420 696e 6974 6961 6c20 ecified initial │ │ │ │ +0000be60: 6964 6561 6c0a 2020 2020 2020 2a20 2267 ideal. * "g │ │ │ │ +0000be70: 726f 6562 6e65 7246 616d 696c 7928 4964 roebnerFamily(Id │ │ │ │ +0000be80: 6561 6c2c 4c69 7374 2922 202d 2d20 7365 eal,List)" -- se │ │ │ │ +0000be90: 6520 2a6e 6f74 6520 6772 6f65 626e 6572 e *note groebner │ │ │ │ +0000bea0: 4661 6d69 6c79 3a0a 2020 2020 2020 2020 Family:. │ │ │ │ +0000beb0: 6772 6f65 626e 6572 4661 6d69 6c79 2c20 groebnerFamily, │ │ │ │ +0000bec0: 2d2d 2063 6f6d 7075 7465 7320 6661 6d69 -- computes fami │ │ │ │ +0000bed0: 6c69 6573 206f 6620 6964 6561 6c73 2077 lies of ideals w │ │ │ │ +0000bee0: 6974 6820 6120 7370 6563 6966 6965 6420 ith a specified │ │ │ │ +0000bef0: 696e 6974 6961 6c0a 2020 2020 2020 2020 initial. │ │ │ │ +0000bf00: 6964 6561 6c0a 2020 2020 2020 2a20 2267 ideal. * "g │ │ │ │ +0000bf10: 726f 6562 6e65 7253 7472 6174 756d 2849 roebnerStratum(I │ │ │ │ +0000bf20: 6465 616c 2922 202d 2d20 7365 6520 2a6e deal)" -- see *n │ │ │ │ +0000bf30: 6f74 6520 6772 6f65 626e 6572 5374 7261 ote groebnerStra │ │ │ │ +0000bf40: 7475 6d3a 2067 726f 6562 6e65 7253 7472 tum: groebnerStr │ │ │ │ +0000bf50: 6174 756d 2c0a 2020 2020 2020 2020 2d2d atum,. -- │ │ │ │ +0000bf60: 2063 6f6d 7075 7465 2074 6865 2069 6465 compute the ide │ │ │ │ +0000bf70: 616c 2077 6865 7265 2061 2067 6976 656e al where a given │ │ │ │ +0000bf80: 2069 7320 6120 4772 6f65 626e 6572 2062 is a Groebner b │ │ │ │ +0000bf90: 6173 6973 0a20 2020 2020 202a 2022 6e6f asis. * "no │ │ │ │ +0000bfa0: 6e6d 696e 696d 616c 4d61 7073 2849 6465 nminimalMaps(Ide │ │ │ │ +0000bfb0: 616c 2922 202d 2d20 7365 6520 2a6e 6f74 al)" -- see *not │ │ │ │ +0000bfc0: 6520 6e6f 6e6d 696e 696d 616c 4d61 7073 e nonminimalMaps │ │ │ │ +0000bfd0: 3a20 6e6f 6e6d 696e 696d 616c 4d61 7073 : nonminimalMaps │ │ │ │ +0000bfe0: 2c20 2d2d 0a20 2020 2020 2020 2066 696e , --. fin │ │ │ │ +0000bff0: 6420 7468 6520 6465 6772 6565 207a 6572 d the degree zer │ │ │ │ +0000c000: 6f20 6d61 7073 2069 6e20 7468 6520 5363 o maps in the Sc │ │ │ │ +0000c010: 6872 6579 6572 2072 6573 6f6c 7574 696f hreyer resolutio │ │ │ │ +0000c020: 6e20 6f66 2061 6e20 6964 6561 6c0a 2020 n of an ideal. │ │ │ │ +0000c030: 2020 2020 2a20 2a6e 6f74 6520 7261 6e64 * *note rand │ │ │ │ +0000c040: 6f6d 506f 696e 744f 6e52 6174 696f 6e61 omPointOnRationa │ │ │ │ +0000c050: 6c56 6172 6965 7479 2849 6465 616c 293a lVariety(Ideal): │ │ │ │ +0000c060: 0a20 2020 2020 2020 2072 616e 646f 6d50 . randomP │ │ │ │ +0000c070: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ +0000c080: 7269 6574 795f 6c70 4964 6561 6c5f 7270 riety_lpIdeal_rp │ │ │ │ +0000c090: 2c20 2d2d 2066 696e 6420 6120 7261 6e64 , -- find a rand │ │ │ │ +0000c0a0: 6f6d 2070 6f69 6e74 206f 6e20 610a 2020 om point on a. │ │ │ │ +0000c0b0: 2020 2020 2020 7661 7269 6574 7920 7468 variety th │ │ │ │ +0000c0c0: 6174 2063 616e 2062 6520 6465 7465 6374 at can be detect │ │ │ │ +0000c0d0: 6564 2074 6f20 6265 2072 6174 696f 6e61 ed to be rationa │ │ │ │ +0000c0e0: 6c0a 2020 2020 2020 2a20 2a6e 6f74 6520 l. * *note │ │ │ │ +0000c0f0: 7261 6e64 6f6d 506f 696e 7473 4f6e 5261 randomPointsOnRa │ │ │ │ +0000c100: 7469 6f6e 616c 5661 7269 6574 7928 4964 tionalVariety(Id │ │ │ │ +0000c110: 6561 6c2c 5a5a 293a 0a20 2020 2020 2020 eal,ZZ):. │ │ │ │ +0000c120: 2072 616e 646f 6d50 6f69 6e74 734f 6e52 randomPointsOnR │ │ │ │ +0000c130: 6174 696f 6e61 6c56 6172 6965 7479 5f6c ationalVariety_l │ │ │ │ +0000c140: 7049 6465 616c 5f63 6d5a 5a5f 7270 2c20 pIdeal_cmZZ_rp, │ │ │ │ +0000c150: 2d2d 2066 696e 6420 7261 6e64 6f6d 2070 -- find random p │ │ │ │ +0000c160: 6f69 6e74 7320 6f6e 0a20 2020 2020 2020 oints on. │ │ │ │ +0000c170: 2061 2076 6172 6965 7479 2074 6861 7420 a variety that │ │ │ │ +0000c180: 6361 6e20 6265 2064 6574 6563 7465 6420 can be detected │ │ │ │ +0000c190: 746f 2062 6520 7261 7469 6f6e 616c 0a20 to be rational. │ │ │ │ +0000c1a0: 2020 2020 202a 2022 736d 616c 6c65 724d * "smallerM │ │ │ │ +0000c1b0: 6f6e 6f6d 6961 6c73 2849 6465 616c 2922 onomials(Ideal)" │ │ │ │ +0000c1c0: 202d 2d20 7365 6520 2a6e 6f74 6520 736d -- see *note sm │ │ │ │ +0000c1d0: 616c 6c65 724d 6f6e 6f6d 6961 6c73 3a0a allerMonomials:. │ │ │ │ +0000c1e0: 2020 2020 2020 2020 736d 616c 6c65 724d smallerM │ │ │ │ +0000c1f0: 6f6e 6f6d 6961 6c73 2c20 2d2d 2072 6574 onomials, -- ret │ │ │ │ +0000c200: 7572 6e73 2074 6865 2073 7461 6e64 6172 urns the standar │ │ │ │ +0000c210: 6420 6d6f 6e6f 6d69 616c 7320 736d 616c d monomials smal │ │ │ │ +0000c220: 6c65 7220 6275 7420 6f66 2074 6865 0a20 ler but of the. │ │ │ │ +0000c230: 2020 2020 2020 2073 616d 6520 6465 6772 same degr │ │ │ │ +0000c240: 6565 2061 7320 6769 7665 6e20 6d6f 6e6f ee as given mono │ │ │ │ +0000c250: 6d69 616c 2873 290a 2020 2020 2020 2a20 mial(s). * │ │ │ │ +0000c260: 2273 6d61 6c6c 6572 4d6f 6e6f 6d69 616c "smallerMonomial │ │ │ │ +0000c270: 7328 4964 6561 6c2c 5269 6e67 456c 656d s(Ideal,RingElem │ │ │ │ +0000c280: 656e 7429 2220 2d2d 2073 6565 202a 6e6f ent)" -- see *no │ │ │ │ +0000c290: 7465 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 te smallerMonomi │ │ │ │ +0000c2a0: 616c 733a 0a20 2020 2020 2020 2073 6d61 als:. sma │ │ │ │ +0000c2b0: 6c6c 6572 4d6f 6e6f 6d69 616c 732c 202d llerMonomials, - │ │ │ │ +0000c2c0: 2d20 7265 7475 726e 7320 7468 6520 7374 - returns the st │ │ │ │ +0000c2d0: 616e 6461 7264 206d 6f6e 6f6d 6961 6c73 andard monomials │ │ │ │ +0000c2e0: 2073 6d61 6c6c 6572 2062 7574 206f 6620 smaller but of │ │ │ │ +0000c2f0: 7468 650a 2020 2020 2020 2020 7361 6d65 the. same │ │ │ │ +0000c300: 2064 6567 7265 6520 6173 2067 6976 656e degree as given │ │ │ │ +0000c310: 206d 6f6e 6f6d 6961 6c28 7329 0a20 2020 monomial(s). │ │ │ │ +0000c320: 2020 202a 2022 7374 616e 6461 7264 4d6f * "standardMo │ │ │ │ +0000c330: 6e6f 6d69 616c 7328 4964 6561 6c29 2220 nomials(Ideal)" │ │ │ │ +0000c340: 2d2d 2073 6565 202a 6e6f 7465 2073 7461 -- see *note sta │ │ │ │ +0000c350: 6e64 6172 644d 6f6e 6f6d 6961 6c73 3a0a ndardMonomials:. │ │ │ │ +0000c360: 2020 2020 2020 2020 7374 616e 6461 7264 standard │ │ │ │ +0000c370: 4d6f 6e6f 6d69 616c 732c 202d 2d20 636f Monomials, -- co │ │ │ │ +0000c380: 6d70 7574 6573 2073 7461 6e64 6172 6420 mputes standard │ │ │ │ +0000c390: 6d6f 6e6f 6d69 616c 730a 2020 2020 2020 monomials. │ │ │ │ +0000c3a0: 2a20 2273 7461 6e64 6172 644d 6f6e 6f6d * "standardMonom │ │ │ │ +0000c3b0: 6961 6c73 284c 6973 742c 4964 6561 6c29 ials(List,Ideal) │ │ │ │ +0000c3c0: 2220 2d2d 2073 6565 202a 6e6f 7465 2073 " -- see *note s │ │ │ │ +0000c3d0: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ +0000c3e0: 3a0a 2020 2020 2020 2020 7374 616e 6461 :. standa │ │ │ │ +0000c3f0: 7264 4d6f 6e6f 6d69 616c 732c 202d 2d20 rdMonomials, -- │ │ │ │ +0000c400: 636f 6d70 7574 6573 2073 7461 6e64 6172 computes standar │ │ │ │ +0000c410: 6420 6d6f 6e6f 6d69 616c 730a 2020 2020 d monomials. │ │ │ │ +0000c420: 2020 2a20 2273 7461 6e64 6172 644d 6f6e * "standardMon │ │ │ │ +0000c430: 6f6d 6961 6c73 285a 5a2c 4964 6561 6c29 omials(ZZ,Ideal) │ │ │ │ +0000c440: 2220 2d2d 2073 6565 202a 6e6f 7465 2073 " -- see *note s │ │ │ │ +0000c450: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ +0000c460: 3a0a 2020 2020 2020 2020 7374 616e 6461 :. standa │ │ │ │ +0000c470: 7264 4d6f 6e6f 6d69 616c 732c 202d 2d20 rdMonomials, -- │ │ │ │ +0000c480: 636f 6d70 7574 6573 2073 7461 6e64 6172 computes standar │ │ │ │ +0000c490: 6420 6d6f 6e6f 6d69 616c 730a 2020 2020 d monomials. │ │ │ │ +0000c4a0: 2020 2a20 2274 6169 6c4d 6f6e 6f6d 6961 * "tailMonomia │ │ │ │ +0000c4b0: 6c73 2849 6465 616c 2922 202d 2d20 7365 ls(Ideal)" -- se │ │ │ │ +0000c4c0: 6520 2a6e 6f74 6520 7461 696c 4d6f 6e6f e *note tailMono │ │ │ │ +0000c4d0: 6d69 616c 733a 2074 6169 6c4d 6f6e 6f6d mials: tailMonom │ │ │ │ +0000c4e0: 6961 6c73 2c20 2d2d 0a20 2020 2020 2020 ials, --. │ │ │ │ +0000c4f0: 2066 696e 6420 7461 696c 206d 6f6e 6f6d find tail monom │ │ │ │ +0000c500: 6961 6c73 0a20 2020 2020 202a 2022 7461 ials. * "ta │ │ │ │ +0000c510: 696c 4d6f 6e6f 6d69 616c 7328 4964 6561 ilMonomials(Idea │ │ │ │ +0000c520: 6c2c 5269 6e67 456c 656d 656e 7429 2220 l,RingElement)" │ │ │ │ +0000c530: 2d2d 2073 6565 202a 6e6f 7465 2074 6169 -- see *note tai │ │ │ │ +0000c540: 6c4d 6f6e 6f6d 6961 6c73 3a0a 2020 2020 lMonomials:. │ │ │ │ +0000c550: 2020 2020 7461 696c 4d6f 6e6f 6d69 616c tailMonomial │ │ │ │ +0000c560: 732c 202d 2d20 6669 6e64 2074 6169 6c20 s, -- find tail │ │ │ │ +0000c570: 6d6f 6e6f 6d69 616c 730a 2020 2a20 5379 monomials. * Sy │ │ │ │ +0000c580: 6d62 6f6c 730a 2020 2020 2020 2a20 2a6e mbols. * *n │ │ │ │ +0000c590: 6f74 6520 416c 6c53 7461 6e64 6172 643a ote AllStandard: │ │ │ │ +0000c5a0: 2041 6c6c 5374 616e 6461 7264 2c20 2d2d AllStandard, -- │ │ │ │ +0000c5b0: 2062 6f6f 6c65 616e 206f 7074 696f 6e20 boolean option │ │ │ │ +0000c5c0: 666f 7220 6465 7465 726d 696e 696e 6720 for determining │ │ │ │ +0000c5d0: 7468 650a 2020 2020 2020 2020 7573 6520 the. use │ │ │ │ +0000c5e0: 6f66 2061 6c6c 2073 7461 6e64 6172 6420 of all standard │ │ │ │ +0000c5f0: 6f72 2073 6d61 6c6c 6572 2073 7461 6e64 or smaller stand │ │ │ │ +0000c600: 6172 6420 6d6f 6e6f 6d69 616c 730a 2020 ard monomials. │ │ │ │ +0000c610: 2020 2020 2a20 2a6e 6f74 6520 4d69 6e69 * *note Mini │ │ │ │ +0000c620: 6d61 6c69 7a65 3a20 4d69 6e69 6d61 6c69 malize: Minimali │ │ │ │ +0000c630: 7a65 2c20 2d2d 2062 6f6f 6c65 616e 206f ze, -- boolean o │ │ │ │ +0000c640: 7074 696f 6e20 6f66 2067 726f 6562 6e65 ption of groebne │ │ │ │ +0000c650: 7253 7472 6174 756d 0a20 2020 2020 2020 rStratum. │ │ │ │ +0000c660: 2028 6375 7272 656e 746c 7920 6120 6e6f (currently a no │ │ │ │ +0000c670: 2d6f 7029 0a0a 466f 7220 7468 6520 7072 -op)..For the pr │ │ │ │ +0000c680: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +0000c690: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +0000c6a0: 206f 626a 6563 7420 2a6e 6f74 6520 4772 object *note Gr │ │ │ │ +0000c6b0: 6f65 626e 6572 5374 7261 7461 3a20 546f oebnerStrata: To │ │ │ │ +0000c6c0: 702c 2069 7320 6120 2a6e 6f74 6520 7061 p, is a *note pa │ │ │ │ +0000c6d0: 636b 6167 653a 0a28 4d61 6361 756c 6179 ckage:.(Macaulay │ │ │ │ +0000c6e0: 3244 6f63 2950 6163 6b61 6765 2c2c 2064 2Doc)Package,, d │ │ │ │ +0000c6f0: 6566 696e 6564 2069 6e20 4772 6f65 626e efined in Groebn │ │ │ │ +0000c700: 6572 5374 7261 7461 2e6d 322e 0a0a 2d2d erStrata.m2...-- │ │ │ │ +0000c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +0000c760: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +0000c770: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +0000c780: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +0000c790: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +0000c7a0: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ +0000c7b0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +0000c7c0: 6167 6573 2f0a 4772 6f65 626e 6572 5374 ages/.GroebnerSt │ │ │ │ +0000c7d0: 7261 7461 2e6d 323a 3435 393a 302e 0a1f rata.m2:459:0... │ │ │ │ +0000c7e0: 0a46 696c 653a 2047 726f 6562 6e65 7253 .File: GroebnerS │ │ │ │ +0000c7f0: 7472 6174 612e 696e 666f 2c20 4e6f 6465 trata.info, Node │ │ │ │ +0000c800: 3a20 416c 6c53 7461 6e64 6172 642c 204e : AllStandard, N │ │ │ │ +0000c810: 6578 743a 2066 696e 6457 6569 6768 7443 ext: findWeightC │ │ │ │ +0000c820: 6f6e 7374 7261 696e 7473 2c20 5072 6576 onstraints, Prev │ │ │ │ +0000c830: 3a20 546f 702c 2055 703a 2054 6f70 0a0a : Top, Up: Top.. │ │ │ │ +0000c840: 416c 6c53 7461 6e64 6172 6420 2d2d 2062 AllStandard -- b │ │ │ │ +0000c850: 6f6f 6c65 616e 206f 7074 696f 6e20 666f oolean option fo │ │ │ │ +0000c860: 7220 6465 7465 726d 696e 696e 6720 7468 r determining th │ │ │ │ +0000c870: 6520 7573 6520 6f66 2061 6c6c 2073 7461 e use of all sta │ │ │ │ +0000c880: 6e64 6172 6420 6f72 2073 6d61 6c6c 6572 ndard or smaller │ │ │ │ +0000c890: 2073 7461 6e64 6172 6420 6d6f 6e6f 6d69 standard monomi │ │ │ │ +0000c8a0: 616c 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a als.************ │ │ │ │ +0000c8b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000c8c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000c8d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000c8e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000c8f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000c900: 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 6970 *******..Descrip │ │ │ │ +0000c910: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +0000c920: 0a0a 5468 6973 2069 7320 616e 206f 7074 ..This is an opt │ │ │ │ +0000c930: 696f 6e61 6c20 696e 7075 7420 666f 7220 ional input for │ │ │ │ +0000c940: 7468 6520 2a6e 6f74 6520 6772 6f65 626e the *note groebn │ │ │ │ +0000c950: 6572 5374 7261 7475 6d3a 2067 726f 6562 erStratum: groeb │ │ │ │ +0000c960: 6e65 7253 7472 6174 756d 2c20 616e 640a nerStratum, and. │ │ │ │ +0000c970: 2a6e 6f74 6520 6772 6f65 626e 6572 4661 *note groebnerFa │ │ │ │ +0000c980: 6d69 6c79 3a20 6772 6f65 626e 6572 4661 mily: groebnerFa │ │ │ │ +0000c990: 6d69 6c79 2c20 6675 6e63 7469 6f6e 732e mily, functions. │ │ │ │ +0000c9a0: 2020 4974 2074 616b 6573 2076 616c 7565 It takes value │ │ │ │ +0000c9b0: 7320 2474 7275 6524 2061 6e64 0a24 6661 s $true$ and.$fa │ │ │ │ +0000c9c0: 6c73 6524 2e20 2057 6865 6e20 6173 7369 lse$. When assi │ │ │ │ +0000c9d0: 676e 6564 2074 6865 2076 616c 7565 2024 gned the value $ │ │ │ │ +0000c9e0: 7472 7565 242c 2074 6865 2066 756e 6374 true$, the funct │ │ │ │ +0000c9f0: 696f 6e73 2061 7265 2063 6f6d 7075 7465 ions are compute │ │ │ │ +0000ca00: 6420 7769 7468 0a72 6573 7065 6374 2074 d with.respect t │ │ │ │ +0000ca10: 6f20 616c 6c20 7374 616e 6461 7264 206d o all standard m │ │ │ │ +0000ca20: 6f6e 6f6d 6961 6c73 2066 6f72 2074 6865 onomials for the │ │ │ │ +0000ca30: 2073 7065 6369 6669 6564 2067 656e 6572 specified gener │ │ │ │ +0000ca40: 6174 6f72 732e 2020 5768 656e 2061 7373 ators. When ass │ │ │ │ +0000ca50: 6967 6e65 640a 7468 6520 7661 6c75 6520 igned.the value │ │ │ │ +0000ca60: 2466 616c 7365 242c 2074 6865 2066 756e $false$, the fun │ │ │ │ +0000ca70: 6374 696f 6e73 2061 7265 2063 6f6d 7075 ctions are compu │ │ │ │ +0000ca80: 7465 6420 7769 7468 2072 6573 7065 6374 ted with respect │ │ │ │ +0000ca90: 2074 6f20 736d 616c 6c65 7220 6d6f 6e6f to smaller mono │ │ │ │ +0000caa0: 6d69 616c 730a 6f66 2074 6865 2073 616d mials.of the sam │ │ │ │ +0000cab0: 6520 6465 6772 6565 2061 7320 7468 6520 e degree as the │ │ │ │ +0000cac0: 7370 6563 6966 6965 6420 6765 6e65 7261 specified genera │ │ │ │ +0000cad0: 746f 7273 2062 7574 2077 6869 6368 2064 tors but which d │ │ │ │ +0000cae0: 6f20 6e6f 7420 6c69 6520 696e 2074 6865 o not lie in the │ │ │ │ +0000caf0: 0a69 6465 616c 2e0a 0a53 6565 2061 6c73 .ideal...See als │ │ │ │ +0000cb00: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0000cb10: 2a6e 6f74 6520 6772 6f65 626e 6572 5374 *note groebnerSt │ │ │ │ +0000cb20: 7261 7475 6d3a 2067 726f 6562 6e65 7253 ratum: groebnerS │ │ │ │ +0000cb30: 7472 6174 756d 2c20 2d2d 2063 6f6d 7075 tratum, -- compu │ │ │ │ +0000cb40: 7465 2074 6865 2069 6465 616c 2077 6865 te the ideal whe │ │ │ │ +0000cb50: 7265 2061 2067 6976 656e 0a20 2020 2069 re a given. i │ │ │ │ +0000cb60: 7320 6120 4772 6f65 626e 6572 2062 6173 s a Groebner bas │ │ │ │ +0000cb70: 6973 0a20 202a 202a 6e6f 7465 2067 726f is. * *note gro │ │ │ │ +0000cb80: 6562 6e65 7246 616d 696c 793a 2067 726f ebnerFamily: gro │ │ │ │ +0000cb90: 6562 6e65 7246 616d 696c 792c 202d 2d20 ebnerFamily, -- │ │ │ │ +0000cba0: 636f 6d70 7574 6573 2066 616d 696c 6965 computes familie │ │ │ │ +0000cbb0: 7320 6f66 2069 6465 616c 7320 7769 7468 s of ideals with │ │ │ │ +0000cbc0: 2061 0a20 2020 2073 7065 6369 6669 6564 a. specified │ │ │ │ +0000cbd0: 2069 6e69 7469 616c 2069 6465 616c 0a20 initial ideal. │ │ │ │ +0000cbe0: 202a 202a 6e6f 7465 2073 6d61 6c6c 6572 * *note smaller │ │ │ │ +0000cbf0: 4d6f 6e6f 6d69 616c 733a 2073 6d61 6c6c Monomials: small │ │ │ │ +0000cc00: 6572 4d6f 6e6f 6d69 616c 732c 202d 2d20 erMonomials, -- │ │ │ │ +0000cc10: 7265 7475 726e 7320 7468 6520 7374 616e returns the stan │ │ │ │ +0000cc20: 6461 7264 206d 6f6e 6f6d 6961 6c73 0a20 dard monomials. │ │ │ │ +0000cc30: 2020 2073 6d61 6c6c 6572 2062 7574 206f smaller but o │ │ │ │ +0000cc40: 6620 7468 6520 7361 6d65 2064 6567 7265 f the same degre │ │ │ │ +0000cc50: 6520 6173 2067 6976 656e 206d 6f6e 6f6d e as given monom │ │ │ │ +0000cc60: 6961 6c28 7329 0a20 202a 202a 6e6f 7465 ial(s). * *note │ │ │ │ +0000cc70: 2073 7461 6e64 6172 644d 6f6e 6f6d 6961 standardMonomia │ │ │ │ +0000cc80: 6c73 3a20 7374 616e 6461 7264 4d6f 6e6f ls: standardMono │ │ │ │ +0000cc90: 6d69 616c 732c 202d 2d20 636f 6d70 7574 mials, -- comput │ │ │ │ +0000cca0: 6573 2073 7461 6e64 6172 6420 6d6f 6e6f es standard mono │ │ │ │ +0000ccb0: 6d69 616c 730a 0a46 756e 6374 696f 6e73 mials..Functions │ │ │ │ +0000ccc0: 2077 6974 6820 6f70 7469 6f6e 616c 2061 with optional a │ │ │ │ +0000ccd0: 7267 756d 656e 7420 6e61 6d65 6420 416c rgument named Al │ │ │ │ +0000cce0: 6c53 7461 6e64 6172 643a 0a3d 3d3d 3d3d lStandard:.===== │ │ │ │ +0000ccf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0000cd00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0000cd10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0000cd20: 2020 2a20 2267 726f 6562 6e65 7246 616d * "groebnerFam │ │ │ │ +0000cd30: 696c 7928 2e2e 2e2c 416c 6c53 7461 6e64 ily(...,AllStand │ │ │ │ +0000cd40: 6172 643d 3e2e 2e2e 2922 202d 2d20 7365 ard=>...)" -- se │ │ │ │ +0000cd50: 6520 2a6e 6f74 6520 6772 6f65 626e 6572 e *note groebner │ │ │ │ +0000cd60: 4661 6d69 6c79 3a0a 2020 2020 6772 6f65 Family:. groe │ │ │ │ +0000cd70: 626e 6572 4661 6d69 6c79 2c20 2d2d 2063 bnerFamily, -- c │ │ │ │ +0000cd80: 6f6d 7075 7465 7320 6661 6d69 6c69 6573 omputes families │ │ │ │ +0000cd90: 206f 6620 6964 6561 6c73 2077 6974 6820 of ideals with │ │ │ │ +0000cda0: 6120 7370 6563 6966 6965 6420 696e 6974 a specified init │ │ │ │ +0000cdb0: 6961 6c0a 2020 2020 6964 6561 6c0a 2020 ial. ideal. │ │ │ │ +0000cdc0: 2a20 2274 6169 6c4d 6f6e 6f6d 6961 6c73 * "tailMonomials │ │ │ │ +0000cdd0: 282e 2e2e 2c41 6c6c 5374 616e 6461 7264 (...,AllStandard │ │ │ │ +0000cde0: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +0000cdf0: 6e6f 7465 2074 6169 6c4d 6f6e 6f6d 6961 note tailMonomia │ │ │ │ +0000ce00: 6c73 3a0a 2020 2020 7461 696c 4d6f 6e6f ls:. tailMono │ │ │ │ +0000ce10: 6d69 616c 732c 202d 2d20 6669 6e64 2074 mials, -- find t │ │ │ │ +0000ce20: 6169 6c20 6d6f 6e6f 6d69 616c 730a 0a46 ail monomials..F │ │ │ │ +0000ce30: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0000ce40: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0000ce50: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0000ce60: 202a 6e6f 7465 2041 6c6c 5374 616e 6461 *note AllStanda │ │ │ │ +0000ce70: 7264 3a20 416c 6c53 7461 6e64 6172 642c rd: AllStandard, │ │ │ │ +0000ce80: 2069 7320 6120 2a6e 6f74 6520 7379 6d62 is a *note symb │ │ │ │ +0000ce90: 6f6c 3a0a 284d 6163 6175 6c61 7932 446f ol:.(Macaulay2Do │ │ │ │ +0000cea0: 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d c)Symbol,...---- │ │ │ │ +0000ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000cec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ced0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000cee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000cef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +0000cf00: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +0000cf10: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +0000cf20: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +0000cf30: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +0000cf40: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +0000cf50: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +0000cf60: 6573 2f0a 4772 6f65 626e 6572 5374 7261 es/.GroebnerStra │ │ │ │ +0000cf70: 7461 2e6d 323a 3130 3633 3a30 2e0a 1f0a ta.m2:1063:0.... │ │ │ │ +0000cf80: 4669 6c65 3a20 4772 6f65 626e 6572 5374 File: GroebnerSt │ │ │ │ +0000cf90: 7261 7461 2e69 6e66 6f2c 204e 6f64 653a rata.info, Node: │ │ │ │ +0000cfa0: 2066 696e 6457 6569 6768 7443 6f6e 7374 findWeightConst │ │ │ │ +0000cfb0: 7261 696e 7473 2c20 4e65 7874 3a20 6669 raints, Next: fi │ │ │ │ +0000cfc0: 6e64 5765 6967 6874 5665 6374 6f72 2c20 ndWeightVector, │ │ │ │ +0000cfd0: 5072 6576 3a20 416c 6c53 7461 6e64 6172 Prev: AllStandar │ │ │ │ +0000cfe0: 642c 2055 703a 2054 6f70 0a0a 6669 6e64 d, Up: Top..find │ │ │ │ +0000cff0: 5765 6967 6874 436f 6e73 7472 6169 6e74 WeightConstraint │ │ │ │ +0000d000: 7320 2d2d 2072 6574 7572 6e73 2061 206d s -- returns a m │ │ │ │ +0000d010: 6174 7269 7820 6f66 2077 6569 6768 7420 atrix of weight │ │ │ │ +0000d020: 636f 6e73 7472 6169 6e74 730a 2a2a 2a2a constraints.**** │ │ │ │ +0000d030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000d040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000d050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000d060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0000d070: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0000d080: 2063 6f6e 7374 7261 696e 7473 203d 2066 constraints = f │ │ │ │ +0000d090: 696e 6457 6569 6768 7443 6f6e 7374 7261 indWeightConstra │ │ │ │ +0000d0a0: 696e 7473 284d 2c20 4c29 0a20 202a 2049 ints(M, L). * I │ │ │ │ +0000d0b0: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ +0000d0c0: 2c20 616e 202a 6e6f 7465 2069 6465 616c , an *note ideal │ │ │ │ +0000d0d0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0000d0e0: 4964 6561 6c2c 2c20 4d20 7368 6f75 6c64 Ideal,, M should │ │ │ │ +0000d0f0: 2062 6520 6120 6d6f 6e6f 6d69 616c 2069 be a monomial i │ │ │ │ +0000d100: 6465 616c 0a20 2020 2020 202a 204c 2c20 deal. * L, │ │ │ │ +0000d110: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ +0000d120: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ +0000d130: 2c2c 2061 206c 6973 7420 6f66 206c 6973 ,, a list of lis │ │ │ │ +0000d140: 7473 206f 6620 7461 696c 206d 6f6e 6f6d ts of tail monom │ │ │ │ +0000d150: 6961 6c73 0a20 202a 204f 7574 7075 7473 ials. * Outputs │ │ │ │ +0000d160: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ +0000d170: 6520 6d61 7472 6978 3a20 284d 6163 6175 e matrix: (Macau │ │ │ │ +0000d180: 6c61 7932 446f 6329 4d61 7472 6978 2c2c lay2Doc)Matrix,, │ │ │ │ +0000d190: 2077 6974 6820 7468 6520 636f 6e73 7472 with the constr │ │ │ │ +0000d1a0: 6169 6e74 7320 6f6e 2061 2077 6569 6768 aints on a weigh │ │ │ │ +0000d1b0: 740a 2020 2020 2020 2020 7665 6374 6f72 t. vector │ │ │ │ +0000d1c0: 2066 6f72 2024 4d24 2074 6f20 6265 2074 for $M$ to be t │ │ │ │ +0000d1d0: 6865 206c 6561 6420 6d6f 6e6f 6d69 616c he lead monomial │ │ │ │ +0000d1e0: 2063 6f6e 7369 6465 7269 6e67 2074 6865 considering the │ │ │ │ +0000d1f0: 2067 6976 656e 2074 6169 6c0a 2020 2020 given tail. │ │ │ │ +0000d200: 2020 2020 6d6f 6e6f 6d69 616c 7320 696e monomials in │ │ │ │ +0000d210: 2024 4c24 2e20 5468 6520 6e75 6d62 6572 $L$. The number │ │ │ │ +0000d220: 206f 6620 726f 7773 206f 6620 7468 6520 of rows of the │ │ │ │ +0000d230: 6d61 7472 6978 2069 7320 7468 6520 6e75 matrix is the nu │ │ │ │ +0000d240: 6d62 6572 206f 660a 2020 2020 2020 2020 mber of. │ │ │ │ +0000d250: 7661 7269 6162 6c65 7320 696e 2074 6865 variables in the │ │ │ │ +0000d260: 2072 696e 6720 6f66 2024 4d24 2c20 5468 ring of $M$, Th │ │ │ │ +0000d270: 6520 6e75 6d62 6572 206f 6620 636f 6c75 e number of colu │ │ │ │ +0000d280: 6d6e 7320 6973 2074 6865 2074 6f74 616c mns is the total │ │ │ │ +0000d290: 2073 697a 650a 2020 2020 2020 2020 6f66 size. of │ │ │ │ +0000d2a0: 2024 4c24 0a0a 4465 7363 7269 7074 696f $L$..Descriptio │ │ │ │ +0000d2b0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b n.===========..+ │ │ │ │ +0000d2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000d310: 6931 203a 2052 203d 205a 5a2f 3332 3030 i1 : R = ZZ/3200 │ │ │ │ +0000d320: 335b 612c 622c 632c 2064 5d3b 2020 2020 3[a,b,c, d]; │ │ │ │ +0000d330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d350: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000d360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000d3b0: 6932 203a 204d 203d 2069 6465 616c 2028 i2 : M = ideal ( │ │ │ │ +0000d3c0: 615e 322c 2061 2a62 2c20 625e 3229 3b20 a^2, a*b, b^2); │ │ │ │ +0000d3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d3f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000d400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d440: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000d450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d450: 6f32 203a 2049 6465 616c 206f 6620 5220 o2 : Ideal of R │ │ │ │ 0000d460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d490: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000d4a0: 6f34 203d 207c 2031 2020 3220 2032 2020 o4 = | 1 2 2 │ │ │ │ -0000d4b0: 3120 2032 2020 3220 2032 2020 3020 2031 1 2 2 2 0 1 │ │ │ │ -0000d4c0: 2020 3120 2030 2020 3120 2031 2020 3120 1 0 1 1 1 │ │ │ │ -0000d4d0: 202d 3120 3020 2030 2020 2d31 2030 2020 -1 0 0 -1 0 │ │ │ │ -0000d4e0: 3020 2030 2020 7c20 2020 2020 207c 0a7c 0 0 | |.| │ │ │ │ -0000d4f0: 2020 2020 207c 2030 2020 2d31 2030 2020 | 0 -1 0 │ │ │ │ -0000d500: 3020 202d 3120 3020 2030 2020 3120 2030 0 -1 0 0 1 0 │ │ │ │ -0000d510: 2020 3120 2031 2020 3020 2031 2020 3120 1 1 0 1 1 │ │ │ │ -0000d520: 2032 2020 3120 2032 2020 3220 2031 2020 2 1 2 2 1 │ │ │ │ -0000d530: 3220 2032 2020 7c20 2020 2020 207c 0a7c 2 2 | |.| │ │ │ │ -0000d540: 2020 2020 207c 202d 3120 2d31 202d 3220 | -1 -1 -2 │ │ │ │ -0000d550: 3020 2030 2020 2d31 2030 2020 2d31 202d 0 0 -1 0 -1 - │ │ │ │ -0000d560: 3120 2d32 2030 2020 3020 202d 3120 3020 1 -2 0 0 -1 0 │ │ │ │ -0000d570: 202d 3120 2d31 202d 3220 3020 2030 2020 -1 -1 -2 0 0 │ │ │ │ -0000d580: 2d31 2030 2020 7c20 2020 2020 207c 0a7c -1 0 | |.| │ │ │ │ -0000d590: 2020 2020 207c 2030 2020 3020 2030 2020 | 0 0 0 │ │ │ │ -0000d5a0: 2d31 202d 3120 2d31 202d 3220 3020 2030 -1 -1 -1 -2 0 0 │ │ │ │ -0000d5b0: 2020 3020 202d 3120 2d31 202d 3120 2d32 0 -1 -1 -1 -2 │ │ │ │ -0000d5c0: 2030 2020 3020 2030 2020 2d31 202d 3120 0 0 0 -1 -1 │ │ │ │ -0000d5d0: 2d31 202d 3220 7c20 2020 2020 207c 0a7c -1 -2 | |.| │ │ │ │ +0000d490: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000d4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000d4f0: 6933 203a 204c 203d 2073 6d61 6c6c 6572 i3 : L = smaller │ │ │ │ +0000d500: 4d6f 6e6f 6d69 616c 7320 4d3b 2020 2020 Monomials M; │ │ │ │ +0000d510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d530: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000d540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000d580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000d590: 6934 203a 206d 6174 203d 2066 696e 6457 i4 : mat = findW │ │ │ │ +0000d5a0: 6569 6768 7443 6f6e 7374 7261 696e 7473 eightConstraints │ │ │ │ +0000d5b0: 284d 2c4c 2920 2020 2020 2020 2020 2020 (M,L) │ │ │ │ +0000d5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000d5d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000d5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d620: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000d630: 2020 2020 2020 2020 2020 2020 2020 3420 4 │ │ │ │ -0000d640: 2020 2020 2020 3231 2020 2020 2020 2020 21 │ │ │ │ -0000d650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d670: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000d680: 6f34 203a 204d 6174 7269 7820 5a5a 2020 o4 : Matrix ZZ │ │ │ │ -0000d690: 3c2d 2d20 5a5a 2020 2020 2020 2020 2020 <-- ZZ │ │ │ │ -0000d6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d6c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000d6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000d710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000d720: 6935 203a 206e 6565 6473 5061 636b 6167 i5 : needsPackag │ │ │ │ -0000d730: 6520 2250 6f6c 7968 6564 7261 2220 2020 e "Polyhedra" │ │ │ │ -0000d740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000d760: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000d630: 6f34 203d 207c 2031 2020 3220 2032 2020 o4 = | 1 2 2 │ │ │ │ +0000d640: 3120 2032 2020 3220 2032 2020 3020 2031 1 2 2 2 0 1 │ │ │ │ +0000d650: 2020 3120 2030 2020 3120 2031 2020 3120 1 0 1 1 1 │ │ │ │ +0000d660: 202d 3120 3020 2030 2020 2d31 2030 2020 -1 0 0 -1 0 │ │ │ │ +0000d670: 3020 2030 2020 7c20 2020 2020 207c 0a7c 0 0 | |.| │ │ │ │ +0000d680: 2020 2020 207c 2030 2020 2d31 2030 2020 | 0 -1 0 │ │ │ │ +0000d690: 3020 202d 3120 3020 2030 2020 3120 2030 0 -1 0 0 1 0 │ │ │ │ +0000d6a0: 2020 3120 2031 2020 3020 2031 2020 3120 1 1 0 1 1 │ │ │ │ +0000d6b0: 2032 2020 3120 2032 2020 3220 2031 2020 2 1 2 2 1 │ │ │ │ +0000d6c0: 3220 2032 2020 7c20 2020 2020 207c 0a7c 2 2 | |.| │ │ │ │ +0000d6d0: 2020 2020 207c 202d 3120 2d31 202d 3220 | -1 -1 -2 │ │ │ │ +0000d6e0: 3020 2030 2020 2d31 2030 2020 2d31 202d 0 0 -1 0 -1 - │ │ │ │ +0000d6f0: 3120 2d32 2030 2020 3020 202d 3120 3020 1 -2 0 0 -1 0 │ │ │ │ +0000d700: 202d 3120 2d31 202d 3220 3020 2030 2020 -1 -1 -2 0 0 │ │ │ │ +0000d710: 2d31 2030 2020 7c20 2020 2020 207c 0a7c -1 0 | |.| │ │ │ │ +0000d720: 2020 2020 207c 2030 2020 3020 2030 2020 | 0 0 0 │ │ │ │ +0000d730: 2d31 202d 3120 2d31 202d 3220 3020 2030 -1 -1 -1 -2 0 0 │ │ │ │ +0000d740: 2020 3020 202d 3120 2d31 202d 3120 2d32 0 -1 -1 -1 -2 │ │ │ │ +0000d750: 2030 2020 3020 2030 2020 2d31 202d 3120 0 0 0 -1 -1 │ │ │ │ +0000d760: 2d31 202d 3220 7c20 2020 2020 207c 0a7c -1 -2 | |.| │ │ │ │ 0000d770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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0a7c |.| │ │ │ │ -0000da40: 6f36 203a 2050 6163 6b61 6765 2020 2020 o6 : Package │ │ │ │ -0000da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000da60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000da80: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000dae0: 6937 203a 2064 7561 6c43 6f6e 6520 706f i7 : dualCone po │ │ │ │ -0000daf0: 7348 756c 6c20 282d 6d61 7429 2020 2020 sHull (-mat) │ │ │ │ +0000da30: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2020 2020 2020 2020 2020 2020 │ │ │ │ -0000dc10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000dc20: 6f37 203a 2043 6f6e 6520 2020 2020 2020 o7 : Cone │ │ │ │ -0000dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000dc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000dc60: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000dcc0: 6938 203a 2072 6179 7320 6f6f 2020 2020 i8 : rays oo │ │ │ │ +0000dc10: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000dc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000de50: 6938 203a 2072 6179 7320 6f6f 2020 2020 i8 : rays oo │ │ │ │ +0000de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000de70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000de80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000de90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000dea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000deb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000dee0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000def0: 2020 2020 2020 2020 2020 2020 2020 3420 4 │ │ │ │ -0000df00: 2020 2020 2020 3420 2020 2020 2020 2020 4 │ │ │ │ +0000def0: 6f38 203d 207c 2030 2030 2030 2020 3020 o8 = | 0 0 0 0 │ │ │ │ +0000df00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0000df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000df30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000df40: 6f38 203a 204d 6174 7269 7820 5a5a 2020 o8 : Matrix ZZ │ │ │ │ -0000df50: 3c2d 2d20 5a5a 2020 2020 2020 2020 2020 <-- ZZ │ │ │ │ +0000df40: 2020 2020 207c 2030 2030 202d 3120 3120 | 0 0 -1 1 │ │ │ │ +0000df50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0000df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000df80: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000dfe0: 6939 203a 2070 6f73 4875 6c6c 206d 6174 i9 : posHull mat │ │ │ │ -0000dff0: 202d 2d20 7365 656d 7320 7772 6f6e 673f -- seems wrong? │ │ │ │ +0000df80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000df90: 2020 2020 207c 2031 2030 2031 2020 3220 | 1 0 1 2 │ │ │ │ +0000dfa0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0000dfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000dfd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000dfe0: 2020 2020 207c 2030 2031 2031 2020 3220 | 0 1 1 2 │ │ │ │ +0000dff0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0000e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e020: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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-0000e120: 6f39 203a 2043 6f6e 6520 2020 2020 2020 o9 : Cone │ │ │ │ -0000e130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e160: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000e1c0: 6931 3020 3a20 636f 6e65 4672 6f6d 4844 i10 : coneFromHD │ │ │ │ -0000e1d0: 6174 6120 7472 616e 7370 6f73 6520 6d61 ata transpose ma │ │ │ │ -0000e1e0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +0000e110: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000e170: 6939 203a 2070 6f73 4875 6c6c 206d 6174 i9 : posHull mat │ │ │ │ +0000e180: 202d 2d20 7365 656d 7320 7772 6f6e 673f -- seems wrong? │ │ │ │ +0000e190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e1b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e200: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e210: 6f39 203d 2043 6f6e 657b 2e2e 2e31 2e2e o9 = Cone{...1.. │ │ │ │ +0000e220: 2e7d 2020 2020 2020 2020 2020 2020 2020 .} │ │ │ │ 0000e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e250: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e260: 6f31 3020 3d20 436f 6e65 7b2e 2e2e 312e o10 = Cone{...1. │ │ │ │ -0000e270: 2e2e 7d20 2020 2020 2020 2020 2020 2020 ..} │ │ │ │ +0000e260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e2a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e2b0: 6f39 203a 2043 6f6e 6520 2020 2020 2020 o9 : Cone │ │ │ │ 0000e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e2f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e300: 6f31 3020 3a20 436f 6e65 2020 2020 2020 o10 : Cone │ │ │ │ -0000e310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e340: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000e3a0: 6931 3120 3a20 7261 7973 206d 6174 2020 i11 : rays mat │ │ │ │ +0000e2f0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000e300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000e350: 6931 3020 3a20 636f 6e65 4672 6f6d 4844 i10 : coneFromHD │ │ │ │ +0000e360: 6174 6120 7472 616e 7370 6f73 6520 6d61 ata transpose ma │ │ │ │ +0000e370: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +0000e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e390: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e3e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e3f0: 6f31 3020 3d20 436f 6e65 7b2e 2e2e 312e o10 = Cone{...1. │ │ │ │ +0000e400: 2e2e 7d20 2020 2020 2020 2020 2020 2020 ..} │ │ │ │ 0000e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e430: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e440: 6f31 3120 3d20 3020 2020 2020 2020 2020 o11 = 0 │ │ │ │ +0000e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e480: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e490: 6f31 3020 3a20 436f 6e65 2020 2020 2020 o10 : Cone │ │ │ │ 0000e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e4d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e4e0: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -0000e4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e520: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e530: 6f31 3120 3a20 4d61 7472 6978 205a 5a20 o11 : Matrix ZZ │ │ │ │ -0000e540: 203c 2d2d 2030 2020 2020 2020 2020 2020 <-- 0 │ │ │ │ +0000e4d0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000e530: 6931 3120 3a20 7261 7973 206d 6174 2020 i11 : rays mat │ │ │ │ +0000e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e570: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000e580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000e5d0: 6931 3220 3a20 6669 6e64 5765 6967 6874 i12 : findWeight │ │ │ │ -0000e5e0: 5665 6374 6f72 284d 2c4c 2920 2020 2020 Vector(M,L) │ │ │ │ +0000e570: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e5c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000e5d0: 6f31 3120 3d20 3020 2020 2020 2020 2020 o11 = 0 │ │ │ │ +0000e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e610: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000e620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e660: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e670: 6f31 3220 3d20 287b 322c 2032 2c20 312c o12 = ({2, 2, 1, │ │ │ │ -0000e680: 2031 7d2c 207b 312c 2031 2c20 322c 2031 1}, {1, 1, 2, 1 │ │ │ │ -0000e690: 2c20 312c 2032 2c20 322c 2031 2c20 312c , 1, 2, 2, 1, 1, │ │ │ │ -0000e6a0: 2032 2c20 312c 2031 2c20 322c 2032 2c20 2, 1, 1, 2, 2, │ │ │ │ -0000e6b0: 312c 2031 2c20 322c 2031 2c20 207c 0a7c 1, 1, 2, 1, |.| │ │ │ │ -0000e6c0: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ -0000e6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -0000e710: 2020 2020 2020 312c 2032 2c20 327d 2920 1, 2, 2}) │ │ │ │ -0000e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e750: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e670: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ +0000e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e6b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000e6c0: 6f31 3120 3a20 4d61 7472 6978 205a 5a20 o11 : Matrix ZZ │ │ │ │ +0000e6d0: 203c 2d2d 2030 2020 2020 2020 2020 2020 <-- 0 │ │ │ │ +0000e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e700: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000e710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000e760: 6931 3220 3a20 6669 6e64 5765 6967 6874 i12 : findWeight │ │ │ │ +0000e770: 5665 6374 6f72 284d 2c4c 2920 2020 2020 Vector(M,L) │ │ │ │ 0000e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000e7b0: 6f31 3220 3a20 5365 7175 656e 6365 2020 o12 : Sequence │ │ │ │ +0000e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000e7f0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000e800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0000e850: 544f 2042 4520 4649 4e49 5348 4544 2121 TO BE FINISHED!! │ │ │ │ -0000e860: 204e 6f74 6520 7468 6174 2074 6865 2066 Note that the f │ │ │ │ -0000e870: 6972 7374 2067 656e 6572 6174 6f72 206c irst generator l │ │ │ │ -0000e880: 6973 7465 6420 666f 7220 244d 2420 6973 isted for $M$ is │ │ │ │ -0000e890: 2024 615e 3224 2c20 616e 6420 7468 650a $a^2$, and the. │ │ │ │ -0000e8a0: 6669 7273 7420 636f 7272 6573 706f 6e64 first correspond │ │ │ │ -0000e8b0: 696e 6720 7374 616e 6461 7264 206d 6f6e ing standard mon │ │ │ │ -0000e8c0: 6f6d 6961 6c20 6973 2024 612a 6324 2e20 omial is $a*c$. │ │ │ │ -0000e8d0: 2054 6865 2064 6966 6665 7265 6e63 6520 The difference │ │ │ │ -0000e8e0: 6f66 2074 6865 7365 2074 776f 0a6d 6f6e of these two.mon │ │ │ │ -0000e8f0: 6f6d 6961 6c73 2065 7870 6f6e 656e 7420 omials exponent │ │ │ │ -0000e900: 7665 6374 6f72 7320 6973 2024 2831 2c30 vectors is $(1,0 │ │ │ │ -0000e910: 2c2d 312c 3029 242e 2020 5468 6973 2076 ,-1,0)$. This v │ │ │ │ -0000e920: 6563 746f 7220 646f 7474 6564 2077 6974 ector dotted wit │ │ │ │ -0000e930: 6820 7468 6520 7765 6967 6874 0a76 6563 h the weight.vec │ │ │ │ -0000e940: 746f 7220 2428 322c 322c 312c 3129 2420 tor $(2,2,1,1)$ │ │ │ │ -0000e950: 6769 7665 7320 7468 6520 7661 6c75 6520 gives the value │ │ │ │ -0000e960: 2431 242c 2077 6869 6368 2069 7320 7468 $1$, which is th │ │ │ │ -0000e970: 6520 6669 7273 7420 7661 6c75 6520 696e e first value in │ │ │ │ -0000e980: 2074 6865 2073 6563 6f6e 640a 6c69 7374 the second.list │ │ │ │ -0000e990: 2e0a 0a4e 6f74 6520 7468 6174 2074 6865 ...Note that the │ │ │ │ -0000e9a0: 2064 6573 6972 6564 2074 6572 6d20 6f72 desired term or │ │ │ │ -0000e9b0: 6465 7269 6e67 2c20 616e 6420 6865 6e63 dering, and henc │ │ │ │ -0000e9c0: 6520 7765 6967 6874 2076 6563 746f 722c e weight vector, │ │ │ │ -0000e9d0: 206d 6179 206e 6f74 2065 7869 7374 2e20 may not exist. │ │ │ │ -0000e9e0: 496e 0a74 6869 7320 6361 7365 2c20 6e75 In.this case, nu │ │ │ │ -0000e9f0: 6c6c 2069 7320 7265 7475 726e 6564 2e0a ll is returned.. │ │ │ │ -0000ea00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0000ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ea30: 2d2d 2d2b 0a7c 6931 3320 3a20 4d20 3d20 ---+.|i13 : M = │ │ │ │ -0000ea40: 6964 6561 6c22 6162 2220 2020 2020 2020 ideal"ab" │ │ │ │ -0000ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ea60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0000ea70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ea80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ea90: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0000eaa0: 3320 3d20 6964 6561 6c28 612a 6229 2020 3 = ideal(a*b) │ │ │ │ -0000eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000eac0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000ead0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0000eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000eb00: 2020 207c 0a7c 6f31 3320 3a20 4964 6561 |.|o13 : Idea │ │ │ │ -0000eb10: 6c20 6f66 2052 2020 2020 2020 2020 2020 l of R │ │ │ │ -0000eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000eb30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0000eb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000eb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000eb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0000eb70: 3420 3a20 4c31 203d 2073 7461 6e64 6172 4 : L1 = standar │ │ │ │ -0000eb80: 644d 6f6e 6f6d 6961 6c73 204d 2020 2020 dMonomials M │ │ │ │ -0000eb90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000eba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0000ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ebd0: 2020 207c 0a7c 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ -0000ebe0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000ebf0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0000ec00: 2020 2020 3220 207c 0a7c 6f31 3420 3d20 2 |.|o14 = │ │ │ │ -0000ec10: 7b7b 6120 2c20 6220 2c20 612a 632c 2062 {{a , b , a*c, b │ │ │ │ -0000ec20: 2a63 2c20 6320 2c20 612a 642c 2062 2a64 *c, c , a*d, b*d │ │ │ │ -0000ec30: 2c20 632a 642c 2064 207d 7d7c 0a7c 2020 , c*d, d }}|.| │ │ │ │ +0000e7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000e800: 6f31 3220 3d20 287b 322c 2032 2c20 312c o12 = ({2, 2, 1, │ │ │ │ +0000e810: 2031 7d2c 207b 312c 2031 2c20 322c 2031 1}, {1, 1, 2, 1 │ │ │ │ +0000e820: 2c20 312c 2032 2c20 322c 2031 2c20 312c , 1, 2, 2, 1, 1, │ │ │ │ +0000e830: 2032 2c20 312c 2031 2c20 322c 2032 2c20 2, 1, 1, 2, 2, │ │ │ │ +0000e840: 312c 2031 2c20 322c 2031 2c20 207c 0a7c 1, 1, 2, 1, |.| │ │ │ │ +0000e850: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ +0000e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +0000e8a0: 2020 2020 2020 312c 2032 2c20 327d 2920 1, 2, 2}) │ │ │ │ +0000e8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e8e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e930: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000e940: 6f31 3220 3a20 5365 7175 656e 6365 2020 o12 : Sequence │ │ │ │ +0000e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000e980: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0000e9e0: 544f 2042 4520 4649 4e49 5348 4544 2121 TO BE FINISHED!! │ │ │ │ +0000e9f0: 204e 6f74 6520 7468 6174 2074 6865 2066 Note that the f │ │ │ │ +0000ea00: 6972 7374 2067 656e 6572 6174 6f72 206c irst generator l │ │ │ │ +0000ea10: 6973 7465 6420 666f 7220 244d 2420 6973 isted for $M$ is │ │ │ │ +0000ea20: 2024 615e 3224 2c20 616e 6420 7468 650a $a^2$, and the. │ │ │ │ +0000ea30: 6669 7273 7420 636f 7272 6573 706f 6e64 first correspond │ │ │ │ +0000ea40: 696e 6720 7374 616e 6461 7264 206d 6f6e ing standard mon │ │ │ │ +0000ea50: 6f6d 6961 6c20 6973 2024 612a 6324 2e20 omial is $a*c$. │ │ │ │ +0000ea60: 2054 6865 2064 6966 6665 7265 6e63 6520 The difference │ │ │ │ +0000ea70: 6f66 2074 6865 7365 2074 776f 0a6d 6f6e of these two.mon │ │ │ │ +0000ea80: 6f6d 6961 6c73 2065 7870 6f6e 656e 7420 omials exponent │ │ │ │ +0000ea90: 7665 6374 6f72 7320 6973 2024 2831 2c30 vectors is $(1,0 │ │ │ │ +0000eaa0: 2c2d 312c 3029 242e 2020 5468 6973 2076 ,-1,0)$. This v │ │ │ │ +0000eab0: 6563 746f 7220 646f 7474 6564 2077 6974 ector dotted wit │ │ │ │ +0000eac0: 6820 7468 6520 7765 6967 6874 0a76 6563 h the weight.vec │ │ │ │ +0000ead0: 746f 7220 2428 322c 322c 312c 3129 2420 tor $(2,2,1,1)$ │ │ │ │ +0000eae0: 6769 7665 7320 7468 6520 7661 6c75 6520 gives the value │ │ │ │ +0000eaf0: 2431 242c 2077 6869 6368 2069 7320 7468 $1$, which is th │ │ │ │ +0000eb00: 6520 6669 7273 7420 7661 6c75 6520 696e e first value in │ │ │ │ +0000eb10: 2074 6865 2073 6563 6f6e 640a 6c69 7374 the second.list │ │ │ │ +0000eb20: 2e0a 0a4e 6f74 6520 7468 6174 2074 6865 ...Note that the │ │ │ │ +0000eb30: 2064 6573 6972 6564 2074 6572 6d20 6f72 desired term or │ │ │ │ +0000eb40: 6465 7269 6e67 2c20 616e 6420 6865 6e63 dering, and henc │ │ │ │ +0000eb50: 6520 7765 6967 6874 2076 6563 746f 722c e weight vector, │ │ │ │ +0000eb60: 206d 6179 206e 6f74 2065 7869 7374 2e20 may not exist. │ │ │ │ +0000eb70: 496e 0a74 6869 7320 6361 7365 2c20 6e75 In.this case, nu │ │ │ │ +0000eb80: 6c6c 2069 7320 7265 7475 726e 6564 2e0a ll is returned.. │ │ │ │ +0000eb90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0000eba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ebb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ebc0: 2d2d 2d2b 0a7c 6931 3320 3a20 4d20 3d20 ---+.|i13 : M = │ │ │ │ +0000ebd0: 6964 6561 6c22 6162 2220 2020 2020 2020 ideal"ab" │ │ │ │ +0000ebe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ebf0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0000ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ec20: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0000ec30: 3320 3d20 6964 6561 6c28 612a 6229 2020 3 = ideal(a*b) │ │ │ │ 0000ec40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ec60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000ec70: 0a7c 6f31 3420 3a20 4c69 7374 2020 2020 .|o14 : List │ │ │ │ +0000ec50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0000ec60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0000ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ec80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ec90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000eca0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0000ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ecd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 -------+.|i15 : │ │ │ │ -0000ece0: 6669 6e64 5765 6967 6874 5665 6374 6f72 findWeightVector │ │ │ │ -0000ecf0: 284d 2c4c 3129 2020 2020 2020 2020 2020 (M,L1) │ │ │ │ -0000ed00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0000ed10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ed20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ed30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0000ed40: 0a0a 5468 6973 2063 6f6d 6d61 6e64 2069 ..This command i │ │ │ │ -0000ed50: 7320 7573 6564 2069 6e20 7468 6520 2a6e s used in the *n │ │ │ │ -0000ed60: 6f74 6520 6772 6f65 626e 6572 4661 6d69 ote groebnerFami │ │ │ │ -0000ed70: 6c79 3a20 6772 6f65 626e 6572 4661 6d69 ly: groebnerFami │ │ │ │ -0000ed80: 6c79 2c20 726f 7574 696e 652e 0a0a 5365 ly, routine...Se │ │ │ │ -0000ed90: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ -0000eda0: 0a20 202a 202a 6e6f 7465 2067 726f 6562 . * *note groeb │ │ │ │ -0000edb0: 6e65 7253 7472 6174 756d 3a20 6772 6f65 nerStratum: groe │ │ │ │ -0000edc0: 626e 6572 5374 7261 7475 6d2c 202d 2d20 bnerStratum, -- │ │ │ │ -0000edd0: 636f 6d70 7574 6520 7468 6520 6964 6561 compute the idea │ │ │ │ -0000ede0: 6c20 7768 6572 6520 6120 6769 7665 6e0a l where a given. │ │ │ │ -0000edf0: 2020 2020 6973 2061 2047 726f 6562 6e65 is a Groebne │ │ │ │ -0000ee00: 7220 6261 7369 730a 2020 2a20 2a6e 6f74 r basis. * *not │ │ │ │ -0000ee10: 6520 6772 6f65 626e 6572 4661 6d69 6c79 e groebnerFamily │ │ │ │ -0000ee20: 3a20 6772 6f65 626e 6572 4661 6d69 6c79 : groebnerFamily │ │ │ │ -0000ee30: 2c20 2d2d 2063 6f6d 7075 7465 7320 6661 , -- computes fa │ │ │ │ -0000ee40: 6d69 6c69 6573 206f 6620 6964 6561 6c73 milies of ideals │ │ │ │ -0000ee50: 2077 6974 6820 610a 2020 2020 7370 6563 with a. spec │ │ │ │ -0000ee60: 6966 6965 6420 696e 6974 6961 6c20 6964 ified initial id │ │ │ │ -0000ee70: 6561 6c0a 0a57 6179 7320 746f 2075 7365 eal..Ways to use │ │ │ │ -0000ee80: 2066 696e 6457 6569 6768 7443 6f6e 7374 findWeightConst │ │ │ │ -0000ee90: 7261 696e 7473 3a0a 3d3d 3d3d 3d3d 3d3d raints:.======== │ │ │ │ -0000eea0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0000eeb0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0000eec0: 2266 696e 6457 6569 6768 7443 6f6e 7374 "findWeightConst │ │ │ │ -0000eed0: 7261 696e 7473 2849 6465 616c 2c4c 6973 raints(Ideal,Lis │ │ │ │ -0000eee0: 7429 220a 0a46 6f72 2074 6865 2070 726f t)"..For the pro │ │ │ │ -0000eef0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -0000ef00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -0000ef10: 6f62 6a65 6374 202a 6e6f 7465 2066 696e object *note fin │ │ │ │ -0000ef20: 6457 6569 6768 7443 6f6e 7374 7261 696e dWeightConstrain │ │ │ │ -0000ef30: 7473 3a20 6669 6e64 5765 6967 6874 436f ts: findWeightCo │ │ │ │ -0000ef40: 6e73 7472 6169 6e74 732c 2069 7320 6120 nstraints, is a │ │ │ │ -0000ef50: 2a6e 6f74 650a 6d65 7468 6f64 2066 756e *note.method fun │ │ │ │ -0000ef60: 6374 696f 6e3a 2028 4d61 6361 756c 6179 ction: (Macaulay │ │ │ │ -0000ef70: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -0000ef80: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ -0000ef90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000efa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000efb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000efc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000efd0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -0000efe0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -0000eff0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -0000f000: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -0000f010: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -0000f020: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ -0000f030: 6c61 7932 2f70 6163 6b61 6765 732f 0a47 lay2/packages/.G │ │ │ │ -0000f040: 726f 6562 6e65 7253 7472 6174 612e 6d32 roebnerStrata.m2 │ │ │ │ -0000f050: 3a36 3931 3a30 2e0a 1f0a 4669 6c65 3a20 :691:0....File: │ │ │ │ -0000f060: 4772 6f65 626e 6572 5374 7261 7461 2e69 GroebnerStrata.i │ │ │ │ -0000f070: 6e66 6f2c 204e 6f64 653a 2066 696e 6457 nfo, Node: findW │ │ │ │ -0000f080: 6569 6768 7456 6563 746f 722c 204e 6578 eightVector, Nex │ │ │ │ -0000f090: 743a 2067 726f 6562 6e65 7246 616d 696c t: groebnerFamil │ │ │ │ -0000f0a0: 792c 2050 7265 763a 2066 696e 6457 6569 y, Prev: findWei │ │ │ │ -0000f0b0: 6768 7443 6f6e 7374 7261 696e 7473 2c20 ghtConstraints, │ │ │ │ -0000f0c0: 5570 3a20 546f 700a 0a66 696e 6457 6569 Up: Top..findWei │ │ │ │ -0000f0d0: 6768 7456 6563 746f 7220 2d2d 2072 6574 ghtVector -- ret │ │ │ │ -0000f0e0: 7572 6e73 2061 2077 6569 6768 7420 7665 urns a weight ve │ │ │ │ -0000f0f0: 6374 6f72 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a ctor.*********** │ │ │ │ -0000f100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000f110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0000f120: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -0000f130: 2020 2020 2020 2877 2c20 6829 203d 2066 (w, h) = f │ │ │ │ -0000f140: 696e 6457 6569 6768 7456 6563 746f 7220 indWeightVector │ │ │ │ -0000f150: 284d 2c20 4c29 0a20 202a 2049 6e70 7574 (M, L). * Input │ │ │ │ -0000f160: 733a 0a20 2020 2020 202a 204d 2c20 616e s:. * M, an │ │ │ │ -0000f170: 202a 6e6f 7465 2069 6465 616c 3a20 284d *note ideal: (M │ │ │ │ -0000f180: 6163 6175 6c61 7932 446f 6329 4964 6561 acaulay2Doc)Idea │ │ │ │ -0000f190: 6c2c 2c20 4d20 7368 6f75 6c64 2062 6520 l,, M should be │ │ │ │ -0000f1a0: 6120 6d6f 6e6f 6d69 616c 2069 6465 616c a monomial ideal │ │ │ │ -0000f1b0: 0a20 2020 2020 202a 204c 2c20 6120 2a6e . * L, a *n │ │ │ │ -0000f1c0: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ -0000f1d0: 6c61 7932 446f 6329 4c69 7374 2c2c 2061 lay2Doc)List,, a │ │ │ │ -0000f1e0: 206c 6973 7420 6f66 206c 6973 7473 206f list of lists o │ │ │ │ -0000f1f0: 6620 7374 616e 6461 7264 0a20 2020 2020 f standard. │ │ │ │ -0000f200: 2020 206d 6f6e 6f6d 6961 6c73 0a20 202a monomials. * │ │ │ │ -0000f210: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0000f220: 2a20 772c 2061 202a 6e6f 7465 206c 6973 * w, a *note lis │ │ │ │ -0000f230: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -0000f240: 294c 6973 742c 2c20 7720 6973 2061 2077 )List,, w is a w │ │ │ │ -0000f250: 6569 6768 7420 7665 6374 6f72 2074 6861 eight vector tha │ │ │ │ -0000f260: 7420 706c 6163 6573 0a20 2020 2020 2020 t places. │ │ │ │ -0000f270: 2074 6865 2073 7065 6369 6669 6564 2067 the specified g │ │ │ │ -0000f280: 656e 6572 6174 6f72 7320 6f66 204d 2067 enerators of M g │ │ │ │ -0000f290: 7265 6174 6572 2074 6861 6e20 7468 6520 reater than the │ │ │ │ -0000f2a0: 636f 7272 6573 706f 6e64 696e 6720 7374 corresponding st │ │ │ │ -0000f2b0: 616e 6461 7264 0a20 2020 2020 2020 206d andard. m │ │ │ │ -0000f2c0: 6f6e 6f6d 6961 6c73 2c20 6966 2070 6f73 onomials, if pos │ │ │ │ -0000f2d0: 7369 626c 652e 2020 4966 206e 6f74 2070 sible. If not p │ │ │ │ -0000f2e0: 6f73 7369 626c 652c 206e 756c 6c20 6973 ossible, null is │ │ │ │ -0000f2f0: 2072 6574 7572 6e65 6420 696e 7374 6561 returned instea │ │ │ │ -0000f300: 6420 6f66 0a20 2020 2020 2020 2028 772c d of. (w, │ │ │ │ -0000f310: 6829 0a20 2020 2020 202a 2068 2c20 6120 h). * h, a │ │ │ │ -0000f320: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ -0000f330: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ -0000f340: 2068 2069 7320 6120 6c69 7374 206f 6620 h is a list of │ │ │ │ -0000f350: 7661 6c75 6573 2066 6f72 2077 0a20 2020 values for w. │ │ │ │ -0000f360: 2020 2020 2064 6f74 7465 6420 7769 7468 dotted with │ │ │ │ -0000f370: 2074 6865 2064 6966 6665 7265 6e63 6520 the difference │ │ │ │ -0000f380: 6f66 2074 6865 2065 7870 6f6e 656e 7420 of the exponent │ │ │ │ -0000f390: 6f66 2074 6865 2065 6163 6820 7374 616e of the each stan │ │ │ │ -0000f3a0: 6461 7264 0a20 2020 2020 2020 206d 6f6e dard. mon │ │ │ │ -0000f3b0: 6f6d 6961 6c20 666f 7220 6561 6368 2067 omial for each g │ │ │ │ -0000f3c0: 656e 6572 6174 6f72 2061 6e64 2074 6865 enerator and the │ │ │ │ -0000f3d0: 2063 6f72 7265 7370 6f6e 6469 6e67 2067 corresponding g │ │ │ │ -0000f3e0: 656e 6572 6174 6f72 2c20 696e 2074 6865 enerator, in the │ │ │ │ -0000f3f0: 0a20 2020 2020 2020 206f 7264 6572 2074 . order t │ │ │ │ -0000f400: 6865 7920 6172 6520 6c69 7374 6564 2069 hey are listed i │ │ │ │ -0000f410: 6e20 4c0a 0a44 6573 6372 6970 7469 6f6e n L..Description │ │ │ │ -0000f420: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 496e .===========..In │ │ │ │ -0000f430: 2074 6865 2066 6972 7374 2065 6e74 7279 the first entry │ │ │ │ -0000f440: 2c20 7468 6973 2063 6f6d 6d61 6e64 2072 , this command r │ │ │ │ -0000f450: 6574 7572 6e73 2061 2077 6569 6768 7420 eturns a weight │ │ │ │ -0000f460: 7665 6374 6f72 2061 7373 6f63 6961 7465 vector associate │ │ │ │ -0000f470: 6420 746f 2061 0a6d 6f6e 6f6d 6961 6c20 d to a.monomial │ │ │ │ -0000f480: 6f72 6465 7220 7468 6174 2070 6c61 6365 order that place │ │ │ │ -0000f490: 7320 7468 6520 6765 6e65 7261 746f 7273 s the generators │ │ │ │ -0000f4a0: 206f 6620 6120 6d6f 6e6f 6d69 616c 2069 of a monomial i │ │ │ │ -0000f4b0: 6465 616c 2024 4d24 2061 6865 6164 206f deal $M$ ahead o │ │ │ │ -0000f4c0: 660a 7374 616e 6461 7264 206d 6f6e 6f6d f.standard monom │ │ │ │ -0000f4d0: 6961 6c73 206f 6620 7468 6520 7361 6d65 ials of the same │ │ │ │ -0000f4e0: 2064 6567 7265 652e 2020 5468 6520 7365 degree. The se │ │ │ │ -0000f4f0: 636f 6e64 2065 6e74 7279 2069 7320 6120 cond entry is a │ │ │ │ -0000f500: 6c69 7374 206f 6620 7661 6c75 6573 0a66 list of values.f │ │ │ │ -0000f510: 6f72 2074 6865 2077 6569 6768 7420 7665 or the weight ve │ │ │ │ -0000f520: 6374 6f72 2064 6f74 7465 6420 7769 7468 ctor dotted with │ │ │ │ -0000f530: 2074 6865 2064 6966 6665 7265 6e63 6520 the difference │ │ │ │ -0000f540: 6f66 2074 6865 2065 7870 6f6e 656e 7420 of the exponent │ │ │ │ -0000f550: 6f66 2065 6163 680a 7374 616e 6461 7264 of each.standard │ │ │ │ -0000f560: 206d 6f6e 6f6d 6961 6c20 666f 7220 6561 monomial for ea │ │ │ │ -0000f570: 6368 2067 656e 6572 6174 6f72 2061 6e64 ch generator and │ │ │ │ -0000f580: 2074 6865 2063 6f72 7265 7370 6f6e 6469 the correspondi │ │ │ │ -0000f590: 6e67 2067 656e 6572 6174 6f72 2e0a 0a2b ng generator...+ │ │ │ │ -0000f5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000f5f0: 6931 203a 2052 203d 205a 5a2f 3332 3030 i1 : R = ZZ/3200 │ │ │ │ -0000f600: 335b 612c 622c 632c 2064 5d3b 2020 2020 3[a,b,c, d]; │ │ │ │ -0000f610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f630: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000f640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000f690: 6932 203a 204d 203d 2069 6465 616c 2028 i2 : M = ideal ( │ │ │ │ -0000f6a0: 615e 322c 2061 2a62 2c20 625e 3229 3b20 a^2, a*b, b^2); │ │ │ │ -0000f6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f6d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -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 2020 2020 2020 │ │ │ │ -0000f720: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000f730: 6f32 203a 2049 6465 616c 206f 6620 5220 o2 : Ideal of R │ │ │ │ -0000f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f770: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000f780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000f7d0: 6933 203a 204c 203d 2073 6d61 6c6c 6572 i3 : L = smaller │ │ │ │ -0000f7e0: 4d6f 6e6f 6d69 616c 7320 4d3b 2020 2020 Monomials M; │ │ │ │ -0000f7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f810: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000f820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0000f870: 6934 203a 2066 696e 6457 6569 6768 7456 i4 : findWeightV │ │ │ │ -0000f880: 6563 746f 7228 4d2c 4c29 2020 2020 2020 ector(M,L) │ │ │ │ +0000ec90: 2020 207c 0a7c 6f31 3320 3a20 4964 6561 |.|o13 : Idea │ │ │ │ +0000eca0: 6c20 6f66 2052 2020 2020 2020 2020 2020 l of R │ │ │ │ +0000ecb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ecc0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0000ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ecf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0000ed00: 3420 3a20 4c31 203d 2073 7461 6e64 6172 4 : L1 = standar │ │ │ │ +0000ed10: 644d 6f6e 6f6d 6961 6c73 204d 2020 2020 dMonomials M │ │ │ │ +0000ed20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0000ed30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0000ed40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ed60: 2020 207c 0a7c 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ +0000ed70: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0000ed80: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0000ed90: 2020 2020 3220 207c 0a7c 6f31 3420 3d20 2 |.|o14 = │ │ │ │ +0000eda0: 7b7b 6120 2c20 6220 2c20 612a 632c 2062 {{a , b , a*c, b │ │ │ │ +0000edb0: 2a63 2c20 6320 2c20 612a 642c 2062 2a64 *c, c , a*d, b*d │ │ │ │ +0000edc0: 2c20 632a 642c 2064 207d 7d7c 0a7c 2020 , c*d, d }}|.| │ │ │ │ +0000edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000edf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0000ee00: 0a7c 6f31 3420 3a20 4c69 7374 2020 2020 .|o14 : List │ │ │ │ +0000ee10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ee30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0000ee40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ee60: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 -------+.|i15 : │ │ │ │ +0000ee70: 6669 6e64 5765 6967 6874 5665 6374 6f72 findWeightVector │ │ │ │ +0000ee80: 284d 2c4c 3129 2020 2020 2020 2020 2020 (M,L1) │ │ │ │ +0000ee90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0000eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0000eed0: 0a0a 5468 6973 2063 6f6d 6d61 6e64 2069 ..This command i │ │ │ │ +0000eee0: 7320 7573 6564 2069 6e20 7468 6520 2a6e s used in the *n │ │ │ │ +0000eef0: 6f74 6520 6772 6f65 626e 6572 4661 6d69 ote groebnerFami │ │ │ │ +0000ef00: 6c79 3a20 6772 6f65 626e 6572 4661 6d69 ly: groebnerFami │ │ │ │ +0000ef10: 6c79 2c20 726f 7574 696e 652e 0a0a 5365 ly, routine...Se │ │ │ │ +0000ef20: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +0000ef30: 0a20 202a 202a 6e6f 7465 2067 726f 6562 . * *note groeb │ │ │ │ +0000ef40: 6e65 7253 7472 6174 756d 3a20 6772 6f65 nerStratum: groe │ │ │ │ +0000ef50: 626e 6572 5374 7261 7475 6d2c 202d 2d20 bnerStratum, -- │ │ │ │ +0000ef60: 636f 6d70 7574 6520 7468 6520 6964 6561 compute the idea │ │ │ │ +0000ef70: 6c20 7768 6572 6520 6120 6769 7665 6e0a l where a given. │ │ │ │ +0000ef80: 2020 2020 6973 2061 2047 726f 6562 6e65 is a Groebne │ │ │ │ +0000ef90: 7220 6261 7369 730a 2020 2a20 2a6e 6f74 r basis. * *not │ │ │ │ +0000efa0: 6520 6772 6f65 626e 6572 4661 6d69 6c79 e groebnerFamily │ │ │ │ +0000efb0: 3a20 6772 6f65 626e 6572 4661 6d69 6c79 : groebnerFamily │ │ │ │ +0000efc0: 2c20 2d2d 2063 6f6d 7075 7465 7320 6661 , -- computes fa │ │ │ │ +0000efd0: 6d69 6c69 6573 206f 6620 6964 6561 6c73 milies of ideals │ │ │ │ +0000efe0: 2077 6974 6820 610a 2020 2020 7370 6563 with a. spec │ │ │ │ +0000eff0: 6966 6965 6420 696e 6974 6961 6c20 6964 ified initial id │ │ │ │ +0000f000: 6561 6c0a 0a57 6179 7320 746f 2075 7365 eal..Ways to use │ │ │ │ +0000f010: 2066 696e 6457 6569 6768 7443 6f6e 7374 findWeightConst │ │ │ │ +0000f020: 7261 696e 7473 3a0a 3d3d 3d3d 3d3d 3d3d raints:.======== │ │ │ │ +0000f030: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0000f040: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0000f050: 2266 696e 6457 6569 6768 7443 6f6e 7374 "findWeightConst │ │ │ │ +0000f060: 7261 696e 7473 2849 6465 616c 2c4c 6973 raints(Ideal,Lis │ │ │ │ +0000f070: 7429 220a 0a46 6f72 2074 6865 2070 726f t)"..For the pro │ │ │ │ +0000f080: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +0000f090: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0000f0a0: 6f62 6a65 6374 202a 6e6f 7465 2066 696e object *note fin │ │ │ │ +0000f0b0: 6457 6569 6768 7443 6f6e 7374 7261 696e dWeightConstrain │ │ │ │ +0000f0c0: 7473 3a20 6669 6e64 5765 6967 6874 436f ts: findWeightCo │ │ │ │ +0000f0d0: 6e73 7472 6169 6e74 732c 2069 7320 6120 nstraints, is a │ │ │ │ +0000f0e0: 2a6e 6f74 650a 6d65 7468 6f64 2066 756e *note.method fun │ │ │ │ +0000f0f0: 6374 696f 6e3a 2028 4d61 6361 756c 6179 ction: (Macaulay │ │ │ │ +0000f100: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +0000f110: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +0000f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f160: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0000f170: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0000f180: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0000f190: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0000f1a0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0000f1b0: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +0000f1c0: 6c61 7932 2f70 6163 6b61 6765 732f 0a47 lay2/packages/.G │ │ │ │ +0000f1d0: 726f 6562 6e65 7253 7472 6174 612e 6d32 roebnerStrata.m2 │ │ │ │ +0000f1e0: 3a36 3931 3a30 2e0a 1f0a 4669 6c65 3a20 :691:0....File: │ │ │ │ +0000f1f0: 4772 6f65 626e 6572 5374 7261 7461 2e69 GroebnerStrata.i │ │ │ │ +0000f200: 6e66 6f2c 204e 6f64 653a 2066 696e 6457 nfo, Node: findW │ │ │ │ +0000f210: 6569 6768 7456 6563 746f 722c 204e 6578 eightVector, Nex │ │ │ │ +0000f220: 743a 2067 726f 6562 6e65 7246 616d 696c t: groebnerFamil │ │ │ │ +0000f230: 792c 2050 7265 763a 2066 696e 6457 6569 y, Prev: findWei │ │ │ │ +0000f240: 6768 7443 6f6e 7374 7261 696e 7473 2c20 ghtConstraints, │ │ │ │ +0000f250: 5570 3a20 546f 700a 0a66 696e 6457 6569 Up: Top..findWei │ │ │ │ +0000f260: 6768 7456 6563 746f 7220 2d2d 2072 6574 ghtVector -- ret │ │ │ │ +0000f270: 7572 6e73 2061 2077 6569 6768 7420 7665 urns a weight ve │ │ │ │ +0000f280: 6374 6f72 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a ctor.*********** │ │ │ │ +0000f290: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000f2a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0000f2b0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0000f2c0: 2020 2020 2020 2877 2c20 6829 203d 2066 (w, h) = f │ │ │ │ +0000f2d0: 696e 6457 6569 6768 7456 6563 746f 7220 indWeightVector │ │ │ │ +0000f2e0: 284d 2c20 4c29 0a20 202a 2049 6e70 7574 (M, L). * Input │ │ │ │ +0000f2f0: 733a 0a20 2020 2020 202a 204d 2c20 616e s:. * M, an │ │ │ │ +0000f300: 202a 6e6f 7465 2069 6465 616c 3a20 284d *note ideal: (M │ │ │ │ +0000f310: 6163 6175 6c61 7932 446f 6329 4964 6561 acaulay2Doc)Idea │ │ │ │ +0000f320: 6c2c 2c20 4d20 7368 6f75 6c64 2062 6520 l,, M should be │ │ │ │ +0000f330: 6120 6d6f 6e6f 6d69 616c 2069 6465 616c a monomial ideal │ │ │ │ +0000f340: 0a20 2020 2020 202a 204c 2c20 6120 2a6e . * L, a *n │ │ │ │ +0000f350: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ +0000f360: 6c61 7932 446f 6329 4c69 7374 2c2c 2061 lay2Doc)List,, a │ │ │ │ +0000f370: 206c 6973 7420 6f66 206c 6973 7473 206f list of lists o │ │ │ │ +0000f380: 6620 7374 616e 6461 7264 0a20 2020 2020 f standard. │ │ │ │ +0000f390: 2020 206d 6f6e 6f6d 6961 6c73 0a20 202a monomials. * │ │ │ │ +0000f3a0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +0000f3b0: 2a20 772c 2061 202a 6e6f 7465 206c 6973 * w, a *note lis │ │ │ │ +0000f3c0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +0000f3d0: 294c 6973 742c 2c20 7720 6973 2061 2077 )List,, w is a w │ │ │ │ +0000f3e0: 6569 6768 7420 7665 6374 6f72 2074 6861 eight vector tha │ │ │ │ +0000f3f0: 7420 706c 6163 6573 0a20 2020 2020 2020 t places. │ │ │ │ +0000f400: 2074 6865 2073 7065 6369 6669 6564 2067 the specified g │ │ │ │ +0000f410: 656e 6572 6174 6f72 7320 6f66 204d 2067 enerators of M g │ │ │ │ +0000f420: 7265 6174 6572 2074 6861 6e20 7468 6520 reater than the │ │ │ │ +0000f430: 636f 7272 6573 706f 6e64 696e 6720 7374 corresponding st │ │ │ │ +0000f440: 616e 6461 7264 0a20 2020 2020 2020 206d andard. m │ │ │ │ +0000f450: 6f6e 6f6d 6961 6c73 2c20 6966 2070 6f73 onomials, if pos │ │ │ │ +0000f460: 7369 626c 652e 2020 4966 206e 6f74 2070 sible. If not p │ │ │ │ +0000f470: 6f73 7369 626c 652c 206e 756c 6c20 6973 ossible, null is │ │ │ │ +0000f480: 2072 6574 7572 6e65 6420 696e 7374 6561 returned instea │ │ │ │ +0000f490: 6420 6f66 0a20 2020 2020 2020 2028 772c d of. (w, │ │ │ │ +0000f4a0: 6829 0a20 2020 2020 202a 2068 2c20 6120 h). * h, a │ │ │ │ +0000f4b0: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ +0000f4c0: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ +0000f4d0: 2068 2069 7320 6120 6c69 7374 206f 6620 h is a list of │ │ │ │ +0000f4e0: 7661 6c75 6573 2066 6f72 2077 0a20 2020 values for w. │ │ │ │ +0000f4f0: 2020 2020 2064 6f74 7465 6420 7769 7468 dotted with │ │ │ │ +0000f500: 2074 6865 2064 6966 6665 7265 6e63 6520 the difference │ │ │ │ +0000f510: 6f66 2074 6865 2065 7870 6f6e 656e 7420 of the exponent │ │ │ │ +0000f520: 6f66 2074 6865 2065 6163 6820 7374 616e of the each stan │ │ │ │ +0000f530: 6461 7264 0a20 2020 2020 2020 206d 6f6e dard. mon │ │ │ │ +0000f540: 6f6d 6961 6c20 666f 7220 6561 6368 2067 omial for each g │ │ │ │ +0000f550: 656e 6572 6174 6f72 2061 6e64 2074 6865 enerator and the │ │ │ │ +0000f560: 2063 6f72 7265 7370 6f6e 6469 6e67 2067 corresponding g │ │ │ │ +0000f570: 656e 6572 6174 6f72 2c20 696e 2074 6865 enerator, in the │ │ │ │ +0000f580: 0a20 2020 2020 2020 206f 7264 6572 2074 . order t │ │ │ │ +0000f590: 6865 7920 6172 6520 6c69 7374 6564 2069 hey are listed i │ │ │ │ +0000f5a0: 6e20 4c0a 0a44 6573 6372 6970 7469 6f6e n L..Description │ │ │ │ +0000f5b0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 496e .===========..In │ │ │ │ +0000f5c0: 2074 6865 2066 6972 7374 2065 6e74 7279 the first entry │ │ │ │ +0000f5d0: 2c20 7468 6973 2063 6f6d 6d61 6e64 2072 , this command r │ │ │ │ +0000f5e0: 6574 7572 6e73 2061 2077 6569 6768 7420 eturns a weight │ │ │ │ +0000f5f0: 7665 6374 6f72 2061 7373 6f63 6961 7465 vector associate │ │ │ │ +0000f600: 6420 746f 2061 0a6d 6f6e 6f6d 6961 6c20 d to a.monomial │ │ │ │ +0000f610: 6f72 6465 7220 7468 6174 2070 6c61 6365 order that place │ │ │ │ +0000f620: 7320 7468 6520 6765 6e65 7261 746f 7273 s the generators │ │ │ │ +0000f630: 206f 6620 6120 6d6f 6e6f 6d69 616c 2069 of a monomial i │ │ │ │ +0000f640: 6465 616c 2024 4d24 2061 6865 6164 206f deal $M$ ahead o │ │ │ │ +0000f650: 660a 7374 616e 6461 7264 206d 6f6e 6f6d f.standard monom │ │ │ │ +0000f660: 6961 6c73 206f 6620 7468 6520 7361 6d65 ials of the same │ │ │ │ +0000f670: 2064 6567 7265 652e 2020 5468 6520 7365 degree. The se │ │ │ │ +0000f680: 636f 6e64 2065 6e74 7279 2069 7320 6120 cond entry is a │ │ │ │ +0000f690: 6c69 7374 206f 6620 7661 6c75 6573 0a66 list of values.f │ │ │ │ +0000f6a0: 6f72 2074 6865 2077 6569 6768 7420 7665 or the weight ve │ │ │ │ +0000f6b0: 6374 6f72 2064 6f74 7465 6420 7769 7468 ctor dotted with │ │ │ │ +0000f6c0: 2074 6865 2064 6966 6665 7265 6e63 6520 the difference │ │ │ │ +0000f6d0: 6f66 2074 6865 2065 7870 6f6e 656e 7420 of the exponent │ │ │ │ +0000f6e0: 6f66 2065 6163 680a 7374 616e 6461 7264 of each.standard │ │ │ │ +0000f6f0: 206d 6f6e 6f6d 6961 6c20 666f 7220 6561 monomial for ea │ │ │ │ +0000f700: 6368 2067 656e 6572 6174 6f72 2061 6e64 ch generator and │ │ │ │ +0000f710: 2074 6865 2063 6f72 7265 7370 6f6e 6469 the correspondi │ │ │ │ +0000f720: 6e67 2067 656e 6572 6174 6f72 2e0a 0a2b ng generator...+ │ │ │ │ +0000f730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000f780: 6931 203a 2052 203d 205a 5a2f 3332 3030 i1 : R = ZZ/3200 │ │ │ │ +0000f790: 335b 612c 622c 632c 2064 5d3b 2020 2020 3[a,b,c, d]; │ │ │ │ +0000f7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000f7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000f7c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000f820: 6932 203a 204d 203d 2069 6465 616c 2028 i2 : M = ideal ( │ │ │ │ +0000f830: 615e 322c 2061 2a62 2c20 625e 3229 3b20 a^2, a*b, b^2); │ │ │ │ +0000f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000f860: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +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 │ │ │ │ 0000f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000f8c0: 6f32 203a 2049 6465 616c 206f 6620 5220 o2 : Ideal of R │ │ │ │ 0000f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f900: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000f910: 6f34 203d 2028 7b32 2c20 322c 2031 2c20 o4 = ({2, 2, 1, │ │ │ │ -0000f920: 317d 2c20 7b31 2c20 312c 2032 2c20 312c 1}, {1, 1, 2, 1, │ │ │ │ -0000f930: 2031 2c20 322c 2032 2c20 312c 2031 2c20 1, 2, 2, 1, 1, │ │ │ │ -0000f940: 322c 2031 2c20 312c 2032 2c20 322c 2031 2, 1, 1, 2, 2, 1 │ │ │ │ -0000f950: 2c20 312c 2032 2c20 312c 2031 2c7c 0a7c , 1, 2, 1, 1,|.| │ │ │ │ -0000f960: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ -0000f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -0000f9b0: 2020 2020 2032 2c20 327d 2920 2020 2020 2, 2}) │ │ │ │ -0000f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000f9f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000fa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000f900: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000f960: 6933 203a 204c 203d 2073 6d61 6c6c 6572 i3 : L = smaller │ │ │ │ +0000f970: 4d6f 6e6f 6d69 616c 7320 4d3b 2020 2020 Monomials M; │ │ │ │ +0000f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000f9a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0000fa00: 6934 203a 2066 696e 6457 6569 6768 7456 i4 : findWeightV │ │ │ │ +0000fa10: 6563 746f 7228 4d2c 4c29 2020 2020 2020 ector(M,L) │ │ │ │ 0000fa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fa40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000fa50: 6f34 203a 2053 6571 7565 6e63 6520 2020 o4 : Sequence │ │ │ │ +0000fa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fa90: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0000faa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000fab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 2d2b 0a0a -------------+.. │ │ │ │ -0000faf0: 4e6f 7465 2074 6861 7420 7468 6520 6669 Note that the fi │ │ │ │ -0000fb00: 7273 7420 6765 6e65 7261 746f 7220 6c69 rst generator li │ │ │ │ -0000fb10: 7374 6564 2066 6f72 2024 4d24 2069 7320 sted for $M$ is │ │ │ │ -0000fb20: 2461 5e32 242c 2061 6e64 2074 6865 2066 $a^2$, and the f │ │ │ │ -0000fb30: 6972 7374 0a63 6f72 7265 7370 6f6e 6469 irst.correspondi │ │ │ │ -0000fb40: 6e67 2073 7461 6e64 6172 6420 6d6f 6e6f ng standard mono │ │ │ │ -0000fb50: 6d69 616c 2069 7320 2461 2a63 242e 2020 mial is $a*c$. │ │ │ │ -0000fb60: 5468 6520 6469 6666 6572 656e 6365 206f The difference o │ │ │ │ -0000fb70: 6620 7468 6573 6520 7477 6f0a 6d6f 6e6f f these two.mono │ │ │ │ -0000fb80: 6d69 616c 7320 6578 706f 6e65 6e74 2076 mials exponent v │ │ │ │ -0000fb90: 6563 746f 7273 2069 7320 2428 312c 302c ectors is $(1,0, │ │ │ │ -0000fba0: 2d31 2c30 2924 2e20 2054 6869 7320 7665 -1,0)$. This ve │ │ │ │ -0000fbb0: 6374 6f72 2064 6f74 7465 6420 7769 7468 ctor dotted with │ │ │ │ -0000fbc0: 2074 6865 2077 6569 6768 740a 7665 6374 the weight.vect │ │ │ │ -0000fbd0: 6f72 2024 2832 2c32 2c31 2c31 2924 2067 or $(2,2,1,1)$ g │ │ │ │ -0000fbe0: 6976 6573 2074 6865 2076 616c 7565 2024 ives the value $ │ │ │ │ -0000fbf0: 3124 2c20 7768 6963 6820 6973 2074 6865 1$, which is the │ │ │ │ -0000fc00: 2066 6972 7374 2076 616c 7565 2069 6e20 first value in │ │ │ │ -0000fc10: 7468 6520 7365 636f 6e64 0a6c 6973 742e the second.list. │ │ │ │ -0000fc20: 0a0a 4e6f 7465 2074 6861 7420 7468 6520 ..Note that the │ │ │ │ -0000fc30: 6465 7369 7265 6420 7465 726d 206f 7264 desired term ord │ │ │ │ -0000fc40: 6572 696e 672c 2061 6e64 2068 656e 6365 ering, and hence │ │ │ │ -0000fc50: 2077 6569 6768 7420 7665 6374 6f72 2c20 weight vector, │ │ │ │ -0000fc60: 6d61 7920 6e6f 7420 6578 6973 742e 2049 may not exist. I │ │ │ │ -0000fc70: 6e0a 7468 6973 2063 6173 652c 206e 756c n.this case, nul │ │ │ │ -0000fc80: 6c20 6973 2072 6574 7572 6e65 642e 0a0a l is returned... │ │ │ │ -0000fc90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0000fca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000fcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000fcc0: 2d2b 0a7c 6935 203a 204d 203d 2069 6465 -+.|i5 : M = ide │ │ │ │ -0000fcd0: 616c 2261 6222 2020 2020 2020 2020 2020 al"ab" │ │ │ │ -0000fce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fcf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0000fd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fd20: 2020 2020 2020 207c 0a7c 6f35 203d 2069 |.|o5 = i │ │ │ │ -0000fd30: 6465 616c 2861 2a62 2920 2020 2020 2020 deal(a*b) │ │ │ │ -0000fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fd50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0000fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fd80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000fd90: 6f35 203a 2049 6465 616c 206f 6620 5220 o5 : Ideal of R │ │ │ │ -0000fda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fdc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0000fdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000fdf0: 2d2d 2d2b 0a7c 6936 203a 204c 3120 3d20 ---+.|i6 : L1 = │ │ │ │ -0000fe00: 7374 616e 6461 7264 4d6f 6e6f 6d69 616c standardMonomial │ │ │ │ -0000fe10: 7320 4d20 2020 2020 2020 2020 2020 2020 s M │ │ │ │ -0000fe20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0000fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fe50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0000fe60: 2020 2020 3220 2020 3220 2020 2020 2020 2 2 │ │ │ │ -0000fe70: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0000fe80: 2020 2020 2020 2020 2032 2020 7c0a 7c6f 2 |.|o │ │ │ │ -0000fe90: 3620 3d20 7b7b 6120 2c20 6220 2c20 612a 6 = {{a , b , a* │ │ │ │ -0000fea0: 632c 2062 2a63 2c20 6320 2c20 612a 642c c, b*c, c , a*d, │ │ │ │ -0000feb0: 2062 2a64 2c20 632a 642c 2064 207d 7d7c b*d, c*d, d }}| │ │ │ │ -0000fec0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0000fa90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000faa0: 6f34 203d 2028 7b32 2c20 322c 2031 2c20 o4 = ({2, 2, 1, │ │ │ │ +0000fab0: 317d 2c20 7b31 2c20 312c 2032 2c20 312c 1}, {1, 1, 2, 1, │ │ │ │ +0000fac0: 2031 2c20 322c 2032 2c20 312c 2031 2c20 1, 2, 2, 1, 1, │ │ │ │ +0000fad0: 322c 2031 2c20 312c 2032 2c20 322c 2031 2, 1, 1, 2, 2, 1 │ │ │ │ +0000fae0: 2c20 312c 2032 2c20 312c 2031 2c7c 0a7c , 1, 2, 1, 1,|.| │ │ │ │ +0000faf0: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ +0000fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +0000fb40: 2020 2020 2032 2c20 327d 2920 2020 2020 2, 2}) │ │ │ │ +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: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000fb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000fbe0: 6f34 203a 2053 6571 7565 6e63 6520 2020 o4 : Sequence │ │ │ │ +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: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0000fc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000fc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000fc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000fc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0000fc80: 4e6f 7465 2074 6861 7420 7468 6520 6669 Note that the fi │ │ │ │ +0000fc90: 7273 7420 6765 6e65 7261 746f 7220 6c69 rst generator li │ │ │ │ +0000fca0: 7374 6564 2066 6f72 2024 4d24 2069 7320 sted for $M$ is │ │ │ │ +0000fcb0: 2461 5e32 242c 2061 6e64 2074 6865 2066 $a^2$, and the f │ │ │ │ +0000fcc0: 6972 7374 0a63 6f72 7265 7370 6f6e 6469 irst.correspondi │ │ │ │ +0000fcd0: 6e67 2073 7461 6e64 6172 6420 6d6f 6e6f ng standard mono │ │ │ │ +0000fce0: 6d69 616c 2069 7320 2461 2a63 242e 2020 mial is $a*c$. │ │ │ │ +0000fcf0: 5468 6520 6469 6666 6572 656e 6365 206f The difference o │ │ │ │ +0000fd00: 6620 7468 6573 6520 7477 6f0a 6d6f 6e6f f these two.mono │ │ │ │ +0000fd10: 6d69 616c 7320 6578 706f 6e65 6e74 2076 mials exponent v │ │ │ │ +0000fd20: 6563 746f 7273 2069 7320 2428 312c 302c ectors is $(1,0, │ │ │ │ +0000fd30: 2d31 2c30 2924 2e20 2054 6869 7320 7665 -1,0)$. This ve │ │ │ │ +0000fd40: 6374 6f72 2064 6f74 7465 6420 7769 7468 ctor dotted with │ │ │ │ +0000fd50: 2074 6865 2077 6569 6768 740a 7665 6374 the weight.vect │ │ │ │ +0000fd60: 6f72 2024 2832 2c32 2c31 2c31 2924 2067 or $(2,2,1,1)$ g │ │ │ │ +0000fd70: 6976 6573 2074 6865 2076 616c 7565 2024 ives the value $ │ │ │ │ +0000fd80: 3124 2c20 7768 6963 6820 6973 2074 6865 1$, which is the │ │ │ │ +0000fd90: 2066 6972 7374 2076 616c 7565 2069 6e20 first value in │ │ │ │ +0000fda0: 7468 6520 7365 636f 6e64 0a6c 6973 742e the second.list. │ │ │ │ +0000fdb0: 0a0a 4e6f 7465 2074 6861 7420 7468 6520 ..Note that the │ │ │ │ +0000fdc0: 6465 7369 7265 6420 7465 726d 206f 7264 desired term ord │ │ │ │ +0000fdd0: 6572 696e 672c 2061 6e64 2068 656e 6365 ering, and hence │ │ │ │ +0000fde0: 2077 6569 6768 7420 7665 6374 6f72 2c20 weight vector, │ │ │ │ +0000fdf0: 6d61 7920 6e6f 7420 6578 6973 742e 2049 may not exist. I │ │ │ │ +0000fe00: 6e0a 7468 6973 2063 6173 652c 206e 756c n.this case, nul │ │ │ │ +0000fe10: 6c20 6973 2072 6574 7572 6e65 642e 0a0a l is returned... │ │ │ │ +0000fe20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0000fe30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000fe40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000fe50: 2d2b 0a7c 6935 203a 204d 203d 2069 6465 -+.|i5 : M = ide │ │ │ │ +0000fe60: 616c 2261 6222 2020 2020 2020 2020 2020 al"ab" │ │ │ │ +0000fe70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000fe80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0000fe90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000fea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000feb0: 2020 2020 2020 207c 0a7c 6f35 203d 2069 |.|o5 = i │ │ │ │ +0000fec0: 6465 616c 2861 2a62 2920 2020 2020 2020 deal(a*b) │ │ │ │ 0000fed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000fef0: 2020 7c0a 7c6f 3620 3a20 4c69 7374 2020 |.|o6 : List │ │ │ │ +0000fee0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0000fef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ff00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ff10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000ff20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0000ff30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ff50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 --------+.|i7 : │ │ │ │ -0000ff60: 6669 6e64 5765 6967 6874 5665 6374 6f72 findWeightVector │ │ │ │ -0000ff70: 284d 2c4c 3129 2020 2020 2020 2020 2020 (M,L1) │ │ │ │ -0000ff80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0000ff90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ffa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0000ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0000ffc0: 0a54 6869 7320 636f 6d6d 616e 6420 6973 .This command is │ │ │ │ -0000ffd0: 2075 7365 6420 696e 2074 6865 202a 6e6f used in the *no │ │ │ │ -0000ffe0: 7465 2067 726f 6562 6e65 7246 616d 696c te groebnerFamil │ │ │ │ -0000fff0: 793a 2067 726f 6562 6e65 7246 616d 696c y: groebnerFamil │ │ │ │ -00010000: 792c 2072 6f75 7469 6e65 2e0a 0a53 6565 y, routine...See │ │ │ │ -00010010: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -00010020: 2020 2a20 2a6e 6f74 6520 6772 6f65 626e * *note groebn │ │ │ │ -00010030: 6572 5374 7261 7475 6d3a 2067 726f 6562 erStratum: groeb │ │ │ │ -00010040: 6e65 7253 7472 6174 756d 2c20 2d2d 2063 nerStratum, -- c │ │ │ │ -00010050: 6f6d 7075 7465 2074 6865 2069 6465 616c ompute the ideal │ │ │ │ -00010060: 2077 6865 7265 2061 2067 6976 656e 0a20 where a given. │ │ │ │ -00010070: 2020 2069 7320 6120 4772 6f65 626e 6572 is a Groebner │ │ │ │ -00010080: 2062 6173 6973 0a20 202a 202a 6e6f 7465 basis. * *note │ │ │ │ -00010090: 2067 726f 6562 6e65 7246 616d 696c 793a groebnerFamily: │ │ │ │ -000100a0: 2067 726f 6562 6e65 7246 616d 696c 792c groebnerFamily, │ │ │ │ -000100b0: 202d 2d20 636f 6d70 7574 6573 2066 616d -- computes fam │ │ │ │ -000100c0: 696c 6965 7320 6f66 2069 6465 616c 7320 ilies of ideals │ │ │ │ -000100d0: 7769 7468 2061 0a20 2020 2073 7065 6369 with a. speci │ │ │ │ -000100e0: 6669 6564 2069 6e69 7469 616c 2069 6465 fied initial ide │ │ │ │ -000100f0: 616c 0a0a 5761 7973 2074 6f20 7573 6520 al..Ways to use │ │ │ │ -00010100: 6669 6e64 5765 6967 6874 5665 6374 6f72 findWeightVector │ │ │ │ -00010110: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00010120: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00010130: 0a20 202a 2022 6669 6e64 5765 6967 6874 . * "findWeight │ │ │ │ -00010140: 5665 6374 6f72 2849 6465 616c 2c4c 6973 Vector(Ideal,Lis │ │ │ │ -00010150: 7429 220a 0a46 6f72 2074 6865 2070 726f t)"..For the pro │ │ │ │ -00010160: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -00010170: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -00010180: 6f62 6a65 6374 202a 6e6f 7465 2066 696e object *note fin │ │ │ │ -00010190: 6457 6569 6768 7456 6563 746f 723a 2066 dWeightVector: f │ │ │ │ -000101a0: 696e 6457 6569 6768 7456 6563 746f 722c indWeightVector, │ │ │ │ -000101b0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ -000101c0: 6f64 0a66 756e 6374 696f 6e3a 2028 4d61 od.function: (Ma │ │ │ │ -000101d0: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -000101e0: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ -000101f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00010240: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00010250: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00010260: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00010270: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00010280: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00010290: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -000102a0: 6765 732f 0a47 726f 6562 6e65 7253 7472 ges/.GroebnerStr │ │ │ │ -000102b0: 6174 612e 6d32 3a36 3337 3a30 2e0a 1f0a ata.m2:637:0.... │ │ │ │ -000102c0: 4669 6c65 3a20 4772 6f65 626e 6572 5374 File: GroebnerSt │ │ │ │ -000102d0: 7261 7461 2e69 6e66 6f2c 204e 6f64 653a rata.info, Node: │ │ │ │ -000102e0: 2067 726f 6562 6e65 7246 616d 696c 792c groebnerFamily, │ │ │ │ -000102f0: 204e 6578 743a 2067 726f 6562 6e65 7253 Next: groebnerS │ │ │ │ -00010300: 7472 6174 756d 2c20 5072 6576 3a20 6669 tratum, Prev: fi │ │ │ │ -00010310: 6e64 5765 6967 6874 5665 6374 6f72 2c20 ndWeightVector, │ │ │ │ -00010320: 5570 3a20 546f 700a 0a67 726f 6562 6e65 Up: Top..groebne │ │ │ │ -00010330: 7246 616d 696c 7920 2d2d 2063 6f6d 7075 rFamily -- compu │ │ │ │ -00010340: 7465 7320 6661 6d69 6c69 6573 206f 6620 tes families of │ │ │ │ -00010350: 6964 6561 6c73 2077 6974 6820 6120 7370 ideals with a sp │ │ │ │ -00010360: 6563 6966 6965 6420 696e 6974 6961 6c20 ecified initial │ │ │ │ -00010370: 6964 6561 6c0a 2a2a 2a2a 2a2a 2a2a 2a2a ideal.********** │ │ │ │ -00010380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00010390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000103a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000103b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000103c0: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ -000103d0: 2020 2020 2020 2020 4a20 3d20 6772 6f65 J = groe │ │ │ │ -000103e0: 626e 6572 4661 6d69 6c79 204d 0a20 2020 bnerFamily M. │ │ │ │ -000103f0: 2020 2020 204a 203d 2067 726f 6562 6e65 J = groebne │ │ │ │ -00010400: 7246 616d 696c 7928 4d2c 204c 290a 2020 rFamily(M, L). │ │ │ │ -00010410: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -00010420: 2a20 4d2c 2061 6e20 2a6e 6f74 6520 6964 * M, an *note id │ │ │ │ -00010430: 6561 6c3a 2028 4d61 6361 756c 6179 3244 eal: (Macaulay2D │ │ │ │ -00010440: 6f63 2949 6465 616c 2c2c 2077 6869 6368 oc)Ideal,, which │ │ │ │ -00010450: 2069 7320 6765 6e65 7261 7465 6420 6279 is generated by │ │ │ │ -00010460: 0a20 2020 2020 2020 206d 6f6e 6f6d 6961 . monomia │ │ │ │ -00010470: 6c73 0a20 2020 2020 202a 204c 2c20 6120 ls. * L, a │ │ │ │ -00010480: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ -00010490: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ -000104a0: 2061 206c 6973 7420 6f66 206c 6973 7473 a list of lists │ │ │ │ -000104b0: 206f 6620 7374 616e 6461 7264 0a20 2020 of standard. │ │ │ │ -000104c0: 2020 2020 206d 6f6e 6f6d 6961 6c73 206f monomials o │ │ │ │ -000104d0: 7220 736d 616c 6c65 7220 7374 616e 6461 r smaller standa │ │ │ │ -000104e0: 7264 206d 6f6e 6f6d 6961 6c73 2066 6f72 rd monomials for │ │ │ │ -000104f0: 2074 6865 2067 656e 6572 6174 6f72 7320 the generators │ │ │ │ -00010500: 6f66 204d 0a20 202a 202a 6e6f 7465 204f of M. * *note O │ │ │ │ -00010510: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ -00010520: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ -00010530: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ -00010540: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ -00010550: 7473 2c3a 0a20 2020 2020 202a 2041 6c6c ts,:. * All │ │ │ │ -00010560: 5374 616e 6461 7264 203d 3e20 6120 2a6e Standard => a *n │ │ │ │ -00010570: 6f74 6520 426f 6f6c 6561 6e20 7661 6c75 ote Boolean valu │ │ │ │ -00010580: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -00010590: 2942 6f6f 6c65 616e 2c2c 2064 6566 6175 )Boolean,, defau │ │ │ │ -000105a0: 6c74 0a20 2020 2020 2020 2076 616c 7565 lt. value │ │ │ │ -000105b0: 2066 616c 7365 0a20 2020 2020 202a 2057 false. * W │ │ │ │ -000105c0: 6569 6768 7473 203d 3e20 6120 2a6e 6f74 eights => a *not │ │ │ │ -000105d0: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ -000105e0: 7932 446f 6329 4c69 7374 2c2c 2064 6566 y2Doc)List,, def │ │ │ │ -000105f0: 6175 6c74 2076 616c 7565 206e 756c 6c0a ault value null. │ │ │ │ -00010600: 2020 2020 2020 2a20 5661 7269 6162 6c65 * Variable │ │ │ │ -00010610: 203d 3e20 6120 2a6e 6f74 6520 7379 6d62 => a *note symb │ │ │ │ -00010620: 6f6c 3a20 284d 6163 6175 6c61 7932 446f ol: (Macaulay2Do │ │ │ │ -00010630: 6329 5379 6d62 6f6c 2c2c 2064 6566 6175 c)Symbol,, defau │ │ │ │ -00010640: 6c74 2076 616c 7565 2074 2c20 6f72 0a20 lt value t, or. │ │ │ │ -00010650: 2020 2020 2020 2061 202a 6e6f 7465 2073 a *note s │ │ │ │ -00010660: 7472 696e 673a 2028 4d61 6361 756c 6179 tring: (Macaulay │ │ │ │ -00010670: 3244 6f63 2953 7472 696e 672c 0a20 202a 2Doc)String,. * │ │ │ │ -00010680: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00010690: 2a20 462c 2061 6e20 2a6e 6f74 6520 6964 * F, an *note id │ │ │ │ -000106a0: 6561 6c3a 2028 4d61 6361 756c 6179 3244 eal: (Macaulay2D │ │ │ │ -000106b0: 6f63 2949 6465 616c 2c2c 2074 6865 2067 oc)Ideal,, the g │ │ │ │ -000106c0: 726f 6562 6e65 7220 6661 6d69 6c79 2c20 roebner family, │ │ │ │ -000106d0: 616e 2069 6465 616c 0a20 2020 2020 2020 an ideal. │ │ │ │ -000106e0: 2069 6e20 7468 6520 706f 6c79 6e6f 6d69 in the polynomi │ │ │ │ -000106f0: 616c 2072 696e 6720 6f76 6572 2074 6865 al ring over the │ │ │ │ -00010700: 206f 7269 6769 6e61 6c20 7661 7269 6162 original variab │ │ │ │ -00010710: 6c65 7320 616e 6420 7468 6520 7061 7261 les and the para │ │ │ │ -00010720: 6d65 7465 7273 0a0a 4465 7363 7269 7074 meters..Descript │ │ │ │ -00010730: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00010740: 0a47 6976 656e 2061 206d 6f6e 6f6d 6961 .Given a monomia │ │ │ │ -00010750: 6c20 6964 6561 6c20 244d 2420 696e 2061 l ideal $M$ in a │ │ │ │ -00010760: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ -00010770: 2024 5224 2c20 7468 6973 2063 6f6d 7075 $R$, this compu │ │ │ │ -00010780: 7465 7320 7468 650a 7061 7261 6d65 7465 tes the.paramete │ │ │ │ -00010790: 7220 6661 6d69 6c69 6573 206f 6620 686f r families of ho │ │ │ │ -000107a0: 6d6f 6765 6e65 6f75 7320 6964 6561 6c73 mogeneous ideals │ │ │ │ -000107b0: 2077 6865 7265 2024 4d24 2063 6f75 6c64 where $M$ could │ │ │ │ -000107c0: 2062 6520 7468 6569 7220 696e 6974 6961 be their initia │ │ │ │ -000107d0: 6c0a 6964 6561 6c2e 2020 5468 6573 6520 l.ideal. These │ │ │ │ -000107e0: 6661 6d69 6c69 6573 2061 7265 206f 6274 families are obt │ │ │ │ -000107f0: 6169 6e65 6420 6672 6f6d 2065 6974 6865 ained from eithe │ │ │ │ -00010800: 7220 7468 6520 7374 616e 6461 7264 206d r the standard m │ │ │ │ -00010810: 6f6e 6f6d 6961 6c73 2074 6f20 7468 650a onomials to the. │ │ │ │ -00010820: 6765 6e65 7261 746f 7273 206f 6620 244d generators of $M │ │ │ │ -00010830: 242c 206f 7220 7468 6520 7374 616e 6461 $, or the standa │ │ │ │ -00010840: 7264 206d 6f6e 6f6d 6961 6c73 2073 6d61 rd monomials sma │ │ │ │ -00010850: 6c6c 6572 2074 6861 6e20 7468 6520 6765 ller than the ge │ │ │ │ -00010860: 6e65 7261 746f 7273 206f 6620 244d 240a nerators of $M$. │ │ │ │ -00010870: 6275 7420 6f66 2074 6865 2073 616d 6520 but of the same │ │ │ │ -00010880: 6465 6772 6565 2061 7320 7468 6573 6520 degree as these │ │ │ │ -00010890: 6765 6e65 7261 746f 7273 2e20 2049 6e20 generators. In │ │ │ │ -000108a0: 7468 6520 666f 726d 6572 2063 6173 6520 the former case │ │ │ │ -000108b0: 7765 206f 6274 6169 6e20 610a 6661 6d69 we obtain a.fami │ │ │ │ -000108c0: 6c79 206f 6620 616c 6c20 6964 6561 6c73 ly of all ideals │ │ │ │ -000108d0: 2077 6865 7265 2024 4d24 2063 6f75 6c64 where $M$ could │ │ │ │ -000108e0: 2062 6520 7468 6569 7220 696e 6974 6961 be their initia │ │ │ │ -000108f0: 6c20 6964 6561 6c2e 2020 496e 2074 6865 l ideal. In the │ │ │ │ -00010900: 206c 6174 7465 720a 6361 7365 2c20 7765 latter.case, we │ │ │ │ -00010910: 206f 6274 6169 6e20 7375 6368 2061 2066 obtain such a f │ │ │ │ -00010920: 616d 696c 7920 7769 7468 2072 6573 7065 amily with respe │ │ │ │ -00010930: 6374 2074 6f20 6120 6769 7665 6e20 7465 ct to a given te │ │ │ │ -00010940: 726d 206f 7264 6572 2e0a 0a2b 2d2d 2d2d rm order...+---- │ │ │ │ -00010950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010990: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ -000109a0: 2052 203d 205a 5a2f 3332 3030 335b 612c R = ZZ/32003[a, │ │ │ │ -000109b0: 622c 632c 645d 3b20 2020 2020 2020 2020 b,c,d]; │ │ │ │ -000109c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000109d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000109e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000109f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010a30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -00010a40: 204d 203d 2069 6465 616c 2028 615e 322c M = ideal (a^2, │ │ │ │ -00010a50: 2061 2a62 2c20 625e 3229 2020 2020 2020 a*b, b^2) │ │ │ │ -00010a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010a80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00010a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010ad0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00010ae0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00010af0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00010b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010b20: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ -00010b30: 2069 6465 616c 2028 6120 2c20 612a 622c ideal (a , a*b, │ │ │ │ -00010b40: 2062 2029 2020 2020 2020 2020 2020 2020 b ) │ │ │ │ +0000ff10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0000ff20: 6f35 203a 2049 6465 616c 206f 6620 5220 o5 : Ideal of R │ │ │ │ +0000ff30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ff40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ff50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0000ff60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ff70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0000ff80: 2d2d 2d2b 0a7c 6936 203a 204c 3120 3d20 ---+.|i6 : L1 = │ │ │ │ +0000ff90: 7374 616e 6461 7264 4d6f 6e6f 6d69 616c standardMonomial │ │ │ │ +0000ffa0: 7320 4d20 2020 2020 2020 2020 2020 2020 s M │ │ │ │ +0000ffb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0000ffc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ffd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0000ffe0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0000fff0: 2020 2020 3220 2020 3220 2020 2020 2020 2 2 │ │ │ │ +00010000: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00010010: 2020 2020 2020 2020 2032 2020 7c0a 7c6f 2 |.|o │ │ │ │ +00010020: 3620 3d20 7b7b 6120 2c20 6220 2c20 612a 6 = {{a , b , a* │ │ │ │ +00010030: 632c 2062 2a63 2c20 6320 2c20 612a 642c c, b*c, c , a*d, │ │ │ │ +00010040: 2062 2a64 2c20 632a 642c 2064 207d 7d7c b*d, c*d, d }}| │ │ │ │ +00010050: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00010060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010080: 2020 7c0a 7c6f 3620 3a20 4c69 7374 2020 |.|o6 : List │ │ │ │ +00010090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000100a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000100b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000100c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000100d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000100e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 --------+.|i7 : │ │ │ │ +000100f0: 6669 6e64 5765 6967 6874 5665 6374 6f72 findWeightVector │ │ │ │ +00010100: 284d 2c4c 3129 2020 2020 2020 2020 2020 (M,L1) │ │ │ │ +00010110: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00010120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00010150: 0a54 6869 7320 636f 6d6d 616e 6420 6973 .This command is │ │ │ │ +00010160: 2075 7365 6420 696e 2074 6865 202a 6e6f used in the *no │ │ │ │ +00010170: 7465 2067 726f 6562 6e65 7246 616d 696c te groebnerFamil │ │ │ │ +00010180: 793a 2067 726f 6562 6e65 7246 616d 696c y: groebnerFamil │ │ │ │ +00010190: 792c 2072 6f75 7469 6e65 2e0a 0a53 6565 y, routine...See │ │ │ │ +000101a0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ +000101b0: 2020 2a20 2a6e 6f74 6520 6772 6f65 626e * *note groebn │ │ │ │ +000101c0: 6572 5374 7261 7475 6d3a 2067 726f 6562 erStratum: groeb │ │ │ │ +000101d0: 6e65 7253 7472 6174 756d 2c20 2d2d 2063 nerStratum, -- c │ │ │ │ +000101e0: 6f6d 7075 7465 2074 6865 2069 6465 616c ompute the ideal │ │ │ │ +000101f0: 2077 6865 7265 2061 2067 6976 656e 0a20 where a given. │ │ │ │ +00010200: 2020 2069 7320 6120 4772 6f65 626e 6572 is a Groebner │ │ │ │ +00010210: 2062 6173 6973 0a20 202a 202a 6e6f 7465 basis. * *note │ │ │ │ +00010220: 2067 726f 6562 6e65 7246 616d 696c 793a groebnerFamily: │ │ │ │ +00010230: 2067 726f 6562 6e65 7246 616d 696c 792c groebnerFamily, │ │ │ │ +00010240: 202d 2d20 636f 6d70 7574 6573 2066 616d -- computes fam │ │ │ │ +00010250: 696c 6965 7320 6f66 2069 6465 616c 7320 ilies of ideals │ │ │ │ +00010260: 7769 7468 2061 0a20 2020 2073 7065 6369 with a. speci │ │ │ │ +00010270: 6669 6564 2069 6e69 7469 616c 2069 6465 fied initial ide │ │ │ │ +00010280: 616c 0a0a 5761 7973 2074 6f20 7573 6520 al..Ways to use │ │ │ │ +00010290: 6669 6e64 5765 6967 6874 5665 6374 6f72 findWeightVector │ │ │ │ +000102a0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +000102b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +000102c0: 0a20 202a 2022 6669 6e64 5765 6967 6874 . * "findWeight │ │ │ │ +000102d0: 5665 6374 6f72 2849 6465 616c 2c4c 6973 Vector(Ideal,Lis │ │ │ │ +000102e0: 7429 220a 0a46 6f72 2074 6865 2070 726f t)"..For the pro │ │ │ │ +000102f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00010300: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00010310: 6f62 6a65 6374 202a 6e6f 7465 2066 696e object *note fin │ │ │ │ +00010320: 6457 6569 6768 7456 6563 746f 723a 2066 dWeightVector: f │ │ │ │ +00010330: 696e 6457 6569 6768 7456 6563 746f 722c indWeightVector, │ │ │ │ +00010340: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ +00010350: 6f64 0a66 756e 6374 696f 6e3a 2028 4d61 od.function: (Ma │ │ │ │ +00010360: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ +00010370: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ +00010380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000103a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000103b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000103c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +000103d0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +000103e0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +000103f0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00010400: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00010410: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ +00010420: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00010430: 6765 732f 0a47 726f 6562 6e65 7253 7472 ges/.GroebnerStr │ │ │ │ +00010440: 6174 612e 6d32 3a36 3337 3a30 2e0a 1f0a ata.m2:637:0.... │ │ │ │ +00010450: 4669 6c65 3a20 4772 6f65 626e 6572 5374 File: GroebnerSt │ │ │ │ +00010460: 7261 7461 2e69 6e66 6f2c 204e 6f64 653a rata.info, Node: │ │ │ │ +00010470: 2067 726f 6562 6e65 7246 616d 696c 792c groebnerFamily, │ │ │ │ +00010480: 204e 6578 743a 2067 726f 6562 6e65 7253 Next: groebnerS │ │ │ │ +00010490: 7472 6174 756d 2c20 5072 6576 3a20 6669 tratum, Prev: fi │ │ │ │ +000104a0: 6e64 5765 6967 6874 5665 6374 6f72 2c20 ndWeightVector, │ │ │ │ +000104b0: 5570 3a20 546f 700a 0a67 726f 6562 6e65 Up: Top..groebne │ │ │ │ +000104c0: 7246 616d 696c 7920 2d2d 2063 6f6d 7075 rFamily -- compu │ │ │ │ +000104d0: 7465 7320 6661 6d69 6c69 6573 206f 6620 tes families of │ │ │ │ +000104e0: 6964 6561 6c73 2077 6974 6820 6120 7370 ideals with a sp │ │ │ │ +000104f0: 6563 6966 6965 6420 696e 6974 6961 6c20 ecified initial │ │ │ │ +00010500: 6964 6561 6c0a 2a2a 2a2a 2a2a 2a2a 2a2a ideal.********** │ │ │ │ +00010510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00010520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00010530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00010540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00010550: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +00010560: 2020 2020 2020 2020 4a20 3d20 6772 6f65 J = groe │ │ │ │ +00010570: 626e 6572 4661 6d69 6c79 204d 0a20 2020 bnerFamily M. │ │ │ │ +00010580: 2020 2020 204a 203d 2067 726f 6562 6e65 J = groebne │ │ │ │ +00010590: 7246 616d 696c 7928 4d2c 204c 290a 2020 rFamily(M, L). │ │ │ │ +000105a0: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +000105b0: 2a20 4d2c 2061 6e20 2a6e 6f74 6520 6964 * M, an *note id │ │ │ │ +000105c0: 6561 6c3a 2028 4d61 6361 756c 6179 3244 eal: (Macaulay2D │ │ │ │ +000105d0: 6f63 2949 6465 616c 2c2c 2077 6869 6368 oc)Ideal,, which │ │ │ │ +000105e0: 2069 7320 6765 6e65 7261 7465 6420 6279 is generated by │ │ │ │ +000105f0: 0a20 2020 2020 2020 206d 6f6e 6f6d 6961 . monomia │ │ │ │ +00010600: 6c73 0a20 2020 2020 202a 204c 2c20 6120 ls. * L, a │ │ │ │ +00010610: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ +00010620: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ +00010630: 2061 206c 6973 7420 6f66 206c 6973 7473 a list of lists │ │ │ │ +00010640: 206f 6620 7374 616e 6461 7264 0a20 2020 of standard. │ │ │ │ +00010650: 2020 2020 206d 6f6e 6f6d 6961 6c73 206f monomials o │ │ │ │ +00010660: 7220 736d 616c 6c65 7220 7374 616e 6461 r smaller standa │ │ │ │ +00010670: 7264 206d 6f6e 6f6d 6961 6c73 2066 6f72 rd monomials for │ │ │ │ +00010680: 2074 6865 2067 656e 6572 6174 6f72 7320 the generators │ │ │ │ +00010690: 6f66 204d 0a20 202a 202a 6e6f 7465 204f of M. * *note O │ │ │ │ +000106a0: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ +000106b0: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ +000106c0: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ +000106d0: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ +000106e0: 7473 2c3a 0a20 2020 2020 202a 2041 6c6c ts,:. * All │ │ │ │ +000106f0: 5374 616e 6461 7264 203d 3e20 6120 2a6e Standard => a *n │ │ │ │ +00010700: 6f74 6520 426f 6f6c 6561 6e20 7661 6c75 ote Boolean valu │ │ │ │ +00010710: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +00010720: 2942 6f6f 6c65 616e 2c2c 2064 6566 6175 )Boolean,, defau │ │ │ │ +00010730: 6c74 0a20 2020 2020 2020 2076 616c 7565 lt. value │ │ │ │ +00010740: 2066 616c 7365 0a20 2020 2020 202a 2057 false. * W │ │ │ │ +00010750: 6569 6768 7473 203d 3e20 6120 2a6e 6f74 eights => a *not │ │ │ │ +00010760: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +00010770: 7932 446f 6329 4c69 7374 2c2c 2064 6566 y2Doc)List,, def │ │ │ │ +00010780: 6175 6c74 2076 616c 7565 206e 756c 6c0a ault value null. │ │ │ │ +00010790: 2020 2020 2020 2a20 5661 7269 6162 6c65 * Variable │ │ │ │ +000107a0: 203d 3e20 6120 2a6e 6f74 6520 7379 6d62 => a *note symb │ │ │ │ +000107b0: 6f6c 3a20 284d 6163 6175 6c61 7932 446f ol: (Macaulay2Do │ │ │ │ +000107c0: 6329 5379 6d62 6f6c 2c2c 2064 6566 6175 c)Symbol,, defau │ │ │ │ +000107d0: 6c74 2076 616c 7565 2074 2c20 6f72 0a20 lt value t, or. │ │ │ │ +000107e0: 2020 2020 2020 2061 202a 6e6f 7465 2073 a *note s │ │ │ │ +000107f0: 7472 696e 673a 2028 4d61 6361 756c 6179 tring: (Macaulay │ │ │ │ +00010800: 3244 6f63 2953 7472 696e 672c 0a20 202a 2Doc)String,. * │ │ │ │ +00010810: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +00010820: 2a20 462c 2061 6e20 2a6e 6f74 6520 6964 * F, an *note id │ │ │ │ +00010830: 6561 6c3a 2028 4d61 6361 756c 6179 3244 eal: (Macaulay2D │ │ │ │ +00010840: 6f63 2949 6465 616c 2c2c 2074 6865 2067 oc)Ideal,, the g │ │ │ │ +00010850: 726f 6562 6e65 7220 6661 6d69 6c79 2c20 roebner family, │ │ │ │ +00010860: 616e 2069 6465 616c 0a20 2020 2020 2020 an ideal. │ │ │ │ +00010870: 2069 6e20 7468 6520 706f 6c79 6e6f 6d69 in the polynomi │ │ │ │ +00010880: 616c 2072 696e 6720 6f76 6572 2074 6865 al ring over the │ │ │ │ +00010890: 206f 7269 6769 6e61 6c20 7661 7269 6162 original variab │ │ │ │ +000108a0: 6c65 7320 616e 6420 7468 6520 7061 7261 les and the para │ │ │ │ +000108b0: 6d65 7465 7273 0a0a 4465 7363 7269 7074 meters..Descript │ │ │ │ +000108c0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +000108d0: 0a47 6976 656e 2061 206d 6f6e 6f6d 6961 .Given a monomia │ │ │ │ +000108e0: 6c20 6964 6561 6c20 244d 2420 696e 2061 l ideal $M$ in a │ │ │ │ +000108f0: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ +00010900: 2024 5224 2c20 7468 6973 2063 6f6d 7075 $R$, this compu │ │ │ │ +00010910: 7465 7320 7468 650a 7061 7261 6d65 7465 tes the.paramete │ │ │ │ +00010920: 7220 6661 6d69 6c69 6573 206f 6620 686f r families of ho │ │ │ │ +00010930: 6d6f 6765 6e65 6f75 7320 6964 6561 6c73 mogeneous ideals │ │ │ │ +00010940: 2077 6865 7265 2024 4d24 2063 6f75 6c64 where $M$ could │ │ │ │ +00010950: 2062 6520 7468 6569 7220 696e 6974 6961 be their initia │ │ │ │ +00010960: 6c0a 6964 6561 6c2e 2020 5468 6573 6520 l.ideal. These │ │ │ │ +00010970: 6661 6d69 6c69 6573 2061 7265 206f 6274 families are obt │ │ │ │ +00010980: 6169 6e65 6420 6672 6f6d 2065 6974 6865 ained from eithe │ │ │ │ +00010990: 7220 7468 6520 7374 616e 6461 7264 206d r the standard m │ │ │ │ +000109a0: 6f6e 6f6d 6961 6c73 2074 6f20 7468 650a onomials to the. │ │ │ │ +000109b0: 6765 6e65 7261 746f 7273 206f 6620 244d generators of $M │ │ │ │ +000109c0: 242c 206f 7220 7468 6520 7374 616e 6461 $, or the standa │ │ │ │ +000109d0: 7264 206d 6f6e 6f6d 6961 6c73 2073 6d61 rd monomials sma │ │ │ │ +000109e0: 6c6c 6572 2074 6861 6e20 7468 6520 6765 ller than the ge │ │ │ │ +000109f0: 6e65 7261 746f 7273 206f 6620 244d 240a nerators of $M$. │ │ │ │ +00010a00: 6275 7420 6f66 2074 6865 2073 616d 6520 but of the same │ │ │ │ +00010a10: 6465 6772 6565 2061 7320 7468 6573 6520 degree as these │ │ │ │ +00010a20: 6765 6e65 7261 746f 7273 2e20 2049 6e20 generators. In │ │ │ │ +00010a30: 7468 6520 666f 726d 6572 2063 6173 6520 the former case │ │ │ │ +00010a40: 7765 206f 6274 6169 6e20 610a 6661 6d69 we obtain a.fami │ │ │ │ +00010a50: 6c79 206f 6620 616c 6c20 6964 6561 6c73 ly of all ideals │ │ │ │ +00010a60: 2077 6865 7265 2024 4d24 2063 6f75 6c64 where $M$ could │ │ │ │ +00010a70: 2062 6520 7468 6569 7220 696e 6974 6961 be their initia │ │ │ │ +00010a80: 6c20 6964 6561 6c2e 2020 496e 2074 6865 l ideal. In the │ │ │ │ +00010a90: 206c 6174 7465 720a 6361 7365 2c20 7765 latter.case, we │ │ │ │ +00010aa0: 206f 6274 6169 6e20 7375 6368 2061 2066 obtain such a f │ │ │ │ +00010ab0: 616d 696c 7920 7769 7468 2072 6573 7065 amily with respe │ │ │ │ +00010ac0: 6374 2074 6f20 6120 6769 7665 6e20 7465 ct to a given te │ │ │ │ +00010ad0: 726d 206f 7264 6572 2e0a 0a2b 2d2d 2d2d rm order...+---- │ │ │ │ +00010ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010b20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ +00010b30: 2052 203d 205a 5a2f 3332 3030 335b 612c R = ZZ/32003[a, │ │ │ │ +00010b40: 622c 632c 645d 3b20 2020 2020 2020 2020 b,c,d]; │ │ │ │ 00010b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010b70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00010b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010bc0: 2020 2020 2020 2020 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ -00010bd0: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ │ │ -00010be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010b70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00010b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010bc0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ +00010bd0: 204d 203d 2069 6465 616c 2028 615e 322c M = ideal (a^2, │ │ │ │ +00010be0: 2061 2a62 2c20 625e 3229 2020 2020 2020 a*b, b^2) │ │ │ │ 00010bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010c10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00010c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010c60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -00010c70: 2046 203d 2067 726f 6562 6e65 7246 616d F = groebnerFam │ │ │ │ -00010c80: 696c 7920 4d20 2020 2020 2020 2020 2020 ily M │ │ │ │ +00010c10: 2020 2020 2020 2020 207c 0a7c 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 2020 2020 2020 │ │ │ │ +00010c60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00010c70: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00010c80: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00010c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010cb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00010cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010cb0: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ +00010cc0: 2069 6465 616c 2028 6120 2c20 612a 622c ideal (a , a*b, │ │ │ │ +00010cd0: 2062 2029 2020 2020 2020 2020 2020 2020 b ) │ │ │ │ 00010ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00010d10: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00010d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010d40: 3220 2020 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ -00010d50: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ -00010d60: 2069 6465 616c 2028 6120 202b 2074 2061 ideal (a + t a │ │ │ │ -00010d70: 2a63 202b 2074 2062 2a63 202b 2074 2061 *c + t b*c + t a │ │ │ │ -00010d80: 2a64 202b 2074 2062 2a64 202b 2074 2063 *d + t b*d + t c │ │ │ │ -00010d90: 2020 2b20 7420 632a 6420 2b20 7420 6420 + t c*d + t d │ │ │ │ -00010da0: 2c20 612a 6220 2b20 207c 0a7c 2020 2020 , a*b + |.| │ │ │ │ -00010db0: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ -00010dc0: 2020 2020 2020 3220 2020 2020 2020 3420 2 4 │ │ │ │ -00010dd0: 2020 2020 2020 3520 2020 2020 2020 3320 5 3 │ │ │ │ -00010de0: 2020 2020 2036 2020 2020 2020 2037 2020 6 7 │ │ │ │ -00010df0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00010e00: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ -00010e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010e40: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 ---------|.| │ │ │ │ +00010d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010d50: 2020 2020 2020 2020 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ +00010d60: 2049 6465 616c 206f 6620 5220 2020 2020 Ideal of R │ │ │ │ +00010d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010da0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00010db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010df0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +00010e00: 2046 203d 2067 726f 6562 6e65 7246 616d F = groebnerFam │ │ │ │ +00010e10: 696c 7920 4d20 2020 2020 2020 2020 2020 ily M │ │ │ │ +00010e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010e40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00010e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010e70: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00010e80: 2020 2020 2020 2020 3220 2020 3220 2020 2 2 │ │ │ │ +00010e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010e90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00010ea0: 2074 2061 2a63 202b 2074 2062 2a63 202b t a*c + t b*c + │ │ │ │ -00010eb0: 2074 2020 612a 6420 2b20 7420 2062 2a64 t a*d + t b*d │ │ │ │ -00010ec0: 202b 2074 2020 6320 202b 2074 2020 632a + t c + t c* │ │ │ │ -00010ed0: 6420 2b20 7420 2064 202c 2062 2020 2b20 d + t d , b + │ │ │ │ -00010ee0: 7420 2061 2a63 202b 207c 0a7c 2020 2020 t a*c + |.| │ │ │ │ -00010ef0: 2020 3820 2020 2020 2020 3920 2020 2020 8 9 │ │ │ │ -00010f00: 2020 3131 2020 2020 2020 2031 3220 2020 11 12 │ │ │ │ -00010f10: 2020 2020 3130 2020 2020 2020 3133 2020 10 13 │ │ │ │ -00010f20: 2020 2020 2031 3420 2020 2020 2020 2020 14 │ │ │ │ -00010f30: 2031 3520 2020 2020 207c 0a7c 2020 2020 15 |.| │ │ │ │ -00010f40: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ -00010f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00010f80: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 ---------|.| │ │ │ │ -00010f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010fb0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00010fc0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00010fd0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00010fe0: 2074 2020 622a 6320 2b20 7420 2061 2a64 t b*c + t a*d │ │ │ │ -00010ff0: 202b 2074 2020 622a 6420 2b20 7420 2063 + t b*d + t c │ │ │ │ -00011000: 2020 2b20 7420 2063 2a64 202b 2074 2020 + t c*d + t │ │ │ │ -00011010: 6420 2920 2020 2020 2020 2020 2020 2020 d ) │ │ │ │ +00010ea0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00010eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010ed0: 3220 2020 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ +00010ee0: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ +00010ef0: 2069 6465 616c 2028 6120 202b 2074 2061 ideal (a + t a │ │ │ │ +00010f00: 2a63 202b 2074 2062 2a63 202b 2074 2061 *c + t b*c + t a │ │ │ │ +00010f10: 2a64 202b 2074 2062 2a64 202b 2074 2063 *d + t b*d + t c │ │ │ │ +00010f20: 2020 2b20 7420 632a 6420 2b20 7420 6420 + t c*d + t d │ │ │ │ +00010f30: 2c20 612a 6220 2b20 207c 0a7c 2020 2020 , a*b + |.| │ │ │ │ +00010f40: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +00010f50: 2020 2020 2020 3220 2020 2020 2020 3420 2 4 │ │ │ │ +00010f60: 2020 2020 2020 3520 2020 2020 2020 3320 5 3 │ │ │ │ +00010f70: 2020 2020 2036 2020 2020 2020 2037 2020 6 7 │ │ │ │ +00010f80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00010f90: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ +00010fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00010fd0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 ---------|.| │ │ │ │ +00010fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00010ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011000: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00011010: 2020 2020 2020 2020 3220 2020 3220 2020 2 2 │ │ │ │ 00011020: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011030: 2020 3136 2020 2020 2020 2031 3820 2020 16 18 │ │ │ │ -00011040: 2020 2020 3139 2020 2020 2020 2031 3720 19 17 │ │ │ │ -00011050: 2020 2020 2032 3020 2020 2020 2020 3231 20 21 │ │ │ │ -00011060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011070: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000110a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000110b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000110c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000110d0: 2020 2020 2020 2020 2020 2020 5a5a 2020 ZZ │ │ │ │ -000110e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000110f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011110: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ -00011120: 2049 6465 616c 206f 6620 2d2d 2d2d 2d5b Ideal of -----[ │ │ │ │ -00011130: 7420 2c20 7420 2e2e 7420 2c20 7420 202c t , t ..t , t , │ │ │ │ -00011140: 2074 2020 2e2e 7420 202c 2074 2020 2c20 t ..t , t , │ │ │ │ -00011150: 7420 202e 2e74 2020 2c20 7420 2e2e 7420 t ..t , t ..t │ │ │ │ -00011160: 2c20 7420 2e2e 7420 2c7c 0a7c 2020 2020 , t ..t ,|.| │ │ │ │ -00011170: 2020 2020 2020 2020 2020 3332 3030 3320 32003 │ │ │ │ -00011180: 2033 2020 2036 2020 2037 2020 2031 3020 3 6 7 10 │ │ │ │ -00011190: 2020 3133 2020 2031 3420 2020 3137 2020 13 14 17 │ │ │ │ -000111a0: 2032 3020 2020 3231 2020 2031 2020 2032 20 21 1 2 │ │ │ │ -000111b0: 2020 2034 2020 2035 207c 0a7c 2d2d 2d2d 4 5 |.|---- │ │ │ │ -000111c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000111d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000111e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000111f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011200: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7420 2e2e ---------|.|t .. │ │ │ │ -00011210: 7420 2c20 7420 202e 2e74 2020 2c20 7420 t , t ..t , t │ │ │ │ -00011220: 202e 2e74 2020 2c20 7420 202e 2e74 2020 ..t , t ..t │ │ │ │ -00011230: 5d5b 612e 2e64 5d20 2020 2020 2020 2020 ][a..d] │ │ │ │ +00011030: 2074 2061 2a63 202b 2074 2062 2a63 202b t a*c + t b*c + │ │ │ │ +00011040: 2074 2020 612a 6420 2b20 7420 2062 2a64 t a*d + t b*d │ │ │ │ +00011050: 202b 2074 2020 6320 202b 2074 2020 632a + t c + t c* │ │ │ │ +00011060: 6420 2b20 7420 2064 202c 2062 2020 2b20 d + t d , b + │ │ │ │ +00011070: 7420 2061 2a63 202b 207c 0a7c 2020 2020 t a*c + |.| │ │ │ │ +00011080: 2020 3820 2020 2020 2020 3920 2020 2020 8 9 │ │ │ │ +00011090: 2020 3131 2020 2020 2020 2031 3220 2020 11 12 │ │ │ │ +000110a0: 2020 2020 3130 2020 2020 2020 3133 2020 10 13 │ │ │ │ +000110b0: 2020 2020 2031 3420 2020 2020 2020 2020 14 │ │ │ │ +000110c0: 2031 3520 2020 2020 207c 0a7c 2020 2020 15 |.| │ │ │ │ +000110d0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ +000110e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000110f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011110: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 ---------|.| │ │ │ │ +00011120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011140: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00011150: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00011160: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011170: 2074 2020 622a 6320 2b20 7420 2061 2a64 t b*c + t a*d │ │ │ │ +00011180: 202b 2074 2020 622a 6420 2b20 7420 2063 + t b*d + t c │ │ │ │ +00011190: 2020 2b20 7420 2063 2a64 202b 2074 2020 + t c*d + t │ │ │ │ +000111a0: 6420 2920 2020 2020 2020 2020 2020 2020 d ) │ │ │ │ +000111b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000111c0: 2020 3136 2020 2020 2020 2031 3820 2020 16 18 │ │ │ │ +000111d0: 2020 2020 3139 2020 2020 2020 2031 3720 19 17 │ │ │ │ +000111e0: 2020 2020 2032 3020 2020 2020 2020 3231 20 21 │ │ │ │ +000111f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011200: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011250: 2020 2020 2020 2020 207c 0a7c 2038 2020 |.| 8 │ │ │ │ -00011260: 2039 2020 2031 3120 2020 3132 2020 2031 9 11 12 1 │ │ │ │ -00011270: 3520 2020 3136 2020 2031 3820 2020 3139 5 16 18 19 │ │ │ │ +00011250: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011260: 2020 2020 2020 2020 2020 2020 5a5a 2020 ZZ │ │ │ │ +00011270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000112a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000112b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000112c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000112d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000112e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000112f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -00011300: 206e 6574 4c69 7374 2046 5f2a 2020 2020 netList F_* │ │ │ │ -00011310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011340: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011390: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000113a0: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +-------------- │ │ │ │ -000113b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000113c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000113d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000113e0: 2d2b 2020 2020 2020 207c 0a7c 2020 2020 -+ |.| │ │ │ │ -000113f0: 207c 2032 2020 2020 2020 2020 2020 2020 | 2 │ │ │ │ -00011400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011410: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -00011420: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00011430: 207c 2020 2020 2020 207c 0a7c 6f34 203d | |.|o4 = │ │ │ │ -00011440: 207c 6120 202b 2074 2061 2a63 202b 2074 |a + t a*c + t │ │ │ │ -00011450: 2062 2a63 202b 2074 2061 2a64 202b 2074 b*c + t a*d + t │ │ │ │ -00011460: 2062 2a64 202b 2074 2063 2020 2b20 7420 b*d + t c + t │ │ │ │ -00011470: 632a 6420 2b20 7420 6420 2020 2020 2020 c*d + t d │ │ │ │ -00011480: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ -00011490: 207c 2020 2020 2020 3120 2020 2020 2020 | 1 │ │ │ │ -000114a0: 3220 2020 2020 2020 3420 2020 2020 2020 2 4 │ │ │ │ -000114b0: 3520 2020 2020 2020 3320 2020 2020 2036 5 3 6 │ │ │ │ -000114c0: 2020 2020 2020 2037 2020 2020 2020 2020 7 │ │ │ │ -000114d0: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ -000114e0: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +-------------- │ │ │ │ -000114f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011520: 2d2b 2020 2020 2020 207c 0a7c 2020 2020 -+ |.| │ │ │ │ -00011530: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00011540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011550: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00011560: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00011570: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ -00011580: 207c 612a 6220 2b20 7420 612a 6320 2b20 |a*b + t a*c + │ │ │ │ -00011590: 7420 622a 6320 2b20 7420 2061 2a64 202b t b*c + t a*d + │ │ │ │ -000115a0: 2074 2020 622a 6420 2b20 7420 2063 2020 t b*d + t c │ │ │ │ -000115b0: 2b20 7420 2063 2a64 202b 2074 2020 6420 + t c*d + t d │ │ │ │ -000115c0: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ -000115d0: 207c 2020 2020 2020 2038 2020 2020 2020 | 8 │ │ │ │ -000115e0: 2039 2020 2020 2020 2031 3120 2020 2020 9 11 │ │ │ │ -000115f0: 2020 3132 2020 2020 2020 2031 3020 2020 12 10 │ │ │ │ -00011600: 2020 2031 3320 2020 2020 2020 3134 2020 13 14 │ │ │ │ +000112a0: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ +000112b0: 2049 6465 616c 206f 6620 2d2d 2d2d 2d5b Ideal of -----[ │ │ │ │ +000112c0: 7420 2c20 7420 2e2e 7420 2c20 7420 202c t , t ..t , t , │ │ │ │ +000112d0: 2074 2020 2e2e 7420 202c 2074 2020 2c20 t ..t , t , │ │ │ │ +000112e0: 7420 202e 2e74 2020 2c20 7420 2e2e 7420 t ..t , t ..t │ │ │ │ +000112f0: 2c20 7420 2e2e 7420 2c7c 0a7c 2020 2020 , t ..t ,|.| │ │ │ │ +00011300: 2020 2020 2020 2020 2020 3332 3030 3320 32003 │ │ │ │ +00011310: 2033 2020 2036 2020 2037 2020 2031 3020 3 6 7 10 │ │ │ │ +00011320: 2020 3133 2020 2031 3420 2020 3137 2020 13 14 17 │ │ │ │ +00011330: 2032 3020 2020 3231 2020 2031 2020 2032 20 21 1 2 │ │ │ │ +00011340: 2020 2034 2020 2035 207c 0a7c 2d2d 2d2d 4 5 |.|---- │ │ │ │ +00011350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011390: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 7420 2e2e ---------|.|t .. │ │ │ │ +000113a0: 7420 2c20 7420 202e 2e74 2020 2c20 7420 t , t ..t , t │ │ │ │ +000113b0: 202e 2e74 2020 2c20 7420 202e 2e74 2020 ..t , t ..t │ │ │ │ +000113c0: 5d5b 612e 2e64 5d20 2020 2020 2020 2020 ][a..d] │ │ │ │ +000113d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000113e0: 2020 2020 2020 2020 207c 0a7c 2038 2020 |.| 8 │ │ │ │ +000113f0: 2039 2020 2031 3120 2020 3132 2020 2031 9 11 12 1 │ │ │ │ +00011400: 3520 2020 3136 2020 2031 3820 2020 3139 5 16 18 19 │ │ │ │ +00011410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011430: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00011440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011480: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ +00011490: 206e 6574 4c69 7374 2046 5f2a 2020 2020 netList F_* │ │ │ │ +000114a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000114b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000114c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000114d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000114e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000114f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011520: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011530: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +-------------- │ │ │ │ +00011540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011570: 2d2b 2020 2020 2020 207c 0a7c 2020 2020 -+ |.| │ │ │ │ +00011580: 207c 2032 2020 2020 2020 2020 2020 2020 | 2 │ │ │ │ +00011590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000115a0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +000115b0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +000115c0: 207c 2020 2020 2020 207c 0a7c 6f34 203d | |.|o4 = │ │ │ │ +000115d0: 207c 6120 202b 2074 2061 2a63 202b 2074 |a + t a*c + t │ │ │ │ +000115e0: 2062 2a63 202b 2074 2061 2a64 202b 2074 b*c + t a*d + t │ │ │ │ +000115f0: 2062 2a64 202b 2074 2063 2020 2b20 7420 b*d + t c + t │ │ │ │ +00011600: 632a 6420 2b20 7420 6420 2020 2020 2020 c*d + t d │ │ │ │ 00011610: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ -00011620: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +-------------- │ │ │ │ -00011630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011660: 2d2b 2020 2020 2020 207c 0a7c 2020 2020 -+ |.| │ │ │ │ -00011670: 207c 2032 2020 2020 2020 2020 2020 2020 | 2 │ │ │ │ -00011680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011690: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -000116a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000116b0: 327c 2020 2020 2020 207c 0a7c 2020 2020 2| |.| │ │ │ │ -000116c0: 207c 6220 202b 2074 2020 612a 6320 2b20 |b + t a*c + │ │ │ │ -000116d0: 7420 2062 2a63 202b 2074 2020 612a 6420 t b*c + t a*d │ │ │ │ -000116e0: 2b20 7420 2062 2a64 202b 2074 2020 6320 + t b*d + t c │ │ │ │ -000116f0: 202b 2074 2020 632a 6420 2b20 7420 2064 + t c*d + t d │ │ │ │ +00011620: 207c 2020 2020 2020 3120 2020 2020 2020 | 1 │ │ │ │ +00011630: 3220 2020 2020 2020 3420 2020 2020 2020 2 4 │ │ │ │ +00011640: 3520 2020 2020 2020 3320 2020 2020 2036 5 3 6 │ │ │ │ +00011650: 2020 2020 2020 2037 2020 2020 2020 2020 7 │ │ │ │ +00011660: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ +00011670: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +-------------- │ │ │ │ +00011680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000116a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000116b0: 2d2b 2020 2020 2020 207c 0a7c 2020 2020 -+ |.| │ │ │ │ +000116c0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000116d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000116e0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +000116f0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 00011700: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ -00011710: 207c 2020 2020 2020 3135 2020 2020 2020 | 15 │ │ │ │ -00011720: 2031 3620 2020 2020 2020 3138 2020 2020 16 18 │ │ │ │ -00011730: 2020 2031 3920 2020 2020 2020 3137 2020 19 17 │ │ │ │ -00011740: 2020 2020 3230 2020 2020 2020 2032 3120 20 21 │ │ │ │ +00011710: 207c 612a 6220 2b20 7420 612a 6320 2b20 |a*b + t a*c + │ │ │ │ +00011720: 7420 622a 6320 2b20 7420 2061 2a64 202b t b*c + t a*d + │ │ │ │ +00011730: 2074 2020 622a 6420 2b20 7420 2063 2020 t b*d + t c │ │ │ │ +00011740: 2b20 7420 2063 2a64 202b 2074 2020 6420 + t c*d + t d │ │ │ │ 00011750: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ -00011760: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +-------------- │ │ │ │ -00011770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000117a0: 2d2b 2020 2020 2020 207c 0a2b 2d2d 2d2d -+ |.+---- │ │ │ │ -000117b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011760: 207c 2020 2020 2020 2038 2020 2020 2020 | 8 │ │ │ │ +00011770: 2039 2020 2020 2020 2031 3120 2020 2020 9 11 │ │ │ │ +00011780: 2020 3132 2020 2020 2020 2031 3020 2020 12 10 │ │ │ │ +00011790: 2020 2031 3320 2020 2020 2020 3134 2020 13 14 │ │ │ │ +000117a0: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ +000117b0: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +-------------- │ │ │ │ 000117c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000117d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000117e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000117f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ -00011800: 2055 203d 2072 696e 6720 4620 2020 2020 U = ring F │ │ │ │ +000117f0: 2d2b 2020 2020 2020 207c 0a7c 2020 2020 -+ |.| │ │ │ │ +00011800: 207c 2032 2020 2020 2020 2020 2020 2020 | 2 │ │ │ │ 00011810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011820: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 00011830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011840: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011890: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ -000118a0: 2055 2020 2020 2020 2020 2020 2020 2020 U │ │ │ │ -000118b0: 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 |.| │ │ │ │ -000118f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011930: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ -00011940: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ -00011950: 2020 2020 2020 2020 2020 2020 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 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00011990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ -000119e0: 2054 203d 2063 6f65 6666 6963 6965 6e74 T = coefficient │ │ │ │ -000119f0: 5269 6e67 2055 2020 2020 2020 2020 2020 Ring U │ │ │ │ +00011840: 327c 2020 2020 2020 207c 0a7c 2020 2020 2| |.| │ │ │ │ +00011850: 207c 6220 202b 2074 2020 612a 6320 2b20 |b + t a*c + │ │ │ │ +00011860: 7420 2062 2a63 202b 2074 2020 612a 6420 t b*c + t a*d │ │ │ │ +00011870: 2b20 7420 2062 2a64 202b 2074 2020 6320 + t b*d + t c │ │ │ │ +00011880: 202b 2074 2020 632a 6420 2b20 7420 2064 + t c*d + t d │ │ │ │ +00011890: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ +000118a0: 207c 2020 2020 2020 3135 2020 2020 2020 | 15 │ │ │ │ +000118b0: 2031 3620 2020 2020 2020 3138 2020 2020 16 18 │ │ │ │ +000118c0: 2020 2031 3920 2020 2020 2020 3137 2020 19 17 │ │ │ │ +000118d0: 2020 2020 3230 2020 2020 2020 2032 3120 20 21 │ │ │ │ +000118e0: 207c 2020 2020 2020 207c 0a7c 2020 2020 | |.| │ │ │ │ +000118f0: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +-------------- │ │ │ │ +00011900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011930: 2d2b 2020 2020 2020 207c 0a2b 2d2d 2d2d -+ |.+---- │ │ │ │ +00011940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011980: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ +00011990: 2055 203d 2072 696e 6720 4620 2020 2020 U = ring F │ │ │ │ +000119a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000119b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000119c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000119d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000119e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000119f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011a20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011a20: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +00011a30: 2055 2020 2020 2020 2020 2020 2020 2020 U │ │ │ │ 00011a40: 2020 2020 2020 2020 2020 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 207c 0a7c 6f36 203d |.|o6 = │ │ │ │ -00011a80: 2054 2020 2020 2020 2020 2020 2020 2020 T │ │ │ │ +00011a70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011ac0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011ac0: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ +00011ad0: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ 00011ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011b10: 2020 2020 2020 2020 207c 0a7c 6f36 203a |.|o6 : │ │ │ │ -00011b20: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ -00011b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011b60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00011b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011bb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a ---------+.|i7 : │ │ │ │ -00011bc0: 2067 656e 7320 5420 2020 2020 2020 2020 gens T │ │ │ │ +00011b10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00011b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011b60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ +00011b70: 2054 203d 2063 6f65 6666 6963 6965 6e74 T = coefficient │ │ │ │ +00011b80: 5269 6e67 2055 2020 2020 2020 2020 2020 Ring U │ │ │ │ +00011b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011bb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011bd0: 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 |.| │ │ │ │ -00011c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011c00: 2020 2020 2020 2020 207c 0a7c 6f36 203d |.|o6 = │ │ │ │ +00011c10: 2054 2020 2020 2020 2020 2020 2020 2020 T │ │ │ │ 00011c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011c50: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ -00011c60: 207b 7420 2c20 7420 2c20 7420 2c20 7420 {t , t , t , t │ │ │ │ -00011c70: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ -00011c80: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ -00011c90: 7420 2c20 7420 2c20 7420 2c20 7420 2c20 t , t , t , t , │ │ │ │ -00011ca0: 7420 2c20 7420 202c 207c 0a7c 2020 2020 t , t , |.| │ │ │ │ -00011cb0: 2020 2033 2020 2036 2020 2037 2020 2031 3 6 7 1 │ │ │ │ -00011cc0: 3020 2020 3133 2020 2031 3420 2020 3137 0 13 14 17 │ │ │ │ -00011cd0: 2020 2032 3020 2020 3231 2020 2031 2020 20 21 1 │ │ │ │ -00011ce0: 2032 2020 2034 2020 2035 2020 2038 2020 2 4 5 8 │ │ │ │ -00011cf0: 2039 2020 2031 3120 207c 0a7c 2020 2020 9 11 |.| │ │ │ │ -00011d00: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ +00011c50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 2020 207c 0a7c 6f36 203a |.|o6 : │ │ │ │ +00011cb0: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +00011cc0: 2020 2020 2020 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 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00011d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011d10: 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d7c 0a7c 2020 2020 ---------|.| │ │ │ │ -00011d50: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ -00011d60: 7420 202c 2074 2020 7d20 2020 2020 2020 t , t } │ │ │ │ +00011d40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a ---------+.|i7 : │ │ │ │ +00011d50: 2067 656e 7320 5420 2020 2020 2020 2020 gens T │ │ │ │ +00011d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011da0: 2020 3132 2020 2031 3520 2020 3136 2020 12 15 16 │ │ │ │ -00011db0: 2031 3820 2020 3139 2020 2020 2020 2020 18 19 │ │ │ │ +00011da0: 2020 2020 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 │ │ │ │ -00011de0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011e00: 2020 2020 2020 2020 2020 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 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ -00011e40: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ -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 │ │ │ │ -00011e80: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00011e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011de0: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ +00011df0: 207b 7420 2c20 7420 2c20 7420 2c20 7420 {t , t , t , t │ │ │ │ +00011e00: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ +00011e10: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ +00011e20: 7420 2c20 7420 2c20 7420 2c20 7420 2c20 t , t , t , t , │ │ │ │ +00011e30: 7420 2c20 7420 202c 207c 0a7c 2020 2020 t , t , |.| │ │ │ │ +00011e40: 2020 2033 2020 2036 2020 2037 2020 2031 3 6 7 1 │ │ │ │ +00011e50: 3020 2020 3133 2020 2031 3420 2020 3137 0 13 14 17 │ │ │ │ +00011e60: 2020 2032 3020 2020 3231 2020 2031 2020 20 21 1 │ │ │ │ +00011e70: 2032 2020 2034 2020 2035 2020 2038 2020 2 4 5 8 │ │ │ │ +00011e80: 2039 2020 2031 3120 207c 0a7c 2020 2020 9 11 |.| │ │ │ │ +00011e90: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ 00011ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2b 0a7c 6938 203a ---------+.|i8 : │ │ │ │ -00011ee0: 2067 656e 7320 5520 2020 2020 2020 2020 gens U │ │ │ │ -00011ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011ed0: 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 ---------|.| │ │ │ │ +00011ee0: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ +00011ef0: 7420 202c 2074 2020 7d20 2020 2020 2020 t , t } │ │ │ │ 00011f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00011f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011f30: 2020 3132 2020 2031 3520 2020 3136 2020 12 15 16 │ │ │ │ +00011f40: 2031 3820 2020 3139 2020 2020 2020 2020 18 19 │ │ │ │ 00011f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011f70: 2020 2020 2020 2020 207c 0a7c 6f38 203d |.|o8 = │ │ │ │ -00011f80: 207b 612c 2062 2c20 632c 2064 7d20 2020 {a, b, c, d} │ │ │ │ +00011f70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011f90: 2020 2020 2020 2020 2020 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 207c 0a7c 2020 2020 |.| │ │ │ │ -00011fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011fc0: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ +00011fd0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 00011fe0: 2020 2020 2020 2020 2020 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 207c 0a7c 6f38 203a |.|o8 : │ │ │ │ -00012020: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ -00012030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012060: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00012070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012080: 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2b 0a0a 4865 7265 ---------+..Here │ │ │ │ -000120c0: 2c20 2446 2420 6973 2074 6865 2066 616d , $F$ is the fam │ │ │ │ -000120d0: 696c 7920 6f66 2068 6f6d 6f67 656e 656f ily of homogeneo │ │ │ │ -000120e0: 7573 2069 6465 616c 7320 6861 7669 6e67 us ideals having │ │ │ │ -000120f0: 2024 4d24 2061 7320 7468 6569 7220 696e $M$ as their in │ │ │ │ -00012100: 6974 6961 6c0a 6964 6561 6c2c 2075 6e64 itial.ideal, und │ │ │ │ -00012110: 6572 2074 6865 2074 6572 6d20 6f72 6465 er the term orde │ │ │ │ -00012120: 7220 6f66 2074 6865 2072 696e 6720 6f66 r of the ring of │ │ │ │ -00012130: 2024 4d24 2e0a 0a54 6865 206f 7074 696f $M$...The optio │ │ │ │ -00012140: 6e61 6c20 6172 6775 6d65 6e74 202a 6e6f nal argument *no │ │ │ │ -00012150: 7465 2041 6c6c 5374 616e 6461 7264 3a20 te AllStandard: │ │ │ │ -00012160: 416c 6c53 7461 6e64 6172 642c 2069 7320 AllStandard, is │ │ │ │ -00012170: 626f 6f6c 6561 6e2c 2074 616b 696e 6720 boolean, taking │ │ │ │ -00012180: 7468 650a 7661 6c75 6520 2474 7275 6524 the.value $true$ │ │ │ │ -00012190: 2074 6f20 636f 6d70 7574 6520 7468 6520 to compute the │ │ │ │ -000121a0: 6661 6d69 6c79 206f 6620 616c 6c20 686f family of all ho │ │ │ │ -000121b0: 6d6f 6765 6e65 6f75 7320 6964 6561 6c73 mogeneous ideals │ │ │ │ -000121c0: 2077 6974 6820 6120 6769 7665 6e0a 696e with a given.in │ │ │ │ -000121d0: 6974 6961 6c20 6964 6561 6c20 616e 6420 itial ideal and │ │ │ │ -000121e0: 7468 6520 7661 6c75 6520 2466 616c 7365 the value $false │ │ │ │ -000121f0: 2420 746f 2063 6f6d 7075 7465 2074 6865 $ to compute the │ │ │ │ -00012200: 2066 616d 696c 7920 7769 7468 2072 6573 family with res │ │ │ │ -00012210: 7065 6374 2074 6f20 610a 6769 7665 6e20 pect to a.given │ │ │ │ -00012220: 6f72 6465 722e 2020 5468 6520 6465 6661 order. The defa │ │ │ │ -00012230: 756c 7420 7661 6c75 6520 666f 7220 7468 ult value for th │ │ │ │ -00012240: 6973 2061 7267 756d 656e 7420 6973 2066 is argument is f │ │ │ │ -00012250: 616c 7365 2e0a 0a49 6620 244c 2420 6973 alse...If $L$ is │ │ │ │ -00012260: 206e 6f74 2067 6976 656e 2c20 7468 656e not given, then │ │ │ │ -00012270: 2069 7420 6973 2063 6f6d 7075 7465 6420 it is computed │ │ │ │ -00012280: 7573 696e 6720 2a6e 6f74 6520 7374 616e using *note stan │ │ │ │ -00012290: 6461 7264 4d6f 6e6f 6d69 616c 733a 0a73 dardMonomials:.s │ │ │ │ -000122a0: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ -000122b0: 2c20 2869 6620 416c 6c53 7461 6e64 6172 , (if AllStandar │ │ │ │ -000122c0: 6420 6973 2074 7275 6529 2c20 6f72 202a d is true), or * │ │ │ │ -000122d0: 6e6f 7465 2073 6d61 6c6c 6572 4d6f 6e6f note smallerMono │ │ │ │ -000122e0: 6d69 616c 733a 0a73 6d61 6c6c 6572 4d6f mials:.smallerMo │ │ │ │ -000122f0: 6e6f 6d69 616c 732c 2028 6966 2041 6c6c nomials, (if All │ │ │ │ -00012300: 5374 616e 6461 7264 2069 7320 6661 6c73 Standard is fals │ │ │ │ -00012310: 6529 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d e)...+---------- │ │ │ │ -00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012360: 2d2d 2d2b 0a7c 6939 203a 204c 203d 2073 ---+.|i9 : L = s │ │ │ │ -00012370: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ -00012380: 204d 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ -00012390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000123c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012400: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00012410: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -00012420: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00012430: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00012440: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00012450: 2020 207c 0a7c 6f39 203d 207b 7b61 2a63 |.|o9 = {{a*c │ │ │ │ -00012460: 2c20 622a 632c 2063 202c 2061 2a64 2c20 , b*c, c , a*d, │ │ │ │ -00012470: 622a 642c 2063 2a64 2c20 6420 7d2c 207b b*d, c*d, d }, { │ │ │ │ -00012480: 612a 632c 2062 2a63 2c20 6320 2c20 612a a*c, b*c, c , a* │ │ │ │ -00012490: 642c 2062 2a64 2c20 632a 642c 2064 207d d, b*d, c*d, d } │ │ │ │ -000124a0: 2c20 207c 0a7c 2020 2020 202d 2d2d 2d2d , |.| ----- │ │ │ │ +00012010: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00012020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012030: 2d2d 2d2d 2d2d 2d2d 2d2d 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 2d2b 0a7c 6938 203a ---------+.|i8 : │ │ │ │ +00012070: 2067 656e 7320 5520 2020 2020 2020 2020 gens U │ │ │ │ +00012080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000120a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000120b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000120c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000120d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000120e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000120f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012100: 2020 2020 2020 2020 207c 0a7c 6f38 203d |.|o8 = │ │ │ │ +00012110: 207b 612c 2062 2c20 632c 2064 7d20 2020 {a, b, c, d} │ │ │ │ +00012120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012150: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00012160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000121a0: 2020 2020 2020 2020 207c 0a7c 6f38 203a |.|o8 : │ │ │ │ +000121b0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ +000121c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000121d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000121e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000121f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00012200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012240: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4865 7265 ---------+..Here │ │ │ │ +00012250: 2c20 2446 2420 6973 2074 6865 2066 616d , $F$ is the fam │ │ │ │ +00012260: 696c 7920 6f66 2068 6f6d 6f67 656e 656f ily of homogeneo │ │ │ │ +00012270: 7573 2069 6465 616c 7320 6861 7669 6e67 us ideals having │ │ │ │ +00012280: 2024 4d24 2061 7320 7468 6569 7220 696e $M$ as their in │ │ │ │ +00012290: 6974 6961 6c0a 6964 6561 6c2c 2075 6e64 itial.ideal, und │ │ │ │ +000122a0: 6572 2074 6865 2074 6572 6d20 6f72 6465 er the term orde │ │ │ │ +000122b0: 7220 6f66 2074 6865 2072 696e 6720 6f66 r of the ring of │ │ │ │ +000122c0: 2024 4d24 2e0a 0a54 6865 206f 7074 696f $M$...The optio │ │ │ │ +000122d0: 6e61 6c20 6172 6775 6d65 6e74 202a 6e6f nal argument *no │ │ │ │ +000122e0: 7465 2041 6c6c 5374 616e 6461 7264 3a20 te AllStandard: │ │ │ │ +000122f0: 416c 6c53 7461 6e64 6172 642c 2069 7320 AllStandard, is │ │ │ │ +00012300: 626f 6f6c 6561 6e2c 2074 616b 696e 6720 boolean, taking │ │ │ │ +00012310: 7468 650a 7661 6c75 6520 2474 7275 6524 the.value $true$ │ │ │ │ +00012320: 2074 6f20 636f 6d70 7574 6520 7468 6520 to compute the │ │ │ │ +00012330: 6661 6d69 6c79 206f 6620 616c 6c20 686f family of all ho │ │ │ │ +00012340: 6d6f 6765 6e65 6f75 7320 6964 6561 6c73 mogeneous ideals │ │ │ │ +00012350: 2077 6974 6820 6120 6769 7665 6e0a 696e with a given.in │ │ │ │ +00012360: 6974 6961 6c20 6964 6561 6c20 616e 6420 itial ideal and │ │ │ │ +00012370: 7468 6520 7661 6c75 6520 2466 616c 7365 the value $false │ │ │ │ +00012380: 2420 746f 2063 6f6d 7075 7465 2074 6865 $ to compute the │ │ │ │ +00012390: 2066 616d 696c 7920 7769 7468 2072 6573 family with res │ │ │ │ +000123a0: 7065 6374 2074 6f20 610a 6769 7665 6e20 pect to a.given │ │ │ │ +000123b0: 6f72 6465 722e 2020 5468 6520 6465 6661 order. The defa │ │ │ │ +000123c0: 756c 7420 7661 6c75 6520 666f 7220 7468 ult value for th │ │ │ │ +000123d0: 6973 2061 7267 756d 656e 7420 6973 2066 is argument is f │ │ │ │ +000123e0: 616c 7365 2e0a 0a49 6620 244c 2420 6973 alse...If $L$ is │ │ │ │ +000123f0: 206e 6f74 2067 6976 656e 2c20 7468 656e not given, then │ │ │ │ +00012400: 2069 7420 6973 2063 6f6d 7075 7465 6420 it is computed │ │ │ │ +00012410: 7573 696e 6720 2a6e 6f74 6520 7374 616e using *note stan │ │ │ │ +00012420: 6461 7264 4d6f 6e6f 6d69 616c 733a 0a73 dardMonomials:.s │ │ │ │ +00012430: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ +00012440: 2c20 2869 6620 416c 6c53 7461 6e64 6172 , (if AllStandar │ │ │ │ +00012450: 6420 6973 2074 7275 6529 2c20 6f72 202a d is true), or * │ │ │ │ +00012460: 6e6f 7465 2073 6d61 6c6c 6572 4d6f 6e6f note smallerMono │ │ │ │ +00012470: 6d69 616c 733a 0a73 6d61 6c6c 6572 4d6f mials:.smallerMo │ │ │ │ +00012480: 6e6f 6d69 616c 732c 2028 6966 2041 6c6c nomials, (if All │ │ │ │ +00012490: 5374 616e 6461 7264 2069 7320 6661 6c73 Standard is fals │ │ │ │ +000124a0: 6529 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d e)...+---------- │ │ │ │ 000124b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000124c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000124d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000124e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000124f0: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020 ---|.| │ │ │ │ -00012500: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00012510: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +000124f0: 2d2d 2d2b 0a7c 6939 203a 204c 203d 2073 ---+.|i9 : L = s │ │ │ │ +00012500: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ +00012510: 204d 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ 00012520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012540: 2020 207c 0a7c 2020 2020 207b 612a 632c |.| {a*c, │ │ │ │ -00012550: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ -00012560: 2a64 2c20 632a 642c 2064 207d 7d20 2020 *d, c*d, d }} │ │ │ │ +00012540: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012590: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000125a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000125b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000125c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000125d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000125e0: 2020 207c 0a7c 6f39 203a 204c 6973 7420 |.|o9 : List │ │ │ │ -000125f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012630: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000125a0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +000125b0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +000125c0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +000125d0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +000125e0: 2020 207c 0a7c 6f39 203d 207b 7b61 2a63 |.|o9 = {{a*c │ │ │ │ +000125f0: 2c20 622a 632c 2063 202c 2061 2a64 2c20 , b*c, c , a*d, │ │ │ │ +00012600: 622a 642c 2063 2a64 2c20 6420 7d2c 207b b*d, c*d, d }, { │ │ │ │ +00012610: 612a 632c 2062 2a63 2c20 6320 2c20 612a a*c, b*c, c , a* │ │ │ │ +00012620: 642c 2062 2a64 2c20 632a 642c 2064 207d d, b*d, c*d, d } │ │ │ │ +00012630: 2c20 207c 0a7c 2020 2020 202d 2d2d 2d2d , |.| ----- │ │ │ │ 00012640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012680: 2d2d 2d2b 0a7c 6931 3020 3a20 4632 203d ---+.|i10 : F2 = │ │ │ │ -00012690: 2067 726f 6562 6e65 7246 616d 696c 7920 groebnerFamily │ │ │ │ -000126a0: 284d 2c20 4c29 2020 2020 2020 2020 2020 (M, L) │ │ │ │ +00012680: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020 ---|.| │ │ │ │ +00012690: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +000126a0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ 000126b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000126c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000126d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000126e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000126f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000126d0: 2020 207c 0a7c 2020 2020 207b 612a 632c |.| {a*c, │ │ │ │ +000126e0: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ +000126f0: 2a64 2c20 632a 642c 2064 207d 7d20 2020 *d, c*d, d }} │ │ │ │ 00012700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012720: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00012730: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00012730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012750: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00012760: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -00012770: 2020 207c 0a7c 6f31 3020 3d20 6964 6561 |.|o10 = idea │ │ │ │ -00012780: 6c20 2861 2020 2b20 7420 612a 6320 2b20 l (a + t a*c + │ │ │ │ -00012790: 7420 622a 6320 2b20 7420 612a 6420 2b20 t b*c + t a*d + │ │ │ │ -000127a0: 7420 622a 6420 2b20 7420 6320 202b 2074 t b*d + t c + t │ │ │ │ -000127b0: 2063 2a64 202b 2074 2064 202c 2061 2a62 c*d + t d , a*b │ │ │ │ -000127c0: 202b 207c 0a7c 2020 2020 2020 2020 2020 + |.| │ │ │ │ -000127d0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ -000127e0: 2032 2020 2020 2020 2034 2020 2020 2020 2 4 │ │ │ │ -000127f0: 2035 2020 2020 2020 2033 2020 2020 2020 5 3 │ │ │ │ -00012800: 3620 2020 2020 2020 3720 2020 2020 2020 6 7 │ │ │ │ -00012810: 2020 207c 0a7c 2020 2020 2020 2d2d 2d2d |.| ---- │ │ │ │ -00012820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012860: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020 ---|.| │ │ │ │ +00012750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012770: 2020 207c 0a7c 6f39 203a 204c 6973 7420 |.|o9 : List │ │ │ │ +00012780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000127a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000127b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000127c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000127d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000127e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000127f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012810: 2d2d 2d2b 0a7c 6931 3020 3a20 4632 203d ---+.|i10 : F2 = │ │ │ │ +00012820: 2067 726f 6562 6e65 7246 616d 696c 7920 groebnerFamily │ │ │ │ +00012830: 284d 2c20 4c29 2020 2020 2020 2020 2020 (M, L) │ │ │ │ +00012840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012860: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00012870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012880: 2020 2020 2020 2020 2020 2020 2020 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2020 2020 2020 2020 |.| │ │ │ │ +00012c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012c70: 2020 207c 0a7c 2020 2038 2020 2039 2020 |.| 8 9 │ │ │ │ -00012c80: 2031 3120 2020 3132 2020 2031 3520 2020 11 12 15 │ │ │ │ -00012c90: 3136 2020 2031 3820 2020 3139 2020 2020 16 18 19 │ │ │ │ +00012c70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012c80: 2020 2020 2020 205a 5a20 2020 2020 2020 ZZ │ │ │ │ +00012c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012cc0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00012cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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-00012eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00012ee0: 7c69 3131 203a 206b 6b20 3d20 5a5a 2f31 |i11 : kk = ZZ/1 │ │ │ │ -00012ef0: 3031 2020 2020 2020 2020 2020 2020 2020 01 │ │ │ │ -00012f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00012f30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00012f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00012f80: 7c6f 3131 203d 206b 6b20 2020 2020 2020 |o11 = kk │ │ │ │ -00012f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00012fd0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00012fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013010: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013020: 7c6f 3131 203a 2051 756f 7469 656e 7452 |o11 : QuotientR │ │ │ │ -00013030: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -00013040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013060: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013070: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00013080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000130a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000130b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000130c0: 7c69 3132 203a 2045 203d 206b 6b5b 612c |i12 : E = kk[a, │ │ │ │ -000130d0: 622c 632c 642c 652c 536b 6577 436f 6d6d b,c,d,e,SkewComm │ │ │ │ -000130e0: 7574 6174 6976 6520 3d3e 2074 7275 655d utative => true] │ │ │ │ +00012cc0: 2020 207c 0a7c 6f31 3020 3a20 4964 6561 |.|o10 : Idea │ │ │ │ +00012cd0: 6c20 6f66 202d 2d2d 2d2d 5b74 202c 2074 l of -----[t , t │ │ │ │ +00012ce0: 202e 2e74 202c 2074 2020 2c20 7420 202e ..t , t , t . │ │ │ │ +00012cf0: 2e74 2020 2c20 7420 202c 2074 2020 2e2e .t , t , t .. │ │ │ │ +00012d00: 7420 202c 2074 202e 2e74 202c 2074 202e t , t ..t , t . │ │ │ │ +00012d10: 2e74 207c 0a7c 2020 2020 2020 2020 2020 .t |.| │ │ │ │ +00012d20: 2020 2020 2033 3230 3033 2020 3320 2020 32003 3 │ │ │ │ +00012d30: 3620 2020 3720 2020 3130 2020 2031 3320 6 7 10 13 │ │ │ │ +00012d40: 2020 3134 2020 2031 3720 2020 3230 2020 14 17 20 │ │ │ │ +00012d50: 2032 3120 2020 3120 2020 3220 2020 3420 21 1 2 4 │ │ │ │ +00012d60: 2020 357c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 5|.|---------- │ │ │ │ +00012d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012db0: 2d2d 2d7c 0a7c 2c20 7420 2e2e 7420 2c20 ---|.|, t ..t , │ │ │ │ +00012dc0: 7420 202e 2e74 2020 2c20 7420 202e 2e74 t ..t , t ..t │ │ │ │ +00012dd0: 2020 2c20 7420 202e 2e74 2020 5d5b 612e , t ..t ][a. │ │ │ │ +00012de0: 2e64 5d20 2020 2020 2020 2020 2020 2020 .d] │ │ │ │ +00012df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e00: 2020 207c 0a7c 2020 2038 2020 2039 2020 |.| 8 9 │ │ │ │ +00012e10: 2031 3120 2020 3132 2020 2031 3520 2020 11 12 15 │ │ │ │ +00012e20: 3136 2020 2031 3820 2020 3139 2020 2020 16 18 19 │ │ │ │ +00012e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00012e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012ea0: 2d2d 2d2b 0a0a 4e6f 7465 2074 6861 7420 ---+..Note that │ │ │ │ +00012eb0: 2446 2420 616e 6420 2446 5f32 2420 6172 $F$ and $F_2$ ar │ │ │ │ +00012ec0: 6520 7468 6520 7361 6d65 2066 616d 696c e the same famil │ │ │ │ +00012ed0: 792c 2069 6e20 7468 6973 2063 6173 652e y, in this case. │ │ │ │ +00012ee0: 0a0a 5468 6973 2066 756e 6374 696f 6e20 ..This function │ │ │ │ +00012ef0: 616c 736f 2077 6f72 6b73 2069 6620 7468 also works if th │ │ │ │ +00012f00: 6520 7269 6e67 206f 6620 244d 2420 6973 e ring of $M$ is │ │ │ │ +00012f10: 2061 2073 6b65 772d 636f 6d6d 7574 6174 a skew-commutat │ │ │ │ +00012f20: 6976 6520 7269 6e67 2e20 5468 6973 0a66 ive ring. This.f │ │ │ │ +00012f30: 756e 6374 696f 6e20 7072 6f64 7563 6573 unction produces │ │ │ │ +00012f40: 2061 2066 616d 696c 7920 6f66 2069 6465 a family of ide │ │ │ │ +00012f50: 616c 7320 696e 2074 6865 2065 7874 6572 als in the exter │ │ │ │ +00012f60: 696f 7220 616c 6765 6272 612e 2053 696d ior algebra. Sim │ │ │ │ +00012f70: 696c 6172 6c79 2c20 2a6e 6f74 650a 6772 ilarly, *note.gr │ │ │ │ +00012f80: 6f65 626e 6572 5374 7261 7475 6d3a 2067 oebnerStratum: g │ │ │ │ +00012f90: 726f 6562 6e65 7253 7472 6174 756d 2c20 roebnerStratum, │ │ │ │ +00012fa0: 616c 6c6f 7773 206f 6e65 2074 6f20 6669 allows one to fi │ │ │ │ +00012fb0: 6e64 2074 6865 2069 6465 616c 2069 6e20 nd the ideal in │ │ │ │ +00012fc0: 7468 6520 636f 6d6d 7574 696e 670a 636f the commuting.co │ │ │ │ +00012fd0: 6566 6669 6369 656e 7420 7661 7269 6162 efficient variab │ │ │ │ +00012fe0: 6c65 7320 6f66 2074 6865 2066 616d 696c les of the famil │ │ │ │ +00012ff0: 7920 6f66 2069 6465 616c 7320 6861 7669 y of ideals havi │ │ │ │ +00013000: 6e67 2024 4d24 2061 7320 6c65 6164 206d ng $M$ as lead m │ │ │ │ +00013010: 6f6e 6f6d 6961 6c0a 6964 6561 6c2e 0a0a onomial.ideal... │ │ │ │ +00013020: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00013030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00013070: 7c69 3131 203a 206b 6b20 3d20 5a5a 2f31 |i11 : kk = ZZ/1 │ │ │ │ +00013080: 3031 2020 2020 2020 2020 2020 2020 2020 01 │ │ │ │ +00013090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000130a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000130b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000130c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000130d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000130e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000130f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013100: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013110: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013110: 7c6f 3131 203d 206b 6b20 2020 2020 2020 |o11 = kk │ │ │ │ 00013120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013150: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013160: 7c6f 3132 203d 2045 2020 2020 2020 2020 |o12 = E │ │ │ │ +00013160: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00013170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000131a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000131b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000131c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000131b0: 7c6f 3131 203a 2051 756f 7469 656e 7452 |o11 : QuotientR │ │ │ │ +000131c0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 000131d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000131e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000131f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013200: 7c6f 3132 203a 2050 6f6c 796e 6f6d 6961 |o12 : Polynomia │ │ │ │ -00013210: 6c52 696e 672c 2035 2073 6b65 7720 636f lRing, 5 skew co │ │ │ │ -00013220: 6d6d 7574 6174 6976 6520 7661 7269 6162 mmutative variab │ │ │ │ -00013230: 6c65 2873 2920 2020 2020 2020 2020 2020 le(s) │ │ │ │ -00013240: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013250: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00013260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000132a0: 7c69 3133 203a 2049 203d 2069 6465 616c |i13 : I = ideal │ │ │ │ -000132b0: 2861 2a64 2c61 2a63 2c61 2a62 2c62 2a64 (a*d,a*c,a*b,b*d │ │ │ │ -000132c0: 2a65 2c62 2a63 2a65 2c62 2a63 2a64 2920 *e,b*c*e,b*c*d) │ │ │ │ +00013200: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00013210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00013250: 7c69 3132 203a 2045 203d 206b 6b5b 612c |i12 : E = kk[a, │ │ │ │ +00013260: 622c 632c 642c 652c 536b 6577 436f 6d6d b,c,d,e,SkewComm │ │ │ │ +00013270: 7574 6174 6976 6520 3d3e 2074 7275 655d utative => true] │ │ │ │ +00013280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013290: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000132a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000132b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000132c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000132d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000132e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000132f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000132f0: 7c6f 3132 203d 2045 2020 2020 2020 2020 |o12 = E │ │ │ │ 00013300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013330: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013340: 7c6f 3133 203d 2069 6465 616c 2028 612a |o13 = ideal (a* │ │ │ │ -00013350: 642c 2061 2a63 2c20 612a 622c 2062 2a64 d, a*c, a*b, b*d │ │ │ │ -00013360: 2a65 2c20 622a 632a 652c 2062 2a63 2a64 *e, b*c*e, b*c*d │ │ │ │ -00013370: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00013340: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013380: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013390: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000133a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000133b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000133c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013390: 7c6f 3132 203a 2050 6f6c 796e 6f6d 6961 |o12 : Polynomia │ │ │ │ +000133a0: 6c52 696e 672c 2035 2073 6b65 7720 636f lRing, 5 skew co │ │ │ │ +000133b0: 6d6d 7574 6174 6976 6520 7661 7269 6162 mmutative variab │ │ │ │ +000133c0: 6c65 2873 2920 2020 2020 2020 2020 2020 le(s) │ │ │ │ 000133d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000133e0: 7c6f 3133 203a 2049 6465 616c 206f 6620 |o13 : Ideal of │ │ │ │ -000133f0: 4520 2020 2020 2020 2020 2020 2020 2020 E │ │ │ │ -00013400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013420: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013430: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00013440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00013480: 7c69 3134 203a 2046 3120 3d20 6772 6f65 |i14 : F1 = groe │ │ │ │ -00013490: 626e 6572 4661 6d69 6c79 2049 2020 2020 bnerFamily I │ │ │ │ +000133e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +000133f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00013430: 7c69 3133 203a 2049 203d 2069 6465 616c |i13 : I = ideal │ │ │ │ +00013440: 2861 2a64 2c61 2a63 2c61 2a62 2c62 2a64 (a*d,a*c,a*b,b*d │ │ │ │ +00013450: 2a65 2c62 2a63 2a65 2c62 2a63 2a64 2920 *e,b*c*e,b*c*d) │ │ │ │ +00013460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013470: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013480: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000134a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000134b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000134c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000134d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000134e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000134f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000134d0: 7c6f 3133 203d 2069 6465 616c 2028 612a |o13 = ideal (a* │ │ │ │ +000134e0: 642c 2061 2a63 2c20 612a 622c 2062 2a64 d, a*c, a*b, b*d │ │ │ │ +000134f0: 2a65 2c20 622a 632a 652c 2062 2a63 2a64 *e, b*c*e, b*c*d │ │ │ │ +00013500: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00013510: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013520: 7c6f 3134 203d 2069 6465 616c 2028 612a |o14 = ideal (a* │ │ │ │ -00013530: 6420 2b20 7420 622a 6420 2b20 7420 612a d + t b*d + t a* │ │ │ │ -00013540: 6520 2b20 7420 632a 6420 2b20 7420 622a e + t c*d + t b* │ │ │ │ -00013550: 6520 2b20 7420 632a 6520 2b20 7420 642a e + t c*e + t d* │ │ │ │ -00013560: 652c 2061 2a63 202b 2074 2062 2a63 7c0a e, a*c + t b*c|. │ │ │ │ -00013570: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00013580: 2020 2020 2031 2020 2020 2020 2033 2020 1 3 │ │ │ │ -00013590: 2020 2020 2032 2020 2020 2020 2034 2020 2 4 │ │ │ │ -000135a0: 2020 2020 2035 2020 2020 2020 2036 2020 5 6 │ │ │ │ -000135b0: 2020 2020 2020 2020 2020 3720 2020 7c0a 7 |. │ │ │ │ -000135c0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +00013520: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013560: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013570: 7c6f 3133 203a 2049 6465 616c 206f 6620 |o13 : Ideal of │ │ │ │ +00013580: 4520 2020 2020 2020 2020 2020 2020 2020 E │ │ │ │ +00013590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000135a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000135b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000135c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000135d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000135e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000135f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00013610: 7c20 2020 2020 202b 2074 2062 2a64 202b | + t b*d + │ │ │ │ -00013620: 2074 2020 612a 6520 2b20 7420 632a 6420 t a*e + t c*d │ │ │ │ -00013630: 2b20 7420 2062 2a65 202b 2074 2020 632a + t b*e + t c* │ │ │ │ -00013640: 6520 2b20 7420 2064 2a65 2c20 612a 6220 e + t d*e, a*b │ │ │ │ -00013650: 2b20 7420 2062 2a63 202b 2020 2020 7c0a + t b*c + |. │ │ │ │ -00013660: 7c20 2020 2020 2020 2020 3820 2020 2020 | 8 │ │ │ │ -00013670: 2020 3130 2020 2020 2020 2039 2020 2020 10 9 │ │ │ │ -00013680: 2020 2031 3120 2020 2020 2020 3132 2020 11 12 │ │ │ │ -00013690: 2020 2020 2031 3320 2020 2020 2020 2020 13 │ │ │ │ -000136a0: 2020 2031 3420 2020 2020 2020 2020 7c0a 14 |. │ │ │ │ -000136b0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -000136c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000136d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000136e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000136f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00013700: 7c20 2020 2020 2074 2020 622a 6420 2b20 | t b*d + │ │ │ │ -00013710: 7420 2061 2a65 202b 2074 2020 632a 6420 t a*e + t c*d │ │ │ │ -00013720: 2b20 7420 2062 2a65 202b 2074 2020 632a + t b*e + t c* │ │ │ │ -00013730: 6520 2b20 7420 2064 2a65 2c20 622a 642a e + t d*e, b*d* │ │ │ │ -00013740: 6520 2b20 7420 2063 2a64 2a65 2c20 7c0a e + t c*d*e, |. │ │ │ │ -00013750: 7c20 2020 2020 2020 3135 2020 2020 2020 | 15 │ │ │ │ -00013760: 2031 3720 2020 2020 2020 3136 2020 2020 17 16 │ │ │ │ -00013770: 2020 2031 3820 2020 2020 2020 3139 2020 18 19 │ │ │ │ -00013780: 2020 2020 2032 3020 2020 2020 2020 2020 20 │ │ │ │ -00013790: 2020 2020 2032 3120 2020 2020 2020 7c0a 21 |. │ │ │ │ -000137a0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -000137b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000137c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000137d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000137e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -000137f0: 7c20 2020 2020 2062 2a63 2a65 202b 2074 | b*c*e + t │ │ │ │ -00013800: 2020 632a 642a 652c 2062 2a63 2a64 202b c*d*e, b*c*d + │ │ │ │ -00013810: 2074 2020 632a 642a 6529 2020 2020 2020 t c*d*e) │ │ │ │ -00013820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013830: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013840: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00013850: 3232 2020 2020 2020 2020 2020 2020 2020 22 │ │ │ │ -00013860: 2020 3233 2020 2020 2020 2020 2020 2020 23 │ │ │ │ -00013870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013880: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013890: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000138a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000138b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000138c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000138d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000138e0: 7c6f 3134 203a 2049 6465 616c 206f 6620 |o14 : Ideal of │ │ │ │ -000138f0: 6b6b 5b74 2020 2e2e 7420 202c 2074 202e kk[t ..t , t . │ │ │ │ -00013900: 2e74 202c 2074 2020 2e2e 7420 202c 2074 .t , t ..t , t │ │ │ │ -00013910: 2020 2c20 7420 202c 2074 202c 2074 202c , t , t , t , │ │ │ │ -00013920: 2074 202c 2074 2020 2c20 7420 202e 7c0a t , t , t .|. │ │ │ │ -00013930: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00013940: 2020 2020 3139 2020 2032 3020 2020 3520 19 20 5 │ │ │ │ -00013950: 2020 3620 2020 3132 2020 2031 3320 2020 6 12 13 │ │ │ │ -00013960: 3136 2020 2031 3820 2020 3220 2020 3420 16 18 2 4 │ │ │ │ -00013970: 2020 3920 2020 3131 2020 2031 3420 7c0a 9 11 14 |. │ │ │ │ -00013980: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ -00013990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000139a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000139b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000139c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -000139d0: 7c2e 7420 202c 2074 2020 2c20 7420 202c |.t , t , t , │ │ │ │ -000139e0: 2074 202c 2074 202c 2074 202e 2e74 202c t , t , t ..t , │ │ │ │ -000139f0: 2074 2020 2c20 7420 202e 2e74 2020 5d5b t , t ..t ][ │ │ │ │ -00013a00: 612e 2e65 5d20 2020 2020 2020 2020 2020 a..e] │ │ │ │ +00013600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00013610: 7c69 3134 203a 2046 3120 3d20 6772 6f65 |i14 : F1 = groe │ │ │ │ +00013620: 626e 6572 4661 6d69 6c79 2049 2020 2020 bnerFamily I │ │ │ │ +00013630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013650: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013660: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000136a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000136b0: 7c6f 3134 203d 2069 6465 616c 2028 612a |o14 = ideal (a* │ │ │ │ +000136c0: 6420 2b20 7420 622a 6420 2b20 7420 612a d + t b*d + t a* │ │ │ │ +000136d0: 6520 2b20 7420 632a 6420 2b20 7420 622a e + t c*d + t b* │ │ │ │ +000136e0: 6520 2b20 7420 632a 6520 2b20 7420 642a e + t c*e + t d* │ │ │ │ +000136f0: 652c 2061 2a63 202b 2074 2062 2a63 7c0a e, a*c + t b*c|. │ │ │ │ +00013700: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013710: 2020 2020 2031 2020 2020 2020 2033 2020 1 3 │ │ │ │ +00013720: 2020 2020 2032 2020 2020 2020 2034 2020 2 4 │ │ │ │ +00013730: 2020 2020 2035 2020 2020 2020 2036 2020 5 6 │ │ │ │ +00013740: 2020 2020 2020 2020 2020 3720 2020 7c0a 7 |. │ │ │ │ +00013750: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +00013760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +000137a0: 7c20 2020 2020 202b 2074 2062 2a64 202b | + t b*d + │ │ │ │ +000137b0: 2074 2020 612a 6520 2b20 7420 632a 6420 t a*e + t c*d │ │ │ │ +000137c0: 2b20 7420 2062 2a65 202b 2074 2020 632a + t b*e + t c* │ │ │ │ +000137d0: 6520 2b20 7420 2064 2a65 2c20 612a 6220 e + t d*e, a*b │ │ │ │ +000137e0: 2b20 7420 2062 2a63 202b 2020 2020 7c0a + t b*c + |. │ │ │ │ +000137f0: 7c20 2020 2020 2020 2020 3820 2020 2020 | 8 │ │ │ │ +00013800: 2020 3130 2020 2020 2020 2039 2020 2020 10 9 │ │ │ │ +00013810: 2020 2031 3120 2020 2020 2020 3132 2020 11 12 │ │ │ │ +00013820: 2020 2020 2031 3320 2020 2020 2020 2020 13 │ │ │ │ +00013830: 2020 2031 3420 2020 2020 2020 2020 7c0a 14 |. │ │ │ │ +00013840: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +00013850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +00013890: 7c20 2020 2020 2074 2020 622a 6420 2b20 | t b*d + │ │ │ │ +000138a0: 7420 2061 2a65 202b 2074 2020 632a 6420 t a*e + t c*d │ │ │ │ +000138b0: 2b20 7420 2062 2a65 202b 2074 2020 632a + t b*e + t c* │ │ │ │ +000138c0: 6520 2b20 7420 2064 2a65 2c20 622a 642a e + t d*e, b*d* │ │ │ │ +000138d0: 6520 2b20 7420 2063 2a64 2a65 2c20 7c0a e + t c*d*e, |. │ │ │ │ +000138e0: 7c20 2020 2020 2020 3135 2020 2020 2020 | 15 │ │ │ │ +000138f0: 2031 3720 2020 2020 2020 3136 2020 2020 17 16 │ │ │ │ +00013900: 2020 2031 3820 2020 2020 2020 3139 2020 18 19 │ │ │ │ +00013910: 2020 2020 2032 3020 2020 2020 2020 2020 20 │ │ │ │ +00013920: 2020 2020 2032 3120 2020 2020 2020 7c0a 21 |. │ │ │ │ +00013930: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +00013940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +00013980: 7c20 2020 2020 2062 2a63 2a65 202b 2074 | b*c*e + t │ │ │ │ +00013990: 2020 632a 642a 652c 2062 2a63 2a64 202b c*d*e, b*c*d + │ │ │ │ +000139a0: 2074 2020 632a 642a 6529 2020 2020 2020 t c*d*e) │ │ │ │ +000139b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000139c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000139d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000139e0: 3232 2020 2020 2020 2020 2020 2020 2020 22 │ │ │ │ +000139f0: 2020 3233 2020 2020 2020 2020 2020 2020 23 │ │ │ │ +00013a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013a10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013a20: 7c20 2031 3520 2020 3137 2020 2032 3320 | 15 17 23 │ │ │ │ -00013a30: 2020 3120 2020 3320 2020 3720 2020 3820 1 3 7 8 │ │ │ │ -00013a40: 2020 3130 2020 2032 3120 2020 3232 2020 10 21 22 │ │ │ │ +00013a20: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013a60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013a70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00013a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00013ac0: 7c69 3135 203a 206e 6574 4c69 7374 2046 |i15 : netList F │ │ │ │ -00013ad0: 315f 2a20 2020 2020 2020 2020 2020 2020 1_* │ │ │ │ -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 7c0a |. │ │ │ │ -00013b10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00013b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013b50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013b60: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -00013b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ -00013bb0: 7c6f 3135 203d 207c 612a 6420 2b20 7420 |o15 = |a*d + t │ │ │ │ -00013bc0: 622a 6420 2b20 7420 612a 6520 2b20 7420 b*d + t a*e + t │ │ │ │ -00013bd0: 632a 6420 2b20 7420 622a 6520 2b20 7420 c*d + t b*e + t │ │ │ │ -00013be0: 632a 6520 2b20 7420 642a 6520 2020 2020 c*e + t d*e │ │ │ │ -00013bf0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00013c00: 7c20 2020 2020 207c 2020 2020 2020 2031 | | 1 │ │ │ │ -00013c10: 2020 2020 2020 2033 2020 2020 2020 2032 3 2 │ │ │ │ -00013c20: 2020 2020 2020 2034 2020 2020 2020 2035 4 5 │ │ │ │ -00013c30: 2020 2020 2020 2036 2020 2020 2020 2020 6 │ │ │ │ -00013c40: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00013c50: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -00013c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ -00013ca0: 7c20 2020 2020 207c 612a 6320 2b20 7420 | |a*c + t │ │ │ │ -00013cb0: 622a 6320 2b20 7420 622a 6420 2b20 7420 b*c + t b*d + t │ │ │ │ -00013cc0: 2061 2a65 202b 2074 2063 2a64 202b 2074 a*e + t c*d + t │ │ │ │ -00013cd0: 2020 622a 6520 2b20 7420 2063 2a65 202b b*e + t c*e + │ │ │ │ -00013ce0: 2074 2020 642a 6520 2020 7c20 2020 7c0a t d*e | |. │ │ │ │ -00013cf0: 7c20 2020 2020 207c 2020 2020 2020 2037 | | 7 │ │ │ │ -00013d00: 2020 2020 2020 2038 2020 2020 2020 2031 8 1 │ │ │ │ -00013d10: 3020 2020 2020 2020 3920 2020 2020 2020 0 9 │ │ │ │ -00013d20: 3131 2020 2020 2020 2031 3220 2020 2020 11 12 │ │ │ │ -00013d30: 2020 3133 2020 2020 2020 7c20 2020 7c0a 13 | |. │ │ │ │ -00013d40: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ -00013d90: 7c20 2020 2020 207c 612a 6220 2b20 7420 | |a*b + t │ │ │ │ -00013da0: 2062 2a63 202b 2074 2020 622a 6420 2b20 b*c + t b*d + │ │ │ │ -00013db0: 7420 2061 2a65 202b 2074 2020 632a 6420 t a*e + t c*d │ │ │ │ -00013dc0: 2b20 7420 2062 2a65 202b 2074 2020 632a + t b*e + t c* │ │ │ │ -00013dd0: 6520 2b20 7420 2064 2a65 7c20 2020 7c0a e + t d*e| |. │ │ │ │ -00013de0: 7c20 2020 2020 207c 2020 2020 2020 2031 | | 1 │ │ │ │ -00013df0: 3420 2020 2020 2020 3135 2020 2020 2020 4 15 │ │ │ │ -00013e00: 2031 3720 2020 2020 2020 3136 2020 2020 17 16 │ │ │ │ -00013e10: 2020 2031 3820 2020 2020 2020 3139 2020 18 19 │ │ │ │ -00013e20: 2020 2020 2032 3020 2020 7c20 2020 7c0a 20 | |. │ │ │ │ -00013e30: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ -00013e80: 7c20 2020 2020 207c 622a 642a 6520 2b20 | |b*d*e + │ │ │ │ -00013e90: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ -00013ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013ec0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00013ed0: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ -00013ee0: 2032 3120 2020 2020 2020 2020 2020 2020 21 │ │ │ │ -00013ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013f10: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00013f20: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -00013f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ -00013f70: 7c20 2020 2020 207c 622a 632a 6520 2b20 | |b*c*e + │ │ │ │ -00013f80: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ -00013f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013fb0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00013fc0: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ -00013fd0: 2032 3220 2020 2020 2020 2020 2020 2020 22 │ │ │ │ -00013fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014000: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00014010: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -00014020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014050: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ -00014060: 7c20 2020 2020 207c 622a 632a 6420 2b20 | |b*c*d + │ │ │ │ -00014070: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ +00013a70: 7c6f 3134 203a 2049 6465 616c 206f 6620 |o14 : Ideal of │ │ │ │ +00013a80: 6b6b 5b74 2020 2e2e 7420 202c 2074 202e kk[t ..t , t . │ │ │ │ +00013a90: 2e74 202c 2074 2020 2e2e 7420 202c 2074 .t , t ..t , t │ │ │ │ +00013aa0: 2020 2c20 7420 202c 2074 202c 2074 202c , t , t , t , │ │ │ │ +00013ab0: 2074 202c 2074 2020 2c20 7420 202e 7c0a t , t , t .|. │ │ │ │ +00013ac0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013ad0: 2020 2020 3139 2020 2032 3020 2020 3520 19 20 5 │ │ │ │ +00013ae0: 2020 3620 2020 3132 2020 2031 3320 2020 6 12 13 │ │ │ │ +00013af0: 3136 2020 2031 3820 2020 3220 2020 3420 16 18 2 4 │ │ │ │ +00013b00: 2020 3920 2020 3131 2020 2031 3420 7c0a 9 11 14 |. │ │ │ │ +00013b10: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ +00013b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +00013b60: 7c2e 7420 202c 2074 2020 2c20 7420 202c |.t , t , t , │ │ │ │ +00013b70: 2074 202c 2074 202c 2074 202e 2e74 202c t , t , t ..t , │ │ │ │ +00013b80: 2074 2020 2c20 7420 202e 2e74 2020 5d5b t , t ..t ][ │ │ │ │ +00013b90: 612e 2e65 5d20 2020 2020 2020 2020 2020 a..e] │ │ │ │ +00013ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013bb0: 7c20 2031 3520 2020 3137 2020 2032 3320 | 15 17 23 │ │ │ │ +00013bc0: 2020 3120 2020 3320 2020 3720 2020 3820 1 3 7 8 │ │ │ │ +00013bd0: 2020 3130 2020 2032 3120 2020 3232 2020 10 21 22 │ │ │ │ +00013be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013bf0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013c00: 2b2d 2d2d 2d2d 2d2d 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 ---------------- │ │ │ │ +00013c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00013c50: 7c69 3135 203a 206e 6574 4c69 7374 2046 |i15 : netList F │ │ │ │ +00013c60: 315f 2a20 2020 2020 2020 2020 2020 2020 1_* │ │ │ │ +00013c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013c90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013ca0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013ce0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013cf0: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ +00013d40: 7c6f 3135 203d 207c 612a 6420 2b20 7420 |o15 = |a*d + t │ │ │ │ +00013d50: 622a 6420 2b20 7420 612a 6520 2b20 7420 b*d + t a*e + t │ │ │ │ +00013d60: 632a 6420 2b20 7420 622a 6520 2b20 7420 c*d + t b*e + t │ │ │ │ +00013d70: 632a 6520 2b20 7420 642a 6520 2020 2020 c*e + t d*e │ │ │ │ +00013d80: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00013d90: 7c20 2020 2020 207c 2020 2020 2020 2031 | | 1 │ │ │ │ +00013da0: 2020 2020 2020 2033 2020 2020 2020 2032 3 2 │ │ │ │ +00013db0: 2020 2020 2020 2034 2020 2020 2020 2035 4 5 │ │ │ │ +00013dc0: 2020 2020 2020 2036 2020 2020 2020 2020 6 │ │ │ │ +00013dd0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00013de0: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ +00013e30: 7c20 2020 2020 207c 612a 6320 2b20 7420 | |a*c + t │ │ │ │ +00013e40: 622a 6320 2b20 7420 622a 6420 2b20 7420 b*c + t b*d + t │ │ │ │ +00013e50: 2061 2a65 202b 2074 2063 2a64 202b 2074 a*e + t c*d + t │ │ │ │ +00013e60: 2020 622a 6520 2b20 7420 2063 2a65 202b b*e + t c*e + │ │ │ │ +00013e70: 2074 2020 642a 6520 2020 7c20 2020 7c0a t d*e | |. │ │ │ │ +00013e80: 7c20 2020 2020 207c 2020 2020 2020 2037 | | 7 │ │ │ │ +00013e90: 2020 2020 2020 2038 2020 2020 2020 2031 8 1 │ │ │ │ +00013ea0: 3020 2020 2020 2020 3920 2020 2020 2020 0 9 │ │ │ │ +00013eb0: 3131 2020 2020 2020 2031 3220 2020 2020 11 12 │ │ │ │ +00013ec0: 2020 3133 2020 2020 2020 7c20 2020 7c0a 13 | |. │ │ │ │ +00013ed0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +00013ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ +00013f20: 7c20 2020 2020 207c 612a 6220 2b20 7420 | |a*b + t │ │ │ │ +00013f30: 2062 2a63 202b 2074 2020 622a 6420 2b20 b*c + t b*d + │ │ │ │ +00013f40: 7420 2061 2a65 202b 2074 2020 632a 6420 t a*e + t c*d │ │ │ │ +00013f50: 2b20 7420 2062 2a65 202b 2074 2020 632a + t b*e + t c* │ │ │ │ +00013f60: 6520 2b20 7420 2064 2a65 7c20 2020 7c0a e + t d*e| |. │ │ │ │ +00013f70: 7c20 2020 2020 207c 2020 2020 2020 2031 | | 1 │ │ │ │ +00013f80: 3420 2020 2020 2020 3135 2020 2020 2020 4 15 │ │ │ │ +00013f90: 2031 3720 2020 2020 2020 3136 2020 2020 17 16 │ │ │ │ +00013fa0: 2020 2031 3820 2020 2020 2020 3139 2020 18 19 │ │ │ │ +00013fb0: 2020 2020 2032 3020 2020 7c20 2020 7c0a 20 | |. │ │ │ │ +00013fc0: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ +00014010: 7c20 2020 2020 207c 622a 642a 6520 2b20 | |b*d*e + │ │ │ │ +00014020: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ +00014030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014050: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014060: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ +00014070: 2032 3120 2020 2020 2020 2020 2020 2020 21 │ │ │ │ 00014080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000140a0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -000140b0: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ -000140c0: 2032 3320 2020 2020 2020 2020 2020 2020 23 │ │ │ │ -000140d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000140e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000140f0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00014100: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ -00014150: 2b2d 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 2b0a --------------+. │ │ │ │ -000141a0: 7c69 3136 203a 2046 3220 3d20 6772 6f65 |i16 : F2 = groe │ │ │ │ -000141b0: 626e 6572 4661 6d69 6c79 2849 2c20 416c bnerFamily(I, Al │ │ │ │ -000141c0: 6c53 7461 6e64 6172 6420 3d3e 2074 7275 lStandard => tru │ │ │ │ -000141d0: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ -000141e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000141f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00014200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000140b0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +000140c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000140d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000140e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000140f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ +00014100: 7c20 2020 2020 207c 622a 632a 6520 2b20 | |b*c*e + │ │ │ │ +00014110: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ +00014120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014140: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014150: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ +00014160: 2032 3220 2020 2020 2020 2020 2020 2020 22 │ │ │ │ +00014170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014190: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +000141a0: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ +000141f0: 7c20 2020 2020 207c 622a 632a 6420 2b20 | |b*c*d + │ │ │ │ +00014200: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ 00014210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014230: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014240: 7c6f 3136 203d 2069 6465 616c 2028 612a |o16 = ideal (a* │ │ │ │ -00014250: 6420 2b20 7420 612a 6520 2b20 7420 622a d + t a*e + t b* │ │ │ │ -00014260: 6320 2b20 7420 622a 6420 2b20 7420 632a c + t b*d + t c* │ │ │ │ -00014270: 6420 2b20 7420 622a 6520 2b20 7420 632a d + t b*e + t c* │ │ │ │ -00014280: 6520 2b20 7420 642a 652c 2061 2a63 7c0a e + t d*e, a*c|. │ │ │ │ -00014290: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000142a0: 2020 2020 2034 2020 2020 2020 2031 2020 4 1 │ │ │ │ -000142b0: 2020 2020 2032 2020 2020 2020 2033 2020 2 3 │ │ │ │ -000142c0: 2020 2020 2035 2020 2020 2020 2036 2020 5 6 │ │ │ │ -000142d0: 2020 2020 2037 2020 2020 2020 2020 7c0a 7 |. │ │ │ │ -000142e0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +00014230: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014240: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ +00014250: 2032 3320 2020 2020 2020 2020 2020 2020 23 │ │ │ │ +00014260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014280: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014290: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +000142a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000142b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000142c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000142d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ +000142e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000142f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00014330: 7c20 2020 2020 202b 2074 2020 612a 6520 | + t a*e │ │ │ │ -00014340: 2b20 7420 622a 6320 2b20 7420 622a 6420 + t b*c + t b*d │ │ │ │ -00014350: 2b20 7420 2063 2a64 202b 2074 2020 622a + t c*d + t b* │ │ │ │ -00014360: 6520 2b20 7420 2063 2a65 202b 2074 2020 e + t c*e + t │ │ │ │ -00014370: 642a 652c 2061 2a62 202b 2020 2020 7c0a d*e, a*b + |. │ │ │ │ -00014380: 7c20 2020 2020 2020 2020 3131 2020 2020 | 11 │ │ │ │ -00014390: 2020 2038 2020 2020 2020 2039 2020 2020 8 9 │ │ │ │ -000143a0: 2020 2031 3020 2020 2020 2020 3132 2020 10 12 │ │ │ │ -000143b0: 2020 2020 2031 3320 2020 2020 2020 3134 13 14 │ │ │ │ +00014320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00014330: 7c69 3136 203a 2046 3220 3d20 6772 6f65 |i16 : F2 = groe │ │ │ │ +00014340: 626e 6572 4661 6d69 6c79 2849 2c20 416c bnerFamily(I, Al │ │ │ │ +00014350: 6c53 7461 6e64 6172 6420 3d3e 2074 7275 lStandard => tru │ │ │ │ +00014360: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ +00014370: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00014380: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00014390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000143a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000143b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000143c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000143d0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -000143e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000143f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00014420: 7c20 2020 2020 2074 2020 612a 6520 2b20 | t a*e + │ │ │ │ -00014430: 7420 2062 2a63 202b 2074 2020 622a 6420 t b*c + t b*d │ │ │ │ -00014440: 2b20 7420 2063 2a64 202b 2074 2020 622a + t c*d + t b* │ │ │ │ -00014450: 6520 2b20 7420 2063 2a65 202b 2074 2020 e + t c*e + t │ │ │ │ -00014460: 642a 652c 2062 2a64 2a65 202b 2020 7c0a d*e, b*d*e + |. │ │ │ │ -00014470: 7c20 2020 2020 2020 3138 2020 2020 2020 | 18 │ │ │ │ -00014480: 2031 3520 2020 2020 2020 3136 2020 2020 15 16 │ │ │ │ -00014490: 2020 2031 3720 2020 2020 2020 3139 2020 17 19 │ │ │ │ -000144a0: 2020 2020 2032 3020 2020 2020 2020 3231 20 21 │ │ │ │ -000144b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000144c0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -000144d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000144e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000144f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00014510: 7c20 2020 2020 2074 2020 632a 642a 652c | t c*d*e, │ │ │ │ -00014520: 2062 2a63 2a65 202b 2074 2020 632a 642a b*c*e + t c*d* │ │ │ │ -00014530: 652c 2062 2a63 2a64 202b 2074 2020 632a e, b*c*d + t c* │ │ │ │ -00014540: 642a 6529 2020 2020 2020 2020 2020 2020 d*e) │ │ │ │ +000143d0: 7c6f 3136 203d 2069 6465 616c 2028 612a |o16 = ideal (a* │ │ │ │ +000143e0: 6420 2b20 7420 612a 6520 2b20 7420 622a d + t a*e + t b* │ │ │ │ +000143f0: 6320 2b20 7420 622a 6420 2b20 7420 632a c + t b*d + t c* │ │ │ │ +00014400: 6420 2b20 7420 622a 6520 2b20 7420 632a d + t b*e + t c* │ │ │ │ +00014410: 6520 2b20 7420 642a 652c 2061 2a63 7c0a e + t d*e, a*c|. │ │ │ │ +00014420: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00014430: 2020 2020 2034 2020 2020 2020 2031 2020 4 1 │ │ │ │ +00014440: 2020 2020 2032 2020 2020 2020 2033 2020 2 3 │ │ │ │ +00014450: 2020 2020 2035 2020 2020 2020 2036 2020 5 6 │ │ │ │ +00014460: 2020 2020 2037 2020 2020 2020 2020 7c0a 7 |. │ │ │ │ +00014470: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +00014480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000144a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000144b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +000144c0: 7c20 2020 2020 202b 2074 2020 612a 6520 | + t a*e │ │ │ │ +000144d0: 2b20 7420 622a 6320 2b20 7420 622a 6420 + t b*c + t b*d │ │ │ │ +000144e0: 2b20 7420 2063 2a64 202b 2074 2020 622a + t c*d + t b* │ │ │ │ +000144f0: 6520 2b20 7420 2063 2a65 202b 2074 2020 e + t c*e + t │ │ │ │ +00014500: 642a 652c 2061 2a62 202b 2020 2020 7c0a d*e, a*b + |. │ │ │ │ +00014510: 7c20 2020 2020 2020 2020 3131 2020 2020 | 11 │ │ │ │ +00014520: 2020 2038 2020 2020 2020 2039 2020 2020 8 9 │ │ │ │ +00014530: 2020 2031 3020 2020 2020 2020 3132 2020 10 12 │ │ │ │ +00014540: 2020 2020 2031 3320 2020 2020 2020 3134 13 14 │ │ │ │ 00014550: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014560: 7c20 2020 2020 2020 3232 2020 2020 2020 | 22 │ │ │ │ -00014570: 2020 2020 2020 2020 2020 3233 2020 2020 23 │ │ │ │ -00014580: 2020 2020 2020 2020 2020 2020 3234 2020 24 │ │ │ │ -00014590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000145a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000145b0: 7c20 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 7c0a |. │ │ │ │ -00014600: 7c6f 3136 203a 2049 6465 616c 206f 6620 |o16 : Ideal of │ │ │ │ -00014610: 6b6b 5b74 2020 2e2e 7420 202c 2074 202e kk[t ..t , t . │ │ │ │ -00014620: 2e74 202c 2074 2020 2e2e 7420 202c 2074 .t , t ..t , t │ │ │ │ -00014630: 2020 2c20 7420 202c 2074 202c 2074 202c , t , t , t , │ │ │ │ -00014640: 2074 2020 2c20 7420 202c 2074 2020 7c0a t , t , t |. │ │ │ │ -00014650: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00014660: 2020 2020 3230 2020 2032 3120 2020 3620 20 21 6 │ │ │ │ -00014670: 2020 3720 2020 3133 2020 2031 3420 2020 7 13 14 │ │ │ │ -00014680: 3137 2020 2031 3920 2020 3320 2020 3520 17 19 3 5 │ │ │ │ -00014690: 2020 3130 2020 2031 3220 2020 3135 7c0a 10 12 15|. │ │ │ │ -000146a0: 7c2d 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 7c0a --------------|. │ │ │ │ -000146f0: 7c2e 2e74 2020 2c20 7420 202c 2074 2020 |..t , t , t │ │ │ │ -00014700: 2c20 7420 2e2e 7420 2c20 7420 2e2e 7420 , t ..t , t ..t │ │ │ │ -00014710: 2c20 7420 2c20 7420 202c 2074 2020 2e2e , t , t , t .. │ │ │ │ -00014720: 7420 205d 5b61 2e2e 655d 2020 2020 2020 t ][a..e] │ │ │ │ +00014560: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +00014570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014580: 2d2d 2d2d 2d2d 2d2d 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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 7c0a --------------|. │ │ │ │ +000146a0: 7c20 2020 2020 2074 2020 632a 642a 652c | t c*d*e, │ │ │ │ +000146b0: 2062 2a63 2a65 202b 2074 2020 632a 642a b*c*e + t c*d* │ │ │ │ +000146c0: 652c 2062 2a63 2a64 202b 2074 2020 632a e, b*c*d + t c* │ │ │ │ +000146d0: 642a 6529 2020 2020 2020 2020 2020 2020 d*e) │ │ │ │ +000146e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000146f0: 7c20 2020 2020 2020 3232 2020 2020 2020 | 22 │ │ │ │ +00014700: 2020 2020 2020 2020 2020 3233 2020 2020 23 │ │ │ │ +00014710: 2020 2020 2020 2020 2020 2020 3234 2020 24 │ │ │ │ +00014720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014730: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014740: 7c20 2020 3136 2020 2031 3820 2020 3234 | 16 18 24 │ │ │ │ -00014750: 2020 2031 2020 2032 2020 2038 2020 2039 1 2 8 9 │ │ │ │ -00014760: 2020 2034 2020 2031 3120 2020 3232 2020 4 11 22 │ │ │ │ -00014770: 2032 3320 2020 2020 2020 2020 2020 2020 23 │ │ │ │ +00014740: 7c20 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 │ │ │ │ 00014780: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014790: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -000147a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000147b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000147c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000147d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000147e0: 7c69 3137 203a 206e 6574 4c69 7374 2046 |i17 : netList F │ │ │ │ -000147f0: 325f 2a20 2020 2020 2020 2020 2020 2020 2_* │ │ │ │ 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│ -000148f0: 622a 6420 2b20 7420 632a 6420 2b20 7420 b*d + t c*d + t │ │ │ │ -00014900: 622a 6520 2b20 7420 632a 6520 2b20 7420 b*e + t c*e + t │ │ │ │ -00014910: 642a 6520 2020 2020 2020 7c20 2020 7c0a d*e | |. │ │ │ │ -00014920: 7c20 2020 2020 207c 2020 2020 2020 2034 | | 4 │ │ │ │ -00014930: 2020 2020 2020 2031 2020 2020 2020 2032 1 2 │ │ │ │ -00014940: 2020 2020 2020 2033 2020 2020 2020 2035 3 5 │ │ │ │ -00014950: 2020 2020 2020 2036 2020 2020 2020 2037 6 7 │ │ │ │ -00014960: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00014970: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ -000149c0: 7c20 2020 2020 207c 612a 6320 2b20 7420 | |a*c + t │ │ │ │ -000149d0: 2061 2a65 202b 2074 2062 2a63 202b 2074 a*e + t b*c + t │ │ │ │ -000149e0: 2062 2a64 202b 2074 2020 632a 6420 2b20 b*d + t c*d + │ │ │ │ -000149f0: 7420 2062 2a65 202b 2074 2020 632a 6520 t b*e + t c*e │ │ │ │ -00014a00: 2b20 7420 2064 2a65 2020 7c20 2020 7c0a + t d*e | |. │ │ │ │ -00014a10: 7c20 2020 2020 207c 2020 2020 2020 2031 | | 1 │ │ │ │ -00014a20: 3120 2020 2020 2020 3820 2020 2020 2020 1 8 │ │ │ │ -00014a30: 3920 2020 2020 2020 3130 2020 2020 2020 9 10 │ │ │ │ -00014a40: 2031 3220 2020 2020 2020 3133 2020 2020 12 13 │ │ │ │ -00014a50: 2020 2031 3420 2020 2020 7c20 2020 7c0a 14 | |. │ │ │ │ -00014a60: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ -00014ab0: 7c20 2020 2020 207c 612a 6220 2b20 7420 | |a*b + t │ │ │ │ -00014ac0: 2061 2a65 202b 2074 2020 622a 6320 2b20 a*e + t b*c + │ │ │ │ -00014ad0: 7420 2062 2a64 202b 2074 2020 632a 6420 t b*d + t c*d │ │ │ │ -00014ae0: 2b20 7420 2062 2a65 202b 2074 2020 632a + t b*e + t c* │ │ │ │ -00014af0: 6520 2b20 7420 2064 2a65 7c20 2020 7c0a e + t d*e| |. │ │ │ │ -00014b00: 7c20 2020 2020 207c 2020 2020 2020 2031 | | 1 │ │ │ │ -00014b10: 3820 2020 2020 2020 3135 2020 2020 2020 8 15 │ │ │ │ -00014b20: 2031 3620 2020 2020 2020 3137 2020 2020 16 17 │ │ │ │ -00014b30: 2020 2031 3920 2020 2020 2020 3230 2020 19 20 │ │ │ │ -00014b40: 2020 2020 2032 3120 2020 7c20 2020 7c0a 21 | |. │ │ │ │ -00014b50: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -00014b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ -00014ba0: 7c20 2020 2020 207c 622a 642a 6520 2b20 | |b*d*e + │ │ │ │ -00014bb0: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ -00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014be0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00014bf0: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ -00014c00: 2032 3220 2020 2020 2020 2020 2020 2020 22 │ │ │ │ -00014c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014c30: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00014c40: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -00014c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ -00014c90: 7c20 2020 2020 207c 622a 632a 6520 2b20 | |b*c*e + │ │ │ │ -00014ca0: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ -00014cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014cd0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00014ce0: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ -00014cf0: 2032 3320 2020 2020 2020 2020 2020 2020 23 │ │ │ │ -00014d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014d20: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00014d30: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -00014d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ -00014d80: 7c20 2020 2020 207c 622a 632a 6420 2b20 | |b*c*d + │ │ │ │ -00014d90: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ +00014790: 7c6f 3136 203a 2049 6465 616c 206f 6620 |o16 : Ideal of │ │ │ │ +000147a0: 6b6b 5b74 2020 2e2e 7420 202c 2074 202e kk[t ..t , t . │ │ │ │ +000147b0: 2e74 202c 2074 2020 2e2e 7420 202c 2074 .t , t ..t , t │ │ │ │ +000147c0: 2020 2c20 7420 202c 2074 202c 2074 202c , t , t , t , │ │ │ │ +000147d0: 2074 2020 2c20 7420 202c 2074 2020 7c0a t , t , t |. │ │ │ │ +000147e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000147f0: 2020 2020 3230 2020 2032 3120 2020 3620 20 21 6 │ │ │ │ +00014800: 2020 3720 2020 3133 2020 2031 3420 2020 7 13 14 │ │ │ │ +00014810: 3137 2020 2031 3920 2020 3320 2020 3520 17 19 3 5 │ │ │ │ +00014820: 2020 3130 2020 2031 3220 2020 3135 7c0a 10 12 15|. │ │ │ │ +00014830: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ +00014840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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+--------------- │ │ │ │ +00014930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00014970: 7c69 3137 203a 206e 6574 4c69 7374 2046 |i17 : netList F │ │ │ │ +00014980: 325f 2a20 2020 2020 2020 2020 2020 2020 2_* │ │ │ │ +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 7c0a |. │ │ │ │ +000149c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000149d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000149e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000149f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014a00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00014a10: 7c20 2020 2020 202b 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 2b20 2020 7c0a ----------+ |. │ │ │ │ +00014a60: 7c6f 3137 203d 207c 612a 6420 2b20 7420 |o17 = |a*d + t │ │ │ │ +00014a70: 612a 6520 2b20 7420 622a 6320 2b20 7420 a*e + t b*c + t │ │ │ │ +00014a80: 622a 6420 2b20 7420 632a 6420 2b20 7420 b*d + t c*d + t │ │ │ │ +00014a90: 622a 6520 2b20 7420 632a 6520 2b20 7420 b*e + t c*e + t │ │ │ │ +00014aa0: 642a 6520 2020 2020 2020 7c20 2020 7c0a d*e | |. │ │ │ │ +00014ab0: 7c20 2020 2020 207c 2020 2020 2020 2034 | | 4 │ │ │ │ +00014ac0: 2020 2020 2020 2031 2020 2020 2020 2032 1 2 │ │ │ │ +00014ad0: 2020 2020 2020 2033 2020 2020 2020 2035 3 5 │ │ │ │ +00014ae0: 2020 2020 2020 2036 2020 2020 2020 2037 6 7 │ │ │ │ +00014af0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014b00: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +00014b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ +00014b50: 7c20 2020 2020 207c 612a 6320 2b20 7420 | |a*c + t │ │ │ │ +00014b60: 2061 2a65 202b 2074 2062 2a63 202b 2074 a*e + t b*c + t │ │ │ │ +00014b70: 2062 2a64 202b 2074 2020 632a 6420 2b20 b*d + t c*d + │ │ │ │ +00014b80: 7420 2062 2a65 202b 2074 2020 632a 6520 t b*e + t c*e │ │ │ │ +00014b90: 2b20 7420 2064 2a65 2020 7c20 2020 7c0a + t d*e | |. │ │ │ │ +00014ba0: 7c20 2020 2020 207c 2020 2020 2020 2031 | | 1 │ │ │ │ +00014bb0: 3120 2020 2020 2020 3820 2020 2020 2020 1 8 │ │ │ │ +00014bc0: 3920 2020 2020 2020 3130 2020 2020 2020 9 10 │ │ │ │ +00014bd0: 2031 3220 2020 2020 2020 3133 2020 2020 12 13 │ │ │ │ +00014be0: 2020 2031 3420 2020 2020 7c20 2020 7c0a 14 | |. │ │ │ │ +00014bf0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +00014c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ +00014c40: 7c20 2020 2020 207c 612a 6220 2b20 7420 | |a*b + t │ │ │ │ +00014c50: 2061 2a65 202b 2074 2020 622a 6320 2b20 a*e + t b*c + │ │ │ │ +00014c60: 7420 2062 2a64 202b 2074 2020 632a 6420 t b*d + t c*d │ │ │ │ +00014c70: 2b20 7420 2062 2a65 202b 2074 2020 632a + t b*e + t c* │ │ │ │ +00014c80: 6520 2b20 7420 2064 2a65 7c20 2020 7c0a e + t d*e| |. │ │ │ │ +00014c90: 7c20 2020 2020 207c 2020 2020 2020 2031 | | 1 │ │ │ │ +00014ca0: 3820 2020 2020 2020 3135 2020 2020 2020 8 15 │ │ │ │ +00014cb0: 2031 3620 2020 2020 2020 3137 2020 2020 16 17 │ │ │ │ +00014cc0: 2020 2031 3920 2020 2020 2020 3230 2020 19 20 │ │ │ │ +00014cd0: 2020 2020 2032 3120 2020 7c20 2020 7c0a 21 | |. │ │ │ │ +00014ce0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +00014cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ +00014d30: 7c20 2020 2020 207c 622a 642a 6520 2b20 | |b*d*e + │ │ │ │ +00014d40: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ +00014d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014d70: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014d80: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ +00014d90: 2032 3220 2020 2020 2020 2020 2020 2020 22 │ │ │ │ 00014da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014dc0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00014dd0: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ -00014de0: 2032 3420 2020 2020 2020 2020 2020 2020 24 │ │ │ │ -00014df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014e10: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ -00014e20: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -00014e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ -00014e70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00014e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00014ec0: 7c69 3138 203a 204a 3220 3d20 7472 696d |i18 : J2 = trim │ │ │ │ -00014ed0: 2067 726f 6562 6e65 7253 7472 6174 756d groebnerStratum │ │ │ │ -00014ee0: 2046 3220 2020 2020 2020 2020 2020 2020 F2 │ │ │ │ -00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014f00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014f10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00014f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014dd0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +00014de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ +00014e20: 7c20 2020 2020 207c 622a 632a 6520 2b20 | |b*c*e + │ │ │ │ +00014e30: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ +00014e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014e60: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014e70: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ +00014e80: 2032 3320 2020 2020 2020 2020 2020 2020 23 │ │ │ │ +00014e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014eb0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014ec0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +00014ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ +00014f10: 7c20 2020 2020 207c 622a 632a 6420 2b20 | |b*c*d + │ │ │ │ +00014f20: 7420 2063 2a64 2a65 2020 2020 2020 2020 t c*d*e │ │ │ │ 00014f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014f50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014f60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00014f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014f50: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014f60: 7c20 2020 2020 207c 2020 2020 2020 2020 | | │ │ │ │ +00014f70: 2032 3420 2020 2020 2020 2020 2020 2020 24 │ │ │ │ 00014f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014fa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014fb0: 7c6f 3138 203d 2069 6465 616c 2028 7420 |o18 = ideal (t │ │ │ │ -00014fc0: 2020 2b20 7420 2074 2020 2b20 7420 2074 + t t + t t │ │ │ │ -00014fd0: 2020 202d 2074 2074 2020 7420 2020 2b20 - t t t + │ │ │ │ -00014fe0: 7420 2074 2020 202d 2074 2074 2020 7420 t t - t t t │ │ │ │ -00014ff0: 202c 2074 2020 2d20 7420 2020 2b20 7c0a , t - t + |. │ │ │ │ -00015000: 7c20 2020 2020 2020 2020 2020 2020 2031 | 1 │ │ │ │ -00015010: 3420 2020 2032 3420 3920 2020 2031 3020 4 24 9 10 │ │ │ │ -00015020: 3131 2020 2020 3920 3131 2032 3220 2020 11 9 11 22 │ │ │ │ -00015030: 2031 3220 3233 2020 2020 3820 3131 2032 12 23 8 11 2 │ │ │ │ -00015040: 3320 2020 3720 2020 2031 3320 2020 7c0a 3 7 13 |. │ │ │ │ -00015050: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -00015060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +00014fa0: 2020 2020 2020 2020 2020 7c20 2020 7c0a | |. │ │ │ │ +00014fb0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +00014fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 2020 7c0a ----------+ |. │ │ │ │ +00015000: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00015010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00015040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00015050: 7c69 3138 203a 204a 3220 3d20 7472 696d |i18 : J2 = trim │ │ │ │ +00015060: 2067 726f 6562 6e65 7253 7472 6174 756d groebnerStratum │ │ │ │ +00015070: 2046 3220 2020 2020 2020 2020 2020 2020 F2 │ │ │ │ +00015080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015090: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000150a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000150b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000150c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000150d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000150e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000150f0: 7c20 2020 2020 2074 2020 7420 202d 2074 | t t - t │ │ │ │ -00015100: 2020 7420 202b 2074 2020 7420 202b 2074 t + t t + t │ │ │ │ -00015110: 2074 2020 202b 2074 2020 7420 2020 2d20 t + t t - │ │ │ │ -00015120: 7420 7420 7420 2020 2d20 7420 7420 2074 t t t - t t t │ │ │ │ -00015130: 2020 202b 2074 2074 2020 202d 2020 7c0a + t t - |. │ │ │ │ -00015140: 7c20 2020 2020 2020 3234 2032 2020 2020 | 24 2 │ │ │ │ -00015150: 3234 2038 2020 2020 3130 2034 2020 2020 24 8 10 4 │ │ │ │ -00015160: 3320 3131 2020 2020 3132 2032 3220 2020 3 11 12 22 │ │ │ │ -00015170: 2039 2034 2032 3220 2020 2032 2031 3120 9 4 22 2 11 │ │ │ │ -00015180: 3232 2020 2020 3520 3233 2020 2020 7c0a 22 5 23 |. │ │ │ │ -00015190: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -000151a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000151b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000151c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000151d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -000151e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000151f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015220: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00015230: 7c20 2020 2020 2074 2074 2074 2020 202d | t t t - │ │ │ │ -00015240: 2074 2074 2020 7420 202c 2074 2020 2b20 t t t , t + │ │ │ │ -00015250: 7420 2074 2020 2d20 7420 7420 202d 2074 t t - t t - t │ │ │ │ -00015260: 2074 2020 202b 2074 2074 2074 2020 202b t + t t t + │ │ │ │ -00015270: 2074 2074 2074 2020 2c20 7420 2020 7c0a t t t , t |. │ │ │ │ -00015280: 7c20 2020 2020 2020 3820 3420 3233 2020 | 8 4 23 │ │ │ │ -00015290: 2020 3120 3131 2032 3320 2020 3620 2020 1 11 23 6 │ │ │ │ -000152a0: 2032 3420 3120 2020 2033 2034 2020 2020 24 1 3 4 │ │ │ │ -000152b0: 3520 3232 2020 2020 3220 3420 3232 2020 5 22 2 4 22 │ │ │ │ -000152c0: 2020 3120 3420 3233 2020 2032 3120 7c0a 1 4 23 21 |. │ │ │ │ -000152d0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -000152e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000152f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00015320: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00015330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015360: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00015370: 7c20 2020 2020 202b 2074 2020 7420 2020 | + t t │ │ │ │ -00015380: 2d20 7420 2074 2020 202b 2074 2020 7420 - t t + t t │ │ │ │ -00015390: 2020 2b20 7420 2074 2020 202d 2074 2020 + t t - t │ │ │ │ -000153a0: 7420 7420 2020 2b20 7420 2074 2074 2020 t t + t t t │ │ │ │ -000153b0: 202b 2074 2020 7420 2074 2020 202d 7c0a + t t t -|. │ │ │ │ -000153c0: 7c20 2020 2020 2020 2020 3130 2031 3820 | 10 18 │ │ │ │ -000153d0: 2020 2031 3020 3234 2020 2020 3136 2032 10 24 16 2 │ │ │ │ -000153e0: 3420 2020 2031 3720 3131 2020 2020 3138 4 17 11 18 │ │ │ │ -000153f0: 2039 2032 3220 2020 2032 3420 3920 3232 9 22 24 9 22 │ │ │ │ -00015400: 2020 2020 3130 2031 3120 3232 2020 7c0a 10 11 22 |. │ │ │ │ -00015410: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -00015420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00015460: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00015470: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -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 2020 7c0a |. │ │ │ │ -000154b0: 7c20 2020 2020 2074 2020 7420 2074 2020 | t t t │ │ │ │ -000154c0: 202d 2074 2074 2020 7420 2020 2d20 7420 - t t t - t │ │ │ │ -000154d0: 2074 2020 202b 2074 2020 7420 2020 2d20 t + t t - │ │ │ │ -000154e0: 7420 2074 2074 2020 202d 2074 2020 7420 t t t - t t │ │ │ │ -000154f0: 2074 2020 202b 2020 2020 2020 2020 7c0a t + |. │ │ │ │ -00015500: 7c20 2020 2020 2020 3136 2031 3120 3232 | 16 11 22 │ │ │ │ -00015510: 2020 2020 3920 3131 2032 3220 2020 2031 9 11 22 1 │ │ │ │ -00015520: 3320 3233 2020 2020 3139 2032 3320 2020 3 23 19 23 │ │ │ │ -00015530: 2031 3820 3820 3233 2020 2020 3135 2031 18 8 23 15 1 │ │ │ │ -00015540: 3120 3233 2020 2020 2020 2020 2020 7c0a 1 23 |. │ │ │ │ -00015550: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -00015560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -000155a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000155b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000155c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000155e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000155f0: 7c20 2020 2020 2074 2020 7420 2074 2020 | t t t │ │ │ │ -00015600: 202d 2074 2074 2020 7420 2074 2020 2c20 - t t t t , │ │ │ │ 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2020 2020 3139 2038 2032 3220 3233 23 19 8 22 23 │ │ │ │ -00017960: 2020 2020 3234 2032 2038 2032 3220 3233 24 2 8 22 23 │ │ │ │ -00017970: 2020 2020 3234 2038 2032 3220 3233 2020 24 8 22 23 │ │ │ │ -00017980: 2020 3130 2032 2034 2032 3220 3233 7c0a 10 2 4 22 23|. │ │ │ │ -00017990: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -000179a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000179b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000179c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000179d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -000179e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000179f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017a20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00017a30: 7c20 2020 2020 202b 2074 2020 7420 7420 | 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7c0a --------------|. │ │ │ │ -00017b20: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00017b30: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00017b40: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00017b50: 2020 2020 2020 2032 2020 2020 3220 2020 2 2 │ │ │ │ -00017b60: 2020 2020 2020 2020 2020 3220 2020 7c0a 2 |. │ │ │ │ -00017b70: 7c20 2020 2020 2074 2074 2074 2020 7420 | t t t t │ │ │ │ -00017b80: 2074 2020 202d 2074 2074 2074 2074 2020 t - t t t t │ │ │ │ -00017b90: 7420 2020 2d20 7420 7420 7420 2074 2020 t - t t t t │ │ │ │ -00017ba0: 7420 2020 2b20 7420 7420 2074 2020 7420 t + t t t t │ │ │ │ -00017bb0: 2020 2b20 7420 2074 2074 2020 202d 7c0a + t t t -|. │ │ │ │ -00017bc0: 7c20 2020 2020 2020 3320 3820 3131 2032 | 3 8 11 2 │ │ │ │ -00017bd0: 3220 3233 2020 2020 3820 3920 3420 3232 2 23 8 9 4 22 │ │ │ │ -00017be0: 2032 3320 2020 2032 2038 2031 3120 3232 23 2 8 11 22 │ │ │ │ -00017bf0: 2032 3320 2020 2038 2031 3120 3232 2032 23 8 11 22 2 │ │ │ │ -00017c00: 3320 2020 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t - t t t t │ │ │ │ -00017cf0: 2020 202d 2020 2020 2020 2020 2020 7c0a - |. │ │ │ │ -00017d00: 7c20 2020 2020 2020 3139 2031 2032 3320 | 19 1 23 │ │ │ │ -00017d10: 2020 2032 3420 3120 3820 3233 2020 2020 24 1 8 23 │ │ │ │ -00017d20: 3130 2031 2034 2032 3320 2020 2031 3620 10 1 4 23 16 │ │ │ │ -00017d30: 3120 3420 3233 2020 2020 3320 3820 3420 1 4 23 3 8 4 │ │ │ │ -00017d40: 3233 2020 2020 2020 2020 2020 2020 7c0a 23 |. │ │ │ │ -00017d50: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -00017d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00017da0: 7c20 2020 2020 2020 2020 2020 2020 2032 | 2 │ │ │ │ -00017db0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -00017dc0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ 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2020 │ │ │ │ -00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017fc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00017fd0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018010: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00018020: 7c6f 3138 203a 2049 6465 616c 206f 6620 |o18 : Ideal of │ │ │ │ -00018030: 6b6b 5b74 2020 2e2e 7420 202c 2074 202e kk[t ..t , t . │ │ │ │ -00018040: 2e74 202c 2074 2020 2e2e 7420 202c 2074 .t , t ..t , t │ │ │ │ -00018050: 2020 2c20 7420 202c 2074 202c 2074 202c , t , t , t , │ │ │ │ -00018060: 2074 2020 2c20 7420 202c 2074 2020 7c0a t , t , t |. │ │ │ │ -00018070: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00018080: 2020 2020 3230 2020 2032 3120 2020 3620 20 21 6 │ │ │ │ -00018090: 2020 3720 2020 3133 2020 2031 3420 2020 7 13 14 │ │ │ │ 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---------------- │ │ │ │ -0001a760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a770: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ -0001a780: 7c20 2020 2020 207c 7420 202d 2074 2020 | |t - t │ │ │ │ -0001a790: 202b 2074 2020 7420 202b 2074 2020 7420 + t t + t t │ │ │ │ -0001a7a0: 202b 2074 2074 2020 202b 2074 2020 7420 + t t + t t │ │ │ │ -0001a7b0: 2020 2d20 7420 7420 2074 2020 202b 2074 - t t t + t │ │ │ │ -0001a7c0: 2074 2074 2020 207c 2020 2020 2020 7c0a t t | |. │ │ │ │ -0001a7d0: 7c20 2020 2020 207c 2037 2020 2020 3139 | | 7 19 │ │ │ │ -0001a7e0: 2020 2020 3234 2032 2020 2020 3136 2034 24 2 16 4 │ │ │ │ -0001a7f0: 2020 2020 3320 3131 2020 2020 3135 2031 3 11 15 1 │ │ │ │ -0001a800: 3120 2020 2032 2031 3120 3232 2020 2020 1 2 11 22 │ │ │ │ -0001a810: 3220 3420 3233 207c 2020 2020 2020 7c0a 2 4 23 | |. │ │ │ │ -0001a820: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -0001a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a860: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ -0001a870: 7c20 2020 2020 207c 7420 202b 2074 2020 | |t + t │ │ │ │ -0001a880: 7420 202d 2074 2074 2020 2d20 7420 7420 t - t t - t t │ │ │ │ -0001a890: 2074 2020 202b 2074 2074 2074 2020 2020 t + t t t │ │ │ │ -0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8b0: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ -0001a8c0: 7c20 2020 2020 207c 2036 2020 2020 3234 | | 6 24 │ │ │ │ -0001a8d0: 2031 2020 2020 3320 3420 2020 2031 2031 1 3 4 1 1 │ │ │ │ -0001a8e0: 3120 3232 2020 2020 3120 3420 3233 2020 1 22 1 4 23 │ │ │ │ -0001a8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a900: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ -0001a910: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -0001a920: 2d2d 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2d2d | +-------- │ │ │ │ -0001aa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001aa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001aa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001aa40: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ -0001aa50: 7c20 2020 2020 207c 7420 2020 2b20 7420 | |t + t │ │ │ │ -0001aa60: 2074 2020 202d 2074 2020 7420 202d 2074 t - t t - t │ │ │ │ -0001aa70: 2020 7420 2074 2020 202b 2074 2020 7420 t t + t t │ │ │ │ -0001aa80: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +0001a5f0: 7c20 2020 2020 202b 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a630: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ +0001a640: 7c20 2020 2020 207c 7420 202d 2074 2074 | |t - t t │ │ │ │ +0001a650: 2020 2d20 7420 7420 2020 2020 2020 2020 - t t │ │ │ │ +0001a660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a680: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ +0001a690: 7c20 2020 2020 207c 2035 2020 2020 3220 | | 5 2 │ │ │ │ +0001a6a0: 3420 2020 2031 2031 3120 2020 2020 2020 4 1 11 │ │ │ │ +0001a6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a6d0: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ +0001a6e0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +0001a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a720: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ +0001a730: 7c20 2020 2020 207c 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│ │ │ +0001a820: 7c20 2020 2020 207c 7420 2020 2d20 7420 | |t - t │ │ │ │ +0001a830: 2020 2b20 7420 2074 2020 2d20 7420 2074 + t t - t t │ │ │ │ +0001a840: 2020 2b20 7420 2074 2020 2b20 7420 2074 + t t + t t │ │ │ │ +0001a850: 2020 202d 2074 2074 2020 7420 2020 2b20 - t t t + │ │ │ │ +0001a860: 7420 7420 7420 207c 2020 2020 2020 7c0a t t t | |. │ │ │ │ +0001a870: 7c20 2020 2020 207c 2031 3320 2020 2031 | | 13 1 │ │ │ │ +0001a880: 3920 2020 2032 3420 3820 2020 2031 3020 9 24 8 10 │ │ │ │ +0001a890: 3420 2020 2031 3620 3420 2020 2031 3520 4 16 4 15 │ │ │ │ +0001a8a0: 3131 2020 2020 3820 3131 2032 3220 2020 11 8 11 22 │ │ │ │ +0001a8b0: 2038 2034 2032 337c 2020 2020 2020 7c0a 8 4 23| |. │ │ │ │ +0001a8c0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +0001a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a900: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ +0001a910: 7c20 2020 2020 207c 7420 202d 2074 2020 | |t - t │ │ │ │ +0001a920: 202b 2074 2020 7420 202b 2074 2020 7420 + t t + t t │ │ │ │ +0001a930: 202b 2074 2074 2020 202b 2074 2020 7420 + t t + t t │ │ │ │ +0001a940: 2020 2d20 7420 7420 2074 2020 202b 2074 - t t t + t │ │ │ │ +0001a950: 2074 2074 2020 207c 2020 2020 2020 7c0a t t | |. │ │ │ │ +0001a960: 7c20 2020 2020 207c 2037 2020 2020 3139 | | 7 19 │ │ │ │ +0001a970: 2020 2020 3234 2032 2020 2020 3136 2034 24 2 16 4 │ │ │ │ +0001a980: 2020 2020 3320 3131 2020 2020 3135 2031 3 11 15 1 │ │ │ │ +0001a990: 3120 2020 2032 2031 3120 3232 2020 2020 1 2 11 22 │ │ │ │ +0001a9a0: 3220 3420 3233 207c 2020 2020 2020 7c0a 2 4 23 | |. │ │ │ │ +0001a9b0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +0001a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001a9f0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ +0001aa00: 7c20 2020 2020 207c 7420 202b 2074 2020 | |t + t │ │ │ │ +0001aa10: 7420 202d 2074 2074 2020 2d20 7420 7420 t - t t - t t │ │ │ │ +0001aa20: 2074 2020 202b 2074 2074 2074 2020 2020 t + t t t │ │ │ │ +0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001aa40: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ +0001aa50: 7c20 2020 2020 207c 2036 2020 2020 3234 | | 6 24 │ │ │ │ +0001aa60: 2031 2020 2020 3320 3420 2020 2031 2031 1 3 4 1 1 │ │ │ │ +0001aa70: 3120 3232 2020 2020 3120 3420 3233 2020 1 22 1 4 23 │ │ │ │ +0001aa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa90: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ -0001aaa0: 7c20 2020 2020 207c 2032 3020 2020 2031 | | 20 1 │ │ │ │ -0001aab0: 3520 3234 2020 2020 3137 2034 2020 2020 5 24 17 4 │ │ │ │ -0001aac0: 3135 2031 3120 3232 2020 2020 3135 2034 15 11 22 15 4 │ │ │ │ -0001aad0: 2032 3320 2020 2020 2020 2020 2020 2020 23 │ │ │ │ -0001aae0: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ -0001aaf0: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ -0001ab00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab30: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ -0001ab40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0001ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001ab90: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -0001aba0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6772 ==.. * *note gr │ │ │ │ -0001abb0: 6f65 626e 6572 5374 7261 7475 6d3a 2067 oebnerStratum: g │ │ │ │ -0001abc0: 726f 6562 6e65 7253 7472 6174 756d 2c20 roebnerStratum, │ │ │ │ -0001abd0: 2d2d 2063 6f6d 7075 7465 2074 6865 2069 -- compute the i │ │ │ │ -0001abe0: 6465 616c 2077 6865 7265 2061 2067 6976 deal where a giv │ │ │ │ -0001abf0: 656e 0a20 2020 2069 7320 6120 4772 6f65 en. is a Groe │ │ │ │ -0001ac00: 626e 6572 2062 6173 6973 0a20 202a 202a bner basis. * * │ │ │ │ -0001ac10: 6e6f 7465 2073 6d61 6c6c 6572 4d6f 6e6f note smallerMono │ │ │ │ -0001ac20: 6d69 616c 733a 2073 6d61 6c6c 6572 4d6f mials: smallerMo │ │ │ │ -0001ac30: 6e6f 6d69 616c 732c 202d 2d20 7265 7475 nomials, -- retu │ │ │ │ -0001ac40: 726e 7320 7468 6520 7374 616e 6461 7264 rns the standard │ │ │ │ -0001ac50: 206d 6f6e 6f6d 6961 6c73 0a20 2020 2073 monomials. s │ │ │ │ -0001ac60: 6d61 6c6c 6572 2062 7574 206f 6620 7468 maller but of th │ │ │ │ -0001ac70: 6520 7361 6d65 2064 6567 7265 6520 6173 e same degree as │ │ │ │ -0001ac80: 2067 6976 656e 206d 6f6e 6f6d 6961 6c28 given monomial( │ │ │ │ -0001ac90: 7329 0a20 202a 202a 6e6f 7465 2073 7461 s). * *note sta │ │ │ │ -0001aca0: 6e64 6172 644d 6f6e 6f6d 6961 6c73 3a20 ndardMonomials: │ │ │ │ -0001acb0: 7374 616e 6461 7264 4d6f 6e6f 6d69 616c standardMonomial │ │ │ │ -0001acc0: 732c 202d 2d20 636f 6d70 7574 6573 2073 s, -- computes s │ │ │ │ -0001acd0: 7461 6e64 6172 6420 6d6f 6e6f 6d69 616c tandard monomial │ │ │ │ -0001ace0: 730a 0a57 6179 7320 746f 2075 7365 2067 s..Ways to use g │ │ │ │ -0001acf0: 726f 6562 6e65 7246 616d 696c 793a 0a3d roebnerFamily:.= │ │ │ │ -0001ad00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001ad10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0001ad20: 2267 726f 6562 6e65 7246 616d 696c 7928 "groebnerFamily( │ │ │ │ -0001ad30: 4964 6561 6c29 220a 2020 2a20 2267 726f Ideal)". * "gro │ │ │ │ -0001ad40: 6562 6e65 7246 616d 696c 7928 4964 6561 ebnerFamily(Idea │ │ │ │ -0001ad50: 6c2c 4c69 7374 2922 0a0a 466f 7220 7468 l,List)"..For th │ │ │ │ -0001ad60: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0001ad70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0001ad80: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0001ad90: 6520 6772 6f65 626e 6572 4661 6d69 6c79 e groebnerFamily │ │ │ │ -0001ada0: 3a20 6772 6f65 626e 6572 4661 6d69 6c79 : groebnerFamily │ │ │ │ -0001adb0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0001adc0: 686f 6420 6675 6e63 7469 6f6e 0a77 6974 hod function.wit │ │ │ │ -0001add0: 6820 6f70 7469 6f6e 733a 2028 4d61 6361 h options: (Maca │ │ │ │ -0001ade0: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -0001adf0: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ -0001ae00: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ -0001ae10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ae20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ae30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ae40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ae50: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -0001ae60: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0001ae70: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -0001ae80: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -0001ae90: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -0001aea0: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -0001aeb0: 6179 322f 7061 636b 6167 6573 2f0a 4772 ay2/packages/.Gr │ │ │ │ -0001aec0: 6f65 626e 6572 5374 7261 7461 2e6d 323a oebnerStrata.m2: │ │ │ │ -0001aed0: 3737 363a 302e 0a1f 0a46 696c 653a 2047 776:0....File: G │ │ │ │ -0001aee0: 726f 6562 6e65 7253 7472 6174 612e 696e roebnerStrata.in │ │ │ │ -0001aef0: 666f 2c20 4e6f 6465 3a20 6772 6f65 626e fo, Node: groebn │ │ │ │ -0001af00: 6572 5374 7261 7475 6d2c 204e 6578 743a erStratum, Next: │ │ │ │ -0001af10: 206c 696e 6561 7250 6172 742c 2050 7265 linearPart, Pre │ │ │ │ -0001af20: 763a 2067 726f 6562 6e65 7246 616d 696c v: groebnerFamil │ │ │ │ -0001af30: 792c 2055 703a 2054 6f70 0a0a 6772 6f65 y, Up: Top..groe │ │ │ │ -0001af40: 626e 6572 5374 7261 7475 6d20 2d2d 2063 bnerStratum -- c │ │ │ │ -0001af50: 6f6d 7075 7465 2074 6865 2069 6465 616c ompute the ideal │ │ │ │ -0001af60: 2077 6865 7265 2061 2067 6976 656e 2069 where a given i │ │ │ │ -0001af70: 7320 6120 4772 6f65 626e 6572 2062 6173 s a Groebner bas │ │ │ │ -0001af80: 6973 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a is.************* │ │ │ │ -0001af90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001afa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001afb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001afc0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0001afd0: 7361 6765 3a20 0a20 2020 2020 2020 204a sage: . J │ │ │ │ -0001afe0: 203d 2067 726f 6562 6e65 7253 7472 6174 = groebnerStrat │ │ │ │ -0001aff0: 756d 2046 0a20 202a 2049 6e70 7574 733a um F. * Inputs: │ │ │ │ -0001b000: 0a20 2020 2020 202a 2046 2c20 616e 202a . * F, an * │ │ │ │ -0001b010: 6e6f 7465 2069 6465 616c 3a20 284d 6163 note ideal: (Mac │ │ │ │ -0001b020: 6175 6c61 7932 446f 6329 4964 6561 6c2c aulay2Doc)Ideal, │ │ │ │ -0001b030: 2c20 416e 2069 6465 616c 2063 6f6e 7374 , An ideal const │ │ │ │ -0001b040: 7275 6374 6564 2066 726f 6d0a 2020 2020 ructed from. │ │ │ │ -0001b050: 2020 2020 2a6e 6f74 6520 6772 6f65 626e *note groebn │ │ │ │ -0001b060: 6572 4661 6d69 6c79 3a20 6772 6f65 626e erFamily: groebn │ │ │ │ -0001b070: 6572 4661 6d69 6c79 2c0a 2020 2a20 2a6e erFamily,. * *n │ │ │ │ -0001b080: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70 ote Optional inp │ │ │ │ -0001b090: 7574 733a 2028 4d61 6361 756c 6179 3244 uts: (Macaulay2D │ │ │ │ -0001b0a0: 6f63 2975 7369 6e67 2066 756e 6374 696f oc)using functio │ │ │ │ -0001b0b0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ -0001b0c0: 2069 6e70 7574 732c 3a0a 2020 2020 2020 inputs,:. │ │ │ │ -0001b0d0: 2a20 4d69 6e69 6d61 6c69 7a65 203d 3e20 * Minimalize => │ │ │ │ -0001b0e0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -0001b0f0: 7565 2066 616c 7365 0a20 202a 204f 7574 ue false. * Out │ │ │ │ -0001b100: 7075 7473 3a0a 2020 2020 2020 2a20 4a2c puts:. * J, │ │ │ │ -0001b110: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a an *note ideal: │ │ │ │ -0001b120: 2028 4d61 6361 756c 6179 3244 6f63 2949 (Macaulay2Doc)I │ │ │ │ -0001b130: 6465 616c 2c2c 2074 6865 2069 6465 616c deal,, the ideal │ │ │ │ -0001b140: 206f 6620 7468 6520 4772 6f65 626e 6572 of the Groebner │ │ │ │ -0001b150: 0a20 2020 2020 2020 2073 7472 6174 756d . stratum │ │ │ │ -0001b160: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0001b170: 3d3d 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e =========..Given │ │ │ │ -0001b180: 2061 206d 6f6e 6f6d 6961 6c20 6964 6561 a monomial idea │ │ │ │ -0001b190: 6c20 244d 242c 2074 6869 7320 636f 6d6d l $M$, this comm │ │ │ │ -0001b1a0: 616e 6420 7265 7475 726e 7320 6120 6661 and returns a fa │ │ │ │ -0001b1b0: 6d69 6c79 206f 6620 6964 6561 6c73 2068 mily of ideals h │ │ │ │ -0001b1c0: 6176 696e 6720 244d 240a 6173 2061 6e20 aving $M$.as an │ │ │ │ -0001b1d0: 696e 6974 6961 6c20 6964 6561 6c2c 2061 initial ideal, a │ │ │ │ -0001b1e0: 6e64 2063 6f6e 6469 7469 6f6e 7320 6f6e nd conditions on │ │ │ │ -0001b1f0: 2074 6865 2070 6172 616d 6574 6572 7320 the parameters │ │ │ │ -0001b200: 736f 2074 6861 7420 7468 6520 6661 6d69 so that the fami │ │ │ │ -0001b210: 6c79 2069 730a 666c 6174 2e20 2049 6620 ly is.flat. If │ │ │ │ -0001b220: 7468 6520 6f70 7469 6f6e 616c 2069 6e70 the optional inp │ │ │ │ -0001b230: 7574 202a 6e6f 7465 2041 6c6c 5374 616e ut *note AllStan │ │ │ │ -0001b240: 6461 7264 3a20 416c 6c53 7461 6e64 6172 dard: AllStandar │ │ │ │ -0001b250: 642c 2069 7320 7370 6563 6966 6965 6420 d, is specified │ │ │ │ -0001b260: 6173 0a24 7472 7565 242c 2074 6865 2066 as.$true$, the f │ │ │ │ -0001b270: 616d 696c 7920 6f66 2061 6c6c 2068 6f6d amily of all hom │ │ │ │ -0001b280: 6f67 656e 656f 7573 2069 6465 616c 7320 ogeneous ideals │ │ │ │ -0001b290: 6861 7669 6e67 2024 4d24 2061 7320 616e having $M$ as an │ │ │ │ -0001b2a0: 2069 6e69 7469 616c 2069 6465 616c 2069 initial ideal i │ │ │ │ -0001b2b0: 730a 636f 6d70 7574 6564 2c20 616e 6420 s.computed, and │ │ │ │ -0001b2c0: 6966 2069 7420 6973 2073 7065 6369 6669 if it is specifi │ │ │ │ -0001b2d0: 6564 2061 7320 2466 616c 7365 2420 7468 ed as $false$ th │ │ │ │ -0001b2e0: 6520 6661 6d69 6c79 206f 6620 616c 6c20 e family of all │ │ │ │ -0001b2f0: 686f 6d6f 6765 6e65 6f75 730a 6964 6561 homogeneous.idea │ │ │ │ -0001b300: 6c73 2068 6176 696e 6720 244d 2420 6173 ls having $M$ as │ │ │ │ -0001b310: 2061 6e20 696e 6974 6961 6c20 6964 6561 an initial idea │ │ │ │ -0001b320: 6c20 7769 7468 2072 6573 7065 6374 2074 l with respect t │ │ │ │ -0001b330: 6f20 7468 6520 6769 7665 6e20 7465 726d o the given term │ │ │ │ -0001b340: 206f 7264 6572 2069 730a 636f 6d70 7574 order is.comput │ │ │ │ -0001b350: 6564 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ed...+---------- │ │ │ │ -0001b360: 2d2d 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b3a0: 2d2d 2d2b 0a7c 6931 203a 2052 203d 205a ---+.|i1 : R = Z │ │ │ │ -0001b3b0: 5a2f 3332 3030 335b 612c 622c 635d 2020 Z/32003[a,b,c] │ │ │ │ -0001b3c0: 2020 2020 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 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b440: 2020 207c 0a7c 6f31 203d 2052 2020 2020 |.|o1 = R │ │ │ │ -0001b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b490: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b4e0: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ -0001b4f0: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ -0001b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b530: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -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 2d2d 2d2d ---------------- │ │ │ │ -0001b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b580: 2d2d 2d2b 0a7c 6932 203a 204d 203d 2069 ---+.|i2 : M = i │ │ │ │ -0001b590: 6465 616c 2028 615e 322c 2061 2a62 2c20 deal (a^2, a*b, │ │ │ │ -0001b5a0: 625e 3229 2020 2020 2020 2020 2020 2020 b^2) │ │ │ │ +0001aaa0: 7c20 2020 2020 202b 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aae0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ +0001aaf0: 7c20 2020 2020 207c 7420 2020 2b20 7420 | |t + t │ │ │ │ +0001ab00: 2074 2020 202b 2074 2020 7420 2020 2d20 t + t t - │ │ │ │ +0001ab10: 7420 2074 2020 7420 2020 2b20 7420 2074 t t t + t t │ │ │ │ +0001ab20: 2074 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +0001ab30: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ +0001ab40: 7c20 2020 2020 207c 2032 3120 2020 2031 | | 21 1 │ │ │ │ +0001ab50: 3620 3234 2020 2020 3137 2031 3120 2020 6 24 17 11 │ │ │ │ +0001ab60: 2031 3620 3131 2032 3220 2020 2031 3620 16 11 22 16 │ │ │ │ +0001ab70: 3420 3233 2020 2020 2020 2020 2020 2020 4 23 │ │ │ │ +0001ab80: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ +0001ab90: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +0001aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001abd0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ +0001abe0: 7c20 2020 2020 207c 7420 2020 2b20 7420 | |t + t │ │ │ │ +0001abf0: 2074 2020 202d 2074 2020 7420 202d 2074 t - t t - t │ │ │ │ +0001ac00: 2020 7420 2074 2020 202b 2074 2020 7420 t t + t t │ │ │ │ +0001ac10: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +0001ac20: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ +0001ac30: 7c20 2020 2020 207c 2032 3020 2020 2031 | | 20 1 │ │ │ │ +0001ac40: 3520 3234 2020 2020 3137 2034 2020 2020 5 24 17 4 │ │ │ │ +0001ac50: 3135 2031 3120 3232 2020 2020 3135 2034 15 11 22 15 4 │ │ │ │ +0001ac60: 2032 3320 2020 2020 2020 2020 2020 2020 23 │ │ │ │ +0001ac70: 2020 2020 2020 207c 2020 2020 2020 7c0a | |. │ │ │ │ +0001ac80: 7c20 2020 2020 202b 2d2d 2d2d 2d2d 2d2d | +-------- │ │ │ │ +0001ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001acb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001acc0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 7c0a -------+ |. │ │ │ │ +0001acd0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001ace0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001ad20: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +0001ad30: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6772 ==.. * *note gr │ │ │ │ +0001ad40: 6f65 626e 6572 5374 7261 7475 6d3a 2067 oebnerStratum: g │ │ │ │ +0001ad50: 726f 6562 6e65 7253 7472 6174 756d 2c20 roebnerStratum, │ │ │ │ +0001ad60: 2d2d 2063 6f6d 7075 7465 2074 6865 2069 -- compute the i │ │ │ │ +0001ad70: 6465 616c 2077 6865 7265 2061 2067 6976 deal where a giv │ │ │ │ +0001ad80: 656e 0a20 2020 2069 7320 6120 4772 6f65 en. is a Groe │ │ │ │ +0001ad90: 626e 6572 2062 6173 6973 0a20 202a 202a bner basis. * * │ │ │ │ +0001ada0: 6e6f 7465 2073 6d61 6c6c 6572 4d6f 6e6f note smallerMono │ │ │ │ +0001adb0: 6d69 616c 733a 2073 6d61 6c6c 6572 4d6f mials: smallerMo │ │ │ │ +0001adc0: 6e6f 6d69 616c 732c 202d 2d20 7265 7475 nomials, -- retu │ │ │ │ +0001add0: 726e 7320 7468 6520 7374 616e 6461 7264 rns the standard │ │ │ │ +0001ade0: 206d 6f6e 6f6d 6961 6c73 0a20 2020 2073 monomials. s │ │ │ │ +0001adf0: 6d61 6c6c 6572 2062 7574 206f 6620 7468 maller but of th │ │ │ │ +0001ae00: 6520 7361 6d65 2064 6567 7265 6520 6173 e same degree as │ │ │ │ +0001ae10: 2067 6976 656e 206d 6f6e 6f6d 6961 6c28 given monomial( │ │ │ │ +0001ae20: 7329 0a20 202a 202a 6e6f 7465 2073 7461 s). * *note sta │ │ │ │ +0001ae30: 6e64 6172 644d 6f6e 6f6d 6961 6c73 3a20 ndardMonomials: │ │ │ │ +0001ae40: 7374 616e 6461 7264 4d6f 6e6f 6d69 616c standardMonomial │ │ │ │ +0001ae50: 732c 202d 2d20 636f 6d70 7574 6573 2073 s, -- computes s │ │ │ │ +0001ae60: 7461 6e64 6172 6420 6d6f 6e6f 6d69 616c tandard monomial │ │ │ │ +0001ae70: 730a 0a57 6179 7320 746f 2075 7365 2067 s..Ways to use g │ │ │ │ +0001ae80: 726f 6562 6e65 7246 616d 696c 793a 0a3d roebnerFamily:.= │ │ │ │ +0001ae90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001aea0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0001aeb0: 2267 726f 6562 6e65 7246 616d 696c 7928 "groebnerFamily( │ │ │ │ +0001aec0: 4964 6561 6c29 220a 2020 2a20 2267 726f Ideal)". * "gro │ │ │ │ +0001aed0: 6562 6e65 7246 616d 696c 7928 4964 6561 ebnerFamily(Idea │ │ │ │ +0001aee0: 6c2c 4c69 7374 2922 0a0a 466f 7220 7468 l,List)"..For th │ │ │ │ +0001aef0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0001af00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0001af10: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0001af20: 6520 6772 6f65 626e 6572 4661 6d69 6c79 e groebnerFamily │ │ │ │ +0001af30: 3a20 6772 6f65 626e 6572 4661 6d69 6c79 : groebnerFamily │ │ │ │ +0001af40: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +0001af50: 686f 6420 6675 6e63 7469 6f6e 0a77 6974 hod function.wit │ │ │ │ +0001af60: 6820 6f70 7469 6f6e 733a 2028 4d61 6361 h options: (Maca │ │ │ │ +0001af70: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +0001af80: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ +0001af90: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ +0001afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001afe0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0001aff0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0001b000: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0001b010: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0001b020: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +0001b030: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +0001b040: 6179 322f 7061 636b 6167 6573 2f0a 4772 ay2/packages/.Gr │ │ │ │ +0001b050: 6f65 626e 6572 5374 7261 7461 2e6d 323a oebnerStrata.m2: │ │ │ │ +0001b060: 3737 363a 302e 0a1f 0a46 696c 653a 2047 776:0....File: G │ │ │ │ +0001b070: 726f 6562 6e65 7253 7472 6174 612e 696e roebnerStrata.in │ │ │ │ +0001b080: 666f 2c20 4e6f 6465 3a20 6772 6f65 626e fo, Node: groebn │ │ │ │ +0001b090: 6572 5374 7261 7475 6d2c 204e 6578 743a erStratum, Next: │ │ │ │ +0001b0a0: 206c 696e 6561 7250 6172 742c 2050 7265 linearPart, Pre │ │ │ │ +0001b0b0: 763a 2067 726f 6562 6e65 7246 616d 696c v: groebnerFamil │ │ │ │ +0001b0c0: 792c 2055 703a 2054 6f70 0a0a 6772 6f65 y, Up: Top..groe │ │ │ │ +0001b0d0: 626e 6572 5374 7261 7475 6d20 2d2d 2063 bnerStratum -- c │ │ │ │ +0001b0e0: 6f6d 7075 7465 2074 6865 2069 6465 616c ompute the ideal │ │ │ │ +0001b0f0: 2077 6865 7265 2061 2067 6976 656e 2069 where a given i │ │ │ │ +0001b100: 7320 6120 4772 6f65 626e 6572 2062 6173 s a Groebner bas │ │ │ │ +0001b110: 6973 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a is.************* │ │ │ │ +0001b120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001b130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001b140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001b150: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +0001b160: 7361 6765 3a20 0a20 2020 2020 2020 204a sage: . J │ │ │ │ +0001b170: 203d 2067 726f 6562 6e65 7253 7472 6174 = groebnerStrat │ │ │ │ +0001b180: 756d 2046 0a20 202a 2049 6e70 7574 733a um F. * Inputs: │ │ │ │ +0001b190: 0a20 2020 2020 202a 2046 2c20 616e 202a . * F, an * │ │ │ │ +0001b1a0: 6e6f 7465 2069 6465 616c 3a20 284d 6163 note ideal: (Mac │ │ │ │ +0001b1b0: 6175 6c61 7932 446f 6329 4964 6561 6c2c aulay2Doc)Ideal, │ │ │ │ +0001b1c0: 2c20 416e 2069 6465 616c 2063 6f6e 7374 , An ideal const │ │ │ │ +0001b1d0: 7275 6374 6564 2066 726f 6d0a 2020 2020 ructed from. │ │ │ │ +0001b1e0: 2020 2020 2a6e 6f74 6520 6772 6f65 626e *note groebn │ │ │ │ +0001b1f0: 6572 4661 6d69 6c79 3a20 6772 6f65 626e erFamily: groebn │ │ │ │ +0001b200: 6572 4661 6d69 6c79 2c0a 2020 2a20 2a6e erFamily,. * *n │ │ │ │ +0001b210: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70 ote Optional inp │ │ │ │ +0001b220: 7574 733a 2028 4d61 6361 756c 6179 3244 uts: (Macaulay2D │ │ │ │ +0001b230: 6f63 2975 7369 6e67 2066 756e 6374 696f oc)using functio │ │ │ │ +0001b240: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ +0001b250: 2069 6e70 7574 732c 3a0a 2020 2020 2020 inputs,:. │ │ │ │ +0001b260: 2a20 4d69 6e69 6d61 6c69 7a65 203d 3e20 * Minimalize => │ │ │ │ +0001b270: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ +0001b280: 7565 2066 616c 7365 0a20 202a 204f 7574 ue false. * Out │ │ │ │ +0001b290: 7075 7473 3a0a 2020 2020 2020 2a20 4a2c puts:. * J, │ │ │ │ +0001b2a0: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a an *note ideal: │ │ │ │ +0001b2b0: 2028 4d61 6361 756c 6179 3244 6f63 2949 (Macaulay2Doc)I │ │ │ │ +0001b2c0: 6465 616c 2c2c 2074 6865 2069 6465 616c deal,, the ideal │ │ │ │ +0001b2d0: 206f 6620 7468 6520 4772 6f65 626e 6572 of the Groebner │ │ │ │ +0001b2e0: 0a20 2020 2020 2020 2073 7472 6174 756d . stratum │ │ │ │ +0001b2f0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0001b300: 3d3d 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e =========..Given │ │ │ │ +0001b310: 2061 206d 6f6e 6f6d 6961 6c20 6964 6561 a monomial idea │ │ │ │ +0001b320: 6c20 244d 242c 2074 6869 7320 636f 6d6d l $M$, this comm │ │ │ │ +0001b330: 616e 6420 7265 7475 726e 7320 6120 6661 and returns a fa │ │ │ │ +0001b340: 6d69 6c79 206f 6620 6964 6561 6c73 2068 mily of ideals h │ │ │ │ +0001b350: 6176 696e 6720 244d 240a 6173 2061 6e20 aving $M$.as an │ │ │ │ +0001b360: 696e 6974 6961 6c20 6964 6561 6c2c 2061 initial ideal, a │ │ │ │ +0001b370: 6e64 2063 6f6e 6469 7469 6f6e 7320 6f6e nd conditions on │ │ │ │ +0001b380: 2074 6865 2070 6172 616d 6574 6572 7320 the parameters │ │ │ │ +0001b390: 736f 2074 6861 7420 7468 6520 6661 6d69 so that the fami │ │ │ │ +0001b3a0: 6c79 2069 730a 666c 6174 2e20 2049 6620 ly is.flat. If │ │ │ │ +0001b3b0: 7468 6520 6f70 7469 6f6e 616c 2069 6e70 the optional inp │ │ │ │ +0001b3c0: 7574 202a 6e6f 7465 2041 6c6c 5374 616e ut *note AllStan │ │ │ │ +0001b3d0: 6461 7264 3a20 416c 6c53 7461 6e64 6172 dard: AllStandar │ │ │ │ +0001b3e0: 642c 2069 7320 7370 6563 6966 6965 6420 d, is specified │ │ │ │ +0001b3f0: 6173 0a24 7472 7565 242c 2074 6865 2066 as.$true$, the f │ │ │ │ +0001b400: 616d 696c 7920 6f66 2061 6c6c 2068 6f6d amily of all hom │ │ │ │ +0001b410: 6f67 656e 656f 7573 2069 6465 616c 7320 ogeneous ideals │ │ │ │ +0001b420: 6861 7669 6e67 2024 4d24 2061 7320 616e having $M$ as an │ │ │ │ +0001b430: 2069 6e69 7469 616c 2069 6465 616c 2069 initial ideal i │ │ │ │ +0001b440: 730a 636f 6d70 7574 6564 2c20 616e 6420 s.computed, and │ │ │ │ +0001b450: 6966 2069 7420 6973 2073 7065 6369 6669 if it is specifi │ │ │ │ +0001b460: 6564 2061 7320 2466 616c 7365 2420 7468 ed as $false$ th │ │ │ │ +0001b470: 6520 6661 6d69 6c79 206f 6620 616c 6c20 e family of all │ │ │ │ +0001b480: 686f 6d6f 6765 6e65 6f75 730a 6964 6561 homogeneous.idea │ │ │ │ +0001b490: 6c73 2068 6176 696e 6720 244d 2420 6173 ls having $M$ as │ │ │ │ +0001b4a0: 2061 6e20 696e 6974 6961 6c20 6964 6561 an initial idea │ │ │ │ +0001b4b0: 6c20 7769 7468 2072 6573 7065 6374 2074 l with respect t │ │ │ │ +0001b4c0: 6f20 7468 6520 6769 7665 6e20 7465 726d o the given term │ │ │ │ +0001b4d0: 206f 7264 6572 2069 730a 636f 6d70 7574 order is.comput │ │ │ │ +0001b4e0: 6564 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ed...+---------- │ │ │ │ +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 2d2d 2d2d ---------------- │ │ │ │ +0001b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b530: 2d2d 2d2b 0a7c 6931 203a 2052 203d 205a ---+.|i1 : R = Z │ │ │ │ +0001b540: 5a2f 3332 3030 335b 612c 622c 635d 2020 Z/32003[a,b,c] │ │ │ │ +0001b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b580: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b5d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b5d0: 2020 207c 0a7c 6f31 203d 2052 2020 2020 |.|o1 = R │ │ │ │ 0001b5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b620: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b630: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +0001b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b670: 2020 207c 0a7c 6f32 203d 2069 6465 616c |.|o2 = ideal │ │ │ │ -0001b680: 2028 6120 2c20 612a 622c 2062 2029 2020 (a , a*b, b ) │ │ │ │ +0001b670: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ +0001b680: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ 0001b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b6c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b710: 2020 207c 0a7c 6f32 203a 2049 6465 616c |.|o2 : Ideal │ │ │ │ -0001b720: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ -0001b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b6c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b710: 2d2d 2d2b 0a7c 6932 203a 204d 203d 2069 ---+.|i2 : M = i │ │ │ │ +0001b720: 6465 616c 2028 615e 322c 2061 2a62 2c20 deal (a^2, a*b, │ │ │ │ +0001b730: 625e 3229 2020 2020 2020 2020 2020 2020 b^2) │ │ │ │ 0001b740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b760: 2020 207c 0a2b 2d2d 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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b7b0: 2d2d 2d2b 0a7c 6933 203a 2046 203d 2067 ---+.|i3 : F = g │ │ │ │ -0001b7c0: 726f 6562 6e65 7246 616d 696c 7920 4d20 roebnerFamily M │ │ │ │ +0001b760: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b7b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b7c0: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ 0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b800: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b800: 2020 207c 0a7c 6f32 203d 2069 6465 616c |.|o2 = ideal │ │ │ │ +0001b810: 2028 6120 2c20 612a 622c 2062 2029 2020 (a , a*b, b ) │ │ │ │ 0001b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b850: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001b860: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001b870: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0001b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b890: 2020 2020 2020 3220 2020 3220 2020 2020 2 2 │ │ │ │ -0001b8a0: 2020 207c 0a7c 6f33 203d 2069 6465 616c |.|o3 = ideal │ │ │ │ -0001b8b0: 2028 6120 202b 2074 2061 2a63 202b 2074 (a + t a*c + t │ │ │ │ -0001b8c0: 2062 2a63 202b 2074 2063 202c 2061 2a62 b*c + t c , a*b │ │ │ │ -0001b8d0: 202b 2074 2061 2a63 202b 2074 2062 2a63 + t a*c + t b*c │ │ │ │ -0001b8e0: 202b 2074 2063 202c 2062 2020 2b20 7420 + t c , b + t │ │ │ │ -0001b8f0: 612a 637c 0a7c 2020 2020 2020 2020 2020 a*c|.| │ │ │ │ -0001b900: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ -0001b910: 3220 2020 2020 2020 3320 2020 2020 2020 2 3 │ │ │ │ -0001b920: 2020 2020 3420 2020 2020 2020 3520 2020 4 5 │ │ │ │ -0001b930: 2020 2020 3620 2020 2020 2020 2020 2037 6 7 │ │ │ │ -0001b940: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ -0001b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b990: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020 ---|.| │ │ │ │ -0001b9a0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0001b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b8a0: 2020 207c 0a7c 6f32 203a 2049 6465 616c |.|o2 : Ideal │ │ │ │ +0001b8b0: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +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 2020 │ │ │ │ +0001b8f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001b900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001b940: 2d2d 2d2b 0a7c 6933 203a 2046 203d 2067 ---+.|i3 : F = g │ │ │ │ +0001b950: 726f 6562 6e65 7246 616d 696c 7920 4d20 roebnerFamily M │ │ │ │ +0001b960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b990: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9e0: 2020 207c 0a7c 2020 2020 202b 2074 2062 |.| + t b │ │ │ │ -0001b9f0: 2a63 202b 2074 2063 2029 2020 2020 2020 *c + t c ) │ │ │ │ -0001ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b9e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b9f0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001ba00: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ 0001ba10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba30: 2020 207c 0a7c 2020 2020 2020 2020 3820 |.| 8 │ │ │ │ -0001ba40: 2020 2020 2020 3920 2020 2020 2020 2020 9 │ │ │ │ -0001ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba80: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001baa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bad0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001bae0: 2020 2020 2020 5a5a 2020 2020 2020 2020 ZZ │ │ │ │ -0001baf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb20: 2020 207c 0a7c 6f33 203a 2049 6465 616c |.|o3 : Ideal │ │ │ │ -0001bb30: 206f 6620 2d2d 2d2d 2d5b 7420 2c20 7420 of -----[t , t │ │ │ │ -0001bb40: 2c20 7420 2c20 7420 2e2e 7420 2c20 7420 , t , t ..t , t │ │ │ │ -0001bb50: 2e2e 7420 2c20 7420 2e2e 7420 5d5b 612e ..t , t ..t ][a. │ │ │ │ -0001bb60: 2e63 5d20 2020 2020 2020 2020 2020 2020 .c] │ │ │ │ -0001bb70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001bb80: 2020 2020 3332 3030 3320 2033 2020 2036 32003 3 6 │ │ │ │ -0001bb90: 2020 2039 2020 2031 2020 2032 2020 2034 9 1 2 4 │ │ │ │ -0001bba0: 2020 2035 2020 2037 2020 2038 2020 2020 5 7 8 │ │ │ │ +0001ba20: 2020 2020 2020 3220 2020 3220 2020 2020 2 2 │ │ │ │ +0001ba30: 2020 207c 0a7c 6f33 203d 2069 6465 616c |.|o3 = ideal │ │ │ │ +0001ba40: 2028 6120 202b 2074 2061 2a63 202b 2074 (a + t a*c + t │ │ │ │ +0001ba50: 2062 2a63 202b 2074 2063 202c 2061 2a62 b*c + t c , a*b │ │ │ │ +0001ba60: 202b 2074 2061 2a63 202b 2074 2062 2a63 + t a*c + t b*c │ │ │ │ +0001ba70: 202b 2074 2063 202c 2062 2020 2b20 7420 + t c , b + t │ │ │ │ +0001ba80: 612a 637c 0a7c 2020 2020 2020 2020 2020 a*c|.| │ │ │ │ +0001ba90: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ +0001baa0: 3220 2020 2020 2020 3320 2020 2020 2020 2 3 │ │ │ │ +0001bab0: 2020 2020 3420 2020 2020 2020 3520 2020 4 5 │ │ │ │ +0001bac0: 2020 2020 3620 2020 2020 2020 2020 2037 6 7 │ │ │ │ +0001bad0: 2020 207c 0a7c 2020 2020 202d 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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bb20: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020 ---|.| │ │ │ │ +0001bb30: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb70: 2020 207c 0a7c 2020 2020 202b 2074 2062 |.| + t b │ │ │ │ +0001bb80: 2a63 202b 2074 2063 2029 2020 2020 2020 *c + t c ) │ │ │ │ +0001bb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bbc0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0001bbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bc10: 2d2d 2d2b 0a7c 6934 203a 204a 203d 2074 ---+.|i4 : J = t │ │ │ │ -0001bc20: 7269 6d20 6772 6f65 626e 6572 5374 7261 rim groebnerStra │ │ │ │ -0001bc30: 7475 6d20 4620 2020 2020 2020 2020 2020 tum F │ │ │ │ +0001bbc0: 2020 207c 0a7c 2020 2020 2020 2020 3820 |.| 8 │ │ │ │ +0001bbd0: 2020 2020 2020 3920 2020 2020 2020 2020 9 │ │ │ │ +0001bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bc10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc60: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bc70: 2020 2020 2020 5a5a 2020 2020 2020 2020 ZZ │ │ │ │ 0001bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bcb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001bcc0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -0001bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bd00: 2020 207c 0a7c 6f34 203d 2069 6465 616c |.|o4 = ideal │ │ │ │ -0001bd10: 2028 7420 202b 2074 2020 2d20 7420 7420 (t + t - t t │ │ │ │ -0001bd20: 202b 2074 2074 2020 2d20 7420 7420 2c20 + t t - t t , │ │ │ │ -0001bd30: 7420 202d 2074 2074 2020 2b20 7420 7420 t - t t + t t │ │ │ │ -0001bd40: 2c20 7420 202b 2074 2074 2020 2d20 7420 , t + t t - t │ │ │ │ -0001bd50: 7420 207c 0a7c 2020 2020 2020 2020 2020 t |.| │ │ │ │ -0001bd60: 2020 2039 2020 2020 3420 2020 2031 2037 9 4 1 7 │ │ │ │ -0001bd70: 2020 2020 3520 3720 2020 2034 2038 2020 5 7 4 8 │ │ │ │ -0001bd80: 2036 2020 2020 3420 3520 2020 2032 2037 6 4 5 2 7 │ │ │ │ -0001bd90: 2020 2033 2020 2020 3220 3420 2020 2031 3 2 4 1 │ │ │ │ -0001bda0: 2035 207c 0a7c 2020 2020 202d 2d2d 2d2d 5 |.| ----- │ │ │ │ -0001bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bdf0: 2d2d 2d7c 0a7c 2020 2020 2020 2020 3220 ---|.| 2 │ │ │ │ +0001bcb0: 2020 207c 0a7c 6f33 203a 2049 6465 616c |.|o3 : Ideal │ │ │ │ +0001bcc0: 206f 6620 2d2d 2d2d 2d5b 7420 2c20 7420 of -----[t , t │ │ │ │ +0001bcd0: 2c20 7420 2c20 7420 2e2e 7420 2c20 7420 , t , t ..t , t │ │ │ │ +0001bce0: 2e2e 7420 2c20 7420 2e2e 7420 5d5b 612e ..t , t ..t ][a. │ │ │ │ +0001bcf0: 2e63 5d20 2020 2020 2020 2020 2020 2020 .c] │ │ │ │ +0001bd00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001bd10: 2020 2020 3332 3030 3320 2033 2020 2036 32003 3 6 │ │ │ │ +0001bd20: 2020 2039 2020 2031 2020 2032 2020 2034 9 1 2 4 │ │ │ │ +0001bd30: 2020 2035 2020 2037 2020 2038 2020 2020 5 7 8 │ │ │ │ +0001bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bd50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001bd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001bda0: 2d2d 2d2b 0a7c 6934 203a 204a 203d 2074 ---+.|i4 : J = t │ │ │ │ +0001bdb0: 7269 6d20 6772 6f65 626e 6572 5374 7261 rim groebnerStra │ │ │ │ +0001bdc0: 7475 6d20 4620 2020 2020 2020 2020 2020 tum F │ │ │ │ +0001bdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bdf0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be40: 2020 207c 0a7c 2020 2020 202b 2074 2020 |.| + t │ │ │ │ -0001be50: 2d20 7420 7420 2920 2020 2020 2020 2020 - t t ) │ │ │ │ +0001be40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001be50: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ 0001be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be90: 2020 207c 0a7c 2020 2020 2020 2020 3520 |.| 5 │ │ │ │ -0001bea0: 2020 2032 2038 2020 2020 2020 2020 2020 2 8 │ │ │ │ -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 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001bef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001bf40: 2020 2020 2020 5a5a 2020 2020 2020 2020 ZZ │ │ │ │ -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 207c 0a7c 6f34 203a 2049 6465 616c |.|o4 : Ideal │ │ │ │ -0001bf90: 206f 6620 2d2d 2d2d 2d5b 7420 2c20 7420 of -----[t , t │ │ │ │ -0001bfa0: 2c20 7420 2c20 7420 2e2e 7420 2c20 7420 , t , t ..t , t │ │ │ │ -0001bfb0: 2e2e 7420 2c20 7420 2e2e 7420 5d20 2020 ..t , t ..t ] │ │ │ │ +0001be90: 2020 207c 0a7c 6f34 203d 2069 6465 616c |.|o4 = ideal │ │ │ │ +0001bea0: 2028 7420 202b 2074 2020 2d20 7420 7420 (t + t - t t │ │ │ │ +0001beb0: 202b 2074 2074 2020 2d20 7420 7420 2c20 + t t - t t , │ │ │ │ +0001bec0: 7420 202d 2074 2074 2020 2b20 7420 7420 t - t t + t t │ │ │ │ +0001bed0: 2c20 7420 202b 2074 2074 2020 2d20 7420 , t + t t - t │ │ │ │ +0001bee0: 7420 207c 0a7c 2020 2020 2020 2020 2020 t |.| │ │ │ │ +0001bef0: 2020 2039 2020 2020 3420 2020 2031 2037 9 4 1 7 │ │ │ │ +0001bf00: 2020 2020 3520 3720 2020 2034 2038 2020 5 7 4 8 │ │ │ │ +0001bf10: 2036 2020 2020 3420 3520 2020 2032 2037 6 4 5 2 7 │ │ │ │ +0001bf20: 2020 2033 2020 2020 3220 3420 2020 2031 3 2 4 1 │ │ │ │ +0001bf30: 2035 207c 0a7c 2020 2020 202d 2d2d 2d2d 5 |.| ----- │ │ │ │ +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 2d2d 2d2d ---------------- │ │ │ │ +0001bf80: 2d2d 2d7c 0a7c 2020 2020 2020 2020 3220 ---|.| 2 │ │ │ │ +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 2020 2020 2020 2020 │ │ │ │ -0001bfd0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001bfe0: 2020 2020 3332 3030 3320 2033 2020 2036 32003 3 6 │ │ │ │ -0001bff0: 2020 2039 2020 2031 2020 2032 2020 2034 9 1 2 4 │ │ │ │ -0001c000: 2020 2035 2020 2037 2020 2038 2020 2020 5 7 8 │ │ │ │ +0001bfd0: 2020 207c 0a7c 2020 2020 202b 2074 2020 |.| + t │ │ │ │ +0001bfe0: 2d20 7420 7420 2920 2020 2020 2020 2020 - t t ) │ │ │ │ +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 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0001c030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c070: 2d2d 2d2b 0a0a 496e 2074 6869 7320 6578 ---+..In this ex │ │ │ │ -0001c080: 616d 706c 652c 2046 2069 7320 7468 6520 ample, F is the │ │ │ │ -0001c090: 756e 6976 6572 7361 6c20 6661 6d69 6c79 universal family │ │ │ │ -0001c0a0: 2c20 616e 6420 4c20 6973 2074 6865 2069 , and L is the i │ │ │ │ -0001c0b0: 6465 616c 2067 6976 696e 6720 7468 650a deal giving the. │ │ │ │ -0001c0c0: 636f 6e64 6974 696f 6e73 206f 6e20 7468 conditions on th │ │ │ │ -0001c0d0: 6520 7061 7261 6d65 7465 7273 2e20 2049 e parameters. I │ │ │ │ -0001c0e0: 6e20 6765 6e65 7261 6c2c 2073 6576 6572 n general, sever │ │ │ │ -0001c0f0: 616c 206f 6620 7468 6520 7061 7261 6d65 al of the parame │ │ │ │ -0001c100: 7465 7273 2061 7265 0a75 6e6e 6563 6573 ters are.unneces │ │ │ │ -0001c110: 7361 7279 2e20 4e6f 7465 2074 6861 7420 sary. Note that │ │ │ │ -0001c120: 4820 6973 2061 6e20 6964 6561 6c20 696e H is an ideal in │ │ │ │ -0001c130: 2061 2072 696e 6720 7769 7468 2066 6172 a ring with far │ │ │ │ -0001c140: 2066 6577 6572 2070 6172 616d 6574 6572 fewer parameter │ │ │ │ -0001c150: 732e 2054 6869 730a 6973 2062 6563 6175 s. This.is becau │ │ │ │ -0001c160: 7365 2061 206d 6178 696d 616c 2073 6574 se a maximal set │ │ │ │ -0001c170: 206f 6620 656c 696d 696e 6162 6c65 2070 of eliminable p │ │ │ │ -0001c180: 6172 616d 6574 6572 7320 6672 6f6d 2074 arameters from t │ │ │ │ -0001c190: 6865 206f 7269 6769 6e61 6c20 6964 6561 he original idea │ │ │ │ -0001c1a0: 6c20 6f66 0a63 6f6e 6469 7469 6f6e 7320 l of.conditions │ │ │ │ -0001c1b0: 6f6e 2070 6172 616d 6574 6572 7320 6861 on parameters ha │ │ │ │ -0001c1c0: 7665 2062 6565 6e20 656c 696d 696e 6174 ve been eliminat │ │ │ │ -0001c1d0: 6564 2e20 2049 6620 7468 6520 6675 6c6c ed. If the full │ │ │ │ -0001c1e0: 2069 6465 616c 2069 6e20 7468 650a 706f ideal in the.po │ │ │ │ -0001c1f0: 6c79 6e6f 6d69 616c 2072 696e 6720 6f76 lynomial ring ov │ │ │ │ -0001c200: 6572 2061 6c6c 2074 6865 2070 6172 616d er all the param │ │ │ │ -0001c210: 6574 6572 7320 6973 2064 6573 6972 6564 eters is desired │ │ │ │ -0001c220: 2c20 7365 7420 7468 6520 6f70 7469 6f6e , set the option │ │ │ │ -0001c230: 616c 2061 7267 756d 656e 740a 244d 696e al argument.$Min │ │ │ │ -0001c240: 696d 616c 697a 6524 2074 6f20 6661 6c73 imalize$ to fals │ │ │ │ -0001c250: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d e...+----------- │ │ │ │ -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 2b0a 7c69 3520 3a20 4a32 203d 2074 --+.|i5 : J2 = t │ │ │ │ -0001c2b0: 7269 6d20 6772 6f65 626e 6572 5374 7261 rim groebnerStra │ │ │ │ -0001c2c0: 7475 6d28 462c 204d 696e 696d 616c 697a tum(F, Minimaliz │ │ │ │ -0001c2d0: 6520 3d3e 2066 616c 7365 2920 2020 2020 e => false) │ │ │ │ -0001c2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c2f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c340: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c350: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -0001c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c390: 2020 7c0a 7c6f 3520 3d20 6964 6561 6c20 |.|o5 = ideal │ │ │ │ -0001c3a0: 2874 2020 2b20 7420 202d 2074 2074 2020 (t + t - t t │ │ │ │ -0001c3b0: 2b20 7420 7420 202d 2074 2074 202c 2074 + t t - t t , t │ │ │ │ -0001c3c0: 2020 2d20 7420 7420 202b 2074 2074 202c - t t + t t , │ │ │ │ -0001c3d0: 2074 2020 2b20 7420 7420 202d 2074 2074 t + t t - t t │ │ │ │ -0001c3e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c3f0: 2020 3920 2020 2034 2020 2020 3120 3720 9 4 1 7 │ │ │ │ -0001c400: 2020 2035 2037 2020 2020 3420 3820 2020 5 7 4 8 │ │ │ │ -0001c410: 3620 2020 2034 2035 2020 2020 3220 3720 6 4 5 2 7 │ │ │ │ -0001c420: 2020 3320 2020 2032 2034 2020 2020 3120 3 2 4 1 │ │ │ │ -0001c430: 3520 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 5 |.| ------ │ │ │ │ -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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c480: 2d2d 7c0a 7c20 2020 2020 2020 2032 2020 --|.| 2 │ │ │ │ +0001c020: 2020 207c 0a7c 2020 2020 2020 2020 3520 |.| 5 │ │ │ │ +0001c030: 2020 2032 2038 2020 2020 2020 2020 2020 2 8 │ │ │ │ +0001c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c070: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c0c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c0d0: 2020 2020 2020 5a5a 2020 2020 2020 2020 ZZ │ │ │ │ +0001c0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c110: 2020 207c 0a7c 6f34 203a 2049 6465 616c |.|o4 : Ideal │ │ │ │ +0001c120: 206f 6620 2d2d 2d2d 2d5b 7420 2c20 7420 of -----[t , t │ │ │ │ +0001c130: 2c20 7420 2c20 7420 2e2e 7420 2c20 7420 , t , t ..t , t │ │ │ │ +0001c140: 2e2e 7420 2c20 7420 2e2e 7420 5d20 2020 ..t , t ..t ] │ │ │ │ +0001c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c160: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c170: 2020 2020 3332 3030 3320 2033 2020 2036 32003 3 6 │ │ │ │ +0001c180: 2020 2039 2020 2031 2020 2032 2020 2034 9 1 2 4 │ │ │ │ +0001c190: 2020 2035 2020 2037 2020 2038 2020 2020 5 7 8 │ │ │ │ +0001c1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c1b0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c200: 2d2d 2d2b 0a0a 496e 2074 6869 7320 6578 ---+..In this ex │ │ │ │ +0001c210: 616d 706c 652c 2046 2069 7320 7468 6520 ample, F is the │ │ │ │ +0001c220: 756e 6976 6572 7361 6c20 6661 6d69 6c79 universal family │ │ │ │ +0001c230: 2c20 616e 6420 4c20 6973 2074 6865 2069 , and L is the i │ │ │ │ +0001c240: 6465 616c 2067 6976 696e 6720 7468 650a deal giving the. │ │ │ │ +0001c250: 636f 6e64 6974 696f 6e73 206f 6e20 7468 conditions on th │ │ │ │ +0001c260: 6520 7061 7261 6d65 7465 7273 2e20 2049 e parameters. I │ │ │ │ +0001c270: 6e20 6765 6e65 7261 6c2c 2073 6576 6572 n general, sever │ │ │ │ +0001c280: 616c 206f 6620 7468 6520 7061 7261 6d65 al of the parame │ │ │ │ +0001c290: 7465 7273 2061 7265 0a75 6e6e 6563 6573 ters are.unneces │ │ │ │ +0001c2a0: 7361 7279 2e20 4e6f 7465 2074 6861 7420 sary. Note that │ │ │ │ +0001c2b0: 4820 6973 2061 6e20 6964 6561 6c20 696e H is an ideal in │ │ │ │ +0001c2c0: 2061 2072 696e 6720 7769 7468 2066 6172 a ring with far │ │ │ │ +0001c2d0: 2066 6577 6572 2070 6172 616d 6574 6572 fewer parameter │ │ │ │ +0001c2e0: 732e 2054 6869 730a 6973 2062 6563 6175 s. This.is becau │ │ │ │ +0001c2f0: 7365 2061 206d 6178 696d 616c 2073 6574 se a maximal set │ │ │ │ +0001c300: 206f 6620 656c 696d 696e 6162 6c65 2070 of eliminable p │ │ │ │ +0001c310: 6172 616d 6574 6572 7320 6672 6f6d 2074 arameters from t │ │ │ │ +0001c320: 6865 206f 7269 6769 6e61 6c20 6964 6561 he original idea │ │ │ │ +0001c330: 6c20 6f66 0a63 6f6e 6469 7469 6f6e 7320 l of.conditions │ │ │ │ +0001c340: 6f6e 2070 6172 616d 6574 6572 7320 6861 on parameters ha │ │ │ │ +0001c350: 7665 2062 6565 6e20 656c 696d 696e 6174 ve been eliminat │ │ │ │ +0001c360: 6564 2e20 2049 6620 7468 6520 6675 6c6c ed. If the full │ │ │ │ +0001c370: 2069 6465 616c 2069 6e20 7468 650a 706f ideal in the.po │ │ │ │ +0001c380: 6c79 6e6f 6d69 616c 2072 696e 6720 6f76 lynomial ring ov │ │ │ │ +0001c390: 6572 2061 6c6c 2074 6865 2070 6172 616d er all the param │ │ │ │ +0001c3a0: 6574 6572 7320 6973 2064 6573 6972 6564 eters is desired │ │ │ │ +0001c3b0: 2c20 7365 7420 7468 6520 6f70 7469 6f6e , set the option │ │ │ │ +0001c3c0: 616c 2061 7267 756d 656e 740a 244d 696e al argument.$Min │ │ │ │ +0001c3d0: 696d 616c 697a 6524 2074 6f20 6661 6c73 imalize$ to fals │ │ │ │ +0001c3e0: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d e...+----------- │ │ │ │ +0001c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c430: 2d2d 2b0a 7c69 3520 3a20 4a32 203d 2074 --+.|i5 : J2 = t │ │ │ │ +0001c440: 7269 6d20 6772 6f65 626e 6572 5374 7261 rim groebnerStra │ │ │ │ +0001c450: 7475 6d28 462c 204d 696e 696d 616c 697a tum(F, Minimaliz │ │ │ │ +0001c460: 6520 3d3e 2066 616c 7365 2920 2020 2020 e => false) │ │ │ │ +0001c470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c480: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4d0: 2020 7c0a 7c20 2020 2020 2b20 7420 202d |.| + t - │ │ │ │ -0001c4e0: 2074 2074 2029 2020 2020 2020 2020 2020 t t ) │ │ │ │ +0001c4d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c4e0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ 0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c520: 2020 7c0a 7c20 2020 2020 2020 2035 2020 |.| 5 │ │ │ │ -0001c530: 2020 3220 3820 2020 2020 2020 2020 2020 2 8 │ │ │ │ -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 2020 │ │ │ │ +0001c520: 2020 7c0a 7c6f 3520 3d20 6964 6561 6c20 |.|o5 = ideal │ │ │ │ +0001c530: 2874 2020 2b20 7420 202d 2074 2074 2020 (t + t - t t │ │ │ │ +0001c540: 2b20 7420 7420 202d 2074 2074 202c 2074 + t t - t t , t │ │ │ │ +0001c550: 2020 2d20 7420 7420 202b 2074 2074 202c - t t + t t , │ │ │ │ +0001c560: 2074 2020 2b20 7420 7420 202d 2074 2074 t + t t - t t │ │ │ │ 0001c570: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c5c0: 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│ │ +0001c5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c610: 2d2d 7c0a 7c20 2020 2020 2020 2032 2020 --|.| 2 │ │ │ │ +0001c620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c660: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c670: 2020 2033 3230 3033 2020 3320 2020 3620 32003 3 6 │ │ │ │ -0001c680: 2020 3920 2020 3120 2020 3220 2020 3420 9 1 2 4 │ │ │ │ -0001c690: 2020 3520 2020 3720 2020 3820 2020 2020 5 7 8 │ │ │ │ +0001c660: 2020 7c0a 7c20 2020 2020 2b20 7420 202d |.| + t - │ │ │ │ +0001c670: 2074 2074 2029 2020 2020 2020 2020 2020 t t ) │ │ │ │ +0001c680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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│ │ +0001c710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c750: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c760: 2020 2020 205a 5a20 2020 2020 2020 2020 ZZ │ │ │ │ 0001c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7a0: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0001c7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c7c0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c7a0: 2020 7c0a 7c6f 3520 3a20 4964 6561 6c20 |.|o5 : Ideal │ │ │ │ +0001c7b0: 6f66 202d 2d2d 2d2d 5b74 202c 2074 202c of -----[t , t , │ │ │ │ +0001c7c0: 2074 202c 2074 202e 2e74 202c 2074 202e t , t ..t , t . │ │ │ │ +0001c7d0: 2e74 202c 2074 202e 2e74 205d 2020 2020 .t , t ..t ] │ │ │ │ 0001c7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7f0: 2020 7c0a 7c20 2020 2020 7c20 2020 2020 |.| | │ │ │ │ -0001c800: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001c810: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ -0001c820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c7f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c800: 2020 2033 3230 3033 2020 3320 2020 3620 32003 3 6 │ │ │ │ +0001c810: 2020 3920 2020 3120 2020 3220 2020 3420 9 1 2 4 │ │ │ │ +0001c820: 2020 3520 2020 3720 2020 3820 2020 2020 5 7 8 │ │ │ │ 0001c830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c840: 2020 7c0a 7c6f 3620 3d20 7c74 2020 2b20 |.|o6 = |t + │ │ │ │ -0001c850: 7420 202d 2074 2074 2020 2b20 7420 7420 t - t t + t t │ │ │ │ -0001c860: 202d 2074 2074 207c 2020 2020 2020 2020 - t t | │ │ │ │ -0001c870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c890: 2020 7c0a 7c20 2020 2020 7c20 3920 2020 |.| | 9 │ │ │ │ -0001c8a0: 2034 2020 2020 3120 3720 2020 2035 2037 4 1 7 5 7 │ │ │ │ -0001c8b0: 2020 2020 3420 387c 2020 2020 2020 2020 4 8| │ │ │ │ +0001c840: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c890: 2d2d 2b0a 7c69 3620 3a20 6e65 744c 6973 --+.|i6 : netLis │ │ │ │ +0001c8a0: 7420 4a5f 2a20 2020 2020 2020 2020 2020 t J_* │ │ │ │ +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 2020 2020 │ │ │ │ -0001c8e0: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0001c8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c900: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0001c8e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c930: 2020 7c0a 7c20 2020 2020 7c74 2020 2d20 |.| |t - │ │ │ │ -0001c940: 7420 7420 202b 2074 2074 2020 2020 2020 t t + t t │ │ │ │ -0001c950: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ +0001c930: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ +0001c940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001c950: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ 0001c960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca20: 2020 7c0a 7c20 2020 2020 7c20 2020 2020 |.| | │ │ │ │ -0001ca30: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -0001ca40: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ +0001ca20: 2020 7c0a 7c20 2020 2020 7c20 3920 2020 |.| | 9 │ │ │ │ +0001ca30: 2034 2020 2020 3120 3720 2020 2035 2037 4 1 7 5 7 │ │ │ │ +0001ca40: 2020 2020 3420 387c 2020 2020 2020 2020 4 8| │ │ │ │ 0001ca50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ca60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ca70: 2020 7c0a 7c20 2020 2020 7c74 2020 2b20 |.| |t + │ │ │ │ -0001ca80: 7420 7420 202d 2074 2074 2020 2b20 7420 t t - t t + t │ │ │ │ -0001ca90: 202d 2074 2074 207c 2020 2020 2020 2020 - t t | │ │ │ │ +0001ca70: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ +0001ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ca90: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ 0001caa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cac0: 2020 7c0a 7c20 2020 2020 7c20 3320 2020 |.| | 3 │ │ │ │ -0001cad0: 2032 2034 2020 2020 3120 3520 2020 2035 2 4 1 5 5 │ │ │ │ -0001cae0: 2020 2020 3220 387c 2020 2020 2020 2020 2 8| │ │ │ │ +0001cac0: 2020 7c0a 7c20 2020 2020 7c74 2020 2d20 |.| |t - │ │ │ │ +0001cad0: 7420 7420 202b 2074 2074 2020 2020 2020 t t + t t │ │ │ │ +0001cae0: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ 0001caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cb10: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0001cb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cb30: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0001cb10: 2020 7c0a 7c20 2020 2020 7c20 3620 2020 |.| | 6 │ │ │ │ +0001cb20: 2034 2035 2020 2020 3220 3720 2020 2020 4 5 2 7 │ │ │ │ +0001cb30: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ 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2020 |.| | │ │ │ │ +0001cbc0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +0001cbd0: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ 0001cbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cc00: 2020 7c0a 7c20 2020 2020 7c74 2020 2b20 |.| |t + │ │ │ │ +0001cc10: 7420 7420 202d 2074 2074 2020 2b20 7420 t t - t t + t │ │ │ │ +0001cc20: 202d 2074 2074 207c 2020 2020 2020 2020 - t t | │ │ │ │ 0001cc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc50: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0001cc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cc70: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0001cc50: 2020 7c0a 7c20 2020 2020 7c20 3320 2020 |.| | 3 │ │ │ │ +0001cc60: 2032 2034 2020 2020 3120 3520 2020 2035 2 4 1 5 5 │ │ │ │ +0001cc70: 2020 2020 3220 387c 2020 2020 2020 2020 2 8| │ │ │ │ 0001cc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cca0: 2020 7c0a 7c20 2020 2020 7c20 2020 2020 |.| | │ │ │ │ -0001ccb0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ccc0: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ +0001cca0: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ +0001ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ccc0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ 0001ccd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ccf0: 2020 7c0a 7c6f 3720 3d20 7c74 2020 2b20 |.|o7 = |t + │ │ │ │ -0001cd00: 7420 202d 2074 2074 2020 2b20 7420 7420 t - t t + t t │ │ │ │ -0001cd10: 202d 2074 2074 207c 2020 2020 2020 2020 - t t | │ │ │ │ -0001cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd40: 2020 7c0a 7c20 2020 2020 7c20 3920 2020 |.| | 9 │ │ │ │ -0001cd50: 2034 2020 2020 3120 3720 2020 2035 2037 4 1 7 5 7 │ │ │ │ -0001cd60: 2020 2020 3420 387c 2020 2020 2020 2020 4 8| │ │ │ │ +0001ccf0: 2020 7c0a 2b2d 2d2d 2d2d 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 3720 3a20 6e65 744c 6973 --+.|i7 : netLis │ │ │ │ +0001cd50: 7420 4a32 5f2a 2020 2020 2020 2020 2020 t J2_* │ │ │ │ +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 2020 2020 │ │ │ │ -0001cd90: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0001cda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cdb0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0001cd90: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cde0: 2020 7c0a 7c20 2020 2020 7c74 2020 2d20 |.| |t - │ │ │ │ -0001cdf0: 7420 7420 202b 2074 2074 2020 2020 2020 t t + t t │ │ │ │ -0001ce00: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ +0001cde0: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ +0001cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ce00: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ 0001ce10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ce20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce30: 2020 7c0a 7c20 2020 2020 7c20 3620 2020 |.| | 6 │ │ │ │ -0001ce40: 2034 2035 2020 2020 3220 3720 2020 2020 4 5 2 7 │ │ │ │ +0001ce30: 2020 7c0a 7c20 2020 2020 7c20 2020 2020 |.| | │ │ │ │ +0001ce40: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001ce50: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ 0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ce70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce80: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0001ce90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cea0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0001ce80: 2020 7c0a 7c6f 3720 3d20 7c74 2020 2b20 |.|o7 = |t + │ │ │ │ +0001ce90: 7420 202d 2074 2074 2020 2b20 7420 7420 t - t t + t t │ │ │ │ +0001cea0: 202d 2074 2074 207c 2020 2020 2020 2020 - t t | │ │ │ │ 0001ceb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ced0: 2020 7c0a 7c20 2020 2020 7c20 2020 2020 |.| | │ │ │ │ -0001cee0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -0001cef0: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ +0001ced0: 2020 7c0a 7c20 2020 2020 7c20 3920 2020 |.| | 9 │ │ │ │ +0001cee0: 2034 2020 2020 3120 3720 2020 2035 2037 4 1 7 5 7 │ │ │ │ +0001cef0: 2020 2020 3420 387c 2020 2020 2020 2020 4 8| │ │ │ │ 0001cf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf20: 2020 7c0a 7c20 2020 2020 7c74 2020 2b20 |.| |t + │ │ │ │ -0001cf30: 7420 7420 202d 2074 2074 2020 2b20 7420 t t - t t + t │ │ │ │ -0001cf40: 202d 2074 2074 207c 2020 2020 2020 2020 - t t | │ │ │ │ +0001cf20: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ +0001cf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001cf40: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ 0001cf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf70: 2020 7c0a 7c20 2020 2020 7c20 3320 2020 |.| | 3 │ │ │ │ -0001cf80: 2032 2034 2020 2020 3120 3520 2020 2035 2 4 1 5 5 │ │ │ │ -0001cf90: 2020 2020 3220 387c 2020 2020 2020 2020 2 8| │ │ │ │ +0001cf70: 2020 7c0a 7c20 2020 2020 7c74 2020 2d20 |.| |t - │ │ │ │ +0001cf80: 7420 7420 202b 2074 2074 2020 2020 2020 t t + t t │ │ │ │ +0001cf90: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ 0001cfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cfc0: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -0001cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cfe0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0001cfc0: 2020 7c0a 7c20 2020 2020 7c20 3620 2020 |.| | 6 │ │ │ │ +0001cfd0: 2034 2035 2020 2020 3220 3720 2020 2020 4 5 2 7 │ │ │ │ +0001cfe0: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ 0001cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d010: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001d010: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ 0001d020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d060: 2d2d 2b0a 0a4e 6f74 6963 6520 7468 6174 --+..Notice that │ │ │ │ -0001d070: 2074 6865 2070 6172 616d 6574 6572 7320 the parameters │ │ │ │ -0001d080: 2474 5f33 242c 2024 745f 3624 2c20 616e $t_3$, $t_6$, an │ │ │ │ -0001d090: 6420 2474 5f39 2420 6172 6520 636c 6561 d $t_9$ are clea │ │ │ │ -0001d0a0: 726c 7920 656c 696d 696e 6162 6c65 2e0a rly eliminable.. │ │ │ │ -0001d0b0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -0001d0c0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6772 ==.. * *note gr │ │ │ │ -0001d0d0: 6f65 626e 6572 4661 6d69 6c79 3a20 6772 oebnerFamily: gr │ │ │ │ -0001d0e0: 6f65 626e 6572 4661 6d69 6c79 2c20 2d2d oebnerFamily, -- │ │ │ │ -0001d0f0: 2063 6f6d 7075 7465 7320 6661 6d69 6c69 computes famili │ │ │ │ -0001d100: 6573 206f 6620 6964 6561 6c73 2077 6974 es of ideals wit │ │ │ │ -0001d110: 6820 610a 2020 2020 7370 6563 6966 6965 h a. specifie │ │ │ │ -0001d120: 6420 696e 6974 6961 6c20 6964 6561 6c0a d initial ideal. │ │ │ │ -0001d130: 0a57 6179 7320 746f 2075 7365 2067 726f .Ways to use gro │ │ │ │ -0001d140: 6562 6e65 7253 7472 6174 756d 3a0a 3d3d ebnerStratum:.== │ │ │ │ -0001d150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d160: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0001d170: 2267 726f 6562 6e65 7253 7472 6174 756d "groebnerStratum │ │ │ │ -0001d180: 2849 6465 616c 2922 0a0a 466f 7220 7468 (Ideal)"..For th │ │ │ │ -0001d190: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0001d1a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0001d1b0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0001d1c0: 6520 6772 6f65 626e 6572 5374 7261 7475 e groebnerStratu │ │ │ │ -0001d1d0: 6d3a 2067 726f 6562 6e65 7253 7472 6174 m: groebnerStrat │ │ │ │ -0001d1e0: 756d 2c20 6973 2061 202a 6e6f 7465 206d um, is a *note m │ │ │ │ -0001d1f0: 6574 686f 6420 6675 6e63 7469 6f6e 0a77 ethod function.w │ │ │ │ -0001d200: 6974 6820 6f70 7469 6f6e 733a 2028 4d61 ith options: (Ma │ │ │ │ -0001d210: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -0001d220: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ -0001d230: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ -0001d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d280: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -0001d290: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -0001d2a0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -0001d2b0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -0001d2c0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -0001d2d0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ -0001d2e0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -0001d2f0: 4772 6f65 626e 6572 5374 7261 7461 2e6d GroebnerStrata.m │ │ │ │ -0001d300: 323a 3832 333a 302e 0a1f 0a46 696c 653a 2:823:0....File: │ │ │ │ -0001d310: 2047 726f 6562 6e65 7253 7472 6174 612e GroebnerStrata. │ │ │ │ -0001d320: 696e 666f 2c20 4e6f 6465 3a20 6c69 6e65 info, Node: line │ │ │ │ -0001d330: 6172 5061 7274 2c20 4e65 7874 3a20 4d69 arPart, Next: Mi │ │ │ │ -0001d340: 6e69 6d61 6c69 7a65 2c20 5072 6576 3a20 nimalize, Prev: │ │ │ │ -0001d350: 6772 6f65 626e 6572 5374 7261 7475 6d2c groebnerStratum, │ │ │ │ -0001d360: 2055 703a 2054 6f70 0a0a 6c69 6e65 6172 Up: Top..linear │ │ │ │ -0001d370: 5061 7274 202d 2d20 7265 7475 726e 7320 Part -- returns │ │ │ │ -0001d380: 6c69 6e65 6172 2070 6172 7420 6f66 2061 linear part of a │ │ │ │ -0001d390: 2070 6f6c 796e 6f6d 6961 6c0a 2a2a 2a2a polynomial.**** │ │ │ │ -0001d3a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d3b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d3c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -0001d3d0: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -0001d3e0: 2020 206c 203d 206c 696e 6561 7250 6172 l = linearPar │ │ │ │ -0001d3f0: 7428 6629 0a20 202a 2049 6e70 7574 733a t(f). * Inputs: │ │ │ │ -0001d400: 0a20 2020 2020 202a 2066 2c20 6120 2a6e . * f, a *n │ │ │ │ -0001d410: 6f74 6520 7269 6e67 2065 6c65 6d65 6e74 ote ring element │ │ │ │ -0001d420: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0001d430: 5269 6e67 456c 656d 656e 742c 2c20 6620 RingElement,, f │ │ │ │ -0001d440: 6973 2061 2070 6f6c 796e 6f6d 6961 6c0a is a polynomial. │ │ │ │ -0001d450: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -0001d460: 2020 202a 206c 2c20 6120 2a6e 6f74 6520 * l, a *note │ │ │ │ -0001d470: 7269 6e67 2065 6c65 6d65 6e74 3a20 284d ring element: (M │ │ │ │ -0001d480: 6163 6175 6c61 7932 446f 6329 5269 6e67 acaulay2Doc)Ring │ │ │ │ -0001d490: 456c 656d 656e 742c 2c20 6c20 6973 2074 Element,, l is t │ │ │ │ -0001d4a0: 6865 2073 756d 206f 660a 2020 2020 2020 he sum of. │ │ │ │ -0001d4b0: 2020 7468 6520 7465 726d 7320 6f66 2066 the terms of f │ │ │ │ -0001d4c0: 2077 686f 7365 206d 6f6e 6f6d 6961 6c20 whose monomial │ │ │ │ -0001d4d0: 6973 2061 2076 6172 6961 626c 650a 0a44 is a variable..D │ │ │ │ -0001d4e0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0001d4f0: 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d ======..+------- │ │ │ │ -0001d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d520: 2d2b 0a7c 6931 203a 2052 203d 205a 5a2f -+.|i1 : R = ZZ/ │ │ │ │ -0001d530: 3332 3030 335b 612c 622c 635d 3b20 2020 32003[a,b,c]; │ │ │ │ -0001d540: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0001d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d570: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2066 -------+.|i2 : f │ │ │ │ -0001d580: 203d 2061 5e32 202b 2033 2a62 202b 2035 = a^2 + 3*b + 5 │ │ │ │ -0001d590: 2a61 2a62 2a63 202b 2032 2a61 202b 625e *a*b*c + 2*a +b^ │ │ │ │ -0001d5a0: 323b 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2;|.+----------- │ │ │ │ -0001d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0001d5d0: 6933 203a 206c 696e 6561 7250 6172 7420 i3 : linearPart │ │ │ │ -0001d5e0: 6620 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ -0001d5f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001d600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d620: 2020 207c 0a7c 6f33 203d 2032 6120 2b20 |.|o3 = 2a + │ │ │ │ -0001d630: 3362 2020 2020 2020 2020 2020 2020 2020 3b │ │ │ │ -0001d640: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d650: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0001d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d670: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ -0001d680: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -0001d690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d6a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0001d6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001d6d0: 0a0a 4966 2077 6520 6861 7665 2061 2070 ..If we have a p │ │ │ │ -0001d6e0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 206f olynomial ring o │ │ │ │ -0001d6f0: 7665 7220 616e 6f74 6865 7220 706f 6c79 ver another poly │ │ │ │ -0001d700: 6e6f 6d69 616c 2072 696e 672c 2074 6865 nomial ring, the │ │ │ │ -0001d710: 2063 6f6d 6d61 6e64 206f 6e6c 790a 6c6f command only.lo │ │ │ │ -0001d720: 6f6b 7320 6174 2074 6865 2065 7870 6f6e oks at the expon │ │ │ │ -0001d730: 656e 7473 206f 6e20 7468 6520 6e65 7720 ents on the new │ │ │ │ -0001d740: 7365 7420 6f66 2076 6172 6961 626c 6573 set of variables │ │ │ │ -0001d750: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -0001d760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d770: 2d2d 2d2d 2b0a 7c69 3420 3a20 5320 3d20 ----+.|i4 : S = │ │ │ │ -0001d780: 5a5a 2f33 3230 3033 5b61 2c62 2c63 5d5b ZZ/32003[a,b,c][ │ │ │ │ -0001d790: 782c 792c 7a5d 3b7c 0a2b 2d2d 2d2d 2d2d x,y,z];|.+------ │ │ │ │ -0001d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -0001d7c0: 3a20 6720 3d20 612a 622a 7820 2b20 332a : g = a*b*x + 3* │ │ │ │ -0001d7d0: 7920 2b20 6120 2b20 625e 3220 207c 0a7c y + a + b^2 |.| │ │ │ │ -0001d7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d030: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0001d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d060: 2020 7c0a 7c20 2020 2020 7c20 2020 2020 |.| | │ │ │ │ +0001d070: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +0001d080: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ +0001d090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d0b0: 2020 7c0a 7c20 2020 2020 7c74 2020 2b20 |.| |t + │ │ │ │ +0001d0c0: 7420 7420 202d 2074 2074 2020 2b20 7420 t t - t t + t │ │ │ │ +0001d0d0: 202d 2074 2074 207c 2020 2020 2020 2020 - t t | │ │ │ │ +0001d0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d100: 2020 7c0a 7c20 2020 2020 7c20 3320 2020 |.| | 3 │ │ │ │ +0001d110: 2032 2034 2020 2020 3120 3520 2020 2035 2 4 1 5 5 │ │ │ │ +0001d120: 2020 2020 3220 387c 2020 2020 2020 2020 2 8| │ │ │ │ +0001d130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d150: 2020 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ +0001d160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d170: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0001d180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d1a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001d1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d1f0: 2d2d 2b0a 0a4e 6f74 6963 6520 7468 6174 --+..Notice that │ │ │ │ +0001d200: 2074 6865 2070 6172 616d 6574 6572 7320 the parameters │ │ │ │ +0001d210: 2474 5f33 242c 2024 745f 3624 2c20 616e $t_3$, $t_6$, an │ │ │ │ +0001d220: 6420 2474 5f39 2420 6172 6520 636c 6561 d $t_9$ are clea │ │ │ │ +0001d230: 726c 7920 656c 696d 696e 6162 6c65 2e0a rly eliminable.. │ │ │ │ +0001d240: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +0001d250: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6772 ==.. * *note gr │ │ │ │ +0001d260: 6f65 626e 6572 4661 6d69 6c79 3a20 6772 oebnerFamily: gr │ │ │ │ +0001d270: 6f65 626e 6572 4661 6d69 6c79 2c20 2d2d oebnerFamily, -- │ │ │ │ +0001d280: 2063 6f6d 7075 7465 7320 6661 6d69 6c69 computes famili │ │ │ │ +0001d290: 6573 206f 6620 6964 6561 6c73 2077 6974 es of ideals wit │ │ │ │ +0001d2a0: 6820 610a 2020 2020 7370 6563 6966 6965 h a. specifie │ │ │ │ +0001d2b0: 6420 696e 6974 6961 6c20 6964 6561 6c0a d initial ideal. │ │ │ │ +0001d2c0: 0a57 6179 7320 746f 2075 7365 2067 726f .Ways to use gro │ │ │ │ +0001d2d0: 6562 6e65 7253 7472 6174 756d 3a0a 3d3d ebnerStratum:.== │ │ │ │ +0001d2e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001d2f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0001d300: 2267 726f 6562 6e65 7253 7472 6174 756d "groebnerStratum │ │ │ │ +0001d310: 2849 6465 616c 2922 0a0a 466f 7220 7468 (Ideal)"..For th │ │ │ │ +0001d320: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0001d330: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0001d340: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0001d350: 6520 6772 6f65 626e 6572 5374 7261 7475 e groebnerStratu │ │ │ │ +0001d360: 6d3a 2067 726f 6562 6e65 7253 7472 6174 m: groebnerStrat │ │ │ │ +0001d370: 756d 2c20 6973 2061 202a 6e6f 7465 206d um, is a *note m │ │ │ │ +0001d380: 6574 686f 6420 6675 6e63 7469 6f6e 0a77 ethod function.w │ │ │ │ +0001d390: 6974 6820 6f70 7469 6f6e 733a 2028 4d61 ith options: (Ma │ │ │ │ +0001d3a0: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ +0001d3b0: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ +0001d3c0: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ +0001d3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d410: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +0001d420: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +0001d430: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +0001d440: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +0001d450: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +0001d460: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +0001d470: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +0001d480: 4772 6f65 626e 6572 5374 7261 7461 2e6d GroebnerStrata.m │ │ │ │ +0001d490: 323a 3832 333a 302e 0a1f 0a46 696c 653a 2:823:0....File: │ │ │ │ +0001d4a0: 2047 726f 6562 6e65 7253 7472 6174 612e GroebnerStrata. │ │ │ │ +0001d4b0: 696e 666f 2c20 4e6f 6465 3a20 6c69 6e65 info, Node: line │ │ │ │ +0001d4c0: 6172 5061 7274 2c20 4e65 7874 3a20 4d69 arPart, Next: Mi │ │ │ │ +0001d4d0: 6e69 6d61 6c69 7a65 2c20 5072 6576 3a20 nimalize, Prev: │ │ │ │ +0001d4e0: 6772 6f65 626e 6572 5374 7261 7475 6d2c groebnerStratum, │ │ │ │ +0001d4f0: 2055 703a 2054 6f70 0a0a 6c69 6e65 6172 Up: Top..linear │ │ │ │ +0001d500: 5061 7274 202d 2d20 7265 7475 726e 7320 Part -- returns │ │ │ │ +0001d510: 6c69 6e65 6172 2070 6172 7420 6f66 2061 linear part of a │ │ │ │ +0001d520: 2070 6f6c 796e 6f6d 6961 6c0a 2a2a 2a2a polynomial.**** │ │ │ │ +0001d530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001d540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001d550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ +0001d560: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ +0001d570: 2020 206c 203d 206c 696e 6561 7250 6172 l = linearPar │ │ │ │ +0001d580: 7428 6629 0a20 202a 2049 6e70 7574 733a t(f). * Inputs: │ │ │ │ +0001d590: 0a20 2020 2020 202a 2066 2c20 6120 2a6e . * f, a *n │ │ │ │ +0001d5a0: 6f74 6520 7269 6e67 2065 6c65 6d65 6e74 ote ring element │ │ │ │ +0001d5b0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0001d5c0: 5269 6e67 456c 656d 656e 742c 2c20 6620 RingElement,, f │ │ │ │ +0001d5d0: 6973 2061 2070 6f6c 796e 6f6d 6961 6c0a is a polynomial. │ │ │ │ +0001d5e0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +0001d5f0: 2020 202a 206c 2c20 6120 2a6e 6f74 6520 * l, a *note │ │ │ │ +0001d600: 7269 6e67 2065 6c65 6d65 6e74 3a20 284d ring element: (M │ │ │ │ +0001d610: 6163 6175 6c61 7932 446f 6329 5269 6e67 acaulay2Doc)Ring │ │ │ │ +0001d620: 456c 656d 656e 742c 2c20 6c20 6973 2074 Element,, l is t │ │ │ │ +0001d630: 6865 2073 756d 206f 660a 2020 2020 2020 he sum of. │ │ │ │ +0001d640: 2020 7468 6520 7465 726d 7320 6f66 2066 the terms of f │ │ │ │ +0001d650: 2077 686f 7365 206d 6f6e 6f6d 6961 6c20 whose monomial │ │ │ │ +0001d660: 6973 2061 2076 6172 6961 626c 650a 0a44 is a variable..D │ │ │ │ +0001d670: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0001d680: 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d ======..+------- │ │ │ │ +0001d690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d6b0: 2d2b 0a7c 6931 203a 2052 203d 205a 5a2f -+.|i1 : R = ZZ/ │ │ │ │ +0001d6c0: 3332 3030 335b 612c 622c 635d 3b20 2020 32003[a,b,c]; │ │ │ │ +0001d6d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001d6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d700: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2066 -------+.|i2 : f │ │ │ │ +0001d710: 203d 2061 5e32 202b 2033 2a62 202b 2035 = a^2 + 3*b + 5 │ │ │ │ +0001d720: 2a61 2a62 2a63 202b 2032 2a61 202b 625e *a*b*c + 2*a +b^ │ │ │ │ +0001d730: 323b 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2;|.+----------- │ │ │ │ +0001d740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001d760: 6933 203a 206c 696e 6561 7250 6172 7420 i3 : linearPart │ │ │ │ +0001d770: 6620 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ +0001d780: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d7b0: 2020 207c 0a7c 6f33 203d 2032 6120 2b20 |.|o3 = 2a + │ │ │ │ +0001d7c0: 3362 2020 2020 2020 2020 2020 2020 2020 3b │ │ │ │ +0001d7d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001d7e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0001d7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d800: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001d810: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0001d820: 2020 207c 0a7c 6f35 203d 2061 2a62 2a78 |.|o5 = a*b*x │ │ │ │ -0001d830: 202b 2033 7920 2b20 6220 202b 2061 2020 + 3y + b + a │ │ │ │ -0001d840: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d860: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ -0001d870: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ -0001d880: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0001d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001d8b0: 0a7c 6936 203a 206c 696e 6561 7250 6172 .|i6 : linearPar │ │ │ │ -0001d8c0: 7420 6720 2020 2020 2020 2020 2020 2020 t g │ │ │ │ -0001d8d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d8f0: 2020 2020 207c 0a7c 6f36 203d 2061 2a62 |.|o6 = a*b │ │ │ │ -0001d900: 2a78 202b 2033 7920 2020 2020 2020 2020 *x + 3y │ │ │ │ -0001d910: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001d920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d930: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0001d940: 203a 2053 2020 2020 2020 2020 2020 2020 : S │ │ │ │ -0001d950: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d960: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0001d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d980: 2d2b 0a0a 4e6f 7465 2074 6861 7420 696e -+..Note that in │ │ │ │ -0001d990: 2074 6865 2070 7265 7669 6f75 7320 6578 the previous ex │ │ │ │ -0001d9a0: 616d 706c 652c 206c 696e 6561 7250 6172 ample, linearPar │ │ │ │ -0001d9b0: 7420 6720 646f 6573 206e 6f74 2069 6e63 t g does not inc │ │ │ │ -0001d9c0: 6c75 6465 2061 2062 7574 2064 6f65 730a lude a but does. │ │ │ │ -0001d9d0: 696e 636c 7564 6520 612a 622a 780a 0a46 include a*b*x..F │ │ │ │ -0001d9e0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -0001d9f0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -0001da00: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -0001da10: 202a 6e6f 7465 206c 696e 6561 7250 6172 *note linearPar │ │ │ │ -0001da20: 743a 206c 696e 6561 7250 6172 742c 2069 t: linearPart, i │ │ │ │ -0001da30: 7320 6120 2a6e 6f74 6520 6675 6e63 7469 s a *note functi │ │ │ │ -0001da40: 6f6e 2063 6c6f 7375 7265 3a0a 284d 6163 on closure:.(Mac │ │ │ │ -0001da50: 6175 6c61 7932 446f 6329 4675 6e63 7469 aulay2Doc)Functi │ │ │ │ -0001da60: 6f6e 436c 6f73 7572 652c 2e0a 0a2d 2d2d onClosure,...--- │ │ │ │ -0001da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -0001dac0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -0001dad0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -0001dae0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -0001daf0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -0001db00: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -0001db10: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -0001db20: 6765 732f 0a47 726f 6562 6e65 7253 7472 ges/.GroebnerStr │ │ │ │ -0001db30: 6174 612e 6d32 3a38 3532 3a30 2e0a 1f0a ata.m2:852:0.... │ │ │ │ -0001db40: 4669 6c65 3a20 4772 6f65 626e 6572 5374 File: GroebnerSt │ │ │ │ -0001db50: 7261 7461 2e69 6e66 6f2c 204e 6f64 653a rata.info, Node: │ │ │ │ -0001db60: 204d 696e 696d 616c 697a 652c 204e 6578 Minimalize, Nex │ │ │ │ -0001db70: 743a 206e 6f6e 6d69 6e69 6d61 6c4d 6170 t: nonminimalMap │ │ │ │ -0001db80: 732c 2050 7265 763a 206c 696e 6561 7250 s, Prev: linearP │ │ │ │ -0001db90: 6172 742c 2055 703a 2054 6f70 0a0a 4d69 art, Up: Top..Mi │ │ │ │ -0001dba0: 6e69 6d61 6c69 7a65 202d 2d20 626f 6f6c nimalize -- bool │ │ │ │ -0001dbb0: 6561 6e20 6f70 7469 6f6e 206f 6620 6772 ean option of gr │ │ │ │ -0001dbc0: 6f65 626e 6572 5374 7261 7475 6d20 2863 oebnerStratum (c │ │ │ │ -0001dbd0: 7572 7265 6e74 6c79 2061 206e 6f2d 6f70 urrently a no-op │ │ │ │ -0001dbe0: 290a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ).************** │ │ │ │ -0001dbf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001dc00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001dc10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001dc20: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ -0001dc30: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -0001dc40: 5468 6973 2069 7320 616e 206f 7074 696f This is an optio │ │ │ │ -0001dc50: 6e61 6c20 696e 7075 7420 666f 7220 7468 nal input for th │ │ │ │ -0001dc60: 6520 2a6e 6f74 6520 6772 6f65 626e 6572 e *note groebner │ │ │ │ -0001dc70: 5374 7261 7475 6d3a 2067 726f 6562 6e65 Stratum: groebne │ │ │ │ -0001dc80: 7253 7472 6174 756d 2c0a 6675 6e63 7469 rStratum,.functi │ │ │ │ -0001dc90: 6f6e 2e20 2049 7420 6973 2069 6e74 656e on. It is inten │ │ │ │ -0001dca0: 6465 6420 746f 2063 6f6e 7472 6f6c 2077 ded to control w │ │ │ │ -0001dcb0: 6865 7468 6572 2065 6c69 6d69 6e61 626c hether eliminabl │ │ │ │ -0001dcc0: 6520 7061 7261 6d65 7465 7273 2061 7265 e parameters are │ │ │ │ -0001dcd0: 0a65 6c69 6d69 6e61 7465 6420 746f 206f .eliminated to o │ │ │ │ -0001dce0: 6274 6169 6e20 7468 6520 4772 6f65 626e btain the Groebn │ │ │ │ -0001dcf0: 6572 2073 7472 6174 756d 2069 6e20 6120 er stratum in a │ │ │ │ -0001dd00: 736d 616c 6c65 7220 7269 6e67 2e0a 0a41 smaller ring...A │ │ │ │ -0001dd10: 7420 7072 6573 656e 7420 7468 6973 206f t present this o │ │ │ │ -0001dd20: 7074 696f 6e20 6973 206e 6f74 2079 6574 ption is not yet │ │ │ │ -0001dd30: 2069 6d70 6c65 6d65 6e74 6564 3a20 7061 implemented: pa │ │ │ │ -0001dd40: 7373 696e 6720 4d69 6e69 6d61 6c69 7a65 ssing Minimalize │ │ │ │ -0001dd50: 203d 3e20 7472 7565 0a70 7269 6e74 7320 => true.prints │ │ │ │ -0001dd60: 6120 7761 726e 696e 6720 616e 6420 6f74 a warning and ot │ │ │ │ -0001dd70: 6865 7277 6973 6520 6265 6861 7665 7320 herwise behaves │ │ │ │ -0001dd80: 6578 6163 746c 7920 6c69 6b65 204d 696e exactly like Min │ │ │ │ -0001dd90: 696d 616c 697a 6520 3d3e 2066 616c 7365 imalize => false │ │ │ │ -0001dda0: 2e20 2053 6565 0a74 6865 2073 6f75 7263 . See.the sourc │ │ │ │ -0001ddb0: 652d 6c65 7665 6c20 6d69 6e69 6d61 6c69 e-level minimali │ │ │ │ -0001ddc0: 7a65 4661 6d69 6c79 2064 7261 6674 2069 zeFamily draft i │ │ │ │ -0001ddd0: 6e20 4772 6f65 626e 6572 5374 7261 7461 n GroebnerStrata │ │ │ │ -0001dde0: 2e6d 3220 666f 7220 7468 6520 6375 7272 .m2 for the curr │ │ │ │ -0001ddf0: 656e 740a 626c 6f63 6b65 7220 2874 6865 ent.blocker (the │ │ │ │ -0001de00: 2070 6172 616d 6574 6572 2072 696e 6720 parameter ring │ │ │ │ -0001de10: 6275 696c 7420 6279 202a 6e6f 7465 2067 built by *note g │ │ │ │ -0001de20: 726f 6562 6e65 7246 616d 696c 793a 2067 roebnerFamily: g │ │ │ │ -0001de30: 726f 6562 6e65 7246 616d 696c 792c 2068 roebnerFamily, h │ │ │ │ -0001de40: 6173 0a61 2073 696e 676c 6520 4752 6576 as.a single GRev │ │ │ │ -0001de50: 4c65 7820 626c 6f63 6b2c 2062 7574 2074 Lex block, but t │ │ │ │ -0001de60: 6865 2070 6172 616d 6574 6572 2d65 6c69 he parameter-eli │ │ │ │ -0001de70: 6d69 6e61 7469 6f6e 206c 6f67 6963 206e mination logic n │ │ │ │ -0001de80: 6565 6473 2061 204c 6578 0a73 7562 626c eeds a Lex.subbl │ │ │ │ -0001de90: 6f63 6b20 746f 2069 6465 6e74 6966 7920 ock to identify │ │ │ │ -0001dea0: 656c 696d 696e 6162 6c65 2070 6172 616d eliminable param │ │ │ │ -0001deb0: 6574 6572 7329 2e0a 0a53 6565 2061 6c73 eters)...See als │ │ │ │ -0001dec0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ -0001ded0: 2a6e 6f74 6520 6772 6f65 626e 6572 5374 *note groebnerSt │ │ │ │ -0001dee0: 7261 7475 6d3a 2067 726f 6562 6e65 7253 ratum: groebnerS │ │ │ │ -0001def0: 7472 6174 756d 2c20 2d2d 2063 6f6d 7075 tratum, -- compu │ │ │ │ -0001df00: 7465 2074 6865 2069 6465 616c 2077 6865 te the ideal whe │ │ │ │ -0001df10: 7265 2061 2067 6976 656e 0a20 2020 2069 re a given. i │ │ │ │ -0001df20: 7320 6120 4772 6f65 626e 6572 2062 6173 s a Groebner bas │ │ │ │ -0001df30: 6973 0a0a 4675 6e63 7469 6f6e 7320 7769 is..Functions wi │ │ │ │ -0001df40: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ -0001df50: 6d65 6e74 206e 616d 6564 204d 696e 696d ment named Minim │ │ │ │ -0001df60: 616c 697a 653a 0a3d 3d3d 3d3d 3d3d 3d3d alize:.========= │ │ │ │ -0001df70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001df80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001df90: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ -0001dfa0: 6772 6f65 626e 6572 5374 7261 7475 6d28 groebnerStratum( │ │ │ │ -0001dfb0: 2e2e 2e2c 4d69 6e69 6d61 6c69 7a65 3d3e ...,Minimalize=> │ │ │ │ -0001dfc0: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f ...)" -- see *no │ │ │ │ -0001dfd0: 7465 2067 726f 6562 6e65 7253 7472 6174 te groebnerStrat │ │ │ │ -0001dfe0: 756d 3a0a 2020 2020 6772 6f65 626e 6572 um:. groebner │ │ │ │ -0001dff0: 5374 7261 7475 6d2c 202d 2d20 636f 6d70 Stratum, -- comp │ │ │ │ -0001e000: 7574 6520 7468 6520 6964 6561 6c20 7768 ute the ideal wh │ │ │ │ -0001e010: 6572 6520 6120 6769 7665 6e20 6973 2061 ere a given is a │ │ │ │ -0001e020: 2047 726f 6562 6e65 7220 6261 7369 730a Groebner basis. │ │ │ │ -0001e030: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -0001e040: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -0001e050: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -0001e060: 6374 202a 6e6f 7465 204d 696e 696d 616c ct *note Minimal │ │ │ │ -0001e070: 697a 653a 204d 696e 696d 616c 697a 652c ize: Minimalize, │ │ │ │ -0001e080: 2069 7320 6120 2a6e 6f74 6520 7379 6d62 is a *note symb │ │ │ │ -0001e090: 6f6c 3a0a 284d 6163 6175 6c61 7932 446f ol:.(Macaulay2Do │ │ │ │ -0001e0a0: 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d c)Symbol,...---- │ │ │ │ -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 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0001e100: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0001e110: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0001e120: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0001e130: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0001e140: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ -0001e150: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0001e160: 6573 2f0a 4772 6f65 626e 6572 5374 7261 es/.GroebnerStra │ │ │ │ -0001e170: 7461 2e6d 323a 3130 3835 3a30 2e0a 1f0a ta.m2:1085:0.... │ │ │ │ -0001e180: 4669 6c65 3a20 4772 6f65 626e 6572 5374 File: GroebnerSt │ │ │ │ -0001e190: 7261 7461 2e69 6e66 6f2c 204e 6f64 653a rata.info, Node: │ │ │ │ -0001e1a0: 206e 6f6e 6d69 6e69 6d61 6c4d 6170 732c nonminimalMaps, │ │ │ │ -0001e1b0: 204e 6578 743a 2072 616e 646f 6d50 6f69 Next: randomPoi │ │ │ │ -0001e1c0: 6e74 4f6e 5261 7469 6f6e 616c 5661 7269 ntOnRationalVari │ │ │ │ -0001e1d0: 6574 795f 6c70 4964 6561 6c5f 7270 2c20 ety_lpIdeal_rp, │ │ │ │ -0001e1e0: 5072 6576 3a20 4d69 6e69 6d61 6c69 7a65 Prev: Minimalize │ │ │ │ -0001e1f0: 2c20 5570 3a20 546f 700a 0a6e 6f6e 6d69 , Up: Top..nonmi │ │ │ │ -0001e200: 6e69 6d61 6c4d 6170 7320 2d2d 2066 696e nimalMaps -- fin │ │ │ │ -0001e210: 6420 7468 6520 6465 6772 6565 207a 6572 d the degree zer │ │ │ │ -0001e220: 6f20 6d61 7073 2069 6e20 7468 6520 5363 o maps in the Sc │ │ │ │ -0001e230: 6872 6579 6572 2072 6573 6f6c 7574 696f hreyer resolutio │ │ │ │ -0001e240: 6e20 6f66 2061 6e20 6964 6561 6c0a 2a2a n of an ideal.** │ │ │ │ -0001e250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001e260: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001e270: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001e280: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001e290: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001e2a0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -0001e2b0: 2020 2020 2020 2843 2c20 4829 203d 206e (C, H) = n │ │ │ │ -0001e2c0: 6f6e 6d69 6e69 6d61 6c4d 6170 7320 490a onminimalMaps I. │ │ │ │ -0001e2d0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -0001e2e0: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520 * I, an *note │ │ │ │ -0001e2f0: 6964 6561 6c3a 2028 4d61 6361 756c 6179 ideal: (Macaulay │ │ │ │ -0001e300: 3244 6f63 2949 6465 616c 2c2c 2069 6e20 2Doc)Ideal,, in │ │ │ │ -0001e310: 6120 706f 6c79 6e6f 6d69 616c 2072 696e a polynomial rin │ │ │ │ -0001e320: 6720 2453 2420 6f76 6572 0a20 2020 2020 g $S$ over. │ │ │ │ -0001e330: 2020 2061 2062 6173 6520 6669 656c 6420 a base field │ │ │ │ -0001e340: 6f72 2063 6f65 6666 6963 6965 6e74 2072 or coefficient r │ │ │ │ -0001e350: 696e 6720 2441 242e 2020 5468 6520 6c65 ing $A$. The le │ │ │ │ -0001e360: 6164 2074 6572 6d73 206f 6620 7468 6520 ad terms of the │ │ │ │ -0001e370: 6765 6e65 7261 746f 7273 0a20 2020 2020 generators. │ │ │ │ -0001e380: 2020 206f 6620 2449 2420 7368 6f75 6c64 of $I$ should │ │ │ │ -0001e390: 2062 6520 7468 6520 696e 6974 6961 6c20 be the initial │ │ │ │ -0001e3a0: 6964 6561 6c20 6f66 2024 4924 2c20 616e ideal of $I$, an │ │ │ │ -0001e3b0: 6420 7368 6f75 6c64 2062 6520 6d6f 6e69 d should be moni │ │ │ │ -0001e3c0: 632e 0a20 202a 204f 7574 7075 7473 3a0a c.. * Outputs:. │ │ │ │ -0001e3d0: 2020 2020 2020 2a20 432c 2061 202a 6e6f * C, a *no │ │ │ │ -0001e3e0: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d te complex: (Com │ │ │ │ -0001e3f0: 706c 6578 6573 2943 6f6d 706c 6578 2c2c plexes)Complex,, │ │ │ │ -0001e400: 2041 2063 6f6d 706c 6578 206f 7665 7220 A complex over │ │ │ │ -0001e410: 6120 706f 6c79 6e6f 6d69 616c 0a20 2020 a polynomial. │ │ │ │ -0001e420: 2020 2020 2072 696e 6720 7768 6572 6520 ring where │ │ │ │ -0001e430: 616e 7920 7061 7261 6d65 7465 7273 2069 any parameters i │ │ │ │ -0001e440: 6e20 7468 6520 6261 7365 2072 696e 6720 n the base ring │ │ │ │ -0001e450: 6172 6520 7365 7420 746f 2068 6176 6520 are set to have │ │ │ │ -0001e460: 6465 6772 6565 2030 2c0a 2020 2020 2020 degree 0,. │ │ │ │ -0001e470: 2020 616e 6420 7468 6520 7661 7269 6162 and the variab │ │ │ │ -0001e480: 6c65 7320 6f66 2074 6865 2072 696e 6720 les of the ring │ │ │ │ -0001e490: 6f66 2024 4924 2061 7265 2073 6574 2074 of $I$ are set t │ │ │ │ -0001e4a0: 6f20 6861 7665 2064 6567 7265 6520 6f6e o have degree on │ │ │ │ -0001e4b0: 652e 0a20 2020 2020 202a 2048 2c20 6120 e.. * H, a │ │ │ │ -0001e4c0: 2a6e 6f74 6520 6861 7368 2074 6162 6c65 *note hash table │ │ │ │ -0001e4d0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0001e4e0: 4861 7368 5461 626c 652c 2c20 5768 6f73 HashTable,, Whos │ │ │ │ -0001e4f0: 6520 6b65 7973 2064 6573 6372 6962 650a e keys describe. │ │ │ │ -0001e500: 2020 2020 2020 2020 7768 6963 6820 7375 which su │ │ │ │ -0001e510: 626d 6174 7269 7820 696e 2074 6865 2072 bmatrix in the r │ │ │ │ -0001e520: 6573 6f6c 7574 696f 6e20 7468 6973 2069 esolution this i │ │ │ │ -0001e530: 732c 2061 6e64 2077 686f 7365 2076 616c s, and whose val │ │ │ │ -0001e540: 7565 7320 6172 6520 7468 6f73 650a 2020 ues are those. │ │ │ │ -0001e550: 2020 2020 2020 7375 626d 6174 7269 6365 submatrice │ │ │ │ -0001e560: 7320 2870 6c61 6365 6420 696e 746f 2074 s (placed into t │ │ │ │ -0001e570: 6865 206f 7269 6769 6e61 6c20 636f 6566 he original coef │ │ │ │ -0001e580: 6669 6369 656e 7420 7269 6e67 2024 4124 ficient ring $A$ │ │ │ │ -0001e590: 290a 0a44 6573 6372 6970 7469 6f6e 0a3d )..Description.= │ │ │ │ -0001e5a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -0001e5b0: 5363 6872 6579 6572 2072 6573 6f6c 7574 Schreyer resolut │ │ │ │ -0001e5c0: 696f 6e20 6f66 2024 4924 2028 7768 6963 ion of $I$ (whic │ │ │ │ -0001e5d0: 6820 6973 2067 656e 6572 616c 6c79 206e h is generally n │ │ │ │ -0001e5e0: 6f6e 2d6d 696e 696d 616c 2920 6973 2063 on-minimal) is c │ │ │ │ -0001e5f0: 6f6d 7075 7465 642e 0a54 6865 206e 6f6e omputed..The non │ │ │ │ -0001e600: 6d69 6e69 6d61 6c20 7061 7274 7320 6172 minimal parts ar │ │ │ │ -0001e610: 6520 7468 6520 7375 626d 6174 7269 6365 e the submatrice │ │ │ │ -0001e620: 7320 696e 2074 6869 7320 7265 736f 6c75 s in this resolu │ │ │ │ -0001e630: 7469 6f6e 2077 6869 6368 2064 6f20 6e6f tion which do no │ │ │ │ -0001e640: 740a 696e 766f 6c76 6520 7468 6520 7661 t.involve the va │ │ │ │ -0001e650: 7269 6162 6c65 7320 696e 2024 5324 2e20 riables in $S$. │ │ │ │ -0001e660: 2054 6865 7920 6172 6520 656c 656d 656e They are elemen │ │ │ │ -0001e670: 7473 2069 6e20 7468 6520 6261 7365 2072 ts in the base r │ │ │ │ -0001e680: 696e 6720 2441 242e 2020 466f 720a 696e ing $A$. For.in │ │ │ │ -0001e690: 7374 616e 6365 2c20 4823 285c 656c 6c2c stance, H#(\ell, │ │ │ │ -0001e6a0: 2064 2920 6973 2074 6865 2073 7562 6d61 d) is the subma │ │ │ │ -0001e6b0: 7472 6978 206f 6620 7468 6520 6d61 7472 trix of the matr │ │ │ │ -0001e6c0: 6978 2066 726f 6d20 2443 5f7b 5c65 6c6c ix from $C_{\ell │ │ │ │ -0001e6d0: 2b31 7d20 5c74 6f0a 435f 7b5c 656c 6c7d +1} \to.C_{\ell} │ │ │ │ -0001e6e0: 2420 7365 6e64 696e 6720 6465 6772 6565 $ sending degree │ │ │ │ -0001e6f0: 2024 6424 2074 6f20 6465 6772 6565 2024 $d$ to degree $ │ │ │ │ -0001e700: 6424 2e0a 0a54 6865 2072 616e 6b73 206f d$...The ranks o │ │ │ │ -0001e710: 6620 7468 6573 6520 6d61 7472 6963 6573 f these matrices │ │ │ │ -0001e720: 2066 6f72 2061 2073 7065 6369 6669 6320 for a specific │ │ │ │ -0001e730: 7061 7261 6d65 7465 7220 7661 6c75 6520 parameter value │ │ │ │ -0001e740: 6465 7465 726d 696e 6520 6578 6163 746c determine exactl │ │ │ │ -0001e750: 790a 7468 6520 6d69 6e69 6d61 6c20 4265 y.the minimal Be │ │ │ │ -0001e760: 7474 6920 7461 626c 6520 666f 7220 7468 tti table for th │ │ │ │ -0001e770: 6520 6964 6561 6c20 2449 242c 2065 7661 e ideal $I$, eva │ │ │ │ -0001e780: 6c75 6174 6564 2061 7420 7468 6174 2070 luated at that p │ │ │ │ -0001e790: 6172 616d 6574 6572 2070 6f69 6e74 2e0a arameter point.. │ │ │ │ -0001e7a0: 0a4e 6f77 2066 6f72 206f 7572 2065 7861 .Now for our exa │ │ │ │ -0001e7b0: 6d70 6c65 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d mple...+-------- │ │ │ │ -0001e7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e800: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b20 -----+.|i1 : kk │ │ │ │ -0001e810: 3d20 5a5a 2f31 3031 3b20 2020 2020 2020 = ZZ/101; │ │ │ │ -0001e820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e850: 2020 2020 207c 0a2b 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 2d2d 2d2d ---------------- │ │ │ │ -0001e8a0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 203d -----+.|i2 : S = │ │ │ │ -0001e8b0: 206b 6b5b 612e 2e64 5d3b 2020 2020 2020 kk[a..d]; │ │ │ │ -0001e8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e8f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e940: 2d2d 2d2d 2d2b 0a7c 6933 203a 2046 203d -----+.|i3 : F = │ │ │ │ -0001e950: 2067 726f 6562 6e65 7246 616d 696c 7920 groebnerFamily │ │ │ │ -0001e960: 6964 6561 6c22 6132 2c61 622c 6163 2c62 ideal"a2,ab,ac,b │ │ │ │ -0001e970: 322c 6263 322c 6333 2220 2020 2020 2020 2,bc2,c3" │ │ │ │ -0001e980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e990: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001e9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d800: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ +0001d810: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0001d820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d830: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001d840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001d860: 0a0a 4966 2077 6520 6861 7665 2061 2070 ..If we have a p │ │ │ │ +0001d870: 6f6c 796e 6f6d 6961 6c20 7269 6e67 206f olynomial ring o │ │ │ │ +0001d880: 7665 7220 616e 6f74 6865 7220 706f 6c79 ver another poly │ │ │ │ +0001d890: 6e6f 6d69 616c 2072 696e 672c 2074 6865 nomial ring, the │ │ │ │ +0001d8a0: 2063 6f6d 6d61 6e64 206f 6e6c 790a 6c6f command only.lo │ │ │ │ +0001d8b0: 6f6b 7320 6174 2074 6865 2065 7870 6f6e oks at the expon │ │ │ │ +0001d8c0: 656e 7473 206f 6e20 7468 6520 6e65 7720 ents on the new │ │ │ │ +0001d8d0: 7365 7420 6f66 2076 6172 6961 626c 6573 set of variables │ │ │ │ +0001d8e0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +0001d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d900: 2d2d 2d2d 2b0a 7c69 3420 3a20 5320 3d20 ----+.|i4 : S = │ │ │ │ +0001d910: 5a5a 2f33 3230 3033 5b61 2c62 2c63 5d5b ZZ/32003[a,b,c][ │ │ │ │ +0001d920: 782c 792c 7a5d 3b7c 0a2b 2d2d 2d2d 2d2d x,y,z];|.+------ │ │ │ │ +0001d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +0001d950: 3a20 6720 3d20 612a 622a 7820 2b20 332a : g = a*b*x + 3* │ │ │ │ +0001d960: 7920 2b20 6120 2b20 625e 3220 207c 0a7c y + a + b^2 |.| │ │ │ │ +0001d970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d990: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001d9a0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +0001d9b0: 2020 207c 0a7c 6f35 203d 2061 2a62 2a78 |.|o5 = a*b*x │ │ │ │ +0001d9c0: 202b 2033 7920 2b20 6220 202b 2061 2020 + 3y + b + a │ │ │ │ +0001d9d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001d9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d9f0: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ +0001da00: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +0001da10: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001da40: 0a7c 6936 203a 206c 696e 6561 7250 6172 .|i6 : linearPar │ │ │ │ +0001da50: 7420 6720 2020 2020 2020 2020 2020 2020 t g │ │ │ │ +0001da60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da80: 2020 2020 207c 0a7c 6f36 203d 2061 2a62 |.|o6 = a*b │ │ │ │ +0001da90: 2a78 202b 2033 7920 2020 2020 2020 2020 *x + 3y │ │ │ │ +0001daa0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dac0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0001dad0: 203a 2053 2020 2020 2020 2020 2020 2020 : S │ │ │ │ +0001dae0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001daf0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001db00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001db10: 2d2b 0a0a 4e6f 7465 2074 6861 7420 696e -+..Note that in │ │ │ │ +0001db20: 2074 6865 2070 7265 7669 6f75 7320 6578 the previous ex │ │ │ │ +0001db30: 616d 706c 652c 206c 696e 6561 7250 6172 ample, linearPar │ │ │ │ +0001db40: 7420 6720 646f 6573 206e 6f74 2069 6e63 t g does not inc │ │ │ │ +0001db50: 6c75 6465 2061 2062 7574 2064 6f65 730a lude a but does. │ │ │ │ +0001db60: 696e 636c 7564 6520 612a 622a 780a 0a46 include a*b*x..F │ │ │ │ +0001db70: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0001db80: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0001db90: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0001dba0: 202a 6e6f 7465 206c 696e 6561 7250 6172 *note linearPar │ │ │ │ +0001dbb0: 743a 206c 696e 6561 7250 6172 742c 2069 t: linearPart, i │ │ │ │ +0001dbc0: 7320 6120 2a6e 6f74 6520 6675 6e63 7469 s a *note functi │ │ │ │ +0001dbd0: 6f6e 2063 6c6f 7375 7265 3a0a 284d 6163 on closure:.(Mac │ │ │ │ +0001dbe0: 6175 6c61 7932 446f 6329 4675 6e63 7469 aulay2Doc)Functi │ │ │ │ +0001dbf0: 6f6e 436c 6f73 7572 652c 2e0a 0a2d 2d2d onClosure,...--- │ │ │ │ +0001dc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +0001dc50: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +0001dc60: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +0001dc70: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +0001dc80: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +0001dc90: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ +0001dca0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +0001dcb0: 6765 732f 0a47 726f 6562 6e65 7253 7472 ges/.GroebnerStr │ │ │ │ +0001dcc0: 6174 612e 6d32 3a38 3532 3a30 2e0a 1f0a ata.m2:852:0.... │ │ │ │ +0001dcd0: 4669 6c65 3a20 4772 6f65 626e 6572 5374 File: GroebnerSt │ │ │ │ +0001dce0: 7261 7461 2e69 6e66 6f2c 204e 6f64 653a rata.info, Node: │ │ │ │ +0001dcf0: 204d 696e 696d 616c 697a 652c 204e 6578 Minimalize, Nex │ │ │ │ +0001dd00: 743a 206e 6f6e 6d69 6e69 6d61 6c4d 6170 t: nonminimalMap │ │ │ │ +0001dd10: 732c 2050 7265 763a 206c 696e 6561 7250 s, Prev: linearP │ │ │ │ +0001dd20: 6172 742c 2055 703a 2054 6f70 0a0a 4d69 art, Up: Top..Mi │ │ │ │ +0001dd30: 6e69 6d61 6c69 7a65 202d 2d20 626f 6f6c nimalize -- bool │ │ │ │ +0001dd40: 6561 6e20 6f70 7469 6f6e 206f 6620 6772 ean option of gr │ │ │ │ +0001dd50: 6f65 626e 6572 5374 7261 7475 6d20 2863 oebnerStratum (c │ │ │ │ +0001dd60: 7572 7265 6e74 6c79 2061 206e 6f2d 6f70 urrently a no-op │ │ │ │ +0001dd70: 290a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ).************** │ │ │ │ +0001dd80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001dd90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001dda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001ddb0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ +0001ddc0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0001ddd0: 5468 6973 2069 7320 616e 206f 7074 696f This is an optio │ │ │ │ +0001dde0: 6e61 6c20 696e 7075 7420 666f 7220 7468 nal input for th │ │ │ │ +0001ddf0: 6520 2a6e 6f74 6520 6772 6f65 626e 6572 e *note groebner │ │ │ │ +0001de00: 5374 7261 7475 6d3a 2067 726f 6562 6e65 Stratum: groebne │ │ │ │ +0001de10: 7253 7472 6174 756d 2c0a 6675 6e63 7469 rStratum,.functi │ │ │ │ +0001de20: 6f6e 2e20 2049 7420 6973 2069 6e74 656e on. It is inten │ │ │ │ +0001de30: 6465 6420 746f 2063 6f6e 7472 6f6c 2077 ded to control w │ │ │ │ +0001de40: 6865 7468 6572 2065 6c69 6d69 6e61 626c hether eliminabl │ │ │ │ +0001de50: 6520 7061 7261 6d65 7465 7273 2061 7265 e parameters are │ │ │ │ +0001de60: 0a65 6c69 6d69 6e61 7465 6420 746f 206f .eliminated to o │ │ │ │ +0001de70: 6274 6169 6e20 7468 6520 4772 6f65 626e btain the Groebn │ │ │ │ +0001de80: 6572 2073 7472 6174 756d 2069 6e20 6120 er stratum in a │ │ │ │ +0001de90: 736d 616c 6c65 7220 7269 6e67 2e0a 0a41 smaller ring...A │ │ │ │ +0001dea0: 7420 7072 6573 656e 7420 7468 6973 206f t present this o │ │ │ │ +0001deb0: 7074 696f 6e20 6973 206e 6f74 2079 6574 ption is not yet │ │ │ │ +0001dec0: 2069 6d70 6c65 6d65 6e74 6564 3a20 7061 implemented: pa │ │ │ │ +0001ded0: 7373 696e 6720 4d69 6e69 6d61 6c69 7a65 ssing Minimalize │ │ │ │ +0001dee0: 203d 3e20 7472 7565 0a70 7269 6e74 7320 => true.prints │ │ │ │ +0001def0: 6120 7761 726e 696e 6720 616e 6420 6f74 a warning and ot │ │ │ │ +0001df00: 6865 7277 6973 6520 6265 6861 7665 7320 herwise behaves │ │ │ │ +0001df10: 6578 6163 746c 7920 6c69 6b65 204d 696e exactly like Min │ │ │ │ +0001df20: 696d 616c 697a 6520 3d3e 2066 616c 7365 imalize => false │ │ │ │ +0001df30: 2e20 2053 6565 0a74 6865 2073 6f75 7263 . See.the sourc │ │ │ │ +0001df40: 652d 6c65 7665 6c20 6d69 6e69 6d61 6c69 e-level minimali │ │ │ │ +0001df50: 7a65 4661 6d69 6c79 2064 7261 6674 2069 zeFamily draft i │ │ │ │ +0001df60: 6e20 4772 6f65 626e 6572 5374 7261 7461 n GroebnerStrata │ │ │ │ +0001df70: 2e6d 3220 666f 7220 7468 6520 6375 7272 .m2 for the curr │ │ │ │ +0001df80: 656e 740a 626c 6f63 6b65 7220 2874 6865 ent.blocker (the │ │ │ │ +0001df90: 2070 6172 616d 6574 6572 2072 696e 6720 parameter ring │ │ │ │ +0001dfa0: 6275 696c 7420 6279 202a 6e6f 7465 2067 built by *note g │ │ │ │ +0001dfb0: 726f 6562 6e65 7246 616d 696c 793a 2067 roebnerFamily: g │ │ │ │ +0001dfc0: 726f 6562 6e65 7246 616d 696c 792c 2068 roebnerFamily, h │ │ │ │ +0001dfd0: 6173 0a61 2073 696e 676c 6520 4752 6576 as.a single GRev │ │ │ │ +0001dfe0: 4c65 7820 626c 6f63 6b2c 2062 7574 2074 Lex block, but t │ │ │ │ +0001dff0: 6865 2070 6172 616d 6574 6572 2d65 6c69 he parameter-eli │ │ │ │ +0001e000: 6d69 6e61 7469 6f6e 206c 6f67 6963 206e mination logic n │ │ │ │ +0001e010: 6565 6473 2061 204c 6578 0a73 7562 626c eeds a Lex.subbl │ │ │ │ +0001e020: 6f63 6b20 746f 2069 6465 6e74 6966 7920 ock to identify │ │ │ │ +0001e030: 656c 696d 696e 6162 6c65 2070 6172 616d eliminable param │ │ │ │ +0001e040: 6574 6572 7329 2e0a 0a53 6565 2061 6c73 eters)...See als │ │ │ │ +0001e050: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0001e060: 2a6e 6f74 6520 6772 6f65 626e 6572 5374 *note groebnerSt │ │ │ │ +0001e070: 7261 7475 6d3a 2067 726f 6562 6e65 7253 ratum: groebnerS │ │ │ │ +0001e080: 7472 6174 756d 2c20 2d2d 2063 6f6d 7075 tratum, -- compu │ │ │ │ +0001e090: 7465 2074 6865 2069 6465 616c 2077 6865 te the ideal whe │ │ │ │ +0001e0a0: 7265 2061 2067 6976 656e 0a20 2020 2069 re a given. i │ │ │ │ +0001e0b0: 7320 6120 4772 6f65 626e 6572 2062 6173 s a Groebner bas │ │ │ │ +0001e0c0: 6973 0a0a 4675 6e63 7469 6f6e 7320 7769 is..Functions wi │ │ │ │ +0001e0d0: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ +0001e0e0: 6d65 6e74 206e 616d 6564 204d 696e 696d ment named Minim │ │ │ │ +0001e0f0: 616c 697a 653a 0a3d 3d3d 3d3d 3d3d 3d3d alize:.========= │ │ │ │ +0001e100: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001e110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001e120: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +0001e130: 6772 6f65 626e 6572 5374 7261 7475 6d28 groebnerStratum( │ │ │ │ +0001e140: 2e2e 2e2c 4d69 6e69 6d61 6c69 7a65 3d3e ...,Minimalize=> │ │ │ │ +0001e150: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f ...)" -- see *no │ │ │ │ +0001e160: 7465 2067 726f 6562 6e65 7253 7472 6174 te groebnerStrat │ │ │ │ +0001e170: 756d 3a0a 2020 2020 6772 6f65 626e 6572 um:. groebner │ │ │ │ +0001e180: 5374 7261 7475 6d2c 202d 2d20 636f 6d70 Stratum, -- comp │ │ │ │ +0001e190: 7574 6520 7468 6520 6964 6561 6c20 7768 ute the ideal wh │ │ │ │ +0001e1a0: 6572 6520 6120 6769 7665 6e20 6973 2061 ere a given is a │ │ │ │ +0001e1b0: 2047 726f 6562 6e65 7220 6261 7369 730a Groebner basis. │ │ │ │ +0001e1c0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0001e1d0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0001e1e0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0001e1f0: 6374 202a 6e6f 7465 204d 696e 696d 616c ct *note Minimal │ │ │ │ +0001e200: 697a 653a 204d 696e 696d 616c 697a 652c ize: Minimalize, │ │ │ │ +0001e210: 2069 7320 6120 2a6e 6f74 6520 7379 6d62 is a *note symb │ │ │ │ +0001e220: 6f6c 3a0a 284d 6163 6175 6c61 7932 446f ol:.(Macaulay2Do │ │ │ │ +0001e230: 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d c)Symbol,...---- │ │ │ │ +0001e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e250: 2d2d 2d2d 2d2d 2d2d 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 2d0a 0a54 6865 -----------..The │ │ │ │ +0001e290: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +0001e2a0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +0001e2b0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +0001e2c0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +0001e2d0: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +0001e2e0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +0001e2f0: 6573 2f0a 4772 6f65 626e 6572 5374 7261 es/.GroebnerStra │ │ │ │ +0001e300: 7461 2e6d 323a 3130 3835 3a30 2e0a 1f0a ta.m2:1085:0.... │ │ │ │ +0001e310: 4669 6c65 3a20 4772 6f65 626e 6572 5374 File: GroebnerSt │ │ │ │ +0001e320: 7261 7461 2e69 6e66 6f2c 204e 6f64 653a rata.info, Node: │ │ │ │ +0001e330: 206e 6f6e 6d69 6e69 6d61 6c4d 6170 732c nonminimalMaps, │ │ │ │ +0001e340: 204e 6578 743a 2072 616e 646f 6d50 6f69 Next: randomPoi │ │ │ │ +0001e350: 6e74 4f6e 5261 7469 6f6e 616c 5661 7269 ntOnRationalVari │ │ │ │ +0001e360: 6574 795f 6c70 4964 6561 6c5f 7270 2c20 ety_lpIdeal_rp, │ │ │ │ +0001e370: 5072 6576 3a20 4d69 6e69 6d61 6c69 7a65 Prev: Minimalize │ │ │ │ +0001e380: 2c20 5570 3a20 546f 700a 0a6e 6f6e 6d69 , Up: Top..nonmi │ │ │ │ +0001e390: 6e69 6d61 6c4d 6170 7320 2d2d 2066 696e nimalMaps -- fin │ │ │ │ +0001e3a0: 6420 7468 6520 6465 6772 6565 207a 6572 d the degree zer │ │ │ │ +0001e3b0: 6f20 6d61 7073 2069 6e20 7468 6520 5363 o maps in the Sc │ │ │ │ +0001e3c0: 6872 6579 6572 2072 6573 6f6c 7574 696f hreyer resolutio │ │ │ │ +0001e3d0: 6e20 6f66 2061 6e20 6964 6561 6c0a 2a2a n of an ideal.** │ │ │ │ +0001e3e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001e3f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001e400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001e410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001e420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001e430: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0001e440: 2020 2020 2020 2843 2c20 4829 203d 206e (C, H) = n │ │ │ │ +0001e450: 6f6e 6d69 6e69 6d61 6c4d 6170 7320 490a onminimalMaps I. │ │ │ │ +0001e460: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0001e470: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520 * I, an *note │ │ │ │ +0001e480: 6964 6561 6c3a 2028 4d61 6361 756c 6179 ideal: (Macaulay │ │ │ │ +0001e490: 3244 6f63 2949 6465 616c 2c2c 2069 6e20 2Doc)Ideal,, in │ │ │ │ +0001e4a0: 6120 706f 6c79 6e6f 6d69 616c 2072 696e a polynomial rin │ │ │ │ +0001e4b0: 6720 2453 2420 6f76 6572 0a20 2020 2020 g $S$ over. │ │ │ │ +0001e4c0: 2020 2061 2062 6173 6520 6669 656c 6420 a base field │ │ │ │ +0001e4d0: 6f72 2063 6f65 6666 6963 6965 6e74 2072 or coefficient r │ │ │ │ +0001e4e0: 696e 6720 2441 242e 2020 5468 6520 6c65 ing $A$. The le │ │ │ │ +0001e4f0: 6164 2074 6572 6d73 206f 6620 7468 6520 ad terms of the │ │ │ │ +0001e500: 6765 6e65 7261 746f 7273 0a20 2020 2020 generators. │ │ │ │ +0001e510: 2020 206f 6620 2449 2420 7368 6f75 6c64 of $I$ should │ │ │ │ +0001e520: 2062 6520 7468 6520 696e 6974 6961 6c20 be the initial │ │ │ │ +0001e530: 6964 6561 6c20 6f66 2024 4924 2c20 616e ideal of $I$, an │ │ │ │ +0001e540: 6420 7368 6f75 6c64 2062 6520 6d6f 6e69 d should be moni │ │ │ │ +0001e550: 632e 0a20 202a 204f 7574 7075 7473 3a0a c.. * Outputs:. │ │ │ │ +0001e560: 2020 2020 2020 2a20 432c 2061 202a 6e6f * C, a *no │ │ │ │ +0001e570: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d te complex: (Com │ │ │ │ +0001e580: 706c 6578 6573 2943 6f6d 706c 6578 2c2c plexes)Complex,, │ │ │ │ +0001e590: 2041 2063 6f6d 706c 6578 206f 7665 7220 A complex over │ │ │ │ +0001e5a0: 6120 706f 6c79 6e6f 6d69 616c 0a20 2020 a polynomial. │ │ │ │ +0001e5b0: 2020 2020 2072 696e 6720 7768 6572 6520 ring where │ │ │ │ +0001e5c0: 616e 7920 7061 7261 6d65 7465 7273 2069 any parameters i │ │ │ │ +0001e5d0: 6e20 7468 6520 6261 7365 2072 696e 6720 n the base ring │ │ │ │ +0001e5e0: 6172 6520 7365 7420 746f 2068 6176 6520 are set to have │ │ │ │ +0001e5f0: 6465 6772 6565 2030 2c0a 2020 2020 2020 degree 0,. │ │ │ │ +0001e600: 2020 616e 6420 7468 6520 7661 7269 6162 and the variab │ │ │ │ +0001e610: 6c65 7320 6f66 2074 6865 2072 696e 6720 les of the ring │ │ │ │ +0001e620: 6f66 2024 4924 2061 7265 2073 6574 2074 of $I$ are set t │ │ │ │ +0001e630: 6f20 6861 7665 2064 6567 7265 6520 6f6e o have degree on │ │ │ │ +0001e640: 652e 0a20 2020 2020 202a 2048 2c20 6120 e.. * H, a │ │ │ │ +0001e650: 2a6e 6f74 6520 6861 7368 2074 6162 6c65 *note hash table │ │ │ │ +0001e660: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0001e670: 4861 7368 5461 626c 652c 2c20 5768 6f73 HashTable,, Whos │ │ │ │ +0001e680: 6520 6b65 7973 2064 6573 6372 6962 650a e keys describe. │ │ │ │ +0001e690: 2020 2020 2020 2020 7768 6963 6820 7375 which su │ │ │ │ +0001e6a0: 626d 6174 7269 7820 696e 2074 6865 2072 bmatrix in the r │ │ │ │ +0001e6b0: 6573 6f6c 7574 696f 6e20 7468 6973 2069 esolution this i │ │ │ │ +0001e6c0: 732c 2061 6e64 2077 686f 7365 2076 616c s, and whose val │ │ │ │ +0001e6d0: 7565 7320 6172 6520 7468 6f73 650a 2020 ues are those. │ │ │ │ +0001e6e0: 2020 2020 2020 7375 626d 6174 7269 6365 submatrice │ │ │ │ +0001e6f0: 7320 2870 6c61 6365 6420 696e 746f 2074 s (placed into t │ │ │ │ +0001e700: 6865 206f 7269 6769 6e61 6c20 636f 6566 he original coef │ │ │ │ +0001e710: 6669 6369 656e 7420 7269 6e67 2024 4124 ficient ring $A$ │ │ │ │ +0001e720: 290a 0a44 6573 6372 6970 7469 6f6e 0a3d )..Description.= │ │ │ │ +0001e730: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0001e740: 5363 6872 6579 6572 2072 6573 6f6c 7574 Schreyer resolut │ │ │ │ +0001e750: 696f 6e20 6f66 2024 4924 2028 7768 6963 ion of $I$ (whic │ │ │ │ +0001e760: 6820 6973 2067 656e 6572 616c 6c79 206e h is generally n │ │ │ │ +0001e770: 6f6e 2d6d 696e 696d 616c 2920 6973 2063 on-minimal) is c │ │ │ │ +0001e780: 6f6d 7075 7465 642e 0a54 6865 206e 6f6e omputed..The non │ │ │ │ +0001e790: 6d69 6e69 6d61 6c20 7061 7274 7320 6172 minimal parts ar │ │ │ │ +0001e7a0: 6520 7468 6520 7375 626d 6174 7269 6365 e the submatrice │ │ │ │ +0001e7b0: 7320 696e 2074 6869 7320 7265 736f 6c75 s in this resolu │ │ │ │ +0001e7c0: 7469 6f6e 2077 6869 6368 2064 6f20 6e6f tion which do no │ │ │ │ +0001e7d0: 740a 696e 766f 6c76 6520 7468 6520 7661 t.involve the va │ │ │ │ +0001e7e0: 7269 6162 6c65 7320 696e 2024 5324 2e20 riables in $S$. │ │ │ │ +0001e7f0: 2054 6865 7920 6172 6520 656c 656d 656e They are elemen │ │ │ │ +0001e800: 7473 2069 6e20 7468 6520 6261 7365 2072 ts in the base r │ │ │ │ +0001e810: 696e 6720 2441 242e 2020 466f 720a 696e ing $A$. For.in │ │ │ │ +0001e820: 7374 616e 6365 2c20 4823 285c 656c 6c2c stance, H#(\ell, │ │ │ │ +0001e830: 2064 2920 6973 2074 6865 2073 7562 6d61 d) is the subma │ │ │ │ +0001e840: 7472 6978 206f 6620 7468 6520 6d61 7472 trix of the matr │ │ │ │ +0001e850: 6978 2066 726f 6d20 2443 5f7b 5c65 6c6c ix from $C_{\ell │ │ │ │ +0001e860: 2b31 7d20 5c74 6f0a 435f 7b5c 656c 6c7d +1} \to.C_{\ell} │ │ │ │ +0001e870: 2420 7365 6e64 696e 6720 6465 6772 6565 $ sending degree │ │ │ │ +0001e880: 2024 6424 2074 6f20 6465 6772 6565 2024 $d$ to degree $ │ │ │ │ +0001e890: 6424 2e0a 0a54 6865 2072 616e 6b73 206f d$...The ranks o │ │ │ │ +0001e8a0: 6620 7468 6573 6520 6d61 7472 6963 6573 f these matrices │ │ │ │ +0001e8b0: 2066 6f72 2061 2073 7065 6369 6669 6320 for a specific │ │ │ │ +0001e8c0: 7061 7261 6d65 7465 7220 7661 6c75 6520 parameter value │ │ │ │ +0001e8d0: 6465 7465 726d 696e 6520 6578 6163 746c determine exactl │ │ │ │ +0001e8e0: 790a 7468 6520 6d69 6e69 6d61 6c20 4265 y.the minimal Be │ │ │ │ +0001e8f0: 7474 6920 7461 626c 6520 666f 7220 7468 tti table for th │ │ │ │ +0001e900: 6520 6964 6561 6c20 2449 242c 2065 7661 e ideal $I$, eva │ │ │ │ +0001e910: 6c75 6174 6564 2061 7420 7468 6174 2070 luated at that p │ │ │ │ +0001e920: 6172 616d 6574 6572 2070 6f69 6e74 2e0a arameter point.. │ │ │ │ +0001e930: 0a4e 6f77 2066 6f72 206f 7572 2065 7861 .Now for our exa │ │ │ │ +0001e940: 6d70 6c65 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d mple...+-------- │ │ │ │ +0001e950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e990: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b20 -----+.|i1 : kk │ │ │ │ +0001e9a0: 3d20 5a5a 2f31 3031 3b20 2020 2020 2020 = ZZ/101; │ │ │ │ 0001e9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001e9f0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ea00: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -0001ea10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ea20: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ea30: 2020 2020 207c 0a7c 6f33 203d 2069 6465 |.|o3 = ide │ │ │ │ -0001ea40: 616c 2028 6120 202b 2074 2062 2a63 202b al (a + t b*c + │ │ │ │ -0001ea50: 2074 2061 2a64 202b 2074 2063 2020 2b20 t a*d + t c + │ │ │ │ -0001ea60: 7420 622a 6420 2b20 7420 632a 6420 2b20 t b*d + t c*d + │ │ │ │ -0001ea70: 7420 6420 2c20 612a 6220 2b20 7420 622a t d , a*b + t b* │ │ │ │ -0001ea80: 6320 2b20 207c 0a7c 2020 2020 2020 2020 c + |.| │ │ │ │ -0001ea90: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ -0001eaa0: 2020 3320 2020 2020 2020 3220 2020 2020 3 2 │ │ │ │ -0001eab0: 2034 2020 2020 2020 2035 2020 2020 2020 4 5 │ │ │ │ -0001eac0: 2036 2020 2020 2020 2020 2020 2037 2020 6 7 │ │ │ │ -0001ead0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d |.| --- │ │ │ │ -0001eae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eb20: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0001eb30: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0001e9e0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ea30: 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 203d -----+.|i2 : S = │ │ │ │ +0001ea40: 206b 6b5b 612e 2e64 5d3b 2020 2020 2020 kk[a..d]; │ │ │ │ +0001ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ea60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ea70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ea80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001ea90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ead0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2046 203d -----+.|i3 : F = │ │ │ │ +0001eae0: 2067 726f 6562 6e65 7246 616d 696c 7920 groebnerFamily │ │ │ │ +0001eaf0: 6964 6561 6c22 6132 2c61 622c 6163 2c62 ideal"a2,ab,ac,b │ │ │ │ +0001eb00: 322c 6263 322c 6333 2220 2020 2020 2020 2,bc2,c3" │ │ │ │ +0001eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eb20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb50: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb70: 2032 2020 207c 0a7c 2020 2020 2074 2061 2 |.| t a │ │ │ │ -0001eb80: 2a64 202b 2074 2063 2020 2b20 7420 2062 *d + t c + t b │ │ │ │ -0001eb90: 2a64 202b 2074 2020 632a 6420 2b20 7420 *d + t c*d + t │ │ │ │ -0001eba0: 2064 202c 2061 2a63 202b 2074 2020 622a d , a*c + t b* │ │ │ │ -0001ebb0: 6320 2b20 7420 2061 2a64 202b 2074 2020 c + t a*d + t │ │ │ │ -0001ebc0: 6320 202b 207c 0a7c 2020 2020 2020 3920 c + |.| 9 │ │ │ │ -0001ebd0: 2020 2020 2020 3820 2020 2020 2031 3020 8 10 │ │ │ │ -0001ebe0: 2020 2020 2020 3131 2020 2020 2020 2031 11 1 │ │ │ │ -0001ebf0: 3220 2020 2020 2020 2020 2020 3133 2020 2 13 │ │ │ │ -0001ec00: 2020 2020 2031 3520 2020 2020 2020 3134 15 14 │ │ │ │ -0001ec10: 2020 2020 207c 0a7c 2020 2020 202d 2d2d |.| --- │ │ │ │ -0001ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec60: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0001ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec80: 2020 2032 2020 2032 2020 2020 2020 2020 2 2 │ │ │ │ -0001ec90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eca0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0001ecb0: 2020 2020 207c 0a7c 2020 2020 2074 2020 |.| t │ │ │ │ -0001ecc0: 622a 6420 2b20 7420 2063 2a64 202b 2074 b*d + t c*d + t │ │ │ │ -0001ecd0: 2020 6420 2c20 6220 202b 2074 2020 622a d , b + t b* │ │ │ │ -0001ece0: 6320 2b20 7420 2061 2a64 202b 2074 2020 c + t a*d + t │ │ │ │ -0001ecf0: 6320 202b 2074 2020 622a 6420 2b20 7420 c + t b*d + t │ │ │ │ -0001ed00: 2063 2a64 207c 0a7c 2020 2020 2020 3136 c*d |.| 16 │ │ │ │ -0001ed10: 2020 2020 2020 2031 3720 2020 2020 2020 17 │ │ │ │ -0001ed20: 3138 2020 2020 2020 2020 2020 3139 2020 18 19 │ │ │ │ -0001ed30: 2020 2020 2032 3120 2020 2020 2020 3230 21 20 │ │ │ │ -0001ed40: 2020 2020 2020 3232 2020 2020 2020 2032 22 2 │ │ │ │ -0001ed50: 3320 2020 207c 0a7c 2020 2020 202d 2d2d 3 |.| --- │ │ │ │ -0001ed60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ed70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ed80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ed90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eda0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -0001edb0: 2020 2032 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ -0001edc0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -0001edd0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0001ede0: 2032 2020 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ -0001edf0: 2020 2033 207c 0a7c 2020 2020 202b 2074 3 |.| + t │ │ │ │ -0001ee00: 2020 6420 2c20 622a 6320 202b 2074 2020 d , b*c + t │ │ │ │ -0001ee10: 622a 632a 6420 2b20 7420 2061 2a64 2020 b*c*d + t a*d │ │ │ │ -0001ee20: 2b20 7420 2063 2064 202b 2074 2020 622a + t c d + t b* │ │ │ │ -0001ee30: 6420 202b 2074 2020 632a 6420 202b 2074 d + t c*d + t │ │ │ │ -0001ee40: 2020 6420 2c7c 0a7c 2020 2020 2020 2020 d ,|.| │ │ │ │ -0001ee50: 3234 2020 2020 2020 2020 2020 2020 3235 24 25 │ │ │ │ -0001ee60: 2020 2020 2020 2020 2032 3720 2020 2020 27 │ │ │ │ -0001ee70: 2020 2032 3620 2020 2020 2020 3238 2020 26 28 │ │ │ │ -0001ee80: 2020 2020 2020 3239 2020 2020 2020 2020 29 │ │ │ │ -0001ee90: 3330 2020 207c 0a7c 2020 2020 202d 2d2d 30 |.| --- │ │ │ │ -0001eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001eee0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3320 -----|.| 3 │ │ │ │ -0001eef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef00: 2020 2032 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ -0001ef10: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0001ef20: 3220 2020 2020 2020 3320 2020 2020 2020 2 3 │ │ │ │ -0001ef30: 2020 2020 207c 0a7c 2020 2020 2063 2020 |.| c │ │ │ │ -0001ef40: 2b20 7420 2062 2a63 2a64 202b 2074 2020 + t b*c*d + t │ │ │ │ -0001ef50: 612a 6420 202b 2074 2020 6320 6420 2b20 a*d + t c d + │ │ │ │ -0001ef60: 7420 2062 2a64 2020 2b20 7420 2063 2a64 t b*d + t c*d │ │ │ │ -0001ef70: 2020 2b20 7420 2064 2029 2020 2020 2020 + t d ) │ │ │ │ -0001ef80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001ef90: 2020 2033 3120 2020 2020 2020 2020 3333 31 33 │ │ │ │ -0001efa0: 2020 2020 2020 2020 3332 2020 2020 2020 32 │ │ │ │ -0001efb0: 2033 3420 2020 2020 2020 2033 3520 2020 34 35 │ │ │ │ -0001efc0: 2020 2020 2033 3620 2020 2020 2020 2020 36 │ │ │ │ -0001efd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f020: 2020 2020 207c 0a7c 6f33 203a 2049 6465 |.|o3 : Ide │ │ │ │ -0001f030: 616c 206f 6620 6b6b 5b74 202c 2074 2020 al of kk[t , t │ │ │ │ -0001f040: 2c20 7420 2c20 7420 202c 2074 2020 2c20 , t , t , t , │ │ │ │ -0001f050: 7420 2c20 7420 202c 2074 2020 2c20 7420 t , t , t , t │ │ │ │ -0001f060: 202c 2074 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ -0001f070: 7420 202c 207c 0a7c 2020 2020 2020 2020 t , |.| │ │ │ │ -0001f080: 2020 2020 2020 2020 2020 3620 2020 3132 6 12 │ │ │ │ -0001f090: 2020 2035 2020 2033 3020 2020 3138 2020 5 30 18 │ │ │ │ -0001f0a0: 2034 2020 2032 3420 2020 3336 2020 2031 4 24 36 1 │ │ │ │ -0001f0b0: 3120 2020 3220 2020 3239 2020 2033 2020 1 2 29 3 │ │ │ │ -0001f0c0: 2031 3020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 10 |.|-------- │ │ │ │ -0001f0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f110: 2d2d 2d2d 2d7c 0a7c 7420 202c 2074 202c -----|.|t , t , │ │ │ │ -0001f120: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ -0001f130: 7420 2c20 7420 202c 2074 202c 2074 2020 t , t , t , t │ │ │ │ -0001f140: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ -0001f150: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -0001f160: 2020 2c20 207c 0a7c 2031 3720 2020 3120 , |.| 17 1 │ │ │ │ -0001f170: 2020 3233 2020 2032 3820 2020 3335 2020 23 28 35 │ │ │ │ -0001f180: 2038 2020 2031 3620 2020 3920 2020 3232 8 16 9 22 │ │ │ │ -0001f190: 2020 2032 3620 2020 3334 2020 2037 2020 26 34 7 │ │ │ │ -0001f1a0: 2031 3420 2020 3237 2020 2031 3520 2020 14 27 15 │ │ │ │ -0001f1b0: 3230 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 20 |.|-------- │ │ │ │ -0001f1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f200: 2d2d 2d2d 2d7c 0a7c 7420 202c 2074 2020 -----|.|t , t │ │ │ │ -0001f210: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -0001f220: 2074 2020 2c20 7420 205d 5b61 2e2e 645d t , t ][a..d] │ │ │ │ -0001f230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f250: 2020 2020 207c 0a7c 2032 3520 2020 3332 |.| 25 32 │ │ │ │ -0001f260: 2020 2031 3320 2020 3231 2020 2033 3320 13 21 33 │ │ │ │ -0001f270: 2020 3139 2020 2033 3120 2020 2020 2020 19 31 │ │ │ │ -0001f280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f2f0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2028 432c -----+.|i4 : (C, │ │ │ │ -0001f300: 2048 2920 3d20 6e6f 6e6d 696e 696d 616c H) = nonminimal │ │ │ │ -0001f310: 4d61 7073 2046 3b20 2020 2020 2020 2020 Maps F; │ │ │ │ -0001f320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f340: 2020 2020 207c 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2020 2020 2020 2037 2020 6 7 │ │ │ │ +0001ec60: 2020 2020 207c 0a7c 2020 2020 202d 2d2d |.| --- │ │ │ │ +0001ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ecb0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +0001ecc0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0001ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ece0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ed00: 2032 2020 207c 0a7c 2020 2020 2074 2061 2 |.| t a │ │ │ │ +0001ed10: 2a64 202b 2074 2063 2020 2b20 7420 2062 *d + t c + t b │ │ │ │ +0001ed20: 2a64 202b 2074 2020 632a 6420 2b20 7420 *d + t c*d + t │ │ │ │ +0001ed30: 2064 202c 2061 2a63 202b 2074 2020 622a d , a*c + t b* │ │ │ │ +0001ed40: 6320 2b20 7420 2061 2a64 202b 2074 2020 c + t a*d + t │ │ │ │ +0001ed50: 6320 202b 207c 0a7c 2020 2020 2020 3920 c + |.| 9 │ │ │ │ +0001ed60: 2020 2020 2020 3820 2020 2020 2031 3020 8 10 │ │ │ │ +0001ed70: 2020 2020 2020 3131 2020 2020 2020 2031 11 1 │ │ │ │ +0001ed80: 3220 2020 2020 2020 2020 2020 3133 2020 2 13 │ │ │ │ +0001ed90: 2020 2020 2031 3520 2020 2020 2020 3134 15 14 │ │ │ │ +0001eda0: 2020 2020 207c 0a7c 2020 2020 202d 2d2d |.| --- │ │ │ │ +0001edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001edd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001edf0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +0001ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ee10: 2020 2032 2020 2032 2020 2020 2020 2020 2 2 │ │ │ │ +0001ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ee30: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001ee40: 2020 2020 207c 0a7c 2020 2020 2074 2020 |.| t │ │ │ │ +0001ee50: 622a 6420 2b20 7420 2063 2a64 202b 2074 b*d + t c*d + t │ │ │ │ +0001ee60: 2020 6420 2c20 6220 202b 2074 2020 622a d , b + t b* │ │ │ │ +0001ee70: 6320 2b20 7420 2061 2a64 202b 2074 2020 c + t a*d + t │ │ │ │ +0001ee80: 6320 202b 2074 2020 622a 6420 2b20 7420 c + t b*d + t │ │ │ │ +0001ee90: 2063 2a64 207c 0a7c 2020 2020 2020 3136 c*d |.| 16 │ │ │ │ +0001eea0: 2020 2020 2020 2031 3720 2020 2020 2020 17 │ │ │ │ +0001eeb0: 3138 2020 2020 2020 2020 2020 3139 2020 18 19 │ │ │ │ +0001eec0: 2020 2020 2032 3120 2020 2020 2020 3230 21 20 │ │ │ │ +0001eed0: 2020 2020 2020 3232 2020 2020 2020 2032 22 2 │ │ │ │ +0001eee0: 3320 2020 207c 0a7c 2020 2020 202d 2d2d 3 |.| --- │ │ │ │ +0001eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ef00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2020 2020 27 │ │ │ │ +0001f000: 2020 2032 3620 2020 2020 2020 3238 2020 26 28 │ │ │ │ +0001f010: 2020 2020 2020 3239 2020 2020 2020 2020 29 │ │ │ │ +0001f020: 3330 2020 207c 0a7c 2020 2020 202d 2d2d 30 |.| --- │ │ │ │ +0001f030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f070: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3320 -----|.| 3 │ │ │ │ +0001f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f090: 2020 2032 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ +0001f0a0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +0001f0b0: 3220 2020 2020 2020 3320 2020 2020 2020 2 3 │ │ │ │ +0001f0c0: 2020 2020 207c 0a7c 2020 2020 2063 2020 |.| c │ │ │ │ +0001f0d0: 2b20 7420 2062 2a63 2a64 202b 2074 2020 + t b*c*d + t │ │ │ │ +0001f0e0: 612a 6420 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2074 2020 2c20 t , t , t , │ │ │ │ +0001f2c0: 7420 2c20 7420 202c 2074 202c 2074 2020 t , t , t , t │ │ │ │ +0001f2d0: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ +0001f2e0: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +0001f2f0: 2020 2c20 207c 0a7c 2031 3720 2020 3120 , |.| 17 1 │ │ │ │ +0001f300: 2020 3233 2020 2032 3820 2020 3335 2020 23 28 35 │ │ │ │ +0001f310: 2038 2020 2031 3620 2020 3920 2020 3232 8 16 9 22 │ │ │ │ +0001f320: 2020 2032 3620 2020 3334 2020 2037 2020 26 34 7 │ │ │ │ +0001f330: 2031 3420 2020 3237 2020 2031 3520 2020 14 27 15 │ │ │ │ +0001f340: 3230 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 20 |.|-------- │ │ │ │ 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f390: 2d2d 2d2d 2d2b 0a7c 6935 203a 2062 6574 -----+.|i5 : bet │ │ │ │ -0001f3a0: 7469 2843 2c20 5765 6967 6874 7320 3d3e ti(C, Weights => │ │ │ │ -0001f3b0: 207b 312c 312c 312c 317d 2920 2020 2020 {1,1,1,1}) │ │ │ │ +0001f390: 2d2d 2d2d 2d7c 0a7c 7420 202c 2074 2020 -----|.|t , t │ │ │ │ +0001f3a0: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +0001f3b0: 2074 2020 2c20 7420 205d 5b61 2e2e 645d t , t ][a..d] │ │ │ │ 0001f3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f3e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001f3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f3e0: 2020 2020 207c 0a7c 2032 3520 2020 3332 |.| 25 32 │ │ │ │ +0001f3f0: 2020 2031 3320 2020 3231 2020 2033 3320 13 21 33 │ │ │ │ +0001f400: 2020 3139 2020 2033 3120 2020 2020 2020 19 31 │ │ │ │ 0001f410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f430: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001f440: 2020 2020 3020 3120 2032 2033 2034 2020 0 1 2 3 4 │ │ │ │ -0001f450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f480: 2020 2020 207c 0a7c 6f35 203d 2074 6f74 |.|o5 = tot │ │ │ │ -0001f490: 616c 3a20 3120 3620 3130 2036 2031 2020 al: 1 6 10 6 1 │ │ │ │ -0001f4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f430: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001f440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f480: 2d2d 2d2d 2d2b 0a7c 6934 203a 2028 432c -----+.|i4 : (C, │ │ │ │ +0001f490: 2048 2920 3d20 6e6f 6e6d 696e 696d 616c H) = nonminimal │ │ │ │ +0001f4a0: 4d61 7073 2046 3b20 2020 2020 2020 2020 Maps F; │ │ │ │ 0001f4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f4d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001f4e0: 2030 3a20 3120 2e20 202e 202e 202e 2020 0: 1 . . . . │ │ │ │ -0001f4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f520: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001f530: 2031 3a20 2e20 3420 2034 2032 202e 2020 1: . 4 4 2 . │ │ │ │ -0001f540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f4d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f500: 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3020 3120 2032 2033 2034 2020 0 1 2 3 4 │ │ │ │ 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 2020 2020 │ │ │ │ -0001f610: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f610: 2020 2020 207c 0a7c 6f35 203d 2074 6f74 |.|o5 = tot │ │ │ │ +0001f620: 616c 3a20 3120 3620 3130 2036 2031 2020 al: 1 6 10 6 1 │ │ │ │ 0001f630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f660: 2020 2020 207c 0a7c 6f35 203a 2042 6574 |.|o5 : Bet │ │ │ │ -0001f670: 7469 5461 6c6c 7920 2020 2020 2020 2020 tiTally │ │ │ │ +0001f660: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001f670: 2030 3a20 3120 2e20 202e 202e 202e 2020 0: 1 . . . . │ │ │ │ 0001f680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f6b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f700: 2d2d 2d2d 2d2b 0a0a 5765 2073 6565 2074 -----+..We see t │ │ │ │ -0001f710: 6861 7420 7468 6572 6520 6172 6520 3420 hat there are 4 │ │ │ │ -0001f720: 6d61 7073 2074 6861 7420 6172 6520 6e6f maps that are no │ │ │ │ -0001f730: 6e6d 696e 696d 616c 2028 6f66 2073 697a nminimal (of siz │ │ │ │ -0001f740: 6573 2024 3220 5c74 696d 6573 2034 242c es $2 \times 4$, │ │ │ │ -0001f750: 2024 350a 5c74 696d 6573 2032 242c 2024 $5.\times 2$, $ │ │ │ │ -0001f760: 3120 5c74 696d 6573 2033 242c 2061 6e64 1 \times 3$, and │ │ │ │ -0001f770: 2024 3120 5c74 696d 6573 2031 2429 2e0a $1 \times 1$).. │ │ │ │ -0001f780: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0001f790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001f7d0: 0a7c 6936 203a 206b 6579 7320 4820 2020 .|i6 : keys H │ │ │ │ +0001f6b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001f6c0: 2031 3a20 2e20 3420 2034 2032 202e 2020 1: . 4 4 2 . │ │ │ │ +0001f6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f700: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001f710: 2032 3a20 2e20 3220 2035 2033 2031 2020 2: . 2 5 3 1 │ │ │ │ +0001f720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f750: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001f760: 2033 3a20 2e20 2e20 2031 2031 202e 2020 3: . . 1 1 . │ │ │ │ +0001f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f7a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001f820: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001f7f0: 2020 2020 207c 0a7c 6f35 203a 2042 6574 |.|o5 : Bet │ │ │ │ +0001f800: 7469 5461 6c6c 7920 2020 2020 2020 2020 tiTally │ │ │ │ +0001f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001f870: 0a7c 6f36 203d 207b 2833 2c20 3429 2c20 .|o6 = {(3, 4), │ │ │ │ -0001f880: 2833 2c20 3529 2c20 2834 2c20 3629 2c20 (3, 5), (4, 6), │ │ │ │ -0001f890: 2832 2c20 3329 7d20 2020 2020 2020 2020 (2, 3)} │ │ │ │ -0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001f8c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f900: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001f910: 0a7c 6f36 203a 204c 6973 7420 2020 2020 .|o6 : List │ │ │ │ -0001f920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001f960: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0001f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001f9b0: 0a7c 6937 203a 2048 2328 322c 3329 2020 .|i7 : H#(2,3) │ │ │ │ +0001f840: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f890: 2d2d 2d2d 2d2b 0a0a 5765 2073 6565 2074 -----+..We see t │ │ │ │ +0001f8a0: 6861 7420 7468 6572 6520 6172 6520 3420 hat there are 4 │ │ │ │ +0001f8b0: 6d61 7073 2074 6861 7420 6172 6520 6e6f maps that are no │ │ │ │ +0001f8c0: 6e6d 696e 696d 616c 2028 6f66 2073 697a nminimal (of siz │ │ │ │ +0001f8d0: 6573 2024 3220 5c74 696d 6573 2034 242c es $2 \times 4$, │ │ │ │ +0001f8e0: 2024 350a 5c74 696d 6573 2032 242c 2024 $5.\times 2$, $ │ │ │ │ +0001f8f0: 3120 5c74 696d 6573 2033 242c 2061 6e64 1 \times 3$, and │ │ │ │ +0001f900: 2024 3120 5c74 696d 6573 2031 2429 2e0a $1 \times 1$).. │ │ │ │ +0001f910: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001f960: 0a7c 6936 203a 206b 6579 7320 4820 2020 .|i6 : keys H │ │ │ │ +0001f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f9a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f9b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f9f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001fa00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001fa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fa00: 0a7c 6f36 203d 207b 2833 2c20 3429 2c20 .|o6 = {(3, 4), │ │ │ │ +0001fa10: 2833 2c20 3529 2c20 2834 2c20 3629 2c20 (3, 5), (4, 6), │ │ │ │ +0001fa20: 2832 2c20 3329 7d20 2020 2020 2020 2020 (2, 3)} │ │ │ │ 0001fa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fa40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001fa50: 0a7c 6f37 203d 207b 337d 207c 202d 745f .|o7 = {3} | -t_ │ │ │ │ -0001fa60: 382d 745f 3230 745f 3133 2020 2020 2020 8-t_20t_13 │ │ │ │ -0001fa70: 745f 3774 5f32 302d 745f 3134 745f 3230 t_7t_20-t_14t_20 │ │ │ │ -0001fa80: 2b74 5f32 3074 5f31 3374 5f31 3920 2020 +t_20t_13t_19 │ │ │ │ +0001fa50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001fa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fa90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001faa0: 0a7c 2020 2020 207b 337d 207c 202d 745f .| {3} | -t_ │ │ │ │ -0001fab0: 372b 745f 3134 2d74 5f31 3374 5f31 3920 7+t_14-t_13t_19 │ │ │ │ -0001fac0: 2d74 5f38 2d74 5f32 3074 5f31 332b 745f -t_8-t_20t_13+t_ │ │ │ │ -0001fad0: 3774 5f31 392d 745f 3134 745f 3139 2b74 7t_19-t_14t_19+t │ │ │ │ -0001fae0: 5f31 3374 5f31 395e 3220 2020 2020 207c _13t_19^2 | │ │ │ │ -0001faf0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0001faa0: 0a7c 6f36 203a 204c 6973 7420 2020 2020 .|o6 : List │ │ │ │ +0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fae0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001faf0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0001fb40: 0a7c 2020 2020 202d 745f 322d 745f 3134 .| -t_2-t_14 │ │ │ │ -0001fb50: 5e32 2b74 5f32 3074 5f31 335e 3220 2020 ^2+t_20t_13^2 │ │ │ │ -0001fb60: 202d 745f 3874 5f31 342b 745f 3174 5f32 -t_8t_14+t_1t_2 │ │ │ │ -0001fb70: 302b 745f 3774 5f32 3074 5f31 3320 2020 0+t_7t_20t_13 │ │ │ │ -0001fb80: 2020 2020 2020 2020 2020 7c20 2020 207c | | │ │ │ │ -0001fb90: 0a7c 2020 2020 202d 745f 312d 3274 5f31 .| -t_1-2t_1 │ │ │ │ -0001fba0: 3474 5f31 332b 745f 3133 5e32 745f 3139 4t_13+t_13^2t_19 │ │ │ │ -0001fbb0: 202d 745f 322d 745f 3774 5f31 342d 745f -t_2-t_7t_14-t_ │ │ │ │ -0001fbc0: 3874 5f31 332b 745f 3174 5f31 392b 745f 8t_13+t_1t_19+t_ │ │ │ │ -0001fbd0: 3774 5f31 3374 5f31 3920 7c20 2020 207c 7t_13t_19 | | │ │ │ │ -0001fbe0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001fbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001fb40: 0a7c 6937 203a 2048 2328 322c 3329 2020 .|i7 : H#(2,3) │ │ │ │ +0001fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fb80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001fb90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001fba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fbd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001fbe0: 0a7c 6f37 203d 207b 337d 207c 202d 745f .|o7 = {3} | -t_ │ │ │ │ +0001fbf0: 382d 745f 3230 745f 3133 2020 2020 2020 8-t_20t_13 │ │ │ │ +0001fc00: 745f 3774 5f32 302d 745f 3134 745f 3230 t_7t_20-t_14t_20 │ │ │ │ +0001fc10: 2b74 5f32 3074 5f31 3374 5f31 3920 2020 +t_20t_13t_19 │ │ │ │ 0001fc20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001fc30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001fc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001fc80: 0a7c 6f37 203a 204d 6174 7269 7820 286b .|o7 : Matrix (k │ │ │ │ -0001fc90: 6b5b 7420 2c20 7420 202c 2074 202c 2074 k[t , t , t , t │ │ │ │ -0001fca0: 2020 2c20 7420 202c 2074 202c 2074 2020 , t , t , t │ │ │ │ -0001fcb0: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ -0001fcc0: 7420 202c 2074 202c 2074 2020 2c20 747c t , t , t , t| │ │ │ │ -0001fcd0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001fce0: 2020 2036 2020 2031 3220 2020 3520 2020 6 12 5 │ │ │ │ -0001fcf0: 3330 2020 2031 3820 2020 3420 2020 3234 30 18 4 24 │ │ │ │ -0001fd00: 2020 2033 3620 2020 3131 2020 2032 2020 36 11 2 │ │ │ │ -0001fd10: 2032 3920 2020 3320 2020 3130 2020 207c 29 3 10 | │ │ │ │ -0001fd20: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -0001fd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0001fc30: 0a7c 2020 2020 207b 337d 207c 202d 745f .| {3} | -t_ │ │ │ │ +0001fc40: 372b 745f 3134 2d74 5f31 3374 5f31 3920 7+t_14-t_13t_19 │ │ │ │ +0001fc50: 2d74 5f38 2d74 5f32 3074 5f31 332b 745f -t_8-t_20t_13+t_ │ │ │ │ +0001fc60: 3774 5f31 392d 745f 3134 745f 3139 2b74 7t_19-t_14t_19+t │ │ │ │ +0001fc70: 5f31 3374 5f31 395e 3220 2020 2020 207c _13t_19^2 | │ │ │ │ +0001fc80: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0001fc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0001fcd0: 0a7c 2020 2020 202d 745f 322d 745f 3134 .| -t_2-t_14 │ │ │ │ +0001fce0: 5e32 2b74 5f32 3074 5f31 335e 3220 2020 ^2+t_20t_13^2 │ │ │ │ +0001fcf0: 202d 745f 3874 5f31 342b 745f 3174 5f32 -t_8t_14+t_1t_2 │ │ │ │ +0001fd00: 302b 745f 3774 5f32 3074 5f31 3320 2020 0+t_7t_20t_13 │ │ │ │ +0001fd10: 2020 2020 2020 2020 2020 7c20 2020 207c | | │ │ │ │ +0001fd20: 0a7c 2020 2020 202d 745f 312d 3274 5f31 .| -t_1-2t_1 │ │ │ │ +0001fd30: 3474 5f31 332b 745f 3133 5e32 745f 3139 4t_13+t_13^2t_19 │ │ │ │ +0001fd40: 202d 745f 322d 745f 3774 5f31 342d 745f -t_2-t_7t_14-t_ │ │ │ │ +0001fd50: 3874 5f31 332b 745f 3174 5f31 392b 745f 8t_13+t_1t_19+t_ │ │ │ │ +0001fd60: 3774 5f31 3374 5f31 3920 7c20 2020 207c 7t_13t_19 | | │ │ │ │ 0001fd70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001fd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fdb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001fdc0: 0a7c 2020 2c20 7420 2c20 7420 202c 2074 .| , t , t , t │ │ │ │ -0001fdd0: 2020 2c20 7420 202c 2074 202c 2074 2020 , t , t , t │ │ │ │ -0001fde0: 2c20 7420 2c20 7420 202c 2074 2020 2c20 , t , t , t , │ │ │ │ -0001fdf0: 7420 202c 2074 202c 2074 2020 2c20 7420 t , t , t , t │ │ │ │ -0001fe00: 202c 2074 2020 2c20 7420 202c 2020 207c , t , t , | │ │ │ │ -0001fe10: 0a7c 3137 2020 2031 2020 2032 3320 2020 .|17 1 23 │ │ │ │ -0001fe20: 3238 2020 2033 3520 2020 3820 2020 3136 28 35 8 16 │ │ │ │ -0001fe30: 2020 2039 2020 2032 3220 2020 3236 2020 9 22 26 │ │ │ │ -0001fe40: 2033 3420 2020 3720 2020 3134 2020 2032 34 7 14 2 │ │ │ │ -0001fe50: 3720 2020 3135 2020 2032 3020 2020 207c 7 15 20 | │ │ │ │ -0001fe60: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -0001fe70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0001feb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001fec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fed0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0001fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fef0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001ff00: 0a7c 7420 202c 2074 2020 2c20 7420 202c .|t , t , t , │ │ │ │ -0001ff10: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ -0001ff20: 7420 205d 2920 203c 2d2d 2028 6b6b 5b74 t ]) <-- (kk[t │ │ │ │ -0001ff30: 202c 2074 2020 2c20 7420 2c20 7420 202c , t , t , t , │ │ │ │ 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, t , │ │ │ │ +000200a0: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ +000200b0: 7420 205d 2920 203c 2d2d 2028 6b6b 5b74 t ]) <-- (kk[t │ │ │ │ +000200c0: 202c 2074 2020 2c20 7420 2c20 7420 202c , t , t , t , │ │ │ │ +000200d0: 2074 2020 2c20 7420 2c20 7420 202c 207c t , t , t , | │ │ │ │ +000200e0: 0a7c 2032 3520 2020 3332 2020 2031 3320 .| 25 32 13 │ │ │ │ +000200f0: 2020 3231 2020 2033 3320 2020 3139 2020 21 33 19 │ │ │ │ +00020100: 2033 3120 2020 2020 2020 2020 2020 2020 31 │ │ │ │ +00020110: 3620 2020 3132 2020 2035 2020 2033 3020 6 12 5 30 │ │ │ │ +00020120: 2020 3138 2020 2034 2020 2032 3420 207c 18 4 24 | │ │ │ │ +00020130: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ +00020140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 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│ │ -00020600: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000205e0: 0a7c 2020 2020 207b 347d 207c 2030 2020 .| {4} | 0 │ │ │ │ +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 207c | │ │ │ │ -00020630: 0a7c 2020 2020 2074 5f37 2d74 5f31 342b .| t_7-t_14+ │ │ │ │ -00020640: 745f 3133 745f 3139 2020 2020 2020 2020 t_13t_19 │ │ │ │ -00020650: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020680: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00020630: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +00020640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020660: 2d2d 2d2d 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│ │ │ +00020720: 0a7c 2020 2020 2074 5f32 2b74 5f37 745f .| t_2+t_7t_ │ │ │ │ +00020730: 3134 2b74 5f38 745f 3133 2d74 5f31 745f 14+t_8t_13-t_1t_ │ │ │ │ +00020740: 3139 2d74 5f37 745f 3133 745f 3139 207c 19-t_7t_13t_19 | │ │ │ │ +00020750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00020770: 0a7c 2020 2020 202d 745f 312d 3274 5f31 .| -t_1-2t_1 │ │ │ │ +00020780: 3474 5f31 332b 745f 3133 5e32 745f 3139 4t_13+t_13^2t_19 │ │ │ │ +00020790: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000207a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000207b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000207c0: 0a7c 2020 2020 2074 5f37 2d74 5f31 342b .| t_7-t_14+ │ │ │ │ +000207d0: 745f 3133 745f 3139 2020 2020 2020 2020 t_13t_19 │ │ │ │ +000207e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000207f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020800: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 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7420 2c20 7420 202c 207c t , t , t , | │ │ │ │ -000209f0: 0a7c 2032 3520 2020 3332 2020 2031 3320 .| 25 32 13 │ │ │ │ -00020a00: 2020 3231 2020 2033 3320 2020 3139 2020 21 33 19 │ │ │ │ -00020a10: 2033 3120 2020 2020 2020 2020 2020 2020 31 │ │ │ │ -00020a20: 3620 2020 3132 2020 2035 2020 2033 3020 6 12 5 30 │ │ │ │ -00020a30: 2020 3138 2020 2034 2020 2032 3420 207c 18 4 24 | │ │ │ │ -00020a40: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -00020a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00020a90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00020aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ad0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020ae0: 0a7c 7420 202c 2074 2020 2c20 7420 2c20 .|t , t , t , │ │ │ │ -00020af0: 7420 202c 2074 202c 2074 2020 2c20 7420 t , t , t , t │ │ │ │ -00020b00: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -00020b10: 2074 2020 2c20 7420 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00020b20: 202c 2074 2020 2c20 7420 202c 2020 207c , t , t , | │ │ │ │ -00020b30: 0a7c 2033 3620 2020 3131 2020 2032 2020 .| 36 11 2 │ │ │ │ -00020b40: 2032 3920 2020 3320 2020 3130 2020 2031 29 3 10 1 │ │ │ │ -00020b50: 3720 2020 3120 2020 3233 2020 2032 3820 7 1 23 28 │ │ │ │ -00020b60: 2020 3335 2020 2038 2020 2031 3620 2020 35 8 16 │ │ │ │ -00020b70: 3920 2020 3232 2020 2032 3620 2020 207c 9 22 26 | │ │ │ │ -00020b80: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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13 2 │ │ │ │ -00020ca0: 3120 2020 3333 2020 2031 3920 2020 3331 1 33 19 31 │ │ │ │ -00020cb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020cc0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00020cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00020d10: 0a7c 6939 203a 2048 2328 332c 3529 2020 .|i9 : H#(3,5) │ │ │ │ -00020d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00020860: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00020870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020880: 2020 2020 2020 2020 2020 2020 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│ │ │ │ -00020e00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00020e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020da0: 2020 3220 2020 2020 2020 2020 2020 207c 2 | │ │ │ │ +00020db0: 0a7c 7420 202c 2074 202c 2074 2020 2c20 .|t , t , t , │ │ │ │ +00020dc0: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00020dd0: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ +00020de0: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ +00020df0: 5d29 2020 2020 2020 2020 2020 2020 207c ]) | │ │ │ │ +00020e00: 0a7c 2033 3420 2020 3720 2020 3134 2020 .| 34 7 14 │ │ │ │ +00020e10: 2032 3720 2020 3135 2020 2032 3020 2020 27 15 20 │ │ │ │ +00020e20: 3235 2020 2033 3220 2020 3133 2020 2032 25 32 13 2 │ │ │ │ +00020e30: 3120 2020 3333 2020 2031 3920 2020 3331 1 33 19 31 │ │ │ │ 00020e40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 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│ │ │ │ -00020ff0: 7420 202c 2074 2020 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ -00021000: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -00021010: 2074 2020 2c20 7420 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00021020: 2020 2c20 7420 202c 2074 2020 2c20 207c , t , t , | │ │ │ │ -00021030: 0a7c 2031 3720 2020 3120 2020 3233 2020 .| 17 1 23 │ │ │ │ -00021040: 2032 3820 2020 3335 2020 2038 2020 2031 28 35 8 1 │ │ │ │ -00021050: 3620 2020 3920 2020 3232 2020 2032 3620 6 9 22 26 │ │ │ │ -00021060: 2020 3334 2020 2037 2020 2031 3420 2020 34 7 14 │ │ │ │ -00021070: 3237 2020 2031 3520 2020 3230 2020 207c 27 15 20 | │ │ │ │ -00021080: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -00021090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000210a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000210b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000210c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -000210d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000210e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000210f0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ -00021100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021110: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021120: 0a7c 7420 202c 2074 2020 2c20 7420 202c .|t , t , t , │ │ │ │ -00021130: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ -00021140: 7420 205d 2920 203c 2d2d 2028 6b6b 5b74 t ]) <-- (kk[t │ │ │ │ -00021150: 202c 2074 2020 2c20 7420 2c20 7420 202c , t , t , t , │ │ │ │ -00021160: 2074 2020 2c20 7420 2c20 7420 202c 207c t , t , t , | │ │ │ │ -00021170: 0a7c 2032 3520 2020 3332 2020 2031 3320 .| 25 32 13 │ │ │ │ -00021180: 2020 3231 2020 2033 3320 2020 3139 2020 21 33 19 │ │ │ │ -00021190: 2033 3120 2020 2020 2020 2020 2020 2020 31 │ │ │ │ -000211a0: 3620 2020 3132 2020 2035 2020 2033 3020 6 12 5 30 │ │ │ │ -000211b0: 2020 3138 2020 2034 2020 2032 3420 207c 18 4 24 | │ │ │ │ -000211c0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -000211d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00021210: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00021220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021250: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021260: 0a7c 7420 202c 2074 2020 2c20 7420 2c20 .|t , t , t , │ │ │ │ -00021270: 7420 202c 2074 202c 2074 2020 2c20 7420 t , t , t , t │ │ │ │ -00021280: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -00021290: 2074 2020 2c20 7420 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -000212a0: 202c 2074 2020 2c20 7420 202c 2020 207c , t , t , | │ │ │ │ -000212b0: 0a7c 2033 3620 2020 3131 2020 2032 2020 .| 36 11 2 │ │ │ │ -000212c0: 2032 3920 2020 3320 2020 3130 2020 2031 29 3 10 1 │ │ │ │ -000212d0: 3720 2020 3120 2020 3233 2020 2032 3820 7 1 23 28 │ │ │ │ -000212e0: 2020 3335 2020 2038 2020 2031 3620 2020 35 8 16 │ │ │ │ -000212f0: 3920 2020 3232 2020 2032 3620 2020 207c 9 22 26 | │ │ │ │ -00021300: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -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 2d7c ---------------| │ │ │ │ -00021350: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00021360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021380: 2020 2020 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---------------- │ │ │ │ -00021470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00021490: 0a7c 6931 3020 3a20 4823 2834 2c36 2920 .|i10 : H#(4,6) │ │ │ │ -000214a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00020fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00020ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021020: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00021030: 0a7c 6f39 203a 204d 6174 7269 7820 286b .|o9 : Matrix (k │ │ │ │ +00021040: 6b5b 7420 2c20 7420 202c 2074 202c 2074 k[t , t , t , t │ │ │ │ +00021050: 2020 2c20 7420 202c 2074 202c 2074 2020 , t , t , t │ │ │ │ +00021060: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ +00021070: 7420 202c 2074 202c 2074 2020 2c20 207c t , t , t , | │ │ │ │ +00021080: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00021090: 2020 2036 2020 2031 3220 2020 3520 2020 6 12 5 │ │ │ │ +000210a0: 3330 2020 2031 3820 2020 3420 2020 3234 30 18 4 24 │ │ │ │ +000210b0: 2020 2033 3620 2020 3131 2020 2032 2020 36 11 2 │ │ │ │ +000210c0: 2032 3920 2020 3320 2020 3130 2020 207c 29 3 10 | │ │ │ │ +000210d0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ +000210e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000210f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00021120: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00021130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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│ │ +00021310: 2020 3231 2020 2033 3320 2020 3139 2020 21 33 19 │ │ │ │ +00021320: 2033 3120 2020 2020 2020 2020 2020 2020 31 │ │ │ │ +00021330: 3620 2020 3132 2020 2035 2020 2033 3020 6 12 5 30 │ │ │ │ +00021340: 2020 3138 2020 2034 2020 2032 3420 207c 18 4 24 | │ │ │ │ +00021350: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ +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 2d7c ---------------| │ │ │ │ +000213a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000213b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000213c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000213d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000213e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000213f0: 0a7c 7420 202c 2074 2020 2c20 7420 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2c20 .|t , t , t , │ │ │ │ +00021540: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00021550: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ +00021560: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ +00021570: 5d29 2020 2020 2020 2020 2020 2020 207c ]) | │ │ │ │ +00021580: 0a7c 2033 3420 2020 3720 2020 3134 2020 .| 34 7 14 │ │ │ │ +00021590: 2032 3720 2020 3135 2020 2032 3020 2020 27 15 20 │ │ │ │ +000215a0: 3235 2020 2033 3220 2020 3133 2020 2032 25 32 13 2 │ │ │ │ +000215b0: 3120 2020 3333 2020 2031 3920 2020 3331 1 33 19 31 │ │ │ │ 000215c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000215d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000215e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021610: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021620: 0a7c 6f31 3020 3a20 4d61 7472 6978 2028 .|o10 : Matrix ( │ │ │ │ -00021630: 6b6b 5b74 202c 2074 2020 2c20 7420 2c20 kk[t , t , t , │ │ │ │ -00021640: 7420 202c 2074 2020 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ -00021650: 202c 2074 2020 2c20 7420 202c 2074 202c , t , t , t , │ │ │ │ -00021660: 2074 2020 2c20 7420 2c20 7420 202c 207c t , t , t , | │ │ │ │ +000215d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000215e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000215f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00021620: 0a7c 6931 3020 3a20 4823 2834 2c36 2920 .|i10 : H#(4,6) │ │ │ │ +00021630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021660: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00021670: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00021680: 2020 2020 3620 2020 3132 2020 2035 2020 6 12 5 │ │ │ │ -00021690: 2033 3020 2020 3138 2020 2034 2020 2032 30 18 4 2 │ │ │ │ -000216a0: 3420 2020 3336 2020 2031 3120 2020 3220 4 36 11 2 │ │ │ │ -000216b0: 2020 3239 2020 2033 2020 2031 3020 207c 29 3 10 | │ │ │ │ -000216c0: 0a7c 2d2d 2d2d 2d2d 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 2d7c ---------------| │ │ │ │ +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 207c | │ │ │ │ +000216c0: 0a7c 6f31 3020 3d20 7b36 7d20 7c20 2d31 .|o10 = {6} | -1 │ │ │ │ +000216d0: 207c 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 207c | │ │ │ │ 00021710: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00021720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021750: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021760: 0a7c 7420 202c 2074 202c 2074 2020 2c20 .|t , t , t , │ │ │ │ -00021770: 7420 202c 2074 2020 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ -00021780: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -00021790: 2074 2020 2c20 7420 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -000217a0: 2020 2c20 7420 202c 2074 2020 2c20 207c , t , t , | │ │ │ │ -000217b0: 0a7c 2031 3720 2020 3120 2020 3233 2020 .| 17 1 23 │ │ │ │ -000217c0: 2032 3820 2020 3335 2020 2038 2020 2031 28 35 8 1 │ │ │ │ -000217d0: 3620 2020 3920 2020 3232 2020 2032 3620 6 9 22 26 │ │ │ │ -000217e0: 2020 3334 2020 2037 2020 2031 3420 2020 34 7 14 │ │ │ │ -000217f0: 3237 2020 2031 3520 2020 3230 2020 207c 27 15 20 | │ │ │ │ -00021800: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -00021810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00021850: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00021860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021870: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ -00021880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021890: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000218a0: 0a7c 7420 202c 2074 2020 2c20 7420 202c .|t , t , t , │ │ │ │ -000218b0: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ -000218c0: 7420 205d 2920 203c 2d2d 2028 6b6b 5b74 t ]) <-- (kk[t │ │ │ │ -000218d0: 202c 2074 2020 2c20 7420 2c20 7420 202c , t , t , t , │ │ │ │ -000218e0: 2074 2020 2c20 7420 2c20 7420 202c 207c t , t , t , | │ │ │ │ -000218f0: 0a7c 2032 3520 2020 3332 2020 2031 3320 .| 25 32 13 │ │ │ │ -00021900: 2020 3231 2020 2033 3320 2020 3139 2020 21 33 19 │ │ │ │ -00021910: 2033 3120 2020 2020 2020 2020 2020 2020 31 │ │ │ │ -00021920: 3620 2020 3132 2020 2035 2020 2033 3020 6 12 5 30 │ │ │ │ -00021930: 2020 3138 2020 2034 2020 2032 3420 207c 18 4 24 | │ │ │ │ -00021940: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -00021950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00021990: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000219a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000219b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000219c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000219d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000219e0: 0a7c 7420 202c 2074 2020 2c20 7420 2c20 .|t , t , t , │ │ │ │ -000219f0: 7420 202c 2074 202c 2074 2020 2c20 7420 t , t , t , t │ │ │ │ -00021a00: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -00021a10: 2074 2020 2c20 7420 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00021a20: 202c 2074 2020 2c20 7420 202c 2020 207c , t , t , | │ │ │ │ -00021a30: 0a7c 2033 3620 2020 3131 2020 2032 2020 .| 36 11 2 │ │ │ │ -00021a40: 2032 3920 2020 3320 2020 3130 2020 2031 29 3 10 1 │ │ │ │ -00021a50: 3720 2020 3120 2020 3233 2020 2032 3820 7 1 23 28 │ │ │ │ -00021a60: 2020 3335 2020 2038 2020 2031 3620 2020 35 8 16 │ │ │ │ -00021a70: 3920 2020 3232 2020 2032 3620 2020 207c 9 22 26 | │ │ │ │ -00021a80: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ -00021a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00021ad0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00021ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021b10: 2020 3120 2020 2020 2020 2020 2020 207c 1 | │ │ │ │ -00021b20: 0a7c 7420 202c 2074 202c 2074 2020 2c20 .|t , t , t , │ │ │ │ -00021b30: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00021b40: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ -00021b50: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ -00021b60: 5d29 2020 2020 2020 2020 2020 2020 207c ]) | │ │ │ │ -00021b70: 0a7c 2033 3420 2020 3720 2020 3134 2020 .| 34 7 14 │ │ │ │ -00021b80: 2032 3720 2020 3135 2020 2032 3020 2020 27 15 20 │ │ │ │ -00021b90: 3235 2020 2033 3220 2020 3133 2020 2032 25 32 13 2 │ │ │ │ -00021ba0: 3120 2020 3333 2020 2031 3920 2020 3331 1 33 19 31 │ │ │ │ -00021bb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00021bc0: 0a2b 2d2d 2d2d 2d2d 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 2d2b ---------------+ │ │ │ │ -00021c10: 0a0a 4c65 7427 7320 696d 706f 7365 2074 ..Let's impose t │ │ │ │ -00021c20: 6865 2063 6f6e 6469 7469 6f6e 2074 6861 he condition tha │ │ │ │ -00021c30: 7420 7468 6520 6d61 7020 4823 2832 2c33 t the map H#(2,3 │ │ │ │ -00021c40: 2920 7661 6e69 7368 6573 2028 736f 2068 ) vanishes (so h │ │ │ │ -00021c50: 6173 2072 616e 6b20 3029 2e20 5468 650a as rank 0). The. │ │ │ │ -00021c60: 4265 7474 6920 6469 6167 7261 6d20 6f66 Betti diagram of │ │ │ │ -00021c70: 2073 7563 6820 6964 6561 6c73 2069 7320 such ideals is │ │ │ │ -00021c80: 6e6f 7420 7468 6520 6f6e 6520 666f 7220 not the one for │ │ │ │ -00021c90: 6120 7365 7420 6f66 2036 2067 656e 6572 a set of 6 gener │ │ │ │ -00021ca0: 6963 2070 6f69 6e74 7320 696e 0a24 5c50 ic points in.$\P │ │ │ │ -00021cb0: 505e 3324 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d P^3$...+-------- │ │ │ │ -00021cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 2d2b 0a7c 6931 3120 3a20 4a20 -----+.|i11 : J │ │ │ │ -00021d10: 3d20 7472 696d 286d 696e 6f72 7328 312c = trim(minors(1, │ │ │ │ -00021d20: 2048 2328 322c 3329 2920 2b20 6772 6f65 H#(2,3)) + groe │ │ │ │ -00021d30: 626e 6572 5374 7261 7475 6d20 4629 3b20 bnerStratum F); │ │ │ │ -00021d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021d50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00021d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021da0: 2020 2020 207c 0a7c 6f31 3120 3a20 4964 |.|o11 : Id │ │ │ │ -00021db0: 6561 6c20 6f66 206b 6b5b 7420 2c20 7420 eal of kk[t , t │ │ │ │ -00021dc0: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -00021dd0: 2074 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00021de0: 2020 2c20 7420 2c20 7420 202c 2074 202c , t , t , t , │ │ │ │ -00021df0: 2074 2020 2c7c 0a7c 2020 2020 2020 2020 t ,|.| │ │ │ │ -00021e00: 2020 2020 2020 2020 2020 2036 2020 2031 6 1 │ │ │ │ -00021e10: 3220 2020 3520 2020 3330 2020 2031 3820 2 5 30 18 │ │ │ │ -00021e20: 2020 3420 2020 3234 2020 2033 3620 2020 4 24 36 │ │ │ │ -00021e30: 3131 2020 2032 2020 2032 3920 2020 3320 11 2 29 3 │ │ │ │ -00021e40: 2020 3130 207c 0a7c 2d2d 2d2d 2d2d 2d2d 10 |.|-------- │ │ │ │ +00021760: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00021770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000217a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000217b0: 0a7c 6f31 3020 3a20 4d61 7472 6978 2028 .|o10 : Matrix ( │ │ │ │ +000217c0: 6b6b 5b74 202c 2074 2020 2c20 7420 2c20 kk[t , t , t , │ │ │ │ +000217d0: 7420 202c 2074 2020 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ +000217e0: 202c 2074 2020 2c20 7420 202c 2074 202c , t , t , t , │ │ │ │ +000217f0: 2074 2020 2c20 7420 2c20 7420 202c 207c t , t , t , | │ │ │ │ +00021800: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00021810: 2020 2020 3620 2020 3132 2020 2035 2020 6 12 5 │ │ │ │ +00021820: 2033 3020 2020 3138 2020 2034 2020 2032 30 18 4 2 │ │ │ │ +00021830: 3420 2020 3336 2020 2031 3120 2020 3220 4 36 11 2 │ │ │ │ +00021840: 2020 3239 2020 2033 2020 2031 3020 207c 29 3 10 | │ │ │ │ +00021850: 0a7c 2d2d 2d2d 2d2d 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 2d7c ---------------| │ │ │ │ +000218a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000218b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000218c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000218d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000218e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000218f0: 0a7c 7420 202c 2074 202c 2074 2020 2c20 .|t , t , t , │ │ │ │ +00021900: 7420 202c 2074 2020 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ +00021910: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +00021920: 2074 2020 2c20 7420 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00021930: 2020 2c20 7420 202c 2074 2020 2c20 207c , t , t , | │ │ │ │ +00021940: 0a7c 2031 3720 2020 3120 2020 3233 2020 .| 17 1 23 │ │ │ │ +00021950: 2032 3820 2020 3335 2020 2038 2020 2031 28 35 8 1 │ │ │ │ +00021960: 3620 2020 3920 2020 3232 2020 2032 3620 6 9 22 26 │ │ │ │ +00021970: 2020 3334 2020 2037 2020 2031 3420 2020 34 7 14 │ │ │ │ +00021980: 3237 2020 2031 3520 2020 3230 2020 207c 27 15 20 | │ │ │ │ +00021990: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ +000219a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000219b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000219c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000219d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +000219e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000219f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021a00: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00021a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021a20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00021a30: 0a7c 7420 202c 2074 2020 2c20 7420 202c .|t , t , t , │ │ │ │ +00021a40: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ +00021a50: 7420 205d 2920 203c 2d2d 2028 6b6b 5b74 t ]) <-- (kk[t │ │ │ │ +00021a60: 202c 2074 2020 2c20 7420 2c20 7420 202c , t , t , t , │ │ │ │ +00021a70: 2074 2020 2c20 7420 2c20 7420 202c 207c t , t , t , | │ │ │ │ +00021a80: 0a7c 2032 3520 2020 3332 2020 2031 3320 .| 25 32 13 │ │ │ │ +00021a90: 2020 3231 2020 2033 3320 2020 3139 2020 21 33 19 │ │ │ │ +00021aa0: 2033 3120 2020 2020 2020 2020 2020 2020 31 │ │ │ │ +00021ab0: 3620 2020 3132 2020 2035 2020 2033 3020 6 12 5 30 │ │ │ │ +00021ac0: 2020 3138 2020 2034 2020 2032 3420 207c 18 4 24 | │ │ │ │ +00021ad0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ +00021ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00021b20: 0a7c 2020 2020 2020 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 207c | │ │ │ │ +00021b70: 0a7c 7420 202c 2074 2020 2c20 7420 2c20 .|t , t , t , │ │ │ │ +00021b80: 7420 202c 2074 202c 2074 2020 2c20 7420 t , t , t , t │ │ │ │ +00021b90: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +00021ba0: 2074 2020 2c20 7420 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00021bb0: 202c 2074 2020 2c20 7420 202c 2020 207c , t , t , | │ │ │ │ +00021bc0: 0a7c 2033 3620 2020 3131 2020 2032 2020 .| 36 11 2 │ │ │ │ +00021bd0: 2032 3920 2020 3320 2020 3130 2020 2031 29 3 10 1 │ │ │ │ +00021be0: 3720 2020 3120 2020 3233 2020 2032 3820 7 1 23 28 │ │ │ │ +00021bf0: 2020 3335 2020 2038 2020 2031 3620 2020 35 8 16 │ │ │ │ +00021c00: 3920 2020 3232 2020 2032 3620 2020 207c 9 22 26 | │ │ │ │ +00021c10: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ +00021c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00021c60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00021c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021ca0: 2020 3120 2020 2020 2020 2020 2020 207c 1 | │ │ │ │ +00021cb0: 0a7c 7420 202c 2074 202c 2074 2020 2c20 .|t , t , t , │ │ │ │ +00021cc0: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00021cd0: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ +00021ce0: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ +00021cf0: 5d29 2020 2020 2020 2020 2020 2020 207c ]) | │ │ │ │ +00021d00: 0a7c 2033 3420 2020 3720 2020 3134 2020 .| 34 7 14 │ │ │ │ +00021d10: 2032 3720 2020 3135 2020 2032 3020 2020 27 15 20 │ │ │ │ +00021d20: 3235 2020 2033 3220 2020 3133 2020 2032 25 32 13 2 │ │ │ │ +00021d30: 3120 2020 3333 2020 2031 3920 2020 3331 1 33 19 31 │ │ │ │ +00021d40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00021d50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00021d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00021da0: 0a0a 4c65 7427 7320 696d 706f 7365 2074 ..Let's impose t │ │ │ │ +00021db0: 6865 2063 6f6e 6469 7469 6f6e 2074 6861 he condition tha │ │ │ │ +00021dc0: 7420 7468 6520 6d61 7020 4823 2832 2c33 t the map H#(2,3 │ │ │ │ +00021dd0: 2920 7661 6e69 7368 6573 2028 736f 2068 ) vanishes (so h │ │ │ │ +00021de0: 6173 2072 616e 6b20 3029 2e20 5468 650a as rank 0). The. │ │ │ │ +00021df0: 4265 7474 6920 6469 6167 7261 6d20 6f66 Betti diagram of │ │ │ │ +00021e00: 2073 7563 6820 6964 6561 6c73 2069 7320 such ideals is │ │ │ │ +00021e10: 6e6f 7420 7468 6520 6f6e 6520 666f 7220 not the one for │ │ │ │ +00021e20: 6120 7365 7420 6f66 2036 2067 656e 6572 a set of 6 gener │ │ │ │ +00021e30: 6963 2070 6f69 6e74 7320 696e 0a24 5c50 ic points in.$\P │ │ │ │ +00021e40: 505e 3324 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d P^3$...+-------- │ │ │ │ 00021e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021e90: 2d2d 2d2d 2d7c 0a7c 7420 202c 2074 202c -----|.|t , t , │ │ │ │ -00021ea0: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ -00021eb0: 7420 2c20 7420 202c 2074 202c 2074 2020 t , t , t , t │ │ │ │ -00021ec0: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ -00021ed0: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00021ee0: 2020 2c20 207c 0a7c 2031 3720 2020 3120 , |.| 17 1 │ │ │ │ -00021ef0: 2020 3233 2020 2032 3820 2020 3335 2020 23 28 35 │ │ │ │ -00021f00: 2038 2020 2031 3620 2020 3920 2020 3232 8 16 9 22 │ │ │ │ -00021f10: 2020 2032 3620 2020 3334 2020 2037 2020 26 34 7 │ │ │ │ -00021f20: 2031 3420 2020 3237 2020 2031 3520 2020 14 27 15 │ │ │ │ -00021f30: 3230 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 20 |.|-------- │ │ │ │ -00021f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021f80: 2d2d 2d2d 2d7c 0a7c 7420 202c 2074 2020 -----|.|t , t │ │ │ │ -00021f90: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -00021fa0: 2074 2020 2c20 7420 205d 2020 2020 2020 t , t ] │ │ │ │ -00021fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021fd0: 2020 2020 207c 0a7c 2032 3520 2020 3332 |.| 25 32 │ │ │ │ -00021fe0: 2020 2031 3320 2020 3231 2020 2033 3320 13 21 33 │ │ │ │ -00021ff0: 2020 3139 2020 2033 3120 2020 2020 2020 19 31 │ │ │ │ -00022000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022020: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00022030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022070: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 636f -----+.|i12 : co │ │ │ │ -00022080: 6d70 734a 203d 2064 6563 6f6d 706f 7365 mpsJ = decompose │ │ │ │ -00022090: 204a 3b20 2020 2020 2020 2020 2020 2020 J; │ │ │ │ -000220a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000220b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000220c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00021e90: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 4a20 -----+.|i11 : J │ │ │ │ +00021ea0: 3d20 7472 696d 286d 696e 6f72 7328 312c = trim(minors(1, │ │ │ │ +00021eb0: 2048 2328 322c 3329 2920 2b20 6772 6f65 H#(2,3)) + groe │ │ │ │ +00021ec0: 626e 6572 5374 7261 7475 6d20 4629 3b20 bnerStratum F); │ │ │ │ +00021ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021ee0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00021ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021f30: 2020 2020 207c 0a7c 6f31 3120 3a20 4964 |.|o11 : Id │ │ │ │ +00021f40: 6561 6c20 6f66 206b 6b5b 7420 2c20 7420 eal of kk[t , t │ │ │ │ +00021f50: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +00021f60: 2074 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00021f70: 2020 2c20 7420 2c20 7420 202c 2074 202c , t , t , t , │ │ │ │ +00021f80: 2074 2020 2c7c 0a7c 2020 2020 2020 2020 t ,|.| │ │ │ │ +00021f90: 2020 2020 2020 2020 2020 2036 2020 2031 6 1 │ │ │ │ +00021fa0: 3220 2020 3520 2020 3330 2020 2031 3820 2 5 30 18 │ │ │ │ +00021fb0: 2020 3420 2020 3234 2020 2033 3620 2020 4 24 36 │ │ │ │ +00021fc0: 3131 2020 2032 2020 2032 3920 2020 3320 11 2 29 3 │ │ │ │ +00021fd0: 2020 3130 207c 0a7c 2d2d 2d2d 2d2d 2d2d 10 |.|-------- │ │ │ │ +00021fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022020: 2d2d 2d2d 2d7c 0a7c 7420 202c 2074 202c -----|.|t , t , │ │ │ │ +00022030: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ +00022040: 7420 2c20 7420 202c 2074 202c 2074 2020 t , t , t , t │ │ │ │ +00022050: 2c20 7420 202c 2074 2020 2c20 7420 2c20 , t , t , t , │ │ │ │ +00022060: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00022070: 2020 2c20 207c 0a7c 2031 3720 2020 3120 , |.| 17 1 │ │ │ │ +00022080: 2020 3233 2020 2032 3820 2020 3335 2020 23 28 35 │ │ │ │ +00022090: 2038 2020 2031 3620 2020 3920 2020 3232 8 16 9 22 │ │ │ │ +000220a0: 2020 2032 3620 2020 3334 2020 2037 2020 26 34 7 │ │ │ │ +000220b0: 2031 3420 2020 3237 2020 2031 3520 2020 14 27 15 │ │ │ │ +000220c0: 3230 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 20 |.|-------- │ │ │ │ 000220d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000220e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000220f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022110: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 2363 -----+.|i13 : #c │ │ │ │ -00022120: 6f6d 7073 4a20 2020 2020 2020 2020 2020 ompsJ │ │ │ │ -00022130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022110: 2d2d 2d2d 2d7c 0a7c 7420 202c 2074 2020 -----|.|t , t │ │ │ │ +00022120: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +00022130: 2074 2020 2c20 7420 205d 2020 2020 2020 t , t ] │ │ │ │ 00022140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022160: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022160: 2020 2020 207c 0a7c 2032 3520 2020 3332 |.| 25 32 │ │ │ │ +00022170: 2020 2031 3320 2020 3231 2020 2033 3320 13 21 33 │ │ │ │ +00022180: 2020 3139 2020 2033 3120 2020 2020 2020 19 31 │ │ │ │ 00022190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000221a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000221b0: 2020 2020 207c 0a7c 6f31 3320 3d20 3220 |.|o13 = 2 │ │ │ │ -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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00022210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022250: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 7074 -----+.|i14 : pt │ │ │ │ -00022260: 3120 3d20 7261 6e64 6f6d 506f 696e 744f 1 = randomPointO │ │ │ │ -00022270: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ -00022280: 2063 6f6d 7073 4a5f 3020 2020 2020 2020 compsJ_0 │ │ │ │ -00022290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000222b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000221b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000221c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000221d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000221e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000221f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022200: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 636f -----+.|i12 : co │ │ │ │ +00022210: 6d70 734a 203d 2064 6563 6f6d 706f 7365 mpsJ = decompose │ │ │ │ +00022220: 204a 3b20 2020 2020 2020 2020 2020 2020 J; │ │ │ │ +00022230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022250: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00022260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000222a0: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 2363 -----+.|i13 : #c │ │ │ │ +000222b0: 6f6d 7073 4a20 2020 2020 2020 2020 2020 ompsJ │ │ │ │ 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 207c 0a7c 6f31 3420 3d20 7c20 |.|o14 = | │ │ │ │ -00022300: 2d34 3820 2d32 3120 2d32 3820 2d33 3220 -48 -21 -28 -32 │ │ │ │ -00022310: 3136 2033 3120 3135 202d 3132 2031 3420 16 31 15 -12 14 │ │ │ │ -00022320: 3520 2d31 3320 3239 202d 3430 202d 3232 5 -13 29 -40 -22 │ │ │ │ -00022330: 2032 2033 3020 3235 202d 3130 2033 202d 2 30 25 -10 3 - │ │ │ │ -00022340: 3120 3320 207c 0a7c 2020 2020 2020 2d2d 1 3 |.| -- │ │ │ │ -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 2d7c 0a7c 2020 2020 2020 2d33 -----|.| -3 │ │ │ │ -000223a0: 3020 3139 202d 3136 202d 3234 202d 3239 0 19 -16 -24 -29 │ │ │ │ -000223b0: 202d 3239 2032 3420 2d32 3920 2d32 3420 -29 24 -29 -24 │ │ │ │ -000223c0: 2d33 3620 3231 202d 3820 2d33 3820 3139 -36 21 -8 -38 19 │ │ │ │ -000223d0: 2033 3920 7c20 2020 2020 2020 2020 2020 39 | │ │ │ │ -000223e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000223f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000222f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00022300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022340: 2020 2020 207c 0a7c 6f31 3320 3d20 3220 |.|o13 = 2 │ │ │ │ +00022350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022390: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000223a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000223b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000223c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000223d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000223e0: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 7074 -----+.|i14 : pt │ │ │ │ +000223f0: 3120 3d20 7261 6e64 6f6d 506f 696e 744f 1 = randomPointO │ │ │ │ +00022400: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ +00022410: 2063 6f6d 7073 4a5f 3020 2020 2020 2020 compsJ_0 │ │ │ │ 00022420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022430: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022440: 2020 2020 2020 2031 2020 2020 2020 2033 1 3 │ │ │ │ -00022450: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +00022440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022480: 2020 2020 207c 0a7c 6f31 3420 3a20 4d61 |.|o14 : Ma │ │ │ │ -00022490: 7472 6978 206b 6b20 203c 2d2d 206b 6b20 trix kk <-- kk │ │ │ │ -000224a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00022480: 2020 2020 207c 0a7c 6f31 3420 3d20 7c20 |.|o14 = | │ │ │ │ +00022490: 2d33 3820 3237 202d 3231 202d 3339 202d -38 27 -21 -39 - │ │ │ │ +000224a0: 3134 2031 3620 3335 202d 3220 3920 2d31 14 16 35 -2 9 -1 │ │ │ │ +000224b0: 3720 2d31 3420 3439 202d 3436 202d 3130 7 -14 49 -46 -10 │ │ │ │ +000224c0: 2033 3120 3232 2031 2031 3920 3120 2d31 31 22 1 19 1 -1 │ │ │ │ +000224d0: 3820 3138 207c 0a7c 2020 2020 2020 2d2d 8 18 |.| -- │ │ │ │ 000224e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000224f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022520: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 7074 -----+.|i15 : pt │ │ │ │ -00022530: 3220 3d20 7261 6e64 6f6d 506f 696e 744f 2 = randomPointO │ │ │ │ -00022540: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ -00022550: 2063 6f6d 7073 4a5f 3120 2020 2020 2020 compsJ_1 │ │ │ │ -00022560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022520: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 3234 -----|.| 24 │ │ │ │ +00022530: 202d 3330 202d 3234 2031 3220 2d32 3920 -30 -24 12 -29 │ │ │ │ +00022540: 2d33 3620 3339 2031 3920 2d38 2032 3120 -36 39 19 -8 21 │ │ │ │ +00022550: 2d31 3620 2d32 3920 2d32 3220 2d32 3920 -16 -29 -22 -29 │ │ │ │ +00022560: 2d33 3820 7c20 2020 2020 2020 2020 2020 -38 | │ │ │ │ 00022570: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00022580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000225a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000225b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000225c0: 2020 2020 207c 0a7c 6f31 3520 3d20 7c20 |.|o15 = | │ │ │ │ -000225d0: 3620 3239 202d 3530 2034 3120 3530 202d 6 29 -50 41 50 - │ │ │ │ -000225e0: 3139 202d 3135 202d 3334 202d 3238 202d 19 -15 -34 -28 - │ │ │ │ -000225f0: 3434 2032 3120 2d34 3520 2d32 3320 2d31 44 21 -45 -23 -1 │ │ │ │ -00022600: 3320 2d32 3320 3235 2033 3120 2d31 3820 3 -23 25 31 -18 │ │ │ │ -00022610: 3435 2032 327c 0a7c 2020 2020 2020 2d2d 45 22|.| -- │ │ │ │ -00022620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022660: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2d34 -----|.| -4 │ │ │ │ -00022670: 3020 3137 202d 3437 2033 3820 2d32 3520 0 17 -47 38 -25 │ │ │ │ -00022680: 2d34 3320 3139 2032 3620 2d34 3720 2d31 -43 19 26 -47 -1 │ │ │ │ -00022690: 3520 3334 2031 3620 3020 2d32 3820 2d33 5 34 16 0 -28 -3 │ │ │ │ -000226a0: 3920 3220 7c20 2020 2020 2020 2020 2020 9 2 | │ │ │ │ -000226b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000226c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000226d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000226e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000225d0: 2020 2020 2020 2031 2020 2020 2020 2033 1 3 │ │ │ │ +000225e0: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +000225f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022610: 2020 2020 207c 0a7c 6f31 3420 3a20 4d61 |.|o14 : Ma │ │ │ │ +00022620: 7472 6978 206b 6b20 203c 2d2d 206b 6b20 trix kk <-- kk │ │ │ │ +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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00022670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000226a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000226b0: 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 7074 -----+.|i15 : pt │ │ │ │ +000226c0: 3220 3d20 7261 6e64 6f6d 506f 696e 744f 2 = randomPointO │ │ │ │ +000226d0: 6e52 6174 696f 6e61 6c56 6172 6965 7479 nRationalVariety │ │ │ │ +000226e0: 2063 6f6d 7073 4a5f 3120 2020 2020 2020 compsJ_1 │ │ │ │ 000226f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022700: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022710: 2020 2020 2020 2031 2020 2020 2020 2033 1 3 │ │ │ │ -00022720: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +00022710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022750: 2020 2020 207c 0a7c 6f31 3520 3a20 4d61 |.|o15 : Ma │ │ │ │ -00022760: 7472 6978 206b 6b20 203c 2d2d 206b 6b20 trix kk <-- kk │ │ │ │ -00022770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000227a0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00022750: 2020 2020 207c 0a7c 6f31 3520 3d20 7c20 |.|o15 = | │ │ │ │ +00022760: 3330 2031 3020 3433 2032 3020 2d33 3920 30 10 43 20 -39 │ │ │ │ +00022770: 3233 202d 3330 2034 3020 2d33 3420 3232 23 -30 40 -34 22 │ │ │ │ +00022780: 2034 3620 2d32 3520 3231 202d 3138 202d 46 -25 21 -18 - │ │ │ │ +00022790: 3335 202d 3120 3231 202d 3339 202d 3435 35 -1 21 -39 -45 │ │ │ │ +000227a0: 2031 3620 207c 0a7c 2020 2020 2020 2d2d 16 |.| -- │ │ │ │ 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 2d2b 0a7c 6931 3620 3a20 4631 -----+.|i16 : F1 │ │ │ │ -00022800: 203d 2073 7562 2846 2c20 2876 6172 7320 = sub(F, (vars │ │ │ │ -00022810: 5329 7c70 7431 2920 2020 2020 2020 2020 S)|pt1) │ │ │ │ -00022820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000227f0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2d33 -----|.| -3 │ │ │ │ +00022800: 3520 2d35 2031 3920 2d34 3720 2d32 3020 5 -5 19 -47 -20 │ │ │ │ +00022810: 2d31 3320 3334 2033 3320 2d32 3820 2d34 -13 34 33 -28 -4 │ │ │ │ +00022820: 3320 3232 2032 2030 202d 3135 202d 3437 3 22 2 0 -15 -47 │ │ │ │ +00022830: 2033 3820 7c20 2020 2020 2020 2020 2020 38 | │ │ │ │ 00022840: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00022850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022890: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000228a0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -000228b0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000228a0: 2020 2020 2020 2031 2020 2020 2020 2033 1 3 │ │ │ │ +000228b0: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ 000228c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000228d0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000228e0: 2020 2020 207c 0a7c 6f31 3620 3d20 6964 |.|o16 = id │ │ │ │ -000228f0: 6561 6c20 2861 2020 2b20 3262 2a63 202b eal (a + 2b*c + │ │ │ │ -00022900: 2035 6320 202b 2032 3961 2a64 202b 2033 5c + 29a*d + 3 │ │ │ │ -00022910: 3162 2a64 202d 2032 3863 2a64 202d 2034 1b*d - 28c*d - 4 │ │ │ │ -00022920: 3864 202c 2061 2a62 202d 2032 3462 2a63 8d , a*b - 24b*c │ │ │ │ -00022930: 202b 2020 207c 0a7c 2020 2020 2020 2d2d + |.| -- │ │ │ │ +000228d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000228e0: 2020 2020 207c 0a7c 6f31 3520 3a20 4d61 |.|o15 : Ma │ │ │ │ +000228f0: 7472 6978 206b 6b20 203c 2d2d 206b 6b20 trix kk <-- kk │ │ │ │ +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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00022940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022980: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -00022990: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000229a0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00022980: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 4631 -----+.|i16 : F1 │ │ │ │ +00022990: 203d 2073 7562 2846 2c20 2876 6172 7320 = sub(F, (vars │ │ │ │ +000229a0: 5329 7c70 7431 2920 2020 2020 2020 2020 S)|pt1) │ │ │ │ 000229b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229c0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000229d0: 2020 2020 207c 0a7c 2020 2020 2020 3363 |.| 3c │ │ │ │ -000229e0: 2020 2b20 3361 2a64 202d 2034 3062 2a64 + 3a*d - 40b*d │ │ │ │ -000229f0: 202b 2031 3463 2a64 202d 2032 3164 202c + 14c*d - 21d , │ │ │ │ -00022a00: 2061 2a63 202b 2032 3162 2a63 202d 2032 a*c + 21b*c - 2 │ │ │ │ -00022a10: 3963 2020 2b20 3234 612a 6420 2d20 622a 9c + 24a*d - b* │ │ │ │ -00022a20: 6420 2d20 207c 0a7c 2020 2020 2020 2d2d d - |.| -- │ │ │ │ -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 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -00022a80: 2020 2020 2020 2020 2032 2020 2032 2020 2 2 │ │ │ │ -00022a90: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ab0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -00022ac0: 3220 2020 207c 0a7c 2020 2020 2020 3232 2 |.| 22 │ │ │ │ -00022ad0: 632a 6420 2b20 3136 6420 2c20 6220 202b c*d + 16d , b + │ │ │ │ -00022ae0: 2031 3962 2a63 202d 2032 3963 2020 2d20 19b*c - 29c - │ │ │ │ -00022af0: 3861 2a64 202d 2033 3062 2a64 202b 2033 8a*d - 30b*d + 3 │ │ │ │ -00022b00: 3063 2a64 202b 2031 3564 202c 2062 2a63 0c*d + 15d , b*c │ │ │ │ -00022b10: 2020 2d20 207c 0a7c 2020 2020 2020 2d2d - |.| -- │ │ │ │ -00022b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022b60: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -00022b70: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00022b80: 2020 2020 2032 2020 2020 2020 2020 3220 2 2 │ │ │ │ -00022b90: 2020 2020 2020 2032 2020 2020 2020 3320 2 3 │ │ │ │ -00022ba0: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -00022bb0: 2020 2032 207c 0a7c 2020 2020 2020 3234 2 |.| 24 │ │ │ │ -00022bc0: 622a 632a 6420 2b20 3139 6320 6420 2d20 b*c*d + 19c d - │ │ │ │ -00022bd0: 3239 612a 6420 202b 2032 3562 2a64 2020 29a*d + 25b*d │ │ │ │ -00022be0: 2d20 3133 632a 6420 202d 2033 3264 202c - 13c*d - 32d , │ │ │ │ -00022bf0: 2063 2020 2b20 3339 622a 632a 6420 2d20 c + 39b*c*d - │ │ │ │ -00022c00: 3336 6320 647c 0a7c 2020 2020 2020 2d2d 36c d|.| -- │ │ │ │ -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 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -00022c60: 2020 2020 2032 2020 2020 2020 2020 3220 2 2 │ │ │ │ -00022c70: 2020 2020 2020 2032 2020 2020 2020 3320 2 3 │ │ │ │ -00022c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ca0: 2020 2020 207c 0a7c 2020 2020 2020 2d20 |.| - │ │ │ │ -00022cb0: 3338 612a 6420 202d 2031 3662 2a64 2020 38a*d - 16b*d │ │ │ │ -00022cc0: 2d20 3130 632a 6420 202d 2031 3264 2029 - 10c*d - 12d ) │ │ │ │ -00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cf0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000229c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000229d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000229e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000229f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022a20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00022a30: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00022a40: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00022a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022a60: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00022a70: 2020 2020 207c 0a7c 6f31 3620 3d20 6964 |.|o16 = id │ │ │ │ +00022a80: 6561 6c20 2861 2020 2b20 3331 622a 6320 eal (a + 31b*c │ │ │ │ +00022a90: 2d20 3137 6320 202b 2034 3961 2a64 202b - 17c + 49a*d + │ │ │ │ +00022aa0: 2031 3662 2a64 202d 2032 3163 2a64 202d 16b*d - 21c*d - │ │ │ │ +00022ab0: 2033 3864 202c 2061 2a62 202b 2031 3262 38d , a*b + 12b │ │ │ │ +00022ac0: 2a63 202b 207c 0a7c 2020 2020 2020 2d2d *c + |.| -- │ │ │ │ +00022ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022b10: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2032 -----|.| 2 │ │ │ │ +00022b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022b30: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00022b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022b50: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00022b60: 2020 2020 207c 0a7c 2020 2020 2020 6320 |.| c │ │ │ │ +00022b70: 202b 2031 3861 2a64 202d 2034 3662 2a64 + 18a*d - 46b*d │ │ │ │ +00022b80: 202b 2039 632a 6420 2b20 3237 6420 2c20 + 9c*d + 27d , │ │ │ │ +00022b90: 612a 6320 2d20 3136 622a 6320 2d20 3239 a*c - 16b*c - 29 │ │ │ │ +00022ba0: 6320 202b 2033 3961 2a64 202d 2031 3862 c + 39a*d - 18b │ │ │ │ +00022bb0: 2a64 202d 207c 0a7c 2020 2020 2020 2d2d *d - |.| -- │ │ │ │ +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 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +00022c10: 2020 2020 2020 2020 2032 2020 2032 2020 2 2 │ │ │ │ +00022c20: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +00022c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022c40: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +00022c50: 2032 2020 207c 0a7c 2020 2020 2020 3130 2 |.| 10 │ │ │ │ +00022c60: 632a 6420 2d20 3134 6420 2c20 6220 202d c*d - 14d , b - │ │ │ │ +00022c70: 2032 3962 2a63 202b 2031 3963 2020 2d20 29b*c + 19c - │ │ │ │ +00022c80: 3239 612a 6420 2b20 3234 622a 6420 2b20 29a*d + 24b*d + │ │ │ │ +00022c90: 3232 632a 6420 2b20 3335 6420 2c20 622a 22c*d + 35d , b* │ │ │ │ +00022ca0: 6320 202d 207c 0a7c 2020 2020 2020 2d2d c - |.| -- │ │ │ │ +00022cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022cf0: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ +00022d00: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00022d10: 2020 2020 3220 2020 2020 2032 2020 2020 2 2 │ │ │ │ +00022d20: 2020 2020 3220 2020 2020 2033 2020 2033 2 3 3 │ │ │ │ 00022d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d40: 2020 2020 207c 0a7c 6f31 3620 3a20 4964 |.|o16 : Id │ │ │ │ -00022d50: 6561 6c20 6f66 2053 2020 2020 2020 2020 eal of S │ │ │ │ -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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00022d40: 3220 2020 207c 0a7c 2020 2020 2020 3862 2 |.| 8b │ │ │ │ +00022d50: 2a63 2a64 202d 2033 3063 2064 202d 2033 *c*d - 30c d - 3 │ │ │ │ +00022d60: 3661 2a64 2020 2b20 622a 6420 202d 2031 6a*d + b*d - 1 │ │ │ │ +00022d70: 3463 2a64 2020 2d20 3339 6420 2c20 6320 4c*d - 39d , c │ │ │ │ +00022d80: 202d 2033 3862 2a63 2a64 202b 2032 3163 - 38b*c*d + 21c │ │ │ │ +00022d90: 2064 202d 207c 0a7c 2020 2020 2020 2d2d d - |.| -- │ │ │ │ 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 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2020 │ │ │ │ 00022e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022e90: 2020 2020 2030 2031 2032 2033 2020 2020 0 1 2 3 │ │ │ │ +00022e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ed0: 2020 2020 207c 0a7c 6f31 3720 3d20 746f |.|o17 = to │ │ │ │ -00022ee0: 7461 6c3a 2031 2036 2038 2033 2020 2020 tal: 1 6 8 3 │ │ │ │ +00022ed0: 2020 2020 207c 0a7c 6f31 3620 3a20 4964 |.|o16 : Id │ │ │ │ +00022ee0: 6561 6c20 6f66 2053 2020 2020 2020 2020 eal of S │ │ │ │ 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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022f30: 2020 303a 2031 202e 202e 202e 2020 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2020 2020 2020 2020 │ │ │ │ 00022fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022fd0: 2020 323a 202e 2032 2034 2032 2020 2020 2: . 2 4 2 │ │ │ │ +00022fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00023020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023020: 2020 2020 2030 2031 2032 2033 2020 2020 0 1 2 3 │ │ │ │ 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 207c 0a7c 6f31 3720 3a20 4265 |.|o17 : Be │ │ │ │ -00023070: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +00023060: 2020 2020 207c 0a7c 6f31 3720 3d20 746f |.|o17 = to │ │ │ │ +00023070: 7461 6c3a 2031 2036 2038 2033 2020 2020 tal: 1 6 8 3 │ │ │ │ 00023080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000230a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000230b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -000230c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000230d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000230e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000230f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023100: 2d2d 2d2d 2d2b 0a7c 6931 3820 3a20 4632 -----+.|i18 : F2 │ │ │ │ -00023110: 203d 2073 7562 2846 2c20 2876 6172 7320 = sub(F, (vars │ │ │ │ -00023120: 5329 7c70 7432 2920 2020 2020 2020 2020 S)|pt2) │ │ │ │ +000230b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000230c0: 2020 303a 2031 202e 202e 202e 2020 2020 0: 1 . . . │ │ │ │ +000230d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 2020 2020 │ │ │ │ +000231c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000231d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000231e0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000231f0: 2020 2020 207c 0a7c 6f31 3820 3d20 6964 |.|o18 = id │ │ │ │ -00023200: 6561 6c20 2861 2020 2d20 3233 622a 6320 eal (a - 23b*c │ │ │ │ -00023210: 2d20 3434 6320 202d 2034 3561 2a64 202d - 44c - 45a*d - │ │ │ │ -00023220: 2031 3962 2a64 202d 2035 3063 2a64 202b 19b*d - 50c*d + │ │ │ │ -00023230: 2036 6420 2c20 612a 6220 2d20 3235 622a 6d , a*b - 25b* │ │ │ │ -00023240: 6320 2b20 207c 0a7c 2020 2020 2020 2d2d c + |.| -- │ │ │ │ +000231e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000231f0: 2020 2020 207c 0a7c 6f31 3720 3a20 4265 |.|o17 : Be │ │ │ │ +00023200: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +00023210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023240: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00023250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023290: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -000232a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000232b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232c0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000232d0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -000232e0: 2020 2020 207c 0a7c 2020 2020 2020 3435 |.| 45 │ │ │ │ -000232f0: 6320 202d 2034 3061 2a64 202d 2032 3362 c - 40a*d - 23b │ │ │ │ -00023300: 2a64 202d 2032 3863 2a64 202b 2032 3964 *d - 28c*d + 29d │ │ │ │ -00023310: 202c 2061 2a63 202b 2031 3662 2a63 202d , a*c + 16b*c - │ │ │ │ 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6420 2b20 3235 632a 6420 + 17b*d + 25c*d │ │ │ │ -00023410: 2d20 3135 6420 2c20 622a 6320 202d 2020 - 15d , b*c - │ │ │ │ -00023420: 2020 2020 207c 0a7c 2020 2020 2020 2d2d |.| -- │ │ │ │ -00023430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023470: 2d2d 2d2d 2d7c 0a7c 2020 2020 2020 2020 -----|.| │ │ │ │ -00023480: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00023490: 2020 2020 2032 2020 2020 2020 2020 3220 2 2 │ │ │ │ -000234a0: 2020 2020 2020 2032 2020 2020 2020 3320 2 3 │ │ │ │ -000234b0: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -000234c0: 2020 3220 207c 0a7c 2020 2020 2020 3135 2 |.| 15 │ │ │ │ -000234d0: 622a 632a 6420 2d20 3437 6320 6420 2b20 b*c*d - 47c d + │ │ │ │ -000234e0: 3139 612a 6420 202b 2033 3162 2a64 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7320 = sub(F, (vars │ │ │ │ +000232b0: 5329 7c70 7432 2920 2020 2020 2020 2020 S)|pt2) │ │ │ │ +000232c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000232d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000232e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000232f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023330: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00023340: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00023350: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00023360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023370: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00023380: 2020 2020 207c 0a7c 6f31 3820 3d20 6964 |.|o18 = id │ │ │ │ +00023390: 6561 6c20 2861 2020 2d20 3335 622a 6320 eal (a - 35b*c │ │ │ │ +000233a0: 2b20 3232 6320 202d 2032 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-----|.| │ │ │ │ +00023700: 2020 2020 2032 2020 2020 2020 2020 3220 2 2 │ │ │ │ +00023710: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ 00023720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023740: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00023750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023740: 2020 2020 207c 0a7c 2020 2020 2020 2d20 |.| - │ │ │ │ +00023750: 3437 622a 6420 202d 2033 3963 2a64 2020 47b*d - 39c*d │ │ │ │ +00023760: 2b20 3430 6420 2920 2020 2020 2020 2020 + 40d ) │ │ │ │ 00023770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023790: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000237a0: 2020 2020 2030 2031 2032 2033 2020 2020 0 1 2 3 │ │ │ │ +000237a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000237b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000237c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000237d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000237e0: 2020 2020 207c 0a7c 6f31 3920 3d20 746f |.|o19 = to │ │ │ │ -000237f0: 7461 6c3a 2031 2036 2038 2033 2020 2020 tal: 1 6 8 3 │ │ │ │ +000237e0: 2020 2020 207c 0a7c 6f31 3820 3a20 4964 |.|o18 : Id │ │ │ │ +000237f0: 6561 6c20 6f66 2053 2020 2020 2020 2020 eal of S │ │ │ │ 00023800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023830: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00023840: 2020 303a 2031 202e 202e 202e 2020 2020 0: 1 . . . │ │ │ │ -00023850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023880: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00023890: 2020 313a 202e 2034 2034 2031 2020 2020 1: . 4 4 1 │ │ │ │ +00023830: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00023840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023880: 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 6265 -----+.|i19 : be │ │ │ │ +00023890: 7474 6920 7265 7320 4632 2020 2020 2020 tti res F2 │ │ │ │ 000238a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000238b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000238c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000238d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000238e0: 2020 323a 202e 2032 2034 2032 2020 2020 2: . 2 4 2 │ │ │ │ +000238e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000238f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023920: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00023930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023930: 2020 2020 2030 2031 2032 2033 2020 2020 0 1 2 3 │ │ │ │ 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 207c 0a7c 6f31 3920 3a20 4265 |.|o19 : Be │ │ │ │ -00023980: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +00023970: 2020 2020 207c 0a7c 6f31 3920 3d20 746f |.|o19 = to │ │ │ │ +00023980: 7461 6c3a 2031 2036 2038 2033 2020 2020 tal: 1 6 8 3 │ │ │ │ 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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -000239d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000239e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000239f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023a10: 2d2d 2d2d 2d2b 0a0a 5768 6174 2061 7265 -----+..What are │ │ │ │ -00023a20: 2074 6865 2069 6465 616c 7320 4631 2061 the ideals F1 a │ │ │ │ -00023a30: 6e64 2046 323f 0a0a 2b2d 2d2d 2d2d 2d2d nd F2?..+------- │ │ │ │ -00023a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023a80: 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a 206e ------+.|i20 : n │ │ │ │ -00023a90: 6574 4c69 7374 2064 6563 6f6d 706f 7365 etList decompose │ │ │ │ -00023aa0: 2046 3120 2020 2020 2020 2020 2020 2020 F1 │ │ │ │ -00023ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000239c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000239d0: 2020 303a 2031 202e 202e 202e 2020 2020 0: 1 . . . │ │ │ │ +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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00023a20: 2020 313a 202e 2034 2034 2031 2020 2020 1: . 4 4 1 │ │ │ │ +00023a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023a60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00023a70: 2020 323a 202e 2032 2034 2032 2020 2020 2: . 2 4 2 │ │ │ │ +00023a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ab0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00023ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ad0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023ad0: 2020 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 2020 │ │ │ │ -00023b20: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ -00023b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023b60: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023b70: 2020 2020 2020 7c0a 7c6f 3230 203d 207c |.|o20 = | │ │ │ │ -00023b80: 6964 6561 6c20 2863 202d 2034 3964 2c20 ideal (c - 49d, │ │ │ │ -00023b90: 6220 2d20 3664 2c20 6120 2d20 3135 6429 b - 6d, a - 15d) │ │ │ │ -00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bb0: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ -00023bc0: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +00023b00: 2020 2020 207c 0a7c 6f31 3920 3a20 4265 |.|o19 : Be │ │ │ │ +00023b10: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00023b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ba0: 2d2d 2d2d 2d2b 0a0a 5768 6174 2061 7265 -----+..What are │ │ │ │ +00023bb0: 2074 6865 2069 6465 616c 7320 4631 2061 the ideals F1 a │ │ │ │ +00023bc0: 6e64 2046 323f 0a0a 2b2d 2d2d 2d2d 2d2d nd F2?..+------- │ │ │ │ 00023bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c00: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023c10: 2020 2020 2020 7c0a 7c20 2020 2020 207c |.| | │ │ │ │ -00023c20: 6964 6561 6c20 2863 202b 2032 3464 2c20 ideal (c + 24d, │ │ │ │ -00023c30: 6220 2b20 3337 642c 2061 202b 2033 3564 b + 37d, a + 35d │ │ │ │ -00023c40: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00023c50: 2020 2020 2020 207c 2020 2020 2020 2020 | │ │ │ │ -00023c60: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ -00023c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ca0: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023cb0: 2020 2020 2020 7c0a 7c20 2020 2020 207c |.| | │ │ │ │ -00023cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ce0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00023cf0: 2020 2020 2032 207c 2020 2020 2020 2020 2 | │ │ │ │ -00023d00: 2020 2020 2020 7c0a 7c20 2020 2020 207c |.| | │ │ │ │ -00023d10: 6964 6561 6c20 2862 202d 2031 3863 202d ideal (b - 18c - │ │ │ │ -00023d20: 2033 3264 2c20 6120 2b20 3436 6320 2b20 32d, a + 46c + │ │ │ │ -00023d30: 3333 642c 2063 2020 2d20 3236 632a 6420 33d, c - 26c*d │ │ │ │ -00023d40: 2d20 3135 6420 297c 2020 2020 2020 2020 - 15d )| │ │ │ │ +00023c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023c10: 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a 206e ------+.|i20 : n │ │ │ │ +00023c20: 6574 4c69 7374 2064 6563 6f6d 706f 7365 etList decompose │ │ │ │ +00023c30: 2046 3120 2020 2020 2020 2020 2020 2020 F1 │ │ │ │ +00023c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023cb0: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023d00: 2d2d 2d2d 2d2d 7c0a 7c6f 3230 203d 207c ------|.|o20 = | │ │ │ │ +00023d10: 6964 6561 6c20 2863 202d 2064 2c20 6220 ideal (c - d, b │ │ │ │ +00023d20: 2b20 3438 642c 2061 202b 2034 3264 2920 + 48d, a + 42d) │ │ │ │ +00023d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d50: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ 00023d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023d90: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023da0: 2020 2020 2020 7c0a 7c20 2020 2020 207c |.| | │ │ │ │ -00023db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023dd0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00023de0: 2020 2020 2032 207c 2020 2020 2020 2020 2 | │ │ │ │ -00023df0: 2020 2020 2020 7c0a 7c20 2020 2020 207c |.| | │ │ │ │ -00023e00: 6964 6561 6c20 2862 202d 2033 3763 202d ideal (b - 37c - │ │ │ │ -00023e10: 2032 3764 2c20 6120 2b20 3431 6320 2b20 27d, a + 41c + │ │ │ │ -00023e20: 3239 642c 2063 2020 2b20 3135 632a 6420 29d, c + 15c*d │ │ │ │ -00023e30: 2b20 3136 6420 297c 2020 2020 2020 2020 + 16d )| │ │ │ │ -00023e40: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ -00023e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e80: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ -00023e90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -00023ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 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 2b0a 7c69 3231 203a 206e ------+.|i21 : n │ │ │ │ -00023ef0: 6574 4c69 7374 2064 6563 6f6d 706f 7365 etList decompose │ │ │ │ -00023f00: 2046 3220 2020 2020 2020 2020 2020 2020 F2 │ │ │ │ -00023f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f30: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00023f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f80: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +00023d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023da0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 207c ------|.| | │ │ │ │ +00023db0: 6964 6561 6c20 2863 202b 2033 3964 2c20 ideal (c + 39d, │ │ │ │ +00023dc0: 6220 2d20 3233 642c 2061 202b 2033 3464 b - 23d, a + 34d │ │ │ │ +00023dd0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00023de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023df0: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e40: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 207c ------|.| | │ │ │ │ +00023e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023e70: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00023e80: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +00023e90: 3220 2020 2020 7c0a 7c20 2020 2020 207c 2 |.| | │ │ │ │ +00023ea0: 6964 6561 6c20 2861 202d 2031 3662 202d ideal (a - 16b - │ │ │ │ +00023eb0: 2032 3963 202b 2031 3064 2c20 622a 6320 29c + 10d, b*c │ │ │ │ +00023ec0: 2d20 3338 6320 202b 2032 3362 2a64 202b - 38c + 23b*d + │ │ │ │ +00023ed0: 2034 3363 2a64 202d 2031 3664 202c 2062 43c*d - 16d , b │ │ │ │ +00023ee0: 2020 2b20 3238 7c0a 7c20 2020 2020 202b + 28|.| + │ │ │ │ +00023ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023f00: 2d2d 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 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +00023f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023f80: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ 00023f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fd0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 207c ------|.| | │ │ │ │ +00023fa0: 2d2d 2d2b 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 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00023fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ff0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00023ff0: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00024000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024010: 2020 2020 2020 2032 2020 2032 2020 2020 2 2 │ │ │ │ -00024020: 2020 2020 2020 7c0a 7c6f 3231 203d 207c |.|o21 = | │ │ │ │ -00024030: 6964 6561 6c20 2861 2a63 202b 2031 3662 ideal (a*c + 16b │ │ │ │ -00024040: 2a63 202d 2034 3363 2020 2b20 3236 612a *c - 43c + 26a* │ │ │ │ -00024050: 6420 2b20 3232 622a 6420 2d20 3133 632a d + 22b*d - 13c* │ │ │ │ -00024060: 6420 2b20 3530 6420 2c20 6220 202d 2033 d + 50d , b - 3 │ │ │ │ -00024070: 3962 2a63 202d 7c0a 7c20 2020 2020 202b 9b*c -|.| + │ │ │ │ -00024080: 2d2d 2d2d 2d2d 2d2d 2d2d 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 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +00024010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024020: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d |.|------- │ │ │ │ +00024030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024040: 2d2d 2d2b 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 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00024080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024090: 2020 207c 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 7c0a 7c2d 2d2d 2d2d 2d2d |.|------- │ │ │ │ 000240d0: 2d2d 2d2d 2d2d 2d2d 2d2d 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 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ -00024120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024160: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2032 2020 ------|.| 2 │ │ │ │ -00024170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024180: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -00024190: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +000240e0: 2d2d 2d2b 2020 2020 2020 2020 2020 2020 ---+ │ │ │ │ +000240f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024110: 2020 2020 2020 7c0a 7c20 3220 2020 2020 |.| 2 │ │ │ │ +00024120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024130: 2032 207c 2020 2020 2020 2020 2020 2020 2 | │ │ │ │ +00024140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024160: 2020 2020 2020 7c0a 7c63 2020 2b20 3235 |.|c + 25 │ │ │ │ +00024170: 622a 6420 2b20 3234 632a 6420 2d20 3338 b*d + 24c*d - 38 │ │ │ │ +00024180: 6420 297c 2020 2020 2020 2020 2020 2020 d )| │ │ │ │ +00024190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000241a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000241b0: 2020 2020 2020 7c0a 7c20 3437 6320 202b |.| 47c + │ │ │ │ -000241c0: 2031 3762 2a64 202b 2032 3563 2a64 202d 17b*d + 25c*d - │ │ │ │ -000241d0: 2031 3564 202c 2061 2a62 202d 2032 3562 15d , a*b - 25b │ │ │ │ -000241e0: 2a63 202b 2034 3563 2020 2d20 3430 612a *c + 45c - 40a* │ │ │ │ -000241f0: 6420 2d20 3233 622a 6420 2d20 3238 632a d - 23b*d - 28c* │ │ │ │ -00024200: 6420 2b20 3239 7c0a 7c2d 2d2d 2d2d 2d2d d + 29|.|------- │ │ │ │ +000241b0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d |.|------- │ │ │ │ +000241c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000241d0: 2d2d 2d2b 2020 2020 2020 2020 2020 2020 ---+ │ │ │ │ +000241e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000241f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024200: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00024210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024250: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ -00024260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024270: 2d2d 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 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ -000242b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000242c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000242d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000242e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000242f0: 2d2d 2d2d 2d2d 7c0a 7c20 3220 2020 3220 ------|.| 2 2 │ │ │ │ -00024300: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -00024310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024320: 2020 2020 2020 2020 2020 2032 2020 2033 2 3 │ │ │ │ -00024330: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00024340: 2020 2020 2020 7c0a 7c64 202c 2061 2020 |.|d , a │ │ │ │ -00024350: 2d20 3233 622a 6320 2d20 3434 6320 202d - 23b*c - 44c - │ │ │ │ -00024360: 2034 3561 2a64 202d 2031 3962 2a64 202d 45a*d - 19b*d - │ │ │ │ -00024370: 2035 3063 2a64 202b 2036 6420 2c20 6320 50c*d + 6d , c │ │ │ │ -00024380: 202b 2032 622a 632a 6420 2b20 3334 6320 + 2b*c*d + 34c │ │ │ │ -00024390: 6420 2d20 3238 7c0a 7c2d 2d2d 2d2d 2d2d d - 28|.|------- │ │ │ │ -000243a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000243b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000243c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000243d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000243e0: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +00024250: 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a 206e ------+.|i21 : n │ │ │ │ +00024260: 6574 4c69 7374 2064 6563 6f6d 706f 7365 etList decompose │ │ │ │ +00024270: 2046 3220 2020 2020 2020 2020 2020 2020 F2 │ │ │ │ +00024280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000242a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000242b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000242c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000242d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000242e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000242f0: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +00024300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024340: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 207c ------|.| | │ │ │ │ +00024350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024360: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +00024370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024380: 2020 2020 2020 3220 2020 3220 2020 2020 2 2 │ │ │ │ +00024390: 2020 2020 2020 7c0a 7c6f 3231 203d 207c |.|o21 = | │ │ │ │ +000243a0: 6964 6561 6c20 2861 2a63 202b 2032 622a ideal (a*c + 2b* │ │ │ │ +000243b0: 6320 2d20 3133 6320 202b 2033 3361 2a64 c - 13c + 33a*d │ │ │ │ +000243c0: 202b 2031 3662 2a64 202d 2031 3863 2a64 + 16b*d - 18c*d │ │ │ │ +000243d0: 202d 2033 3964 202c 2062 2020 2d20 3437 - 39d , b - 47 │ │ │ │ +000243e0: 622a 6320 2d20 7c0a 7c20 2020 2020 202b b*c - |.| + │ │ │ │ 000243f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024430: 2d2d 2d2d 2d2d 7c0a 7c2d 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024480: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020 ------|.| 2 │ │ │ │ -00024490: 2020 2020 2032 2020 2020 2020 2020 3220 2 2 │ │ │ │ -000244a0: 2020 2020 2033 2020 2020 2032 2020 2020 3 2 │ │ │ │ -000244b0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -000244c0: 2020 2020 2020 3220 2020 2020 2020 2032 2 2 │ │ │ │ -000244d0: 2020 2020 2020 7c0a 7c61 2a64 2020 2b20 |.|a*d + │ │ │ │ -000244e0: 3338 622a 6420 202d 2031 3863 2a64 2020 38b*d - 18c*d │ │ │ │ -000244f0: 2d20 3334 6420 2c20 622a 6320 202d 2031 - 34d , b*c - 1 │ │ │ │ -00024500: 3562 2a63 2a64 202d 2034 3763 2064 202b 5b*c*d - 47c d + │ │ │ │ -00024510: 2031 3961 2a64 2020 2b20 3331 622a 6420 19a*d + 31b*d │ │ │ │ -00024520: 202b 2032 3163 7c0a 7c2d 2d2d 2d2d 2d2d + 21c|.|------- │ │ │ │ -00024530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024540: 2d2d 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 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +00024480: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +00024490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000244a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000244b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000244c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000244d0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 3220 2020 ------|.| 2 │ │ │ │ +000244e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000244f0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00024500: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00024510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024520: 2020 2032 2020 7c0a 7c32 3863 2020 2d20 2 |.|28c - │ │ │ │ +00024530: 3562 2a64 202d 2063 2a64 202d 2033 3064 5b*d - c*d - 30d │ │ │ │ +00024540: 202c 2061 2a62 202d 2032 3062 2a63 202d , a*b - 20b*c - │ │ │ │ +00024550: 2034 3563 2020 2d20 3335 612a 6420 2b20 45c - 35a*d + │ │ │ │ +00024560: 3231 622a 6420 2d20 3334 632a 6420 2b20 21b*d - 34c*d + │ │ │ │ +00024570: 3130 6420 2c20 7c0a 7c2d 2d2d 2d2d 2d2d 10d , |.|------- │ │ │ │ 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000245c0: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ -000245d0: 2d2d 2d2d 2b20 2020 2020 2020 2020 2020 ----+ │ │ │ │ -000245e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000245f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024610: 2020 2020 2020 7c0a 7c20 2032 2020 2020 |.| 2 │ │ │ │ -00024620: 2020 3320 7c20 2020 2020 2020 2020 2020 3 | │ │ │ │ -00024630: 2020 2020 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 7c0a 7c2a 6420 202b 2034 |.|*d + 4 │ │ │ │ -00024670: 3164 2029 7c20 2020 2020 2020 2020 2020 1d )| │ │ │ │ +000245d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000245e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000245f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024610: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +00024620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024660: 2d2d 2d2d 2d2d 7c0a 7c20 3220 2020 2020 ------|.| 2 │ │ │ │ +00024670: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ 00024680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000246a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000246b0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d |.|------- │ │ │ │ -000246c0: 2d2d 2d2d 2b20 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 2020 2020 2020 2020 2020 │ │ │ │ -00024700: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00024690: 2020 2020 2020 2020 3220 2020 3320 2020 2 3 │ │ │ │ +000246a0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +000246b0: 2020 2020 2020 7c0a 7c61 2020 2d20 3335 |.|a - 35 │ │ │ │ +000246c0: 622a 6320 2b20 3232 6320 202d 2032 3561 b*c + 22c - 25a │ │ │ │ +000246d0: 2a64 202b 2032 3362 2a64 202b 2034 3363 *d + 23b*d + 43c │ │ │ │ +000246e0: 2a64 202b 2033 3064 202c 2063 2020 2b20 *d + 30d , c + │ │ │ │ +000246f0: 3338 622a 632a 6420 2b20 3232 6320 6420 38b*c*d + 22c d │ │ │ │ +00024700: 2d20 3135 612a 7c0a 7c2d 2d2d 2d2d 2d2d - 15a*|.|------- │ │ │ │ 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024750: 2d2d 2d2d 2d2d 2b0a 0a57 6520 6361 6e20 ------+..We can │ │ │ │ -00024760: 6465 7465 726d 696e 6520 7768 6174 2074 determine what t │ │ │ │ -00024770: 6865 7365 2072 6570 7265 7365 6e74 2e20 hese represent. │ │ │ │ -00024780: 204f 6e65 2073 686f 756c 6420 6265 2061 One should be a │ │ │ │ -00024790: 2073 6574 206f 6620 3620 706f 696e 7473 set of 6 points │ │ │ │ -000247a0: 2c20 7768 6572 650a 3520 6c69 6520 6f6e , where.5 lie on │ │ │ │ -000247b0: 2061 2070 6c61 6e65 2e20 2054 6865 206f a plane. The o │ │ │ │ -000247c0: 7468 6572 2073 686f 756c 6420 6265 2036 ther should be 6 │ │ │ │ -000247d0: 2070 6f69 6e74 7320 7769 7468 2033 2070 points with 3 p │ │ │ │ -000247e0: 6f69 6e74 7320 6f6e 206f 6e65 206c 696e oints on one lin │ │ │ │ -000247f0: 652c 2061 6e64 0a74 6865 206f 7468 6572 e, and.the other │ │ │ │ -00024800: 2033 2070 6f69 6e74 7320 6f6e 2061 2073 3 points on a s │ │ │ │ -00024810: 6b65 7720 6c69 6e65 2e0a 0a53 6565 2061 kew line...See a │ │ │ │ -00024820: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00024830: 2a20 2a6e 6f74 6520 7261 6e64 6f6d 506f * *note randomPo │ │ │ │ -00024840: 696e 744f 6e52 6174 696f 6e61 6c56 6172 intOnRationalVar │ │ │ │ -00024850: 6965 7479 3a0a 2020 2020 7261 6e64 6f6d iety:. random │ │ │ │ -00024860: 506f 696e 744f 6e52 6174 696f 6e61 6c56 PointOnRationalV │ │ │ │ -00024870: 6172 6965 7479 5f6c 7049 6465 616c 5f72 ariety_lpIdeal_r │ │ │ │ -00024880: 702c 202d 2d20 6669 6e64 2061 2072 616e p, -- find a ran │ │ │ │ -00024890: 646f 6d20 706f 696e 7420 6f6e 2061 0a20 dom point on a. │ │ │ │ -000248a0: 2020 2076 6172 6965 7479 2074 6861 7420 variety that │ │ │ │ -000248b0: 6361 6e20 6265 2064 6574 6563 7465 6420 can be detected │ │ │ │ -000248c0: 746f 2062 6520 7261 7469 6f6e 616c 0a0a to be rational.. │ │ │ │ -000248d0: 5761 7973 2074 6f20 7573 6520 6e6f 6e6d Ways to use nonm │ │ │ │ -000248e0: 696e 696d 616c 4d61 7073 3a0a 3d3d 3d3d inimalMaps:.==== │ │ │ │ -000248f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00024900: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6e6f =======.. * "no │ │ │ │ -00024910: 6e6d 696e 696d 616c 4d61 7073 2849 6465 nminimalMaps(Ide │ │ │ │ -00024920: 616c 2922 0a0a 466f 7220 7468 6520 7072 al)"..For the pr │ │ │ │ -00024930: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -00024940: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -00024950: 206f 626a 6563 7420 2a6e 6f74 6520 6e6f object *note no │ │ │ │ -00024960: 6e6d 696e 696d 616c 4d61 7073 3a20 6e6f nminimalMaps: no │ │ │ │ -00024970: 6e6d 696e 696d 616c 4d61 7073 2c20 6973 nminimalMaps, is │ │ │ │ -00024980: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -00024990: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ -000249a0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -000249b0: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ -000249c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000249d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000249e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000249f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024a00: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00024a10: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00024a20: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -00024a30: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -00024a40: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -00024a50: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -00024a60: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -00024a70: 2f0a 4772 6f65 626e 6572 5374 7261 7461 /.GroebnerStrata │ │ │ │ -00024a80: 2e6d 323a 3130 3330 3a30 2e0a 1f0a 4669 .m2:1030:0....Fi │ │ │ │ -00024a90: 6c65 3a20 4772 6f65 626e 6572 5374 7261 le: GroebnerStra │ │ │ │ -00024aa0: 7461 2e69 6e66 6f2c 204e 6f64 653a 2072 ta.info, Node: r │ │ │ │ -00024ab0: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ -00024ac0: 6f6e 616c 5661 7269 6574 795f 6c70 4964 onalVariety_lpId │ │ │ │ -00024ad0: 6561 6c5f 7270 2c20 4e65 7874 3a20 7261 eal_rp, Next: ra │ │ │ │ -00024ae0: 6e64 6f6d 506f 696e 7473 4f6e 5261 7469 ndomPointsOnRati │ │ │ │ -00024af0: 6f6e 616c 5661 7269 6574 795f 6c70 4964 onalVariety_lpId │ │ │ │ -00024b00: 6561 6c5f 636d 5a5a 5f72 702c 2050 7265 eal_cmZZ_rp, Pre │ │ │ │ -00024b10: 763a 206e 6f6e 6d69 6e69 6d61 6c4d 6170 v: nonminimalMap │ │ │ │ -00024b20: 732c 2055 703a 2054 6f70 0a0a 7261 6e64 s, Up: Top..rand │ │ │ │ -00024b30: 6f6d 506f 696e 744f 6e52 6174 696f 6e61 omPointOnRationa │ │ │ │ -00024b40: 6c56 6172 6965 7479 2849 6465 616c 2920 lVariety(Ideal) │ │ │ │ -00024b50: 2d2d 2066 696e 6420 6120 7261 6e64 6f6d -- find a random │ │ │ │ -00024b60: 2070 6f69 6e74 206f 6e20 6120 7661 7269 point on a vari │ │ │ │ -00024b70: 6574 7920 7468 6174 2063 616e 2062 6520 ety that can be │ │ │ │ -00024b80: 6465 7465 6374 6564 2074 6f20 6265 2072 detected to be r │ │ │ │ -00024b90: 6174 696f 6e61 6c0a 2a2a 2a2a 2a2a 2a2a ational.******** │ │ │ │ -00024ba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024be0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024bf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024c00: 2a2a 2a0a 0a20 202a 2046 756e 6374 696f ***.. * Functio │ │ │ │ -00024c10: 6e3a 202a 6e6f 7465 2072 616e 646f 6d50 n: *note randomP │ │ │ │ -00024c20: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ -00024c30: 7269 6574 793a 0a20 2020 2072 616e 646f riety:. rando │ │ │ │ -00024c40: 6d50 6f69 6e74 4f6e 5261 7469 6f6e 616c mPointOnRational │ │ │ │ -00024c50: 5661 7269 6574 795f 6c70 4964 6561 6c5f Variety_lpIdeal_ │ │ │ │ -00024c60: 7270 2c0a 2020 2a20 5573 6167 653a 200a rp,. * Usage: . │ │ │ │ -00024c70: 2020 2020 2020 2020 7261 6e64 6f6d 506f randomPo │ │ │ │ -00024c80: 696e 744f 6e52 6174 696f 6e61 6c56 6172 intOnRationalVar │ │ │ │ -00024c90: 6965 7479 2049 0a20 2020 2020 2020 2072 iety I. r │ │ │ │ -00024ca0: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ -00024cb0: 6f6e 616c 5661 7269 6574 790a 2020 2a20 onalVariety. * │ │ │ │ -00024cc0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -00024cd0: 492c 2061 6e20 2a6e 6f74 6520 6964 6561 I, an *note idea │ │ │ │ -00024ce0: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ -00024cf0: 2949 6465 616c 2c2c 2041 6e20 6964 6561 )Ideal,, An idea │ │ │ │ -00024d00: 6c20 696e 2061 2070 6f6c 796e 6f6d 6961 l in a polynomia │ │ │ │ -00024d10: 6c20 7269 6e67 0a20 2020 2020 2020 2024 l ring. $ │ │ │ │ -00024d20: 5324 206f 7665 7220 6120 6669 656c 642c S$ over a field, │ │ │ │ -00024d30: 2077 6869 6368 2064 6566 696e 6573 2061 which defines a │ │ │ │ -00024d40: 2070 7269 6d65 2069 6465 616c 0a20 202a prime ideal. * │ │ │ │ -00024d50: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00024d60: 2a20 6120 2a6e 6f74 6520 6d61 7472 6978 * a *note matrix │ │ │ │ -00024d70: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00024d80: 4d61 7472 6978 2c2c 2041 206f 6e65 2072 Matrix,, A one r │ │ │ │ -00024d90: 6f77 206d 6174 7269 7820 6f76 6572 2074 ow matrix over t │ │ │ │ -00024da0: 6865 2062 6173 650a 2020 2020 2020 2020 he base. │ │ │ │ -00024db0: 6669 656c 6420 6f66 2024 5324 2c20 7265 field of $S$, re │ │ │ │ -00024dc0: 7072 6573 656e 7469 6e67 2061 2072 616e presenting a ran │ │ │ │ -00024dd0: 646f 6d6c 7920 6368 6f73 656e 2070 6f69 domly chosen poi │ │ │ │ -00024de0: 6e74 206f 6e20 7468 6520 7a65 726f 206c nt on the zero l │ │ │ │ -00024df0: 6f63 7573 206f 660a 2020 2020 2020 2020 ocus of. │ │ │ │ -00024e00: 2449 242e 2020 6e75 6c6c 2069 7320 7265 $I$. null is re │ │ │ │ -00024e10: 7475 726e 6564 2069 6e20 7468 6520 6361 turned in the ca │ │ │ │ -00024e20: 7365 2077 6865 6e20 7468 6520 726f 7574 se when the rout │ │ │ │ -00024e30: 696e 6520 6361 6e6e 6f74 2064 6574 6572 ine cannot deter │ │ │ │ -00024e40: 6d69 6e65 2069 660a 2020 2020 2020 2020 mine if. │ │ │ │ -00024e50: 7468 6520 7661 7269 6574 7920 6973 2072 the variety is r │ │ │ │ -00024e60: 6174 696f 6e61 6c20 616e 6420 6972 7265 ational and irre │ │ │ │ -00024e70: 6475 6369 626c 652e 0a0a 4465 7363 7269 ducible...Descri │ │ │ │ -00024e80: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00024e90: 3d0a 0a41 7320 6120 6669 7273 7420 6578 =..As a first ex │ │ │ │ -00024ea0: 616d 706c 652c 2077 6520 6669 6e64 2061 ample, we find a │ │ │ │ -00024eb0: 2072 616e 646f 6d20 706f 696e 7420 6f6e random point on │ │ │ │ -00024ec0: 2074 6865 2056 6572 6f6e 6573 6520 7375 the Veronese su │ │ │ │ -00024ed0: 7266 6163 6520 696e 2024 5c50 505e 3524 rface in $\PP^5$ │ │ │ │ -00024ee0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -00024ef0: 2d2d 2d2d 2d2d 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: 2d2b 0a7c 6931 203a 206b 6b20 3d20 5a5a -+.|i1 : kk = ZZ │ │ │ │ -00024f40: 2f31 3031 3b20 2020 2020 2020 2020 2020 /101; │ │ │ │ -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: 207c 0a2b 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fd0: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b -+.|i2 : S = kk[ │ │ │ │ -00024fe0: 612e 2e66 5d3b 2020 2020 2020 2020 2020 a..f]; │ │ │ │ -00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025020: 207c 0a2b 2d2d 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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025070: 2d2b 0a7c 6933 203a 2049 203d 206d 696e -+.|i3 : I = min │ │ │ │ -00025080: 6f72 7328 322c 2067 656e 6572 6963 5379 ors(2, genericSy │ │ │ │ -00025090: 6d6d 6574 7269 634d 6174 7269 7828 532c mmetricMatrix(S, │ │ │ │ -000250a0: 2033 2929 2020 2020 2020 2020 2020 2020 3)) │ │ │ │ -000250b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000250d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025100: 2020 2020 2020 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---------------- │ │ │ │ -000251f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025200: 2d7c 0a7c 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: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00025240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025250: 207c 0a7c 2020 2020 2063 2a65 202b 2062 |.| c*e + b │ │ │ │ -00025260: 2a66 2c20 2d20 632a 6420 2b20 622a 652c *f, - c*d + b*e, │ │ │ │ -00025270: 202d 2063 2a65 202b 2062 2a66 2c20 2d20 - c*e + b*f, - │ │ │ │ -00025280: 6520 202b 2064 2a66 2920 2020 2020 2020 e + d*f) │ │ │ │ -00025290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000252a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000252b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024750: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +00024760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000247a0: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +000247b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000247c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000247d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000247e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000247f0: 2d2d 2d2d 2d2d 7c0a 7c20 3220 2020 2020 ------|.| 2 │ │ │ │ +00024800: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00024810: 2020 2033 2020 2020 2032 2020 2020 2020 3 2 │ │ │ │ +00024820: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00024830: 2020 2020 3220 2020 2020 2020 2032 2020 2 2 │ │ │ │ +00024840: 2020 2020 2020 7c0a 7c64 2020 2d20 3437 |.|d - 47 │ │ │ │ +00024850: 622a 6420 202d 2033 3963 2a64 2020 2b20 b*d - 39c*d + │ │ │ │ +00024860: 3430 6420 2c20 622a 6320 202d 2034 3362 40d , b*c - 43b │ │ │ │ +00024870: 2a63 2a64 202b 2031 3963 2064 202b 2033 *c*d + 19c d + 3 │ │ │ │ +00024880: 3461 2a64 2020 2b20 3231 622a 6420 202b 4a*d + 21b*d + │ │ │ │ +00024890: 2034 3663 2a64 7c0a 7c2d 2d2d 2d2d 2d2d 46c*d|.|------- │ │ │ │ +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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000248e0: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +000248f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024930: 2d2d 2d2d 2d2d 7c0a 7c2d 2d2d 2d2d 2d2d ------|.|------- │ │ │ │ +00024940: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ +00024950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024980: 2020 2020 2020 7c0a 7c32 2020 2020 2020 |.|2 │ │ │ │ +00024990: 3320 7c20 2020 2020 2020 2020 2020 2020 3 | │ │ │ │ +000249a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000249b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000249c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000249d0: 2020 2020 2020 7c0a 7c20 202b 2032 3064 |.| + 20d │ │ │ │ +000249e0: 2029 7c20 2020 2020 2020 2020 2020 2020 )| │ │ │ │ +000249f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a20: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d |.|------- │ │ │ │ +00024a30: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ +00024a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a70: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00024a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024ac0: 2d2d 2d2d 2d2d 2b0a 0a57 6520 6361 6e20 ------+..We can │ │ │ │ +00024ad0: 6465 7465 726d 696e 6520 7768 6174 2074 determine what t │ │ │ │ +00024ae0: 6865 7365 2072 6570 7265 7365 6e74 2e20 hese represent. │ │ │ │ +00024af0: 204f 6e65 2073 686f 756c 6420 6265 2061 One should be a │ │ │ │ +00024b00: 2073 6574 206f 6620 3620 706f 696e 7473 set of 6 points │ │ │ │ +00024b10: 2c20 7768 6572 650a 3520 6c69 6520 6f6e , where.5 lie on │ │ │ │ +00024b20: 2061 2070 6c61 6e65 2e20 2054 6865 206f a plane. The o │ │ │ │ +00024b30: 7468 6572 2073 686f 756c 6420 6265 2036 ther should be 6 │ │ │ │ +00024b40: 2070 6f69 6e74 7320 7769 7468 2033 2070 points with 3 p │ │ │ │ +00024b50: 6f69 6e74 7320 6f6e 206f 6e65 206c 696e oints on one lin │ │ │ │ +00024b60: 652c 2061 6e64 0a74 6865 206f 7468 6572 e, and.the other │ │ │ │ +00024b70: 2033 2070 6f69 6e74 7320 6f6e 2061 2073 3 points on a s │ │ │ │ +00024b80: 6b65 7720 6c69 6e65 2e0a 0a53 6565 2061 kew line...See a │ │ │ │ +00024b90: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +00024ba0: 2a20 2a6e 6f74 6520 7261 6e64 6f6d 506f * *note randomPo │ │ │ │ +00024bb0: 696e 744f 6e52 6174 696f 6e61 6c56 6172 intOnRationalVar │ │ │ │ +00024bc0: 6965 7479 3a0a 2020 2020 7261 6e64 6f6d iety:. random │ │ │ │ +00024bd0: 506f 696e 744f 6e52 6174 696f 6e61 6c56 PointOnRationalV │ │ │ │ +00024be0: 6172 6965 7479 5f6c 7049 6465 616c 5f72 ariety_lpIdeal_r │ │ │ │ +00024bf0: 702c 202d 2d20 6669 6e64 2061 2072 616e p, -- find a ran │ │ │ │ +00024c00: 646f 6d20 706f 696e 7420 6f6e 2061 0a20 dom point on a. │ │ │ │ +00024c10: 2020 2076 6172 6965 7479 2074 6861 7420 variety that │ │ │ │ +00024c20: 6361 6e20 6265 2064 6574 6563 7465 6420 can be detected │ │ │ │ +00024c30: 746f 2062 6520 7261 7469 6f6e 616c 0a0a to be rational.. │ │ │ │ +00024c40: 5761 7973 2074 6f20 7573 6520 6e6f 6e6d Ways to use nonm │ │ │ │ +00024c50: 696e 696d 616c 4d61 7073 3a0a 3d3d 3d3d inimalMaps:.==== │ │ │ │ +00024c60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00024c70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6e6f =======.. * "no │ │ │ │ +00024c80: 6e6d 696e 696d 616c 4d61 7073 2849 6465 nminimalMaps(Ide │ │ │ │ +00024c90: 616c 2922 0a0a 466f 7220 7468 6520 7072 al)"..For the pr │ │ │ │ +00024ca0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +00024cb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +00024cc0: 206f 626a 6563 7420 2a6e 6f74 6520 6e6f object *note no │ │ │ │ +00024cd0: 6e6d 696e 696d 616c 4d61 7073 3a20 6e6f nminimalMaps: no │ │ │ │ +00024ce0: 6e6d 696e 696d 616c 4d61 7073 2c20 6973 nminimalMaps, is │ │ │ │ +00024cf0: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00024d00: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ +00024d10: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +00024d20: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00024d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00024d70: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00024d80: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00024d90: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00024da0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00024db0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00024dc0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ +00024dd0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00024de0: 2f0a 4772 6f65 626e 6572 5374 7261 7461 /.GroebnerStrata │ │ │ │ +00024df0: 2e6d 323a 3130 3330 3a30 2e0a 1f0a 4669 .m2:1030:0....Fi │ │ │ │ +00024e00: 6c65 3a20 4772 6f65 626e 6572 5374 7261 le: GroebnerStra │ │ │ │ +00024e10: 7461 2e69 6e66 6f2c 204e 6f64 653a 2072 ta.info, Node: r │ │ │ │ +00024e20: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ +00024e30: 6f6e 616c 5661 7269 6574 795f 6c70 4964 onalVariety_lpId │ │ │ │ +00024e40: 6561 6c5f 7270 2c20 4e65 7874 3a20 7261 eal_rp, Next: ra │ │ │ │ +00024e50: 6e64 6f6d 506f 696e 7473 4f6e 5261 7469 ndomPointsOnRati │ │ │ │ +00024e60: 6f6e 616c 5661 7269 6574 795f 6c70 4964 onalVariety_lpId │ │ │ │ +00024e70: 6561 6c5f 636d 5a5a 5f72 702c 2050 7265 eal_cmZZ_rp, Pre │ │ │ │ +00024e80: 763a 206e 6f6e 6d69 6e69 6d61 6c4d 6170 v: nonminimalMap │ │ │ │ +00024e90: 732c 2055 703a 2054 6f70 0a0a 7261 6e64 s, Up: Top..rand │ │ │ │ +00024ea0: 6f6d 506f 696e 744f 6e52 6174 696f 6e61 omPointOnRationa │ │ │ │ +00024eb0: 6c56 6172 6965 7479 2849 6465 616c 2920 lVariety(Ideal) │ │ │ │ +00024ec0: 2d2d 2066 696e 6420 6120 7261 6e64 6f6d -- find a random │ │ │ │ +00024ed0: 2070 6f69 6e74 206f 6e20 6120 7661 7269 point on a vari │ │ │ │ +00024ee0: 6574 7920 7468 6174 2063 616e 2062 6520 ety that can be │ │ │ │ +00024ef0: 6465 7465 6374 6564 2074 6f20 6265 2072 detected to be r │ │ │ │ +00024f00: 6174 696f 6e61 6c0a 2a2a 2a2a 2a2a 2a2a ational.******** │ │ │ │ +00024f10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024f30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024f40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024f50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024f60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00024f70: 2a2a 2a0a 0a20 202a 2046 756e 6374 696f ***.. * Functio │ │ │ │ +00024f80: 6e3a 202a 6e6f 7465 2072 616e 646f 6d50 n: *note randomP │ │ │ │ +00024f90: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ +00024fa0: 7269 6574 793a 0a20 2020 2072 616e 646f riety:. rando │ │ │ │ +00024fb0: 6d50 6f69 6e74 4f6e 5261 7469 6f6e 616c mPointOnRational │ │ │ │ +00024fc0: 5661 7269 6574 795f 6c70 4964 6561 6c5f Variety_lpIdeal_ │ │ │ │ +00024fd0: 7270 2c0a 2020 2a20 5573 6167 653a 200a rp,. * Usage: . │ │ │ │ +00024fe0: 2020 2020 2020 2020 7261 6e64 6f6d 506f randomPo │ │ │ │ +00024ff0: 696e 744f 6e52 6174 696f 6e61 6c56 6172 intOnRationalVar │ │ │ │ +00025000: 6965 7479 2049 0a20 2020 2020 2020 2072 iety I. r │ │ │ │ +00025010: 616e 646f 6d50 6f69 6e74 4f6e 5261 7469 andomPointOnRati │ │ │ │ +00025020: 6f6e 616c 5661 7269 6574 790a 2020 2a20 onalVariety. * │ │ │ │ +00025030: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ +00025040: 492c 2061 6e20 2a6e 6f74 6520 6964 6561 I, an *note idea │ │ │ │ +00025050: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ +00025060: 2949 6465 616c 2c2c 2041 6e20 6964 6561 )Ideal,, An idea │ │ │ │ +00025070: 6c20 696e 2061 2070 6f6c 796e 6f6d 6961 l in a polynomia │ │ │ │ +00025080: 6c20 7269 6e67 0a20 2020 2020 2020 2024 l ring. $ │ │ │ │ +00025090: 5324 206f 7665 7220 6120 6669 656c 642c S$ over a field, │ │ │ │ +000250a0: 2077 6869 6368 2064 6566 696e 6573 2061 which defines a │ │ │ │ +000250b0: 2070 7269 6d65 2069 6465 616c 0a20 202a prime ideal. * │ │ │ │ +000250c0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +000250d0: 2a20 6120 2a6e 6f74 6520 6d61 7472 6978 * a *note matrix │ │ │ │ +000250e0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +000250f0: 4d61 7472 6978 2c2c 2041 206f 6e65 2072 Matrix,, A one r │ │ │ │ +00025100: 6f77 206d 6174 7269 7820 6f76 6572 2074 ow matrix over t │ │ │ │ +00025110: 6865 2062 6173 650a 2020 2020 2020 2020 he base. │ │ │ │ +00025120: 6669 656c 6420 6f66 2024 5324 2c20 7265 field of $S$, re │ │ │ │ +00025130: 7072 6573 656e 7469 6e67 2061 2072 616e presenting a ran │ │ │ │ +00025140: 646f 6d6c 7920 6368 6f73 656e 2070 6f69 domly chosen poi │ │ │ │ +00025150: 6e74 206f 6e20 7468 6520 7a65 726f 206c nt on the zero l │ │ │ │ +00025160: 6f63 7573 206f 660a 2020 2020 2020 2020 ocus of. │ │ │ │ +00025170: 2449 242e 2020 6e75 6c6c 2069 7320 7265 $I$. null is re │ │ │ │ +00025180: 7475 726e 6564 2069 6e20 7468 6520 6361 turned in the ca │ │ │ │ +00025190: 7365 2077 6865 6e20 7468 6520 726f 7574 se when the rout │ │ │ │ +000251a0: 696e 6520 6361 6e6e 6f74 2064 6574 6572 ine cannot deter │ │ │ │ +000251b0: 6d69 6e65 2069 660a 2020 2020 2020 2020 mine if. │ │ │ │ +000251c0: 7468 6520 7661 7269 6574 7920 6973 2072 the variety is r │ │ │ │ +000251d0: 6174 696f 6e61 6c20 616e 6420 6972 7265 ational and irre │ │ │ │ +000251e0: 6475 6369 626c 652e 0a0a 4465 7363 7269 ducible...Descri │ │ │ │ +000251f0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +00025200: 3d0a 0a41 7320 6120 6669 7273 7420 6578 =..As a first ex │ │ │ │ +00025210: 616d 706c 652c 2077 6520 6669 6e64 2061 ample, we find a │ │ │ │ +00025220: 2072 616e 646f 6d20 706f 696e 7420 6f6e random point on │ │ │ │ +00025230: 2074 6865 2056 6572 6f6e 6573 6520 7375 the Veronese su │ │ │ │ +00025240: 7266 6163 6520 696e 2024 5c50 505e 3524 rface in $\PP^5$ │ │ │ │ +00025250: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +00025260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000252a0: 2d2b 0a7c 6931 203a 206b 6b20 3d20 5a5a -+.|i1 : kk = ZZ │ │ │ │ +000252b0: 2f31 3031 3b20 2020 2020 2020 2020 2020 /101; │ │ │ │ 000252c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000252f0: 207c 0a7c 6f33 203a 2049 6465 616c 206f |.|o3 : Ideal o │ │ │ │ -00025300: 6620 5320 2020 2020 2020 2020 2020 2020 f S │ │ │ │ -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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00025350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025390: 2d2b 0a7c 6934 203a 2070 7420 3d20 7261 -+.|i4 : pt = ra │ │ │ │ -000253a0: 6e64 6f6d 506f 696e 744f 6e52 6174 696f ndomPointOnRatio │ │ │ │ -000253b0: 6e61 6c56 6172 6965 7479 2049 2020 2020 nalVariety I │ │ │ │ -000253c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000253d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000253e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000253f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000252f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00025300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025340: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b -+.|i2 : S = kk[ │ │ │ │ +00025350: 612e 2e66 5d3b 2020 2020 2020 2020 2020 a..f]; │ │ │ │ +00025360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025390: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +000253a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000253b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000253c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000253d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000253e0: 2d2b 0a7c 6933 203a 2049 203d 206d 696e -+.|i3 : I = min │ │ │ │ +000253f0: 6f72 7328 322c 2067 656e 6572 6963 5379 ors(2, genericSy │ │ │ │ +00025400: 6d6d 6574 7269 634d 6174 7269 7828 532c mmetricMatrix(S, │ │ │ │ +00025410: 2033 2929 2020 2020 2020 2020 2020 2020 3)) │ │ │ │ 00025420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025430: 207c 0a7c 6f34 203d 207c 202d 3235 2032 |.|o4 = | -25 2 │ │ │ │ -00025440: 3020 2d33 3020 2d31 3620 3234 202d 3336 0 -30 -16 24 -36 │ │ │ │ -00025450: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00025430: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025480: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025490: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000254e0: 2020 3120 2020 2020 2020 3620 2020 2020 1 6 │ │ │ │ -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: 207c 0a7c 6f34 203a 204d 6174 7269 7820 |.|o4 : Matrix │ │ │ │ -00025530: 6b6b 2020 3c2d 2d20 6b6b 2020 2020 2020 kk <-- kk │ │ │ │ -00025540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025570: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00025580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000255a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000255b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000255c0: 2d2b 0a7c 6935 203a 2073 7562 2849 2c20 -+.|i5 : sub(I, │ │ │ │ -000255d0: 7074 2920 3d3d 2030 2020 2020 2020 2020 pt) == 0 │ │ │ │ -000255e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000255f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000254c0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000254d0: 207c 0a7c 6f33 203d 2069 6465 616c 2028 |.|o3 = ideal ( │ │ │ │ +000254e0: 2d20 6220 202b 2061 2a64 2c20 2d20 622a - b + a*d, - b* │ │ │ │ +000254f0: 6320 2b20 612a 652c 202d 2063 2a64 202b c + a*e, - c*d + │ │ │ │ +00025500: 2062 2a65 2c20 2d20 622a 6320 2b20 612a b*e, - b*c + a* │ │ │ │ +00025510: 652c 202d 2063 2020 2b20 612a 662c 202d e, - c + a*f, - │ │ │ │ +00025520: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ +00025530: 2d2d 2d2d 2d2d 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: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ +00025580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000255a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000255b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000255c0: 207c 0a7c 2020 2020 2063 2a65 202b 2062 |.| c*e + b │ │ │ │ +000255d0: 2a66 2c20 2d20 632a 6420 2b20 622a 652c *f, - c*d + b*e, │ │ │ │ +000255e0: 202d 2063 2a65 202b 2062 2a66 2c20 2d20 - c*e + b*f, - │ │ │ │ +000255f0: 6520 202b 2064 2a66 2920 2020 2020 2020 e + d*f) │ │ │ │ 00025600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025610: 207c 0a7c 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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025660: 207c 0a7c 6f35 203d 2074 7275 6520 2020 |.|o5 = true │ │ │ │ -00025670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025660: 207c 0a7c 6f33 203a 2049 6465 616c 206f |.|o3 : Ideal o │ │ │ │ +00025670: 6620 5320 2020 2020 2020 2020 2020 2020 f S │ │ │ │ 00025680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000256a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000256b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 000256c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000256d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000256e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000256f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025700: 2d2b 0a2b 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025750: 2d2b 0a7c 6936 203a 2053 203d 206b 6b5b -+.|i6 : S = kk[ │ │ │ │ -00025760: 612e 2e64 5d3b 2020 2020 2020 2020 2020 a..d]; │ │ │ │ +00025700: 2d2b 0a7c 6934 203a 2070 7420 3d20 7261 -+.|i4 : pt = ra │ │ │ │ +00025710: 6e64 6f6d 506f 696e 744f 6e52 6174 696f ndomPointOnRatio │ │ │ │ +00025720: 6e61 6c56 6172 6965 7479 2049 2020 2020 nalVariety I │ │ │ │ +00025730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025750: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000257a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -000257b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000257c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000257d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000257e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000257f0: 2d2b 0a7c 6937 203a 2046 203d 2067 726f -+.|i7 : F = gro │ │ │ │ -00025800: 6562 6e65 7246 616d 696c 7920 6964 6561 ebnerFamily idea │ │ │ │ -00025810: 6c22 6132 2c61 622c 6163 2c62 3222 2020 l"a2,ab,ac,b2" │ │ │ │ +000257a0: 207c 0a7c 6f34 203d 207c 2031 2034 3920 |.|o4 = | 1 49 │ │ │ │ +000257b0: 3234 202d 3233 202d 3336 202d 3330 207c 24 -23 -36 -30 | │ │ │ │ +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: 207c 0a7c 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 2020 2020 2020 2020 2020 │ │ │ │ 00025830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025840: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025850: 2020 3120 2020 2020 2020 3620 2020 2020 1 6 │ │ │ │ 00025860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025890: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000258a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000258b0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -000258c0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00025890: 207c 0a7c 6f34 203a 204d 6174 7269 7820 |.|o4 : Matrix │ │ │ │ +000258a0: 6b6b 2020 3c2d 2d20 6b6b 2020 2020 2020 kk <-- kk │ │ │ │ +000258b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000258c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000258d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000258e0: 207c 0a7c 6f37 203d 2069 6465 616c 2028 |.|o7 = ideal ( │ │ │ │ -000258f0: 6120 202b 2074 2062 2a63 202b 2074 2061 a + t b*c + t a │ │ │ │ -00025900: 2a64 202b 2074 2063 2020 2b20 7420 622a *d + t c + t b* │ │ │ │ -00025910: 6420 2b20 7420 632a 6420 2b20 7420 6420 d + t c*d + t d │ │ │ │ -00025920: 2c20 612a 6220 2b20 7420 622a 6320 2b20 , a*b + t b*c + │ │ │ │ -00025930: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025940: 2020 2020 2020 3120 2020 2020 2020 3320 1 3 │ │ │ │ -00025950: 2020 2020 2020 3220 2020 2020 2034 2020 2 4 │ │ │ │ -00025960: 2020 2020 2035 2020 2020 2020 2036 2020 5 6 │ │ │ │ -00025970: 2020 2020 2020 2020 2037 2020 2020 2020 7 │ │ │ │ -00025980: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00025990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000259a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000259b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000259c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000259d0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -000259e0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -000259f0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +000258e0: 207c 0a2b 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: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025930: 2d2b 0a7c 6935 203a 2073 7562 2849 2c20 -+.|i5 : sub(I, │ │ │ │ +00025940: 7074 2920 3d3d 2030 2020 2020 2020 2020 pt) == 0 │ │ │ │ +00025950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025980: 207c 0a7c 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 2020 2020 2020 2020 2020 │ │ │ │ +000259c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000259d0: 207c 0a7c 6f35 203d 2074 7275 6520 2020 |.|o5 = true │ │ │ │ +000259e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000259f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025a10: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -00025a20: 207c 0a7c 2020 2020 2074 2061 2a64 202b |.| t a*d + │ │ │ │ -00025a30: 2074 2063 2020 2b20 7420 2062 2a64 202b t c + t b*d + │ │ │ │ -00025a40: 2074 2020 632a 6420 2b20 7420 2064 202c t c*d + t d , │ │ │ │ -00025a50: 2061 2a63 202b 2074 2020 622a 6320 2b20 a*c + t b*c + │ │ │ │ -00025a60: 7420 2061 2a64 202b 2074 2020 6320 202b t a*d + t c + │ │ │ │ -00025a70: 207c 0a7c 2020 2020 2020 3920 2020 2020 |.| 9 │ │ │ │ -00025a80: 2020 3820 2020 2020 2031 3020 2020 2020 8 10 │ │ │ │ -00025a90: 2020 3131 2020 2020 2020 2031 3220 2020 11 12 │ │ │ │ -00025aa0: 2020 2020 2020 2020 3133 2020 2020 2020 13 │ │ │ │ -00025ab0: 2031 3520 2020 2020 2020 3134 2020 2020 15 14 │ │ │ │ -00025ac0: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00025ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025b10: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -00025b20: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00025b30: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00025b40: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -00025b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025b60: 207c 0a7c 2020 2020 2074 2020 622a 6420 |.| t b*d │ │ │ │ -00025b70: 2b20 7420 2063 2a64 202b 2074 2020 6420 + t c*d + t d │ │ │ │ -00025b80: 2c20 6220 202b 2074 2020 622a 6320 2b20 , b + t b*c + │ │ │ │ -00025b90: 7420 2061 2a64 202b 2074 2020 6320 202b t a*d + t c + │ │ │ │ -00025ba0: 2074 2020 622a 6420 2b20 7420 2063 2a64 t b*d + t c*d │ │ │ │ -00025bb0: 207c 0a7c 2020 2020 2020 3136 2020 2020 |.| 16 │ │ │ │ -00025bc0: 2020 2031 3720 2020 2020 2020 3138 2020 17 18 │ │ │ │ -00025bd0: 2020 2020 2020 2020 3139 2020 2020 2020 19 │ │ │ │ -00025be0: 2032 3120 2020 2020 2020 3230 2020 2020 21 20 │ │ │ │ -00025bf0: 2020 3232 2020 2020 2020 2032 3320 2020 22 23 │ │ │ │ -00025c00: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00025c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025c50: 2d7c 0a7c 2020 2020 2020 2020 2020 2032 -|.| 2 │ │ │ │ -00025c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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: 207c 0a7c 2020 2020 202b 2074 2020 6420 |.| + t d │ │ │ │ -00025cb0: 2920 2020 2020 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: 207c 0a7c 2020 2020 2020 2020 3234 2020 |.| 24 │ │ │ │ -00025d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00025a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025a70: 2d2b 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -+.+------------ │ │ │ │ +00025a80: 2d2d 2d2d 2d2d 2d2d 2d2d 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: 2d2b 0a7c 6936 203a 2053 203d 206b 6b5b -+.|i6 : S = kk[ │ │ │ │ +00025ad0: 612e 2e64 5d3b 2020 2020 2020 2020 2020 a..d]; │ │ │ │ +00025ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00025b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025b60: 2d2b 0a7c 6937 203a 2046 203d 2067 726f -+.|i7 : F = gro │ │ │ │ +00025b70: 6562 6e65 7246 616d 696c 7920 6964 6561 ebnerFamily idea │ │ │ │ +00025b80: 6c22 6132 2c61 622c 6163 2c62 3222 2020 l"a2,ab,ac,b2" │ │ │ │ +00025b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025bb0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c00: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025c10: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00025c20: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00025c30: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00025c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c50: 207c 0a7c 6f37 203d 2069 6465 616c 2028 |.|o7 = ideal ( │ │ │ │ +00025c60: 6120 202b 2074 2062 2a63 202b 2074 2061 a + t b*c + t a │ │ │ │ +00025c70: 2a64 202b 2074 2063 2020 2b20 7420 622a *d + t c + t b* │ │ │ │ +00025c80: 6420 2b20 7420 632a 6420 2b20 7420 6420 d + t c*d + t d │ │ │ │ +00025c90: 2c20 612a 6220 2b20 7420 622a 6320 2b20 , a*b + t b*c + │ │ │ │ +00025ca0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025cb0: 2020 2020 2020 3120 2020 2020 2020 3320 1 3 │ │ │ │ +00025cc0: 2020 2020 2020 3220 2020 2020 2034 2020 2 4 │ │ │ │ +00025cd0: 2020 2020 2035 2020 2020 2020 2036 2020 5 6 │ │ │ │ +00025ce0: 2020 2020 2020 2020 2037 2020 2020 2020 7 │ │ │ │ +00025cf0: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ +00025d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025d40: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ +00025d50: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00025d60: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ 00025d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d90: 207c 0a7c 6f37 203a 2049 6465 616c 206f |.|o7 : Ideal o │ │ │ │ -00025da0: 6620 6b6b 5b74 202c 2074 202c 2074 2020 f kk[t , t , t │ │ │ │ -00025db0: 2c20 7420 2c20 7420 2c20 7420 202c 2074 , t , t , t , t │ │ │ │ -00025dc0: 2020 2c20 7420 202c 2074 202c 2074 202c , t , t , t , │ │ │ │ -00025dd0: 2074 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -00025de0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00025df0: 2020 2020 2020 3620 2020 3520 2020 3132 6 5 12 │ │ │ │ -00025e00: 2020 2032 2020 2034 2020 2031 3120 2020 2 4 11 │ │ │ │ -00025e10: 3138 2020 2032 3420 2020 3120 2020 3320 18 24 1 3 │ │ │ │ -00025e20: 2020 3820 2020 3130 2020 2031 3720 2020 8 10 17 │ │ │ │ -00025e30: 327c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2|.|------------ │ │ │ │ +00025d80: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00025d90: 207c 0a7c 2020 2020 2074 2061 2a64 202b |.| t a*d + │ │ │ │ +00025da0: 2074 2063 2020 2b20 7420 2062 2a64 202b t c + t b*d + │ │ │ │ +00025db0: 2074 2020 632a 6420 2b20 7420 2064 202c t c*d + t d , │ │ │ │ +00025dc0: 2061 2a63 202b 2074 2020 622a 6320 2b20 a*c + t b*c + │ │ │ │ +00025dd0: 7420 2061 2a64 202b 2074 2020 6320 202b t a*d + t c + │ │ │ │ +00025de0: 207c 0a7c 2020 2020 2020 3920 2020 2020 |.| 9 │ │ │ │ +00025df0: 2020 3820 2020 2020 2031 3020 2020 2020 8 10 │ │ │ │ +00025e00: 2020 3131 2020 2020 2020 2031 3220 2020 11 12 │ │ │ │ +00025e10: 2020 2020 2020 2020 3133 2020 2020 2020 13 │ │ │ │ +00025e20: 2031 3520 2020 2020 2020 3134 2020 2020 15 14 │ │ │ │ +00025e30: 207c 0a7c 2020 2020 202d 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e80: 2d7c 0a7c 202c 2074 202c 2074 202c 2074 -|.| , t , t , t │ │ │ │ -00025e90: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ -00025ea0: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ -00025eb0: 2c20 7420 205d 5b61 2e2e 645d 2020 2020 , t ][a..d] │ │ │ │ +00025e80: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ +00025e90: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00025ea0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00025eb0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ 00025ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025ed0: 207c 0a7c 3320 2020 3720 2020 3920 2020 |.|3 7 9 │ │ │ │ -00025ee0: 3134 2020 2031 3620 2020 3230 2020 2032 14 16 20 2 │ │ │ │ -00025ef0: 3220 2020 3133 2020 2031 3520 2020 3139 2 13 15 19 │ │ │ │ -00025f00: 2020 2032 3120 2020 2020 2020 2020 2020 21 │ │ │ │ -00025f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025f20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00025f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025f70: 2d2b 0a7c 6938 203a 204a 203d 2067 726f -+.|i8 : J = gro │ │ │ │ -00025f80: 6562 6e65 7253 7472 6174 756d 2046 2020 ebnerStratum F │ │ │ │ -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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00025ed0: 207c 0a7c 2020 2020 2074 2020 622a 6420 |.| t b*d │ │ │ │ +00025ee0: 2b20 7420 2063 2a64 202b 2074 2020 6420 + t c*d + t d │ │ │ │ +00025ef0: 2c20 6220 202b 2074 2020 622a 6320 2b20 , b + t b*c + │ │ │ │ +00025f00: 7420 2061 2a64 202b 2074 2020 6320 202b t a*d + t c + │ │ │ │ +00025f10: 2074 2020 622a 6420 2b20 7420 2063 2a64 t b*d + t c*d │ │ │ │ +00025f20: 207c 0a7c 2020 2020 2020 3136 2020 2020 |.| 16 │ │ │ │ +00025f30: 2020 2031 3720 2020 2020 2020 3138 2020 17 18 │ │ │ │ +00025f40: 2020 2020 2020 2020 3139 2020 2020 2020 19 │ │ │ │ +00025f50: 2032 3120 2020 2020 2020 3230 2020 2020 21 20 │ │ │ │ +00025f60: 2020 3232 2020 2020 2020 2032 3320 2020 22 23 │ │ │ │ +00025f70: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ +00025f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025fc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2032 -|.| 2 │ │ │ │ 00025fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026010: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00026020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026010: 207c 0a7c 2020 2020 202b 2074 2020 6420 |.| + t d │ │ │ │ +00026020: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00026030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026060: 207c 0a7c 6f38 203d 2069 6465 616c 2028 |.|o8 = ideal ( │ │ │ │ -00026070: 2d20 7420 202b 2074 2020 202d 2074 2020 - t + t - t │ │ │ │ -00026080: 7420 202c 202d 2074 2020 2d20 7420 2074 t , - t - t t │ │ │ │ -00026090: 2020 2c20 2d20 7420 2020 2b20 7420 2020 , - t + t │ │ │ │ -000260a0: 2b20 7420 7420 2020 2d20 7420 2074 2020 + t t - t t │ │ │ │ +00026060: 207c 0a7c 2020 2020 2020 2020 3234 2020 |.| 24 │ │ │ │ +00026070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000260a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000260b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000260c0: 2020 2037 2020 2020 3134 2020 2020 3133 7 14 13 │ │ │ │ -000260d0: 2031 3920 2020 2020 3820 2020 2032 3020 19 8 20 │ │ │ │ -000260e0: 3133 2020 2020 2031 3020 2020 2031 3720 13 10 17 │ │ │ │ -000260f0: 2020 2039 2031 3320 2020 2032 3220 3133 9 13 22 13 │ │ │ │ -00026100: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00026110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026150: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -00026160: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -00026170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000261a0: 207c 0a7c 2020 2020 202d 2074 2074 2020 |.| - t t │ │ │ │ -000261b0: 202d 2074 2020 7420 2020 2b20 7420 2074 - t t + t t │ │ │ │ -000261c0: 2020 2c20 2d20 7420 2074 2020 202b 2074 , - t t + t │ │ │ │ -000261d0: 2020 7420 2074 2020 2c20 2d20 7420 2020 t t , - t │ │ │ │ -000261e0: 2b20 7420 7420 2020 2d20 7420 2074 2020 + t t - t t │ │ │ │ -000261f0: 207c 0a7c 2020 2020 2020 2020 3720 3135 |.| 7 15 │ │ │ │ -00026200: 2020 2020 3136 2031 3920 2020 2031 3320 16 19 13 │ │ │ │ -00026210: 3231 2020 2020 2031 3620 3231 2020 2020 21 16 21 │ │ │ │ -00026220: 3133 2031 3520 3231 2020 2020 2031 3120 13 15 21 11 │ │ │ │ -00026230: 2020 2039 2031 3420 2020 2031 3620 3230 9 14 16 20 │ │ │ │ -00026240: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00026250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026290: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -000262a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262e0: 207c 0a7c 2020 2020 202d 2074 2020 7420 |.| - t t │ │ │ │ -000262f0: 2020 2d20 7420 7420 2020 2b20 7420 2074 - t t + t t │ │ │ │ -00026300: 2020 7420 202c 2074 2020 202b 2074 2074 t , t + t t │ │ │ │ -00026310: 2020 202d 2074 2020 7420 2020 2d20 7420 - t t - t │ │ │ │ -00026320: 2074 2020 202b 2074 2020 7420 2074 2020 t + t t t │ │ │ │ -00026330: 2c7c 0a7c 2020 2020 2020 2020 3233 2031 ,|.| 23 1 │ │ │ │ -00026340: 3320 2020 2038 2031 3520 2020 2031 3420 3 8 15 14 │ │ │ │ -00026350: 3133 2032 3120 2020 3138 2020 2020 3920 13 21 18 9 │ │ │ │ -00026360: 3136 2020 2020 3136 2032 3220 2020 2031 16 16 22 1 │ │ │ │ -00026370: 3020 3135 2020 2020 3136 2031 3320 3231 0 15 16 13 21 │ │ │ │ -00026380: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00026390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000263a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000263b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000263c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000263d0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -000263e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000263f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026420: 207c 0a7c 2020 2020 202d 2074 2020 202b |.| - t + │ │ │ │ -00026430: 2074 2020 7420 202d 2074 2020 7420 2020 t t - t t │ │ │ │ -00026440: 2d20 7420 2074 2020 202d 2074 2020 7420 - t t - t t │ │ │ │ -00026450: 2020 2b20 7420 2074 2020 7420 202c 2074 + t t t , t │ │ │ │ -00026460: 2020 7420 202d 2074 2020 7420 2020 2d20 t - t t - │ │ │ │ -00026470: 207c 0a7c 2020 2020 2020 2020 3132 2020 |.| 12 │ │ │ │ -00026480: 2020 3137 2039 2020 2020 3233 2031 3620 17 9 23 16 │ │ │ │ -00026490: 2020 2032 3420 3133 2020 2020 3131 2031 24 13 11 1 │ │ │ │ -000264a0: 3520 2020 2031 3720 3133 2032 3120 2020 5 17 13 21 │ │ │ │ -000264b0: 3138 2039 2020 2020 3234 2031 3620 2020 18 9 24 16 │ │ │ │ -000264c0: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -000264d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000264e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000264f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026510: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -00026520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026530: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00026540: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00026550: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00026560: 207c 0a7c 2020 2020 2074 2020 7420 2020 |.| t t │ │ │ │ -00026570: 2b20 7420 2074 2020 7420 202c 202d 2074 + t t t , - t │ │ │ │ -00026580: 2020 2d20 3274 2020 7420 2020 2b20 7420 - 2t t + t │ │ │ │ -00026590: 2074 2020 2c20 2d20 7420 202d 2074 2020 t , - t - t │ │ │ │ -000265a0: 202b 2074 2020 7420 202c 202d 2074 2020 + t t , - t │ │ │ │ -000265b0: 2d7c 0a7c 2020 2020 2020 3132 2031 3520 -|.| 12 15 │ │ │ │ -000265c0: 2020 2031 3820 3133 2032 3120 2020 2020 18 13 21 │ │ │ │ -000265d0: 3120 2020 2020 3134 2031 3320 2020 2031 1 14 13 1 │ │ │ │ -000265e0: 3320 3139 2020 2020 2032 2020 2020 3134 3 19 2 14 │ │ │ │ -000265f0: 2020 2020 3230 2031 3320 2020 2020 3420 20 13 4 │ │ │ │ -00026600: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00026610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026650: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -00026660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026680: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00026690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000266a0: 207c 0a7c 2020 2020 2074 2074 2020 202d |.| t t - │ │ │ │ -000266b0: 2074 2020 7420 2020 2b20 7420 7420 2020 t t + t t │ │ │ │ -000266c0: 2d20 3274 2020 7420 2020 2b20 7420 2074 - 2t t + t t │ │ │ │ -000266d0: 2020 202d 2074 2074 2020 202b 2074 2074 - t t + t t │ │ │ │ -000266e0: 2020 7420 2020 2b20 7420 2074 2020 7420 t + t t t │ │ │ │ -000266f0: 207c 0a7c 2020 2020 2020 3720 3136 2020 |.| 7 16 │ │ │ │ -00026700: 2020 3134 2031 3620 2020 2033 2031 3320 14 16 3 13 │ │ │ │ -00026710: 2020 2020 3137 2031 3320 2020 2032 3220 17 13 22 │ │ │ │ -00026720: 3133 2020 2020 3120 3135 2020 2020 3720 13 1 15 7 │ │ │ │ -00026730: 3133 2031 3520 2020 2031 3420 3133 2031 13 15 14 13 1 │ │ │ │ -00026740: 357c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 5|.| ------- │ │ │ │ -00026750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026790: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -000267a0: 2020 2020 2020 2020 3320 2020 2020 2020 3 │ │ │ │ -000267b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000267c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000267d0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -000267e0: 207c 0a7c 2020 2020 202b 2074 2020 7420 |.| + t t │ │ │ │ -000267f0: 2074 2020 202d 2074 2020 7420 202c 2074 t - t t , t │ │ │ │ -00026800: 2020 202d 2074 2074 2020 202d 2074 2020 - t t - t │ │ │ │ -00026810: 7420 2020 2b20 7420 7420 2074 2020 202b t + t t t + │ │ │ │ -00026820: 2074 2020 7420 2020 2b20 2020 2020 2020 t t + │ │ │ │ -00026830: 207c 0a7c 2020 2020 2020 2020 3136 2031 |.| 16 1 │ │ │ │ -00026840: 3320 3139 2020 2020 3133 2032 3120 2020 3 19 13 21 │ │ │ │ -00026850: 3138 2020 2020 3920 3136 2020 2020 3137 18 9 16 17 │ │ │ │ -00026860: 2031 3520 2020 2039 2031 3320 3135 2020 15 9 13 15 │ │ │ │ -00026870: 2020 3134 2031 3520 2020 2020 2020 2020 14 15 │ │ │ │ -00026880: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00026890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000268a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000268b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000268c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000268d0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -000268e0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -000268f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026920: 207c 0a7c 2020 2020 2074 2020 7420 2074 |.| t t t │ │ │ │ -00026930: 2020 202d 2074 2020 7420 2074 2020 2c20 - t t t , │ │ │ │ -00026940: 2d20 7420 202b 2074 2074 2020 202d 2032 - t + t t - 2 │ │ │ │ -00026950: 7420 2074 2020 202d 2074 2074 2020 202b t t - t t + │ │ │ │ -00026960: 2074 2020 7420 2074 2020 202b 2020 2020 t t t + │ │ │ │ -00026970: 207c 0a7c 2020 2020 2020 3136 2031 3320 |.| 16 13 │ │ │ │ -00026980: 3231 2020 2020 3133 2031 3520 3231 2020 21 13 15 21 │ │ │ │ -00026990: 2020 2035 2020 2020 3320 3134 2020 2020 5 3 14 │ │ │ │ -000269a0: 2031 3720 3134 2020 2020 3820 3136 2020 17 14 8 16 │ │ │ │ -000269b0: 2020 3136 2032 3020 3133 2020 2020 2020 16 20 13 │ │ │ │ -000269c0: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -000269d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000269e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000269f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026a10: 2d7c 0a7c 2020 2020 2020 2020 2032 2020 -|.| 2 │ │ │ │ -00026a20: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00026a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a40: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00026a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a60: 207c 0a7c 2020 2020 2074 2020 7420 2020 |.| t t │ │ │ │ -00026a70: 2d20 7420 7420 2020 2b20 7420 2074 2020 - t t + t t │ │ │ │ -00026a80: 202b 2074 2074 2020 7420 2020 2d20 7420 + t t t - t │ │ │ │ -00026a90: 2074 2020 7420 202c 2074 2074 2020 202d t t , t t - │ │ │ │ -00026aa0: 2074 2020 7420 2020 2d20 7420 2074 2020 t t - t t │ │ │ │ -00026ab0: 207c 0a7c 2020 2020 2020 3233 2031 3320 |.| 23 13 │ │ │ │ -00026ac0: 2020 2032 2031 3520 2020 2031 3420 3135 2 15 14 15 │ │ │ │ -00026ad0: 2020 2020 3820 3133 2031 3520 2020 2031 8 13 15 1 │ │ │ │ -00026ae0: 3420 3133 2032 3120 2020 3320 3136 2020 4 13 21 3 16 │ │ │ │ -00026af0: 2020 3130 2031 3620 2020 2031 3720 3136 10 16 17 16 │ │ │ │ -00026b00: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00026b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026b50: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -00026b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b90: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 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│ │ -00026d70: 2031 3520 2020 2031 3720 3134 2031 3520 15 17 14 15 │ │ │ │ -00026d80: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00026d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026dd0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -00026de0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00026df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026e20: 207c 0a7c 2020 2020 2074 2020 7420 2074 |.| t t t │ │ │ │ -00026e30: 2020 202d 2074 2020 7420 2074 2020 2c20 - t t t , │ │ │ │ -00026e40: 7420 2074 2020 2d20 7420 2074 2020 202d t t - t t - │ │ │ │ -00026e50: 2074 2020 7420 2020 2b20 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│ │ │ │ -00026f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026f60: 207c 0a7c 2020 2020 2074 2020 7420 2074 |.| t t t │ │ │ │ -00026f70: 2020 202b 2074 2020 7420 2074 2020 202d + t t t - │ │ │ │ -00026f80: 2074 2020 7420 2074 2020 2c20 2d20 7420 t t t , - t │ │ │ │ -00026f90: 202d 2074 2020 7420 2020 2b20 7420 7420 - t t + t t │ │ │ │ -00026fa0: 2020 2d20 7420 2074 2020 202b 2020 2020 - t t + │ │ │ │ -00026fb0: 207c 0a7c 2020 2020 2020 3138 2031 3420 |.| 18 14 │ │ │ │ -00026fc0: 3135 2020 2020 3132 2031 3320 3135 2020 15 12 13 15 │ │ │ │ -00026fd0: 2020 3138 2031 3320 3231 2020 2020 2038 18 13 21 8 │ │ │ │ -00026fe0: 2020 2020 3230 2031 3320 2020 2037 2031 20 13 7 1 │ │ │ │ -00026ff0: 3920 2020 2031 3420 3139 2020 2020 2020 9 14 19 │ │ │ │ -00027000: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00027010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027020: 2d2d 2d2d 2d2d 2d2d 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│ │ │ │ -00027200: 2074 2020 202b 2074 2020 7420 2074 2020 t + t t t │ │ │ │ -00027210: 202b 2074 2074 2020 7420 2020 2b20 7420 + t t t + t │ │ │ │ -00027220: 2074 2020 202d 2074 2074 2020 202d 2020 t - t t - │ │ │ │ -00027230: 207c 0a7c 2020 2020 2020 3230 2031 3320 |.| 20 13 │ │ │ │ -00027240: 3135 2020 2020 3130 2031 3920 2020 2031 15 10 19 1 │ │ │ │ -00027250: 3720 3139 2020 2020 3232 2031 3320 3139 7 19 22 13 19 │ │ │ │ -00027260: 2020 2020 3720 3135 2031 3920 2020 2031 7 15 19 1 │ │ │ │ -00027270: 3620 3139 2020 2020 3120 3231 2020 2020 6 19 1 21 │ │ │ │ -00027280: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00027290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000272a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000272b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000272c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000272d0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ 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│ │ │ -00027770: 2039 2020 2020 3233 2031 3620 2020 2020 9 23 16 │ │ │ │ -00027780: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ -00027790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000277a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000277b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000277c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000277d0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ -000277e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027820: 207c 0a7c 2020 2020 2074 2020 7420 2074 |.| t t t │ │ │ │ -00027830: 2020 202d 2074 2020 7420 2020 2b20 7420 - t t + t │ │ │ │ -00027840: 2074 2020 7420 2020 2b20 7420 2074 2020 t t + t t │ │ │ │ -00027850: 7420 2020 2d20 7420 7420 2020 2d20 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2020 2020 3134 2020 2020 3133 7 14 13 │ │ │ │ +00026440: 2031 3920 2020 2020 3820 2020 2032 3020 19 8 20 │ │ │ │ +00026450: 3133 2020 2020 2031 3020 2020 2031 3720 13 10 17 │ │ │ │ +00026460: 2020 2039 2031 3320 2020 2032 3220 3133 9 13 22 13 │ │ │ │ +00026470: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ +00026480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000264a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000264b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000264c0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ +000264d0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +000264e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000264f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026510: 207c 0a7c 2020 2020 202d 2074 2074 2020 |.| - t t │ │ │ │ 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2d2d 2d2d 2d2d ---------------- │ │ │ │ +000289c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000289d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000289e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000289f0: 2d7c 0a7c 2020 2020 2074 2074 2020 7420 -|.| t t t │ │ │ │ +00028a00: 2074 2020 2c20 7420 2074 2020 2b20 7420 t , t t + t │ │ │ │ +00028a10: 2074 2020 2d20 7420 2074 2020 2d20 7420 t - t t - t │ │ │ │ +00028a20: 2074 2020 202d 2074 2020 7420 2020 2b20 t - t t + │ │ │ │ +00028a30: 7420 7420 2020 2d20 7420 7420 202b 2020 t t - t t + │ │ │ │ +00028a40: 207c 0a7c 2020 2020 2020 3720 3136 2031 |.| 7 16 1 │ │ │ │ +00028a50: 3320 3231 2020 2032 3420 3120 2020 2031 3 21 24 1 1 │ │ │ │ +00028a60: 3120 3320 2020 2031 3820 3820 2020 2031 1 3 18 8 1 │ │ │ │ +00028a70: 3120 3130 2020 2020 3131 2031 3720 2020 1 10 11 17 │ │ │ │ +00028a80: 2034 2032 3320 2020 2035 2039 2020 2020 4 23 5 9 │ │ │ │ +00028a90: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ +00028aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ae0: 2d7c 0a7c 2020 2020 2074 2020 7420 7420 -|.| t t t │ │ │ │ +00028af0: 2020 2b20 7420 2074 2074 2020 202b 2074 + t t t + t │ │ │ │ +00028b00: 2074 2020 7420 2020 2b20 7420 2074 2074 t t + t t t │ │ │ │ +00028b10: 2020 202d 2074 2074 2020 7420 2020 2d20 - t t t - │ │ │ │ +00028b20: 7420 2074 2074 2020 7420 202c 202d 2020 t t t t , - │ │ │ │ +00028b30: 207c 0a7c 2020 2020 2020 3233 2037 2031 |.| 23 7 1 │ │ │ │ +00028b40: 3620 2020 2032 3420 3720 3133 2020 2020 6 24 7 13 │ │ │ │ +00028b50: 3820 3137 2031 3520 2020 2031 3120 3720 8 17 15 11 7 │ │ │ │ +00028b60: 3135 2020 2020 3120 3137 2032 3120 2020 15 1 17 21 │ │ │ │ +00028b70: 2031 3720 3720 3133 2032 3120 2020 2020 17 7 13 21 │ │ │ │ +00028b80: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ +00028b90: 2d2d 2d2d 2d2d 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: 2d7c 0a7c 2020 2020 2074 2020 7420 2020 -|.| t t │ │ │ │ +00028be0: 2b20 7420 7420 2020 2b20 7420 2074 2020 + t t + t t │ │ │ │ +00028bf0: 2d20 7420 2074 2020 202d 2074 2074 2020 - t t - t t │ │ │ │ +00028c00: 2b20 7420 2074 2074 2020 202b 2074 2020 + t t t + t │ │ │ │ +00028c10: 7420 7420 2020 2b20 7420 2074 2074 2020 t t + t t t │ │ │ │ +00028c20: 207c 0a7c 2020 2020 2020 3131 2031 3820 |.| 11 18 │ │ │ │ +00028c30: 2020 2034 2032 3420 2020 2031 3220 3320 4 24 12 3 │ │ │ │ +00028c40: 2020 2031 3220 3130 2020 2020 3620 3920 12 10 6 9 │ │ │ │ +00028c50: 2020 2032 3420 3720 3136 2020 2020 3138 24 7 16 18 │ │ │ │ +00028c60: 2038 2031 3520 2020 2031 3220 3720 3135 8 15 12 7 15 │ │ │ │ +00028c70: 207c 0a7c 2020 2020 202d 2d2d 2d2d 2d2d |.| ------- │ │ │ │ 00028c80: 2d2d 2d2d 2d2d 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: 2d2b 0a7c 6931 3020 3a20 636f 6d70 734a -+.|i10 : compsJ │ │ │ │ -00028cd0: 203d 2063 6f6d 7073 4a2f 7472 696d 3b20 = compsJ/trim; │ │ │ │ -00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028cc0: 2d7c 0a7c 2020 2020 202d 2074 2020 7420 -|.| - t t │ │ │ │ +00028cd0: 7420 2020 2d20 7420 2074 2074 2020 7420 t - t t t t │ │ │ │ +00028ce0: 2029 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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00028d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028d60: 2d2b 0a7c 6931 3120 3a20 2363 6f6d 7073 -+.|i11 : #comps │ │ │ │ -00028d70: 4a20 3d3d 2032 2020 2020 2020 2020 2020 J == 2 │ │ │ │ +00028d10: 207c 0a7c 2020 2020 2020 2020 3138 2031 |.| 18 1 │ │ │ │ +00028d20: 2032 3120 2020 2031 3820 3720 3133 2032 21 18 7 13 2 │ │ │ │ +00028d30: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00028d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028d60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028db0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00028dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e00: 207c 0a7c 6f31 3120 3d20 7472 7565 2020 |.|o11 = true │ │ │ │ -00028e10: 2020 2020 2020 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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00028db0: 207c 0a7c 6f38 203a 2049 6465 616c 206f |.|o8 : Ideal o │ │ │ │ +00028dc0: 6620 6b6b 5b74 202c 2074 202c 2074 2020 f kk[t , t , t │ │ │ │ +00028dd0: 2c20 7420 2c20 7420 2c20 7420 202c 2074 , t , t , t , t │ │ │ │ +00028de0: 2020 2c20 7420 202c 2074 202c 2074 202c , t , t , t , │ │ │ │ +00028df0: 2074 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +00028e00: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028e10: 2020 2020 2020 3620 2020 3520 2020 3132 6 5 12 │ │ │ │ +00028e20: 2020 2032 2020 2034 2020 2031 3120 2020 2 4 11 │ │ │ │ +00028e30: 3138 2020 2032 3420 2020 3120 2020 3320 18 24 1 3 │ │ │ │ +00028e40: 2020 3820 2020 3130 2020 2031 3720 2020 8 10 17 │ │ │ │ +00028e50: 327c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2|.|------------ │ │ │ │ 00028e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ea0: 2d2b 0a7c 6931 3220 3a20 636f 6d70 734a -+.|i12 : compsJ │ │ │ │ -00028eb0: 2f64 696d 2020 2020 2020 2020 2020 2020 /dim │ │ │ │ -00028ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ea0: 2d7c 0a7c 202c 2074 202c 2074 202c 2074 -|.| , t , t , t │ │ │ │ +00028eb0: 2020 2c20 7420 202c 2074 2020 2c20 7420 , t , t , t │ │ │ │ +00028ec0: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ +00028ed0: 2c20 7420 205d 2020 2020 2020 2020 2020 , t ] │ │ │ │ 00028ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ef0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00028f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ef0: 207c 0a7c 3320 2020 3720 2020 3920 2020 |.|3 7 9 │ │ │ │ +00028f00: 3134 2020 2031 3620 2020 3230 2020 2032 14 16 20 2 │ │ │ │ +00028f10: 3220 2020 3133 2020 2031 3520 2020 3139 2 13 15 19 │ │ │ │ +00028f20: 2020 2032 3120 2020 2020 2020 2020 2020 21 │ │ │ │ 00028f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f40: 207c 0a7c 6f31 3220 3d20 7b31 312c 2038 |.|o12 = {11, 8 │ │ │ │ -00028f50: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ -00028f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00028fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028f40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00028f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028f90: 2d2b 0a7c 6939 203a 2063 6f6d 7073 4a20 -+.|i9 : compsJ │ │ │ │ +00028fa0: 3d20 6465 636f 6d70 6f73 6520 4a3b 2020 = decompose J; │ │ │ │ 00028fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028fe0: 207c 0a7c 6f31 3220 3a20 4c69 7374 2020 |.|o12 : List │ │ │ │ -00028ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029030: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00029040: 2d2d 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: 2d2b 0a0a 5468 6572 6520 6172 6520 3220 -+..There are 2 │ │ │ │ -00029090: 636f 6d70 6f6e 656e 7473 2e20 2057 6520 components. We │ │ │ │ -000290a0: 6174 7465 6d70 7420 746f 2066 696e 6420 attempt to find │ │ │ │ -000290b0: 6120 706f 696e 7420 6f6e 2074 6865 2066 a point on the f │ │ │ │ -000290c0: 6972 7374 2063 6f6d 706f 6e65 6e74 0a0a irst component.. │ │ │ │ -000290d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -000290e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000290f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00029120: 7c69 3133 203a 2070 7431 203d 2072 616e |i13 : pt1 = ran │ │ │ │ -00029130: 646f 6d50 6f69 6e74 4f6e 5261 7469 6f6e domPointOnRation │ │ │ │ -00029140: 616c 5661 7269 6574 7920 636f 6d70 734a alVariety compsJ │ │ │ │ -00029150: 5f30 2020 2020 2020 2020 2020 2020 2020 _0 │ │ │ │ -00029160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00029170: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00028fe0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00028ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029030: 2d2b 0a7c 6931 3020 3a20 636f 6d70 734a -+.|i10 : compsJ │ │ │ │ +00029040: 203d 2063 6f6d 7073 4a2f 7472 696d 3b20 = compsJ/trim; │ │ │ │ +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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00029090: 2d2d 2d2d 2d2d 2d2d 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: 2d2b 0a7c 6931 3120 3a20 2363 6f6d 7073 -+.|i11 : #comps │ │ │ │ +000290e0: 4a20 3d3d 2032 2020 2020 2020 2020 2020 J == 2 │ │ │ │ +000290f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029120: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00029130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029170: 207c 0a7c 6f31 3120 3d20 7472 7565 2020 |.|o11 = true │ │ │ │ 00029180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000291a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000291b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000291c0: 7c6f 3133 203d 207c 202d 3436 202d 3131 |o13 = | -46 -11 │ │ │ │ -000291d0: 2032 3820 3439 202d 3438 202d 3236 2033 28 49 -48 -26 3 │ │ │ │ -000291e0: 3820 2d34 3620 2d34 3520 3139 202d 3332 8 -46 -45 19 -32 │ │ │ │ -000291f0: 2031 3720 2d31 3020 2d32 3920 3137 202d 17 -10 -29 17 - │ │ │ │ -00029200: 3820 2d32 3920 2d39 202d 3232 2020 7c0a 8 -29 -9 -22 |. │ │ │ │ -00029210: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -00029220: 2d2d 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 7c0a --------------|. │ │ │ │ -00029260: 7c20 2020 2020 202d 3234 202d 3239 202d | -24 -29 - │ │ │ │ -00029270: 3338 2031 3920 2d31 3620 7c20 2020 2020 38 19 -16 | │ │ │ │ +000291b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000291c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +000291d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000291e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000291f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029210: 2d2b 0a7c 6931 3220 3a20 636f 6d70 734a -+.|i12 : compsJ │ │ │ │ +00029220: 2f64 696d 2020 2020 2020 2020 2020 2020 /dim │ │ │ │ +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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00029270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000292a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000292b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000292c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000292a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000292b0: 207c 0a7c 6f31 3220 3d20 7b31 312c 2038 |.|o12 = {11, 8 │ │ │ │ +000292c0: 7d20 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 7c0a |. │ │ │ │ -00029300: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00029310: 3120 2020 2020 2020 3234 2020 2020 2020 1 24 │ │ │ │ +000292f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029300: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00029310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029340: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00029350: 7c6f 3133 203a 204d 6174 7269 7820 6b6b |o13 : Matrix kk │ │ │ │ -00029360: 2020 3c2d 2d20 6b6b 2020 2020 2020 2020 <-- kk │ │ │ │ +00029340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029350: 207c 0a7c 6f31 3220 3a20 4c69 7374 2020 |.|o12 : List │ │ │ │ +00029360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029390: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000293a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00029390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000293a0: 207c 0a2b 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 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000293e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000293f0: 7c69 3134 203a 2046 3120 3d20 7375 6228 |i14 : F1 = sub( │ │ │ │ -00029400: 462c 2028 7661 7273 2053 297c 7074 3129 F, (vars S)|pt1) │ │ │ │ -00029410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029430: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00029440: 7c20 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 2020 2020 2020 │ │ │ │ -00029480: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00029490: 7c20 2020 2020 2020 2020 2020 2020 2032 | 2 │ │ │ │ -000294a0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -000294b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000294c0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +000293e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000293f0: 2d2b 0a0a 5468 6572 6520 6172 6520 3220 -+..There are 2 │ │ │ │ +00029400: 636f 6d70 6f6e 656e 7473 2e20 2057 6520 components. We │ │ │ │ +00029410: 6174 7465 6d70 7420 746f 2066 696e 6420 attempt to find │ │ │ │ +00029420: 6120 706f 696e 7420 6f6e 2074 6865 2066 a point on the f │ │ │ │ +00029430: 6972 7374 2063 6f6d 706f 6e65 6e74 0a0a irst component.. │ │ │ │ +00029440: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00029450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00029490: 7c69 3133 203a 2070 7431 203d 2072 616e |i13 : pt1 = ran │ │ │ │ +000294a0: 646f 6d50 6f69 6e74 4f6e 5261 7469 6f6e domPointOnRation │ │ │ │ +000294b0: 616c 5661 7269 6574 7920 636f 6d70 734a alVariety compsJ │ │ │ │ +000294c0: 5f30 2020 2020 2020 2020 2020 2020 2020 _0 │ │ │ │ 000294d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000294e0: 7c6f 3134 203d 2069 6465 616c 2028 6120 |o14 = ideal (a │ │ │ │ -000294f0: 202d 2034 3562 2a63 202b 2034 3963 2020 - 45b*c + 49c │ │ │ │ -00029500: 2b20 3139 612a 6420 2d20 3438 622a 6420 + 19a*d - 48b*d │ │ │ │ -00029510: 2d20 3131 632a 6420 2d20 3436 6420 2c20 - 11c*d - 46d , │ │ │ │ -00029520: 612a 6220 2b20 3137 622a 6320 2d20 7c0a a*b + 17b*c - |. │ │ │ │ -00029530: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -00029540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00029580: 7c20 2020 2020 2020 2020 3220 2020 2020 | 2 │ │ │ │ -00029590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000295a0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -000295b0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -000295c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000295d0: 7c20 2020 2020 2033 3263 2020 2d20 3861 | 32c - 8a │ │ │ │ -000295e0: 2a64 202b 2031 3762 2a64 202d 2032 3663 *d + 17b*d - 26c │ │ │ │ -000295f0: 2a64 202b 2032 3864 202c 2061 2a63 202d *d + 28d , a*c - │ │ │ │ -00029600: 2032 3962 2a63 202d 2032 3963 2020 2d20 29b*c - 29c - │ │ │ │ -00029610: 3338 612a 6420 2d20 3962 2a64 202d 7c0a 38a*d - 9b*d -|. │ │ │ │ -00029620: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -00029630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +000294e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000294f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029520: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029530: 7c6f 3133 203d 207c 202d 3432 2031 3220 |o13 = | -42 12 │ │ │ │ +00029540: 2d39 202d 3436 202d 3137 202d 3431 202d -9 -46 -17 -41 - │ │ │ │ +00029550: 3233 202d 3620 3132 202d 3338 2032 3120 23 -6 12 -38 21 │ │ │ │ +00029560: 2d31 3120 3139 202d 3239 202d 3320 2d32 -11 19 -29 -3 -2 │ │ │ │ +00029570: 3220 2d31 3620 2d31 3320 2d38 2020 7c0a 2 -16 -13 -8 |. │ │ │ │ +00029580: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +00029590: 2d2d 2d2d 2d2d 2d2d 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 7c0a --------------|. │ │ │ │ +000295d0: 7c20 2020 2020 2031 3920 2d31 3020 2d32 | 19 -10 -2 │ │ │ │ +000295e0: 3920 2d32 3920 2d32 3420 7c20 2020 2020 9 -29 -24 | │ │ │ │ +000295f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029610: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029620: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00029630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029660: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00029670: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00029680: 2020 3220 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ -00029690: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00029680: 3120 2020 2020 2020 3234 2020 2020 2020 1 24 │ │ │ │ +00029690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000296a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000296b0: 2020 2020 3220 2020 2020 2020 2020 7c0a 2 |. │ │ │ │ -000296c0: 7c20 2020 2020 2031 3063 2a64 202b 2033 | 10c*d + 3 │ │ │ │ -000296d0: 3864 202c 2062 2020 2b20 3139 622a 6320 8d , b + 19b*c │ │ │ │ -000296e0: 2d20 3232 6320 202d 2031 3661 2a64 202d - 22c - 16a*d - │ │ │ │ -000296f0: 2032 3462 2a64 202d 2032 3963 2a64 202d 24b*d - 29c*d - │ │ │ │ -00029700: 2034 3664 2029 2020 2020 2020 2020 7c0a 46d ) |. │ │ │ │ -00029710: 7c20 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 2020 2020 2020 │ │ │ │ -00029750: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00029760: 7c6f 3134 203a 2049 6465 616c 206f 6620 |o14 : Ideal of │ │ │ │ -00029770: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +000296b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000296c0: 7c6f 3133 203a 204d 6174 7269 7820 6b6b |o13 : Matrix kk │ │ │ │ +000296d0: 2020 3c2d 2d20 6b6b 2020 2020 2020 2020 <-- kk │ │ │ │ +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 7c0a |. │ │ │ │ +00029710: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00029720: 2d2d 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 2b0a --------------+. │ │ │ │ +00029760: 7c69 3134 203a 2046 3120 3d20 7375 6228 |i14 : F1 = sub( │ │ │ │ +00029770: 462c 2028 7661 7273 2053 297c 7074 3129 F, (vars S)|pt1) │ │ │ │ 00029780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000297a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000297b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -000297c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000297d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000297e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000297f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00029800: 7c69 3135 203a 2064 6563 6f6d 706f 7365 |i15 : decompose │ │ │ │ -00029810: 2046 3120 2020 2020 2020 2020 2020 2020 F1 │ │ │ │ +000297b0: 7c20 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 │ │ │ │ +000297e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000297f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029800: 7c20 2020 2020 2020 2020 2020 2020 2032 | 2 │ │ │ │ +00029810: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ 00029820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029830: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ 00029840: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00029850: 7c20 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 2020 2020 2020 │ │ │ │ -00029890: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000298a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000298b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000298c0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000298d0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000298e0: 2020 2020 2020 2020 2032 2020 2020 7c0a 2 |. │ │ │ │ -000298f0: 7c6f 3135 203d 207b 6964 6561 6c20 2861 |o15 = {ideal (a │ │ │ │ -00029900: 202d 2032 3962 202d 2032 3963 202d 2064 - 29b - 29c - d │ │ │ │ -00029910: 2c20 6220 202b 2031 3962 2a63 202d 2032 , b + 19b*c - 2 │ │ │ │ -00029920: 3263 2020 2b20 3137 622a 6420 2b20 3132 2c + 17b*d + 12 │ │ │ │ -00029930: 632a 6420 2b20 3339 6420 292c 2020 7c0a c*d + 39d ), |. │ │ │ │ -00029940: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ -00029950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -00029990: 7c20 2020 2020 2069 6465 616c 2028 6320 | ideal (c │ │ │ │ -000299a0: 2d20 3338 642c 2062 202d 2034 3964 2c20 - 38d, b - 49d, │ │ │ │ -000299b0: 6120 2b20 3431 6429 7d20 2020 2020 2020 a + 41d)} │ │ │ │ -000299c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000299d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029850: 7c6f 3134 203d 2069 6465 616c 2028 6120 |o14 = ideal (a │ │ │ │ +00029860: 202b 2031 3262 2a63 202d 2034 3663 2020 + 12b*c - 46c │ │ │ │ +00029870: 2d20 3338 612a 6420 2d20 3137 622a 6420 - 38a*d - 17b*d │ │ │ │ +00029880: 2b20 3132 632a 6420 2d20 3432 6420 2c20 + 12c*d - 42d , │ │ │ │ +00029890: 612a 6220 2d20 3362 2a63 202b 2020 7c0a a*b - 3b*c + |. │ │ │ │ +000298a0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +000298b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000298c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000298d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000298e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +000298f0: 7c20 2020 2020 2020 2020 3220 2020 2020 | 2 │ │ │ │ +00029900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029910: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00029920: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +00029930: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029940: 7c20 2020 2020 2032 3163 2020 2d20 3232 | 21c - 22 │ │ │ │ +00029950: 612a 6420 2d20 3131 622a 6420 2d20 3431 a*d - 11b*d - 41 │ │ │ │ +00029960: 632a 6420 2d20 3964 202c 2061 2a63 202d c*d - 9d , a*c - │ │ │ │ +00029970: 2031 3062 2a63 202d 2031 3663 2020 2d20 10b*c - 16c - │ │ │ │ +00029980: 3239 612a 6420 2d20 3133 622a 6420 7c0a 29a*d - 13b*d |. │ │ │ │ +00029990: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +000299a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000299b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000299c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000299d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ 000299e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000299f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000299f0: 2020 2020 3220 2020 3220 2020 2020 2020 2 2 │ │ │ │ +00029a00: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ 00029a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029a20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00029a30: 7c6f 3135 203a 204c 6973 7420 2020 2020 |o15 : List │ │ │ │ -00029a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029a70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00029a80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00029a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00029ad0: 0a57 6520 6174 7465 6d70 7420 746f 2066 .We attempt to f │ │ │ │ -00029ae0: 696e 6420 6120 706f 696e 7420 6f6e 2074 ind a point on t │ │ │ │ -00029af0: 6865 2073 6563 6f6e 6420 636f 6d70 6f6e he second compon │ │ │ │ -00029b00: 656e 7420 696e 2070 6172 616d 6574 6572 ent in parameter │ │ │ │ -00029b10: 2073 7061 6365 2c20 616e 6420 6974 730a space, and its. │ │ │ │ -00029b20: 636f 7272 6573 706f 6e64 696e 6720 6964 corresponding id │ │ │ │ -00029b30: 6561 6c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d eal...+--------- │ │ │ │ +00029a20: 2020 2020 3220 2020 2020 2020 2020 7c0a 2 |. │ │ │ │ +00029a30: 7c20 2020 2020 202b 2031 3963 2a64 202d | + 19c*d - │ │ │ │ +00029a40: 2032 3364 202c 2062 2020 2d20 3239 622a 23d , b - 29b* │ │ │ │ +00029a50: 6320 2d20 3863 2020 2d20 3234 612a 6420 c - 8c - 24a*d │ │ │ │ +00029a60: 2b20 3139 622a 6420 2d20 3239 632a 6420 + 19b*d - 29c*d │ │ │ │ +00029a70: 2d20 3664 2029 2020 2020 2020 2020 7c0a - 6d ) |. │ │ │ │ +00029a80: 7c20 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 2020 2020 2020 │ │ │ │ +00029ac0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029ad0: 7c6f 3134 203a 2049 6465 616c 206f 6620 |o14 : Ideal of │ │ │ │ +00029ae0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00029af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029b10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029b20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00029b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b80: 2d2d 2d2d 2b0a 7c69 3136 203a 2070 7432 ----+.|i16 : pt2 │ │ │ │ -00029b90: 203d 2072 616e 646f 6d50 6f69 6e74 4f6e = randomPointOn │ │ │ │ -00029ba0: 5261 7469 6f6e 616c 5661 7269 6574 7920 RationalVariety │ │ │ │ -00029bb0: 636f 6d70 734a 5f31 2020 2020 2020 2020 compsJ_1 │ │ │ │ -00029bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00029b70: 7c69 3135 203a 2064 6563 6f6d 706f 7365 |i15 : decompose │ │ │ │ +00029b80: 2046 3120 2020 2020 2020 2020 2020 2020 F1 │ │ │ │ +00029b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029bb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029bc0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00029bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c20: 2020 2020 7c0a 7c6f 3136 203d 207c 202d |.|o16 = | - │ │ │ │ -00029c30: 3820 3238 2032 3520 3330 202d 3132 202d 8 28 25 30 -12 - │ │ │ │ -00029c40: 3432 202d 3230 202d 3335 202d 3720 2d33 42 -20 -35 -7 -3 │ │ │ │ -00029c50: 3720 2d35 2034 3520 3139 2032 3020 2d32 7 -5 45 19 20 -2 │ │ │ │ -00029c60: 3420 3334 2033 3920 3231 202d 3437 202d 4 34 39 21 -47 - │ │ │ │ -00029c70: 3339 2020 7c0a 7c20 2020 2020 202d 2d2d 39 |.| --- │ │ │ │ -00029c80: 2d2d 2d2d 2d2d 2d2d 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 7c0a 7c20 2020 2020 202d 3133 ----|.| -13 │ │ │ │ -00029cd0: 202d 3138 2033 3420 3020 7c20 2020 2020 -18 34 0 | │ │ │ │ -00029ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00029d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029c10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00029c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c30: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00029c40: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00029c50: 2020 2020 2020 2020 2032 2020 2020 7c0a 2 |. │ │ │ │ +00029c60: 7c6f 3135 203d 207b 6964 6561 6c20 2861 |o15 = {ideal (a │ │ │ │ +00029c70: 202d 2031 3062 202d 2031 3663 202d 2034 - 10b - 16c - 4 │ │ │ │ +00029c80: 3164 2c20 6220 202d 2032 3962 2a63 202d 1d, b - 29b*c - │ │ │ │ +00029c90: 2038 6320 202d 2031 3962 2a64 202d 2039 8c - 19b*d - 9 │ │ │ │ +00029ca0: 632a 6420 2b20 3230 6420 292c 2020 7c0a c*d + 20d ), |. │ │ │ │ +00029cb0: 7c20 2020 2020 202d 2d2d 2d2d 2d2d 2d2d | --------- │ │ │ │ +00029cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +00029d00: 7c20 2020 2020 2069 6465 616c 2028 6320 | ideal (c │ │ │ │ +00029d10: 2d20 3239 642c 2062 202b 2031 3664 2c20 - 29d, b + 16d, │ │ │ │ +00029d20: 6120 2d20 3230 6429 7d20 2020 2020 2020 a - 20d)} │ │ │ │ 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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00029d70: 2020 2020 2020 3120 2020 2020 2020 3234 1 24 │ │ │ │ +00029d40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029d50: 7c20 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 2020 2020 2020 │ │ │ │ -00029d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029db0: 2020 2020 7c0a 7c6f 3136 203a 204d 6174 |.|o16 : Mat │ │ │ │ -00029dc0: 7269 7820 6b6b 2020 3c2d 2d20 6b6b 2020 rix kk <-- kk │ │ │ │ +00029d90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029da0: 7c6f 3135 203a 204c 6973 7420 2020 2020 |o15 : List │ │ │ │ +00029db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00029de0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00029df0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00029e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 2b0a 7c69 3137 203a 2046 3220 ----+.|i17 : F2 │ │ │ │ -00029e60: 3d20 7375 6228 462c 2028 7661 7273 2053 = sub(F, (vars S │ │ │ │ -00029e70: 297c 7074 3229 2020 2020 2020 2020 2020 )|pt2) │ │ │ │ -00029e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00029eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ef0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00029f00: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00029f10: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00029f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029f30: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00029f40: 2020 2020 7c0a 7c6f 3137 203d 2069 6465 |.|o17 = ide │ │ │ │ -00029f50: 616c 2028 6120 202d 2037 622a 6320 2b20 al (a - 7b*c + │ │ │ │ -00029f60: 3330 6320 202d 2033 3761 2a64 202d 2031 30c - 37a*d - 1 │ │ │ │ -00029f70: 3262 2a64 202b 2032 3863 2a64 202d 2038 2b*d + 28c*d - 8 │ │ │ │ -00029f80: 6420 2c20 612a 6220 2d20 3234 622a 6320 d , a*b - 24b*c │ │ │ │ -00029f90: 2d20 2020 7c0a 7c20 2020 2020 202d 2d2d - |.| --- │ │ │ │ -00029fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029fe0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2032 ----|.| 2 │ │ │ │ -00029ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a000: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -0002a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a020: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002a030: 2020 2020 7c0a 7c20 2020 2020 2035 6320 |.| 5c │ │ │ │ -0002a040: 202b 2033 3461 2a64 202b 2034 3562 2a64 + 34a*d + 45b*d │ │ │ │ -0002a050: 202d 2034 3263 2a64 202b 2032 3564 202c - 42c*d + 25d , │ │ │ │ -0002a060: 2061 2a63 202d 2031 3362 2a63 202b 2033 a*c - 13b*c + 3 │ │ │ │ -0002a070: 3963 2020 2d20 3138 612a 6420 2b20 3231 9c - 18a*d + 21 │ │ │ │ -0002a080: 622a 6420 7c0a 7c20 2020 2020 202d 2d2d b*d |.| --- │ │ │ │ -0002a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0d0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ -0002a0e0: 2020 2020 2020 2020 2020 3220 2020 3220 2 2 │ │ │ │ -0002a0f0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00029e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00029e40: 0a57 6520 6174 7465 6d70 7420 746f 2066 .We attempt to f │ │ │ │ +00029e50: 696e 6420 6120 706f 696e 7420 6f6e 2074 ind a point on t │ │ │ │ +00029e60: 6865 2073 6563 6f6e 6420 636f 6d70 6f6e he second compon │ │ │ │ +00029e70: 656e 7420 696e 2070 6172 616d 6574 6572 ent in parameter │ │ │ │ +00029e80: 2073 7061 6365 2c20 616e 6420 6974 730a space, and its. │ │ │ │ +00029e90: 636f 7272 6573 706f 6e64 696e 6720 6964 corresponding id │ │ │ │ +00029ea0: 6561 6c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d eal...+--------- │ │ │ │ +00029eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029ef0: 2d2d 2d2d 2b0a 7c69 3136 203a 2070 7432 ----+.|i16 : pt2 │ │ │ │ +00029f00: 203d 2072 616e 646f 6d50 6f69 6e74 4f6e = randomPointOn │ │ │ │ +00029f10: 5261 7469 6f6e 616c 5661 7269 6574 7920 RationalVariety │ │ │ │ +00029f20: 636f 6d70 734a 5f31 2020 2020 2020 2020 compsJ_1 │ │ │ │ +00029f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00029f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f90: 2020 2020 7c0a 7c6f 3136 203d 207c 202d |.|o16 = | - │ │ │ │ +00029fa0: 3620 2d31 3320 3231 202d 3135 202d 3331 6 -13 21 -15 -31 │ │ │ │ +00029fb0: 2036 202d 3920 2d31 2035 3020 2d33 3920 6 -9 -1 50 -39 │ │ │ │ +00029fc0: 2d31 3820 3231 202d 3339 2033 3120 2d33 -18 21 -39 31 -3 │ │ │ │ +00029fd0: 3120 3434 202d 3138 202d 3133 202d 3437 1 44 -18 -13 -47 │ │ │ │ +00029fe0: 2032 3120 7c0a 7c20 2020 2020 202d 2d2d 21 |.| --- │ │ │ │ +00029ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a030: 2d2d 2d2d 7c0a 7c20 2020 2020 2033 3420 ----|.| 34 │ │ │ │ +0002a040: 3139 2033 3920 3020 7c20 2020 2020 2020 19 39 0 | │ │ │ │ +0002a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a080: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a0d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a0e0: 2020 2020 2020 3120 2020 2020 2020 3234 1 24 │ │ │ │ +0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a110: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0002a120: 2020 2020 7c0a 7c20 2020 2020 202b 2031 |.| + 1 │ │ │ │ -0002a130: 3963 2a64 202d 2032 3064 202c 2062 2020 9c*d - 20d , b │ │ │ │ -0002a140: 2b20 3334 622a 6320 2d20 3437 6320 202d + 34b*c - 47c - │ │ │ │ -0002a150: 2033 3962 2a64 202b 2032 3063 2a64 202d 39b*d + 20c*d - │ │ │ │ -0002a160: 2033 3564 2029 2020 2020 2020 2020 2020 35d ) │ │ │ │ -0002a170: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1c0: 2020 2020 7c0a 7c6f 3137 203a 2049 6465 |.|o17 : Ide │ │ │ │ -0002a1d0: 616c 206f 6620 5320 2020 2020 2020 2020 al of S │ │ │ │ -0002a1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a120: 2020 2020 7c0a 7c6f 3136 203a 204d 6174 |.|o16 : Mat │ │ │ │ +0002a130: 7269 7820 6b6b 2020 3c2d 2d20 6b6b 2020 rix kk <-- kk │ │ │ │ +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 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a1c0: 2d2d 2d2d 2b0a 7c69 3137 203a 2046 3220 ----+.|i17 : F2 │ │ │ │ +0002a1d0: 3d20 7375 6228 462c 2028 7661 7273 2053 = sub(F, (vars S │ │ │ │ +0002a1e0: 297c 7074 3229 2020 2020 2020 2020 2020 )|pt2) │ │ │ │ 0002a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a210: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0002a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a260: 2d2d 2d2d 2b0a 7c69 3138 203a 2064 6563 ----+.|i18 : dec │ │ │ │ -0002a270: 6f6d 706f 7365 2046 3220 2020 2020 2020 ompose F2 │ │ │ │ -0002a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a260: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a270: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a280: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 0002a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a300: 2020 2020 7c0a 7c6f 3138 203d 207b 6964 |.|o18 = {id │ │ │ │ -0002a310: 6561 6c20 2862 202d 2031 3263 202b 2034 eal (b - 12c + 4 │ │ │ │ -0002a320: 3864 2c20 6120 2d20 3136 6320 2b20 6429 8d, a - 16c + d) │ │ │ │ -0002a330: 2c20 6964 6561 6c20 2862 202b 2034 3663 , ideal (b + 46c │ │ │ │ -0002a340: 202b 2031 3464 2c20 6120 2b20 3331 6320 + 14d, a + 31c │ │ │ │ -0002a350: 2d20 2020 7c0a 7c20 2020 2020 202d 2d2d - |.| --- │ │ │ │ -0002a360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a3a0: 2d2d 2d2d 7c0a 7c20 2020 2020 2035 6429 ----|.| 5d) │ │ │ │ -0002a3b0: 7d20 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 7c20 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 7c6f 3138 203a 204c 6973 |.|o18 : Lis │ │ │ │ -0002a450: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -0002a460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a2a0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a2b0: 2020 2020 7c0a 7c6f 3137 203d 2069 6465 |.|o17 = ide │ │ │ │ +0002a2c0: 616c 2028 6120 202b 2035 3062 2a63 202d al (a + 50b*c - │ │ │ │ +0002a2d0: 2031 3563 2020 2d20 3339 612a 6420 2d20 15c - 39a*d - │ │ │ │ +0002a2e0: 3331 622a 6420 2d20 3133 632a 6420 2d20 31b*d - 13c*d - │ │ │ │ +0002a2f0: 3664 202c 2061 2a62 202d 2033 3162 2a63 6d , a*b - 31b*c │ │ │ │ +0002a300: 202d 2020 7c0a 7c20 2020 2020 202d 2d2d - |.| --- │ │ │ │ +0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a350: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +0002a360: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a370: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +0002a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a390: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a3a0: 2020 2020 7c0a 7c20 2020 2020 2031 3863 |.| 18c │ │ │ │ +0002a3b0: 2020 2b20 3434 612a 6420 2b20 3231 622a + 44a*d + 21b* │ │ │ │ +0002a3c0: 6420 2b20 3663 2a64 202b 2032 3164 202c d + 6c*d + 21d , │ │ │ │ +0002a3d0: 2061 2a63 202b 2033 3462 2a63 202d 2031 a*c + 34b*c - 1 │ │ │ │ +0002a3e0: 3863 2020 2b20 3139 612a 6420 2d20 3133 8c + 19a*d - 13 │ │ │ │ +0002a3f0: 622a 6420 7c0a 7c20 2020 2020 202d 2d2d b*d |.| --- │ │ │ │ +0002a400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a440: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +0002a450: 2020 2020 2020 2020 2032 2020 2032 2020 2 2 │ │ │ │ +0002a460: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ 0002a470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a490: 2020 2020 7c0a 2b2d 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 2d2d ---------------- │ │ │ │ -0002a4e0: 2d2d 2d2d 2b0a 0a49 7420 7475 726e 7320 ----+..It turns │ │ │ │ -0002a4f0: 6f75 7420 7468 6174 2074 6869 7320 6973 out that this is │ │ │ │ -0002a500: 2074 6865 2069 6465 616c 206f 6620 3220 the ideal of 2 │ │ │ │ -0002a510: 736b 6577 206c 696e 6573 2c20 6a75 7374 skew lines, just │ │ │ │ -0002a520: 206e 6f74 2064 6566 696e 6564 206f 7665 not defined ove │ │ │ │ -0002a530: 7220 7468 6973 0a66 6965 6c64 2e0a 0a43 r this.field...C │ │ │ │ -0002a540: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 5468 aveat.======..Th │ │ │ │ -0002a550: 6973 2072 6f75 7469 6e65 2065 7870 6563 is routine expec │ │ │ │ -0002a560: 7473 2074 6865 2069 6e70 7574 2074 6f20 ts the input to │ │ │ │ -0002a570: 7265 7072 6573 656e 7420 616e 2069 7272 represent an irr │ │ │ │ -0002a580: 6564 7563 6962 6c65 2076 6172 6965 7479 educible variety │ │ │ │ -0002a590: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -0002a5a0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2072 ===.. * *note r │ │ │ │ -0002a5b0: 616e 646f 6d50 6f69 6e74 734f 6e52 6174 andomPointsOnRat │ │ │ │ -0002a5c0: 696f 6e61 6c56 6172 6965 7479 3a0a 2020 ionalVariety:. │ │ │ │ -0002a5d0: 2020 7261 6e64 6f6d 506f 696e 7473 4f6e randomPointsOn │ │ │ │ -0002a5e0: 5261 7469 6f6e 616c 5661 7269 6574 795f RationalVariety_ │ │ │ │ -0002a5f0: 6c70 4964 6561 6c5f 636d 5a5a 5f72 702c lpIdeal_cmZZ_rp, │ │ │ │ -0002a600: 202d 2d20 6669 6e64 2072 616e 646f 6d20 -- find random │ │ │ │ -0002a610: 706f 696e 7473 206f 6e20 610a 2020 2020 points on a. │ │ │ │ -0002a620: 7661 7269 6574 7920 7468 6174 2063 616e variety that can │ │ │ │ -0002a630: 2062 6520 6465 7465 6374 6564 2074 6f20 be detected to │ │ │ │ -0002a640: 6265 2072 6174 696f 6e61 6c0a 0a57 6179 be rational..Way │ │ │ │ -0002a650: 7320 746f 2075 7365 2074 6869 7320 6d65 s to use this me │ │ │ │ -0002a660: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d thod:.========== │ │ │ │ -0002a670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -0002a680: 2020 2a20 2a6e 6f74 6520 7261 6e64 6f6d * *note random │ │ │ │ -0002a690: 506f 696e 744f 6e52 6174 696f 6e61 6c56 PointOnRationalV │ │ │ │ -0002a6a0: 6172 6965 7479 2849 6465 616c 293a 0a20 ariety(Ideal):. │ │ │ │ -0002a6b0: 2020 2072 616e 646f 6d50 6f69 6e74 4f6e randomPointOn │ │ │ │ -0002a6c0: 5261 7469 6f6e 616c 5661 7269 6574 795f RationalVariety_ │ │ │ │ -0002a6d0: 6c70 4964 6561 6c5f 7270 2c20 2d2d 2066 lpIdeal_rp, -- f │ │ │ │ -0002a6e0: 696e 6420 6120 7261 6e64 6f6d 2070 6f69 ind a random poi │ │ │ │ -0002a6f0: 6e74 206f 6e20 610a 2020 2020 7661 7269 nt on a. vari │ │ │ │ -0002a700: 6574 7920 7468 6174 2063 616e 2062 6520 ety that can be │ │ │ │ -0002a710: 6465 7465 6374 6564 2074 6f20 6265 2072 detected to be r │ │ │ │ -0002a720: 6174 696f 6e61 6c0a 2d2d 2d2d 2d2d 2d2d ational.-------- │ │ │ │ +0002a480: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a490: 2020 2020 7c0a 7c20 2020 2020 202d 2033 |.| - 3 │ │ │ │ +0002a4a0: 3963 2a64 202d 2039 6420 2c20 6220 202b 9c*d - 9d , b + │ │ │ │ +0002a4b0: 2033 3962 2a63 202d 2034 3763 2020 2b20 39b*c - 47c + │ │ │ │ +0002a4c0: 3231 622a 6420 2b20 3331 632a 6420 2d20 21b*d + 31c*d - │ │ │ │ +0002a4d0: 6420 2920 2020 2020 2020 2020 2020 2020 d ) │ │ │ │ +0002a4e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +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 2020 │ │ │ │ +0002a530: 2020 2020 7c0a 7c6f 3137 203a 2049 6465 |.|o17 : Ide │ │ │ │ +0002a540: 616c 206f 6620 5320 2020 2020 2020 2020 al of S │ │ │ │ +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 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a5d0: 2d2d 2d2d 2b0a 7c69 3138 203a 2064 6563 ----+.|i18 : dec │ │ │ │ +0002a5e0: 6f6d 706f 7365 2046 3220 2020 2020 2020 ompose F2 │ │ │ │ +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 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 |.| │ │ │ │ +0002a680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a690: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +0002a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6b0: 2020 2020 3220 2020 3220 2020 2020 2020 2 2 │ │ │ │ +0002a6c0: 2020 2020 7c0a 7c6f 3138 203d 207b 6964 |.|o18 = {id │ │ │ │ +0002a6d0: 6561 6c20 2861 2a63 202b 2033 3462 2a63 eal (a*c + 34b*c │ │ │ │ +0002a6e0: 202d 2031 3863 2020 2b20 3139 612a 6420 - 18c + 19a*d │ │ │ │ +0002a6f0: 2d20 3133 622a 6420 2d20 3339 632a 6420 - 13b*d - 39c*d │ │ │ │ +0002a700: 2d20 3964 202c 2062 2020 2b20 3339 622a - 9d , b + 39b* │ │ │ │ +0002a710: 6320 2d20 7c0a 7c20 2020 2020 202d 2d2d c - |.| --- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a770: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -0002a780: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -0002a790: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -0002a7a0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -0002a7b0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -0002a7c0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ -0002a7d0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -0002a7e0: 4772 6f65 626e 6572 5374 7261 7461 2e6d GroebnerStrata.m │ │ │ │ -0002a7f0: 323a 3935 373a 302e 0a1f 0a46 696c 653a 2:957:0....File: │ │ │ │ -0002a800: 2047 726f 6562 6e65 7253 7472 6174 612e GroebnerStrata. │ │ │ │ -0002a810: 696e 666f 2c20 4e6f 6465 3a20 7261 6e64 info, Node: rand │ │ │ │ -0002a820: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ -0002a830: 616c 5661 7269 6574 795f 6c70 4964 6561 alVariety_lpIdea │ │ │ │ -0002a840: 6c5f 636d 5a5a 5f72 702c 204e 6578 743a l_cmZZ_rp, Next: │ │ │ │ -0002a850: 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 616c smallerMonomial │ │ │ │ -0002a860: 732c 2050 7265 763a 2072 616e 646f 6d50 s, Prev: randomP │ │ │ │ -0002a870: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ -0002a880: 7269 6574 795f 6c70 4964 6561 6c5f 7270 riety_lpIdeal_rp │ │ │ │ -0002a890: 2c20 5570 3a20 546f 700a 0a72 616e 646f , Up: Top..rando │ │ │ │ -0002a8a0: 6d50 6f69 6e74 734f 6e52 6174 696f 6e61 mPointsOnRationa │ │ │ │ -0002a8b0: 6c56 6172 6965 7479 2849 6465 616c 2c5a lVariety(Ideal,Z │ │ │ │ -0002a8c0: 5a29 202d 2d20 6669 6e64 2072 616e 646f Z) -- find rando │ │ │ │ -0002a8d0: 6d20 706f 696e 7473 206f 6e20 6120 7661 m points on a va │ │ │ │ -0002a8e0: 7269 6574 7920 7468 6174 2063 616e 2062 riety that can b │ │ │ │ -0002a8f0: 6520 6465 7465 6374 6564 2074 6f20 6265 e detected to be │ │ │ │ -0002a900: 2072 6174 696f 6e61 6c0a 2a2a 2a2a 2a2a rational.****** │ │ │ │ -0002a910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -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 0a0a 2020 2a20 4675 ********.. * Fu │ │ │ │ -0002a980: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7261 nction: *note ra │ │ │ │ -0002a990: 6e64 6f6d 506f 696e 7473 4f6e 5261 7469 ndomPointsOnRati │ │ │ │ -0002a9a0: 6f6e 616c 5661 7269 6574 793a 0a20 2020 onalVariety:. │ │ │ │ -0002a9b0: 2072 616e 646f 6d50 6f69 6e74 734f 6e52 randomPointsOnR │ │ │ │ -0002a9c0: 6174 696f 6e61 6c56 6172 6965 7479 5f6c ationalVariety_l │ │ │ │ -0002a9d0: 7049 6465 616c 5f63 6d5a 5a5f 7270 2c0a pIdeal_cmZZ_rp,. │ │ │ │ -0002a9e0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -0002a9f0: 2020 2020 7261 6e64 6f6d 506f 696e 7473 randomPoints │ │ │ │ -0002aa00: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ -0002aa10: 7928 492c 206e 290a 2020 2020 2020 2020 y(I, n). │ │ │ │ -0002aa20: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ -0002aa30: 696f 6e61 6c56 6172 6965 7479 0a20 202a ionalVariety. * │ │ │ │ -0002aa40: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -0002aa50: 2049 2c20 616e 202a 6e6f 7465 2069 6465 I, an *note ide │ │ │ │ -0002aa60: 616c 3a20 284d 6163 6175 6c61 7932 446f al: (Macaulay2Do │ │ │ │ -0002aa70: 6329 4964 6561 6c2c 2c20 416e 2069 6465 c)Ideal,, An ide │ │ │ │ -0002aa80: 616c 2069 6e20 6120 706f 6c79 6e6f 6d69 al in a polynomi │ │ │ │ -0002aa90: 616c 2072 696e 670a 2020 2020 2020 2020 al ring. │ │ │ │ -0002aaa0: 2453 2420 6f76 6572 2061 2066 6965 6c64 $S$ over a field │ │ │ │ -0002aab0: 2c20 7768 6963 6820 6465 6669 6e65 7320 , which defines │ │ │ │ -0002aac0: 6120 7072 696d 6520 6964 6561 6c0a 2020 a prime ideal. │ │ │ │ -0002aad0: 2020 2020 2a20 6e2c 2061 6e20 2a6e 6f74 * n, an *not │ │ │ │ -0002aae0: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ -0002aaf0: 756c 6179 3244 6f63 295a 5a2c 2c20 5468 ulay2Doc)ZZ,, Th │ │ │ │ -0002ab00: 6520 6e75 6d62 6572 206f 6620 706f 696e e number of poin │ │ │ │ -0002ab10: 7473 2074 6f0a 2020 2020 2020 2020 6765 ts to. ge │ │ │ │ -0002ab20: 6e65 7261 7465 0a20 202a 204f 7574 7075 nerate. * Outpu │ │ │ │ -0002ab30: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e ts:. * a *n │ │ │ │ -0002ab40: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ -0002ab50: 6c61 7932 446f 6329 4c69 7374 2c2c 2041 lay2Doc)List,, A │ │ │ │ -0002ab60: 206c 6973 7420 6f66 2024 6e24 206f 6e65 list of $n$ one │ │ │ │ -0002ab70: 2072 6f77 206d 6174 7269 6365 7320 6f76 row matrices ov │ │ │ │ -0002ab80: 6572 0a20 2020 2020 2020 2074 6865 2062 er. the b │ │ │ │ -0002ab90: 6173 6520 6669 656c 6420 6f66 2024 5324 ase field of $S$ │ │ │ │ -0002aba0: 2c20 7468 6174 2061 7265 2072 616e 646f , that are rando │ │ │ │ -0002abb0: 6d6c 7920 6368 6f73 656e 2070 6f69 6e74 mly chosen point │ │ │ │ -0002abc0: 7320 6f6e 2024 4924 2e20 206e 756c 6c20 s on $I$. null │ │ │ │ -0002abd0: 6973 0a20 2020 2020 2020 2072 6574 7572 is. retur │ │ │ │ -0002abe0: 6e65 6420 696e 2074 6865 2063 6173 6520 ned in the case │ │ │ │ -0002abf0: 7768 656e 2074 6865 2072 6f75 7469 6e65 when the routine │ │ │ │ -0002ac00: 2063 616e 6e6f 7420 6465 7465 726d 696e cannot determin │ │ │ │ -0002ac10: 6520 6966 2074 6865 2076 6172 6965 7479 e if the variety │ │ │ │ -0002ac20: 0a20 2020 2020 2020 2069 7320 7261 7469 . is rati │ │ │ │ -0002ac30: 6f6e 616c 2061 6e64 2069 7272 6564 7563 onal and irreduc │ │ │ │ -0002ac40: 6962 6c65 2e0a 0a44 6573 6372 6970 7469 ible...Descripti │ │ │ │ -0002ac50: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -0002ac60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002ac70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ac80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002acb0: 7c69 3120 3a20 6b6b 203d 205a 5a2f 3130 |i1 : kk = ZZ/10 │ │ │ │ -0002acc0: 313b 2020 2020 2020 2020 2020 2020 2020 1; │ │ │ │ -0002acd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002acf0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ad00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002ad10: 2d2d 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 2b0a --------------+. │ │ │ │ -0002ad50: 7c69 3220 3a20 5320 3d20 6b6b 5b61 2e2e |i2 : S = kk[a.. │ │ │ │ -0002ad60: 665d 3b20 2020 2020 2020 2020 2020 2020 f]; │ │ │ │ -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 7c0a |. │ │ │ │ -0002ada0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002adb0: 2d2d 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 2b0a --------------+. │ │ │ │ -0002adf0: 7c69 3320 3a20 4920 3d20 6d69 6e6f 7273 |i3 : I = minors │ │ │ │ -0002ae00: 2832 2c20 6765 6e65 7269 6353 796d 6d65 (2, genericSymme │ │ │ │ -0002ae10: 7472 6963 4d61 7472 6978 2853 2c20 3329 tricMatrix(S, 3) │ │ │ │ -0002ae20: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0002ae30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002ae40: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -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 7c0a |. │ │ │ │ -0002ae90: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002aea0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002aeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aed0: 2020 2032 2020 2020 2020 2020 2020 7c0a 2 |. │ │ │ │ -0002aee0: 7c6f 3320 3d20 6964 6561 6c20 282d 2062 |o3 = ideal (- b │ │ │ │ -0002aef0: 2020 2b20 612a 642c 202d 2062 2a63 202b + a*d, - b*c + │ │ │ │ -0002af00: 2061 2a65 2c20 2d20 632a 6420 2b20 622a a*e, - c*d + b* │ │ │ │ -0002af10: 652c 202d 2062 2a63 202b 2061 2a65 2c20 e, - b*c + a*e, │ │ │ │ -0002af20: 2d20 6320 202b 2061 2a66 2c20 2d20 7c0a - c + a*f, - |. │ │ │ │ -0002af30: 7c20 2020 2020 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 7c0a --------------|. │ │ │ │ -0002af80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afa0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -0002afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002afd0: 7c20 2020 2020 632a 6520 2b20 622a 662c | c*e + b*f, │ │ │ │ -0002afe0: 202d 2063 2a64 202b 2062 2a65 2c20 2d20 - c*d + b*e, - │ │ │ │ -0002aff0: 632a 6520 2b20 622a 662c 202d 2065 2020 c*e + b*f, - e │ │ │ │ -0002b000: 2b20 642a 6629 2020 2020 2020 2020 2020 + d*f) │ │ │ │ -0002b010: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b020: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b060: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b070: 7c6f 3320 3a20 4964 6561 6c20 6f66 2053 |o3 : Ideal of S │ │ │ │ -0002b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b0c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002b110: 7c69 3420 3a20 7074 7320 3d20 7261 6e64 |i4 : pts = rand │ │ │ │ -0002b120: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ -0002b130: 616c 5661 7269 6574 7928 492c 2034 2920 alVariety(I, 4) │ │ │ │ -0002b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b150: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b160: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b1b0: 7c6f 3420 3d20 7b7c 202d 3235 2032 3020 |o4 = {| -25 20 │ │ │ │ -0002b1c0: 2d33 3020 2d31 3620 3234 202d 3336 207c -30 -16 24 -36 | │ │ │ │ -0002b1d0: 2c20 7c20 3139 202d 3239 2031 3920 3233 , | 19 -29 19 23 │ │ │ │ -0002b1e0: 202d 3239 2031 3920 7c2c 207c 202d 3434 -29 19 |, | -44 │ │ │ │ -0002b1f0: 2034 3620 2d38 2037 202d 3130 2020 7c0a 46 -8 7 -10 |. │ │ │ │ -0002b200: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ +0002a760: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +0002a770: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a780: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a790: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +0002a7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a7b0: 2020 2020 7c0a 7c20 2020 2020 2034 3763 |.| 47c │ │ │ │ +0002a7c0: 2020 2b20 3231 622a 6420 2b20 3331 632a + 21b*d + 31c* │ │ │ │ +0002a7d0: 6420 2d20 6420 2c20 612a 6220 2d20 3331 d - d , a*b - 31 │ │ │ │ +0002a7e0: 622a 6320 2d20 3138 6320 202b 2034 3461 b*c - 18c + 44a │ │ │ │ +0002a7f0: 2a64 202b 2032 3162 2a64 202b 2036 632a *d + 21b*d + 6c* │ │ │ │ +0002a800: 6420 2b20 7c0a 7c20 2020 2020 202d 2d2d d + |.| --- │ │ │ │ +0002a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a850: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +0002a860: 3220 2020 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ +0002a870: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a890: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a8a0: 2020 2020 7c0a 7c20 2020 2020 2032 3164 |.| 21d │ │ │ │ +0002a8b0: 202c 2061 2020 2b20 3530 622a 6320 2d20 , a + 50b*c - │ │ │ │ +0002a8c0: 3135 6320 202d 2033 3961 2a64 202d 2033 15c - 39a*d - 3 │ │ │ │ +0002a8d0: 3162 2a64 202d 2031 3363 2a64 202d 2036 1b*d - 13c*d - 6 │ │ │ │ +0002a8e0: 6420 297d 2020 2020 2020 2020 2020 2020 d )} │ │ │ │ +0002a8f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a940: 2020 2020 7c0a 7c6f 3138 203a 204c 6973 |.|o18 : Lis │ │ │ │ +0002a950: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +0002a960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a990: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a9e0: 2d2d 2d2d 2b0a 0a49 7420 7475 726e 7320 ----+..It turns │ │ │ │ +0002a9f0: 6f75 7420 7468 6174 2074 6869 7320 6973 out that this is │ │ │ │ +0002aa00: 2074 6865 2069 6465 616c 206f 6620 3220 the ideal of 2 │ │ │ │ +0002aa10: 736b 6577 206c 696e 6573 2c20 6a75 7374 skew lines, just │ │ │ │ +0002aa20: 206e 6f74 2064 6566 696e 6564 206f 7665 not defined ove │ │ │ │ +0002aa30: 7220 7468 6973 0a66 6965 6c64 2e0a 0a43 r this.field...C │ │ │ │ +0002aa40: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 5468 aveat.======..Th │ │ │ │ +0002aa50: 6973 2072 6f75 7469 6e65 2065 7870 6563 is routine expec │ │ │ │ +0002aa60: 7473 2074 6865 2069 6e70 7574 2074 6f20 ts the input to │ │ │ │ +0002aa70: 7265 7072 6573 656e 7420 616e 2069 7272 represent an irr │ │ │ │ +0002aa80: 6564 7563 6962 6c65 2076 6172 6965 7479 educible variety │ │ │ │ +0002aa90: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +0002aaa0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2072 ===.. * *note r │ │ │ │ +0002aab0: 616e 646f 6d50 6f69 6e74 734f 6e52 6174 andomPointsOnRat │ │ │ │ +0002aac0: 696f 6e61 6c56 6172 6965 7479 3a0a 2020 ionalVariety:. │ │ │ │ +0002aad0: 2020 7261 6e64 6f6d 506f 696e 7473 4f6e randomPointsOn │ │ │ │ +0002aae0: 5261 7469 6f6e 616c 5661 7269 6574 795f RationalVariety_ │ │ │ │ +0002aaf0: 6c70 4964 6561 6c5f 636d 5a5a 5f72 702c lpIdeal_cmZZ_rp, │ │ │ │ +0002ab00: 202d 2d20 6669 6e64 2072 616e 646f 6d20 -- find random │ │ │ │ +0002ab10: 706f 696e 7473 206f 6e20 610a 2020 2020 points on a. │ │ │ │ +0002ab20: 7661 7269 6574 7920 7468 6174 2063 616e variety that can │ │ │ │ +0002ab30: 2062 6520 6465 7465 6374 6564 2074 6f20 be detected to │ │ │ │ +0002ab40: 6265 2072 6174 696f 6e61 6c0a 0a57 6179 be rational..Way │ │ │ │ +0002ab50: 7320 746f 2075 7365 2074 6869 7320 6d65 s to use this me │ │ │ │ +0002ab60: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d thod:.========== │ │ │ │ +0002ab70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0002ab80: 2020 2a20 2a6e 6f74 6520 7261 6e64 6f6d * *note random │ │ │ │ +0002ab90: 506f 696e 744f 6e52 6174 696f 6e61 6c56 PointOnRationalV │ │ │ │ +0002aba0: 6172 6965 7479 2849 6465 616c 293a 0a20 ariety(Ideal):. │ │ │ │ +0002abb0: 2020 2072 616e 646f 6d50 6f69 6e74 4f6e randomPointOn │ │ │ │ +0002abc0: 5261 7469 6f6e 616c 5661 7269 6574 795f RationalVariety_ │ │ │ │ +0002abd0: 6c70 4964 6561 6c5f 7270 2c20 2d2d 2066 lpIdeal_rp, -- f │ │ │ │ +0002abe0: 696e 6420 6120 7261 6e64 6f6d 2070 6f69 ind a random poi │ │ │ │ +0002abf0: 6e74 206f 6e20 610a 2020 2020 7661 7269 nt on a. vari │ │ │ │ +0002ac00: 6574 7920 7468 6174 2063 616e 2062 6520 ety that can be │ │ │ │ +0002ac10: 6465 7465 6374 6564 2074 6f20 6265 2072 detected to be r │ │ │ │ +0002ac20: 6174 696f 6e61 6c0a 2d2d 2d2d 2d2d 2d2d ational.-------- │ │ │ │ +0002ac30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ac40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ac50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ac60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ac70: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +0002ac80: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +0002ac90: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +0002aca0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +0002acb0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +0002acc0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +0002acd0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +0002ace0: 4772 6f65 626e 6572 5374 7261 7461 2e6d GroebnerStrata.m │ │ │ │ +0002acf0: 323a 3935 373a 302e 0a1f 0a46 696c 653a 2:957:0....File: │ │ │ │ +0002ad00: 2047 726f 6562 6e65 7253 7472 6174 612e GroebnerStrata. │ │ │ │ +0002ad10: 696e 666f 2c20 4e6f 6465 3a20 7261 6e64 info, Node: rand │ │ │ │ +0002ad20: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ +0002ad30: 616c 5661 7269 6574 795f 6c70 4964 6561 alVariety_lpIdea │ │ │ │ +0002ad40: 6c5f 636d 5a5a 5f72 702c 204e 6578 743a l_cmZZ_rp, Next: │ │ │ │ +0002ad50: 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 616c smallerMonomial │ │ │ │ +0002ad60: 732c 2050 7265 763a 2072 616e 646f 6d50 s, Prev: randomP │ │ │ │ +0002ad70: 6f69 6e74 4f6e 5261 7469 6f6e 616c 5661 ointOnRationalVa │ │ │ │ +0002ad80: 7269 6574 795f 6c70 4964 6561 6c5f 7270 riety_lpIdeal_rp │ │ │ │ +0002ad90: 2c20 5570 3a20 546f 700a 0a72 616e 646f , Up: Top..rando │ │ │ │ +0002ada0: 6d50 6f69 6e74 734f 6e52 6174 696f 6e61 mPointsOnRationa │ │ │ │ +0002adb0: 6c56 6172 6965 7479 2849 6465 616c 2c5a lVariety(Ideal,Z │ │ │ │ +0002adc0: 5a29 202d 2d20 6669 6e64 2072 616e 646f Z) -- find rando │ │ │ │ +0002add0: 6d20 706f 696e 7473 206f 6e20 6120 7661 m points on a va │ │ │ │ +0002ade0: 7269 6574 7920 7468 6174 2063 616e 2062 riety that can b │ │ │ │ +0002adf0: 6520 6465 7465 6374 6564 2074 6f20 6265 e detected to be │ │ │ │ +0002ae00: 2072 6174 696f 6e61 6c0a 2a2a 2a2a 2a2a rational.****** │ │ │ │ +0002ae10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ae20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ae30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ae40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ae50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ae60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ae70: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 ********.. * Fu │ │ │ │ +0002ae80: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7261 nction: *note ra │ │ │ │ +0002ae90: 6e64 6f6d 506f 696e 7473 4f6e 5261 7469 ndomPointsOnRati │ │ │ │ +0002aea0: 6f6e 616c 5661 7269 6574 793a 0a20 2020 onalVariety:. │ │ │ │ +0002aeb0: 2072 616e 646f 6d50 6f69 6e74 734f 6e52 randomPointsOnR │ │ │ │ +0002aec0: 6174 696f 6e61 6c56 6172 6965 7479 5f6c ationalVariety_l │ │ │ │ +0002aed0: 7049 6465 616c 5f63 6d5a 5a5f 7270 2c0a pIdeal_cmZZ_rp,. │ │ │ │ +0002aee0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0002aef0: 2020 2020 7261 6e64 6f6d 506f 696e 7473 randomPoints │ │ │ │ +0002af00: 4f6e 5261 7469 6f6e 616c 5661 7269 6574 OnRationalVariet │ │ │ │ +0002af10: 7928 492c 206e 290a 2020 2020 2020 2020 y(I, n). │ │ │ │ +0002af20: 7261 6e64 6f6d 506f 696e 744f 6e52 6174 randomPointOnRat │ │ │ │ +0002af30: 696f 6e61 6c56 6172 6965 7479 0a20 202a ionalVariety. * │ │ │ │ +0002af40: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +0002af50: 2049 2c20 616e 202a 6e6f 7465 2069 6465 I, an *note ide │ │ │ │ +0002af60: 616c 3a20 284d 6163 6175 6c61 7932 446f al: (Macaulay2Do │ │ │ │ +0002af70: 6329 4964 6561 6c2c 2c20 416e 2069 6465 c)Ideal,, An ide │ │ │ │ +0002af80: 616c 2069 6e20 6120 706f 6c79 6e6f 6d69 al in a polynomi │ │ │ │ +0002af90: 616c 2072 696e 670a 2020 2020 2020 2020 al ring. │ │ │ │ +0002afa0: 2453 2420 6f76 6572 2061 2066 6965 6c64 $S$ over a field │ │ │ │ +0002afb0: 2c20 7768 6963 6820 6465 6669 6e65 7320 , which defines │ │ │ │ +0002afc0: 6120 7072 696d 6520 6964 6561 6c0a 2020 a prime ideal. │ │ │ │ +0002afd0: 2020 2020 2a20 6e2c 2061 6e20 2a6e 6f74 * n, an *not │ │ │ │ +0002afe0: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +0002aff0: 756c 6179 3244 6f63 295a 5a2c 2c20 5468 ulay2Doc)ZZ,, Th │ │ │ │ +0002b000: 6520 6e75 6d62 6572 206f 6620 706f 696e e number of poin │ │ │ │ +0002b010: 7473 2074 6f0a 2020 2020 2020 2020 6765 ts to. ge │ │ │ │ +0002b020: 6e65 7261 7465 0a20 202a 204f 7574 7075 nerate. * Outpu │ │ │ │ +0002b030: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e ts:. * a *n │ │ │ │ +0002b040: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ +0002b050: 6c61 7932 446f 6329 4c69 7374 2c2c 2041 lay2Doc)List,, A │ │ │ │ +0002b060: 206c 6973 7420 6f66 2024 6e24 206f 6e65 list of $n$ one │ │ │ │ +0002b070: 2072 6f77 206d 6174 7269 6365 7320 6f76 row matrices ov │ │ │ │ +0002b080: 6572 0a20 2020 2020 2020 2074 6865 2062 er. the b │ │ │ │ +0002b090: 6173 6520 6669 656c 6420 6f66 2024 5324 ase field of $S$ │ │ │ │ +0002b0a0: 2c20 7468 6174 2061 7265 2072 616e 646f , that are rando │ │ │ │ +0002b0b0: 6d6c 7920 6368 6f73 656e 2070 6f69 6e74 mly chosen point │ │ │ │ +0002b0c0: 7320 6f6e 2024 4924 2e20 206e 756c 6c20 s on $I$. null │ │ │ │ +0002b0d0: 6973 0a20 2020 2020 2020 2072 6574 7572 is. retur │ │ │ │ +0002b0e0: 6e65 6420 696e 2074 6865 2063 6173 6520 ned in the case │ │ │ │ +0002b0f0: 7768 656e 2074 6865 2072 6f75 7469 6e65 when the routine │ │ │ │ +0002b100: 2063 616e 6e6f 7420 6465 7465 726d 696e cannot determin │ │ │ │ +0002b110: 6520 6966 2074 6865 2076 6172 6965 7479 e if the variety │ │ │ │ +0002b120: 0a20 2020 2020 2020 2069 7320 7261 7469 . is rati │ │ │ │ +0002b130: 6f6e 616c 2061 6e64 2069 7272 6564 7563 onal and irreduc │ │ │ │ +0002b140: 6962 6c65 2e0a 0a44 6573 6372 6970 7469 ible...Descripti │ │ │ │ +0002b150: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0002b160: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002b170: 2d2d 2d2d 2d2d 2d2d 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 2b0a --------------+. │ │ │ │ +0002b1b0: 7c69 3120 3a20 6b6b 203d 205a 5a2f 3130 |i1 : kk = ZZ/10 │ │ │ │ +0002b1c0: 313b 2020 2020 2020 2020 2020 2020 2020 1; │ │ │ │ +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 7c0a |. │ │ │ │ +0002b200: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0002b210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -0002b250: 7c20 2020 2020 2d32 3920 7c2c 207c 2038 | -29 |, | 8 │ │ │ │ -0002b260: 2034 3120 2d32 3420 3436 202d 3232 202d 41 -24 46 -22 - │ │ │ │ -0002b270: 3239 207c 7d20 2020 2020 2020 2020 2020 29 |} │ │ │ │ +0002b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002b250: 7c69 3220 3a20 5320 3d20 6b6b 5b61 2e2e |i2 : S = kk[a.. │ │ │ │ +0002b260: 665d 3b20 2020 2020 2020 2020 2020 2020 f]; │ │ │ │ +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 7c0a |. │ │ │ │ -0002b2a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b2f0: 7c6f 3420 3a20 4c69 7374 2020 2020 2020 |o4 : List │ │ │ │ -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 │ │ │ │ +0002b2a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002b2b0: 2d2d 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 2b0a --------------+. │ │ │ │ +0002b2f0: 7c69 3320 3a20 4920 3d20 6d69 6e6f 7273 |i3 : I = minors │ │ │ │ +0002b300: 2832 2c20 6765 6e65 7269 6353 796d 6d65 (2, genericSymme │ │ │ │ +0002b310: 7472 6963 4d61 7472 6978 2853 2c20 3329 tricMatrix(S, 3) │ │ │ │ +0002b320: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0002b330: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b340: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002b390: 7c69 3520 3a20 666f 7220 7020 696e 2070 |i5 : for p in p │ │ │ │ -0002b3a0: 7473 206c 6973 7420 7375 6228 492c 2070 ts list sub(I, p │ │ │ │ -0002b3b0: 2920 3d3d 2030 2020 2020 2020 2020 2020 ) == 0 │ │ │ │ +0002b340: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b380: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b390: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002b3a0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +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 7c0a |. │ │ │ │ -0002b3e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b420: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b430: 7c6f 3520 3d20 7b74 7275 652c 2074 7275 |o5 = {true, tru │ │ │ │ -0002b440: 652c 2074 7275 652c 2074 7275 657d 2020 e, true, true} │ │ │ │ -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 7c0a |. │ │ │ │ +0002b3d0: 2020 2032 2020 2020 2020 2020 2020 7c0a 2 |. │ │ │ │ +0002b3e0: 7c6f 3320 3d20 6964 6561 6c20 282d 2062 |o3 = ideal (- b │ │ │ │ +0002b3f0: 2020 2b20 612a 642c 202d 2062 2a63 202b + a*d, - b*c + │ │ │ │ +0002b400: 2061 2a65 2c20 2d20 632a 6420 2b20 622a a*e, - c*d + b* │ │ │ │ +0002b410: 652c 202d 2062 2a63 202b 2061 2a65 2c20 e, - b*c + a*e, │ │ │ │ +0002b420: 2d20 6320 202b 2061 2a66 2c20 2d20 7c0a - c + a*f, - |. │ │ │ │ +0002b430: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ +0002b440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ 0002b480: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b4a0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ 0002b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b4c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b4d0: 7c6f 3520 3a20 4c69 7374 2020 2020 2020 |o5 : List │ │ │ │ -0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b4d0: 7c20 2020 2020 632a 6520 2b20 622a 662c | c*e + b*f, │ │ │ │ +0002b4e0: 202d 2063 2a64 202b 2062 2a65 2c20 2d20 - c*d + b*e, - │ │ │ │ +0002b4f0: 632a 6520 2b20 622a 662c 202d 2065 2020 c*e + b*f, - e │ │ │ │ +0002b500: 2b20 642a 6629 2020 2020 2020 2020 2020 + d*f) │ │ │ │ 0002b510: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b520: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002b530: 2d2d 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 2b0a --------------+. │ │ │ │ -0002b570: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002b580: 2d2d 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 2b0a --------------+. │ │ │ │ -0002b5c0: 7c69 3620 3a20 5320 3d20 6b6b 5b61 2e2e |i6 : S = kk[a.. │ │ │ │ -0002b5d0: 645d 3b20 2020 2020 2020 2020 2020 2020 d]; │ │ │ │ -0002b5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b600: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b610: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002b620: 2d2d 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 2b0a --------------+. │ │ │ │ -0002b660: 7c69 3720 3a20 4620 3d20 6772 6f65 626e |i7 : F = groebn │ │ │ │ -0002b670: 6572 4661 6d69 6c79 2069 6465 616c 2261 erFamily ideal"a │ │ │ │ -0002b680: 322c 6162 2c61 632c 6232 2220 2020 2020 2,ab,ac,b2" │ │ │ │ +0002b520: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b560: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b570: 7c6f 3320 3a20 4964 6561 6c20 6f66 2053 |o3 : Ideal of S │ │ │ │ +0002b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b5b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b5c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002b610: 7c69 3420 3a20 7074 7320 3d20 7261 6e64 |i4 : pts = rand │ │ │ │ +0002b620: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ +0002b630: 616c 5661 7269 6574 7928 492c 2034 2920 alVariety(I, 4) │ │ │ │ +0002b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 2020 │ │ │ │ 0002b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b6f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b700: 7c20 2020 2020 2020 2020 2020 2020 3220 | 2 │ │ │ │ -0002b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b720: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b730: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -0002b740: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b750: 7c6f 3720 3d20 6964 6561 6c20 2861 2020 |o7 = ideal (a │ │ │ │ -0002b760: 2b20 7420 622a 6320 2b20 7420 612a 6420 + t b*c + t a*d │ │ │ │ -0002b770: 2b20 7420 6320 202b 2074 2062 2a64 202b + t c + t b*d + │ │ │ │ -0002b780: 2074 2063 2a64 202b 2074 2064 202c 2061 t c*d + t d , a │ │ │ │ -0002b790: 2a62 202b 2074 2062 2a63 202b 2020 7c0a *b + t b*c + |. │ │ │ │ +0002b6b0: 7c6f 3420 3d20 7b7c 202d 3235 2032 3020 |o4 = {| -25 20 │ │ │ │ +0002b6c0: 2d33 3020 2d31 3620 3234 202d 3336 207c -30 -16 24 -36 | │ │ │ │ +0002b6d0: 2c20 7c20 3139 202d 3239 2031 3920 3233 , | 19 -29 19 23 │ │ │ │ +0002b6e0: 202d 3239 2031 3920 7c2c 207c 202d 3434 -29 19 |, | -44 │ │ │ │ +0002b6f0: 2034 3620 2d38 2037 202d 3130 2020 7c0a 46 -8 7 -10 |. │ │ │ │ +0002b700: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ +0002b710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +0002b750: 7c20 2020 2020 2d32 3920 7c2c 207c 2038 | -29 |, | 8 │ │ │ │ +0002b760: 2034 3120 2d32 3420 3436 202d 3232 202d 41 -24 46 -22 - │ │ │ │ +0002b770: 3239 207c 7d20 2020 2020 2020 2020 2020 29 |} │ │ │ │ +0002b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b790: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0002b7a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b7b0: 2020 2031 2020 2020 2020 2033 2020 2020 1 3 │ │ │ │ -0002b7c0: 2020 2032 2020 2020 2020 3420 2020 2020 2 4 │ │ │ │ -0002b7d0: 2020 3520 2020 2020 2020 3620 2020 2020 5 6 │ │ │ │ -0002b7e0: 2020 2020 2020 3720 2020 2020 2020 7c0a 7 |. │ │ │ │ -0002b7f0: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ -0002b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -0002b840: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b850: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b860: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -0002b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b880: 2020 2020 2020 2020 2020 3220 2020 7c0a 2 |. │ │ │ │ -0002b890: 7c20 2020 2020 7420 612a 6420 2b20 7420 | t a*d + t │ │ │ │ -0002b8a0: 6320 202b 2074 2020 622a 6420 2b20 7420 c + t b*d + t │ │ │ │ -0002b8b0: 2063 2a64 202b 2074 2020 6420 2c20 612a c*d + t d , a* │ │ │ │ -0002b8c0: 6320 2b20 7420 2062 2a63 202b 2074 2020 c + t b*c + t │ │ │ │ -0002b8d0: 612a 6420 2b20 7420 2063 2020 2b20 7c0a a*d + t c + |. │ │ │ │ -0002b8e0: 7c20 2020 2020 2039 2020 2020 2020 2038 | 9 8 │ │ │ │ -0002b8f0: 2020 2020 2020 3130 2020 2020 2020 2031 10 1 │ │ │ │ -0002b900: 3120 2020 2020 2020 3132 2020 2020 2020 1 12 │ │ │ │ -0002b910: 2020 2020 2031 3320 2020 2020 2020 3135 13 15 │ │ │ │ -0002b920: 2020 2020 2020 2031 3420 2020 2020 7c0a 14 |. │ │ │ │ -0002b930: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ -0002b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b7e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b7f0: 7c6f 3420 3a20 4c69 7374 2020 2020 2020 |o4 : List │ │ │ │ +0002b800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b830: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b840: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002b850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002b880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002b890: 7c69 3520 3a20 666f 7220 7020 696e 2070 |i5 : for p in p │ │ │ │ +0002b8a0: 7473 206c 6973 7420 7375 6228 492c 2070 ts list sub(I, p │ │ │ │ +0002b8b0: 2920 3d3d 2030 2020 2020 2020 2020 2020 ) == 0 │ │ │ │ +0002b8c0: 2020 2020 2020 2020 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 7c0a |. │ │ │ │ +0002b930: 7c6f 3520 3d20 7b74 7275 652c 2074 7275 |o5 = {true, tru │ │ │ │ +0002b940: 652c 2074 7275 652c 2074 7275 657d 2020 e, true, true} │ │ │ │ +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 7c0a |. │ │ │ │ 0002b980: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b990: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -0002b9a0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002b9b0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b9c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b9d0: 7c20 2020 2020 7420 2062 2a64 202b 2074 | t b*d + t │ │ │ │ -0002b9e0: 2020 632a 6420 2b20 7420 2064 202c 2062 c*d + t d , b │ │ │ │ -0002b9f0: 2020 2b20 7420 2062 2a63 202b 2074 2020 + t b*c + t │ │ │ │ -0002ba00: 612a 6420 2b20 7420 2063 2020 2b20 7420 a*d + t c + t │ │ │ │ -0002ba10: 2062 2a64 202b 2074 2020 632a 6420 7c0a b*d + t c*d |. │ │ │ │ -0002ba20: 7c20 2020 2020 2031 3620 2020 2020 2020 | 16 │ │ │ │ -0002ba30: 3137 2020 2020 2020 2031 3820 2020 2020 17 18 │ │ │ │ -0002ba40: 2020 2020 2031 3920 2020 2020 2020 3231 19 21 │ │ │ │ -0002ba50: 2020 2020 2020 2032 3020 2020 2020 2032 20 2 │ │ │ │ -0002ba60: 3220 2020 2020 2020 3233 2020 2020 7c0a 2 23 |. │ │ │ │ -0002ba70: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ +0002b9d0: 7c6f 3520 3a20 4c69 7374 2020 2020 2020 |o5 : List │ │ │ │ +0002b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ba10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002ba20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002ba30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ba40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ba50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ba60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002ba70: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0002ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -0002bac0: 7c20 2020 2020 2020 2020 2020 3220 2020 | 2 │ │ │ │ -0002bad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002bac0: 7c69 3620 3a20 5320 3d20 6b6b 5b61 2e2e |i6 : S = kk[a.. │ │ │ │ +0002bad0: 645d 3b20 2020 2020 2020 2020 2020 2020 d]; │ │ │ │ 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 7c0a |. │ │ │ │ -0002bb10: 7c20 2020 2020 2b20 7420 2064 2029 2020 | + t d ) │ │ │ │ -0002bb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002bb60: 7c20 2020 2020 2020 2032 3420 2020 2020 | 24 │ │ │ │ -0002bb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 3720 3a20 4620 3d20 6772 6f65 626e |i7 : F = groebn │ │ │ │ +0002bb70: 6572 4661 6d69 6c79 2069 6465 616c 2261 erFamily ideal"a │ │ │ │ +0002bb80: 322c 6162 2c61 632c 6232 2220 2020 2020 2,ab,ac,b2" │ │ │ │ 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 7c0a |. │ │ │ │ -0002bc00: 7c6f 3720 3a20 4964 6561 6c20 6f66 206b |o7 : Ideal of k │ │ │ │ -0002bc10: 6b5b 7420 2c20 7420 2c20 7420 202c 2074 k[t , t , t , t │ │ │ │ -0002bc20: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -0002bc30: 2074 2020 2c20 7420 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ -0002bc40: 2c20 7420 202c 2074 2020 2c20 7420 7c0a , t , t , t |. │ │ │ │ -0002bc50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002bc60: 2020 2036 2020 2035 2020 2031 3220 2020 6 5 12 │ │ │ │ -0002bc70: 3220 2020 3420 2020 3131 2020 2031 3820 2 4 11 18 │ │ │ │ -0002bc80: 2020 3234 2020 2031 2020 2033 2020 2038 24 1 3 8 │ │ │ │ -0002bc90: 2020 2031 3020 2020 3137 2020 2032 7c0a 10 17 2|. │ │ │ │ -0002bca0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ -0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -0002bcf0: 7c20 2c20 7420 2c20 7420 2c20 7420 202c | , t , t , t , │ │ │ │ -0002bd00: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ -0002bd10: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ -0002bd20: 2020 5d5b 612e 2e64 5d20 2020 2020 2020 ][a..d] │ │ │ │ -0002bd30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002bd40: 7c33 2020 2037 2020 2039 2020 2031 3420 |3 7 9 14 │ │ │ │ -0002bd50: 2020 3136 2020 2032 3020 2020 3232 2020 16 20 22 │ │ │ │ -0002bd60: 2031 3320 2020 3135 2020 2031 3920 2020 13 15 19 │ │ │ │ -0002bd70: 3231 2020 2020 2020 2020 2020 2020 2020 21 │ │ │ │ -0002bd80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002bd90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002bde0: 7c69 3820 3a20 4a20 3d20 6772 6f65 626e |i8 : J = groebn │ │ │ │ -0002bdf0: 6572 5374 7261 7475 6d20 463b 2020 2020 erStratum F; │ │ │ │ -0002be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002be30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002be80: 7c6f 3820 3a20 4964 6561 6c20 6f66 206b |o8 : Ideal of k │ │ │ │ -0002be90: 6b5b 7420 2c20 7420 2c20 7420 202c 2074 k[t , t , t , t │ │ │ │ -0002bea0: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -0002beb0: 2074 2020 2c20 7420 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ -0002bec0: 2c20 7420 202c 2074 2020 2c20 2020 7c0a , t , t , |. │ │ │ │ -0002bed0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002bee0: 2020 2036 2020 2035 2020 2031 3220 2020 6 5 12 │ │ │ │ -0002bef0: 3220 2020 3420 2020 3131 2020 2031 3820 2 4 11 18 │ │ │ │ -0002bf00: 2020 3234 2020 2031 2020 2033 2020 2038 24 1 3 8 │ │ │ │ -0002bf10: 2020 2031 3020 2020 3137 2020 2020 7c0a 10 17 |. │ │ │ │ -0002bf20: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ -0002bf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bf60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ -0002bf70: 7c74 2020 2c20 7420 2c20 7420 2c20 7420 |t , t , t , t │ │ │ │ -0002bf80: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ -0002bf90: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ -0002bfa0: 2074 2020 5d20 2020 2020 2020 2020 2020 t ] │ │ │ │ -0002bfb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002bfc0: 7c20 3233 2020 2037 2020 2039 2020 2031 | 23 7 9 1 │ │ │ │ -0002bfd0: 3420 2020 3136 2020 2032 3020 2020 3232 4 16 20 22 │ │ │ │ -0002bfe0: 2020 2031 3320 2020 3135 2020 2031 3920 13 15 19 │ │ │ │ -0002bff0: 2020 3231 2020 2020 2020 2020 2020 2020 21 │ │ │ │ +0002bc00: 7c20 2020 2020 2020 2020 2020 2020 3220 | 2 │ │ │ │ +0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc20: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +0002bc30: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +0002bc40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002bc50: 7c6f 3720 3d20 6964 6561 6c20 2861 2020 |o7 = ideal (a │ │ │ │ +0002bc60: 2b20 7420 622a 6320 2b20 7420 612a 6420 + t b*c + t a*d │ │ │ │ +0002bc70: 2b20 7420 6320 202b 2074 2062 2a64 202b + t c + t b*d + │ │ │ │ +0002bc80: 2074 2063 2a64 202b 2074 2064 202c 2061 t c*d + t d , a │ │ │ │ +0002bc90: 2a62 202b 2074 2062 2a63 202b 2020 7c0a *b + t b*c + |. │ │ │ │ +0002bca0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002bcb0: 2020 2031 2020 2020 2020 2033 2020 2020 1 3 │ │ │ │ +0002bcc0: 2020 2032 2020 2020 2020 3420 2020 2020 2 4 │ │ │ │ +0002bcd0: 2020 3520 2020 2020 2020 3620 2020 2020 5 6 │ │ │ │ +0002bce0: 2020 2020 2020 3720 2020 2020 2020 7c0a 7 |. │ │ │ │ +0002bcf0: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ +0002bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +0002bd40: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002bd50: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002bd60: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0002bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bd80: 2020 2020 2020 2020 2020 3220 2020 7c0a 2 |. │ │ │ │ +0002bd90: 7c20 2020 2020 7420 612a 6420 2b20 7420 | t a*d + t │ │ │ │ +0002bda0: 6320 202b 2074 2020 622a 6420 2b20 7420 c + t b*d + t │ │ │ │ +0002bdb0: 2063 2a64 202b 2074 2020 6420 2c20 612a c*d + t d , a* │ │ │ │ +0002bdc0: 6320 2b20 7420 2062 2a63 202b 2074 2020 c + t b*c + t │ │ │ │ +0002bdd0: 612a 6420 2b20 7420 2063 2020 2b20 7c0a a*d + t c + |. │ │ │ │ +0002bde0: 7c20 2020 2020 2039 2020 2020 2020 2038 | 9 8 │ │ │ │ +0002bdf0: 2020 2020 2020 3130 2020 2020 2020 2031 10 1 │ │ │ │ +0002be00: 3120 2020 2020 2020 3132 2020 2020 2020 1 12 │ │ │ │ +0002be10: 2020 2020 2031 3320 2020 2020 2020 3135 13 15 │ │ │ │ +0002be20: 2020 2020 2020 2031 3420 2020 2020 7c0a 14 |. │ │ │ │ +0002be30: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ +0002be40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +0002be80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002be90: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +0002bea0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002beb0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002bec0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002bed0: 7c20 2020 2020 7420 2062 2a64 202b 2074 | t b*d + t │ │ │ │ +0002bee0: 2020 632a 6420 2b20 7420 2064 202c 2062 c*d + t d , b │ │ │ │ +0002bef0: 2020 2b20 7420 2062 2a63 202b 2074 2020 + t b*c + t │ │ │ │ +0002bf00: 612a 6420 2b20 7420 2063 2020 2b20 7420 a*d + t c + t │ │ │ │ +0002bf10: 2062 2a64 202b 2074 2020 632a 6420 7c0a b*d + t c*d |. │ │ │ │ +0002bf20: 7c20 2020 2020 2031 3620 2020 2020 2020 | 16 │ │ │ │ +0002bf30: 3137 2020 2020 2020 2031 3820 2020 2020 17 18 │ │ │ │ +0002bf40: 2020 2020 2031 3920 2020 2020 2020 3231 19 21 │ │ │ │ +0002bf50: 2020 2020 2020 2032 3020 2020 2020 2032 20 2 │ │ │ │ +0002bf60: 3220 2020 2020 2020 3233 2020 2020 7c0a 2 23 |. │ │ │ │ +0002bf70: 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d | ---------- │ │ │ │ +0002bf80: 2d2d 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 7c0a --------------|. │ │ │ │ +0002bfc0: 7c20 2020 2020 2020 2020 2020 3220 2020 | 2 │ │ │ │ +0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c000: 2020 2020 2020 2020 2020 2020 2020 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 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002c060: 7c69 3920 3a20 636f 6d70 734a 203d 2064 |i9 : compsJ = d │ │ │ │ -0002c070: 6563 6f6d 706f 7365 204a 3b20 2020 2020 ecompose J; │ │ │ │ +0002c010: 7c20 2020 2020 2b20 7420 2064 2029 2020 | + t d ) │ │ │ │ +0002c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c060: 7c20 2020 2020 2020 2032 3420 2020 2020 | 24 │ │ │ │ +0002c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c0a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c0b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002c100: 7c69 3130 203a 2063 6f6d 7073 4a20 3d20 |i10 : compsJ = │ │ │ │ -0002c110: 636f 6d70 734a 2f74 7269 6d3b 2020 2020 compsJ/trim; │ │ │ │ -0002c120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c140: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c150: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002c160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002c1a0: 7c69 3131 203a 2023 636f 6d70 734a 203d |i11 : #compsJ = │ │ │ │ -0002c1b0: 3d20 3220 2020 2020 2020 2020 2020 2020 = 2 │ │ │ │ -0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c1f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002c200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002c0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c100: 7c6f 3720 3a20 4964 6561 6c20 6f66 206b |o7 : Ideal of k │ │ │ │ +0002c110: 6b5b 7420 2c20 7420 2c20 7420 202c 2074 k[t , t , t , t │ │ │ │ +0002c120: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +0002c130: 2074 2020 2c20 7420 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ +0002c140: 2c20 7420 202c 2074 2020 2c20 7420 7c0a , t , t , t |. │ │ │ │ +0002c150: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002c160: 2020 2036 2020 2035 2020 2031 3220 2020 6 5 12 │ │ │ │ +0002c170: 3220 2020 3420 2020 3131 2020 2031 3820 2 4 11 18 │ │ │ │ +0002c180: 2020 3234 2020 2031 2020 2033 2020 2038 24 1 3 8 │ │ │ │ +0002c190: 2020 2031 3020 2020 3137 2020 2032 7c0a 10 17 2|. │ │ │ │ +0002c1a0: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ +0002c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +0002c1f0: 7c20 2c20 7420 2c20 7420 2c20 7420 202c | , t , t , t , │ │ │ │ +0002c200: 2074 2020 2c20 7420 202c 2074 2020 2c20 t , t , t , │ │ │ │ +0002c210: 7420 202c 2074 2020 2c20 7420 202c 2074 t , t , t , t │ │ │ │ +0002c220: 2020 5d5b 612e 2e64 5d20 2020 2020 2020 ][a..d] │ │ │ │ 0002c230: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c240: 7c6f 3131 203d 2074 7275 6520 2020 2020 |o11 = true │ │ │ │ -0002c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c240: 7c33 2020 2037 2020 2039 2020 2031 3420 |3 7 9 14 │ │ │ │ +0002c250: 2020 3136 2020 2032 3020 2020 3232 2020 16 20 22 │ │ │ │ +0002c260: 2031 3320 2020 3135 2020 2031 3920 2020 13 15 19 │ │ │ │ +0002c270: 3231 2020 2020 2020 2020 2020 2020 2020 21 │ │ │ │ 0002c280: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0002c290: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0002c2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002c2e0: 7c69 3132 203a 2063 6f6d 7073 4a2f 6469 |i12 : compsJ/di │ │ │ │ -0002c2f0: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ +0002c2e0: 7c69 3820 3a20 4a20 3d20 6772 6f65 626e |i8 : J = groebn │ │ │ │ +0002c2f0: 6572 5374 7261 7475 6d20 463b 2020 2020 erStratum F; │ │ │ │ 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 7c0a |. │ │ │ │ 0002c330: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c370: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c380: 7c6f 3132 203d 207b 3131 2c20 387d 2020 |o12 = {11, 8} │ │ │ │ -0002c390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c3c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c380: 7c6f 3820 3a20 4964 6561 6c20 6f66 206b |o8 : Ideal of k │ │ │ │ +0002c390: 6b5b 7420 2c20 7420 2c20 7420 202c 2074 k[t , t , t , t │ │ │ │ +0002c3a0: 202c 2074 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +0002c3b0: 2074 2020 2c20 7420 2c20 7420 2c20 7420 t , t , t , t │ │ │ │ +0002c3c0: 2c20 7420 202c 2074 2020 2c20 2020 7c0a , t , t , |. │ │ │ │ 0002c3d0: 7c20 2020 2020 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 2020 7c0a |. │ │ │ │ -0002c420: 7c6f 3132 203a 204c 6973 7420 2020 2020 |o12 : List │ │ │ │ -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 7c0a |. │ │ │ │ -0002c470: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002c480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002c4c0: 0a54 6865 7265 2061 7265 2032 2063 6f6d .There are 2 com │ │ │ │ -0002c4d0: 706f 6e65 6e74 732e 2020 5765 2061 7474 ponents. We att │ │ │ │ -0002c4e0: 656d 7074 2074 6f20 6669 6e64 2070 6f69 empt to find poi │ │ │ │ -0002c4f0: 6e74 7320 6f6e 2065 6163 6820 6f66 2074 nts on each of t │ │ │ │ -0002c500: 6865 7365 2074 776f 0a63 6f6d 706f 6e65 hese two.compone │ │ │ │ -0002c510: 6e74 732e 2057 6520 6172 6520 7375 6363 nts. We are succ │ │ │ │ -0002c520: 6573 7366 756c 2e20 2054 6869 7320 696e essful. This in │ │ │ │ -0002c530: 6469 6361 7465 7320 7468 6174 2074 6865 dicates that the │ │ │ │ -0002c540: 2063 6f72 7265 7370 6f6e 6469 6e67 2076 corresponding v │ │ │ │ -0002c550: 6172 6965 7469 6573 0a61 7265 2062 6f74 arieties.are bot │ │ │ │ -0002c560: 6820 7261 7469 6f6e 616c 2e20 416c 736f h rational. Also │ │ │ │ -0002c570: 2c20 6966 2077 6520 6361 6e20 6669 6e64 , if we can find │ │ │ │ -0002c580: 206f 6e65 2070 6f69 6e74 2c20 7765 2063 one point, we c │ │ │ │ -0002c590: 616e 2066 696e 6420 6173 206d 616e 7920 an find as many │ │ │ │ -0002c5a0: 6173 2077 650a 7761 6e74 2e0a 0a2b 2d2d as we.want...+-- │ │ │ │ -0002c5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c3e0: 2020 2036 2020 2035 2020 2031 3220 2020 6 5 12 │ │ │ │ +0002c3f0: 3220 2020 3420 2020 3131 2020 2031 3820 2 4 11 18 │ │ │ │ +0002c400: 2020 3234 2020 2031 2020 2033 2020 2038 24 1 3 8 │ │ │ │ +0002c410: 2020 2031 3020 2020 3137 2020 2020 7c0a 10 17 |. │ │ │ │ +0002c420: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ +0002c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ +0002c470: 7c74 2020 2c20 7420 2c20 7420 2c20 7420 |t , t , t , t │ │ │ │ +0002c480: 202c 2074 2020 2c20 7420 202c 2074 2020 , t , t , t │ │ │ │ +0002c490: 2c20 7420 202c 2074 2020 2c20 7420 202c , t , t , t , │ │ │ │ +0002c4a0: 2074 2020 5d20 2020 2020 2020 2020 2020 t ] │ │ │ │ +0002c4b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c4c0: 7c20 3233 2020 2037 2020 2039 2020 2031 | 23 7 9 1 │ │ │ │ +0002c4d0: 3420 2020 3136 2020 2032 3020 2020 3232 4 16 20 22 │ │ │ │ +0002c4e0: 2020 2031 3320 2020 3135 2020 2031 3920 13 15 19 │ │ │ │ +0002c4f0: 2020 3231 2020 2020 2020 2020 2020 2020 21 │ │ │ │ +0002c500: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c510: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002c520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002c560: 7c69 3920 3a20 636f 6d70 734a 203d 2064 |i9 : compsJ = d │ │ │ │ +0002c570: 6563 6f6d 706f 7365 204a 3b20 2020 2020 ecompose J; │ │ │ │ +0002c580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c5a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c5b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0002c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0002c600: 3320 3a20 6e65 744c 6973 7420 7261 6e64 3 : netList rand │ │ │ │ -0002c610: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ -0002c620: 616c 5661 7269 6574 7928 636f 6d70 734a alVariety(compsJ │ │ │ │ -0002c630: 5f30 2c20 3130 2920 2020 2020 2020 2020 _0, 10) │ │ │ │ -0002c640: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c690: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002c6a0: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -0002c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 6f31 -----------|.|o1 │ │ │ │ -0002c6f0: 3320 3d20 7c7c 202d 3620 2d32 3620 3432 3 = || -6 -26 42 │ │ │ │ -0002c700: 202d 3337 202d 3233 2032 3820 3239 202d -37 -23 28 29 - │ │ │ │ -0002c710: 3138 202d 3320 2d31 3620 3520 3233 2033 18 -3 -16 5 23 3 │ │ │ │ -0002c720: 3420 3139 202d 3332 202d 3133 202d 3338 4 19 -32 -13 -38 │ │ │ │ -0002c730: 2031 3520 2d31 3820 3231 207c 0a7c 2020 15 -18 21 |.| │ │ │ │ -0002c740: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -0002c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0002c790: 2020 2020 7c7c 2031 3420 2d33 3420 2d33 || 14 -34 -3 │ │ │ │ -0002c7a0: 3320 3920 3237 2031 3120 3332 2034 3120 3 9 27 11 32 41 │ │ │ │ -0002c7b0: 3420 2d32 3820 3135 202d 3336 2032 2031 4 -28 15 -36 2 1 │ │ │ │ -0002c7c0: 3620 2d32 3120 2d34 3820 2d31 3520 2d32 6 -21 -48 -15 -2 │ │ │ │ -0002c7d0: 3020 2d33 3420 3338 2034 357c 0a7c 2020 0 -34 38 45|.| │ │ │ │ -0002c7e0: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -0002c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0002c830: 2020 2020 7c7c 2033 3220 3335 2031 3120 || 32 35 11 │ │ │ │ -0002c840: 2d39 202d 3237 2031 3520 3020 2d31 3220 -9 -27 15 0 -12 │ │ │ │ -0002c850: 3720 3139 2032 3420 3720 3135 202d 3233 7 19 24 7 15 -23 │ │ │ │ -0002c860: 2032 3820 2d31 3120 3437 202d 3430 202d 28 -11 47 -40 - │ │ │ │ -0002c870: 3137 2037 2034 3320 3339 207c 0a7c 2020 17 7 43 39 |.| │ │ │ │ -0002c880: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -0002c890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0002c8d0: 2020 2020 7c7c 202d 3439 202d 3820 2d35 || -49 -8 -5 │ │ │ │ -0002c8e0: 202d 3435 202d 3336 202d 3520 3437 202d -45 -36 -5 47 - │ │ │ │ -0002c8f0: 3231 202d 3334 2033 3520 2d32 3520 3332 21 -34 35 -25 32 │ │ │ │ -0002c900: 2033 3320 3430 2033 3520 3120 3336 2031 33 40 35 1 36 1 │ │ │ │ -0002c910: 202d 3238 202d 3338 2034 367c 0a7c 2020 -28 -38 46|.| │ │ │ │ -0002c920: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -0002c930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0002c970: 2020 2020 7c7c 2033 3520 3134 202d 3920 || 35 14 -9 │ │ │ │ -0002c980: 2d32 3220 2d31 3420 3139 202d 3130 2034 -22 -14 19 -10 4 │ │ │ │ -0002c990: 3820 3233 202d 3437 2035 3020 3920 3220 8 23 -47 50 9 2 │ │ │ │ -0002c9a0: 3239 202d 3337 202d 3133 2032 3220 3220 29 -37 -13 22 2 │ │ │ │ -0002c9b0: 2d33 3720 2d37 2031 3520 2d7c 0a7c 2020 -37 -7 15 -|.| │ │ │ │ -0002c9c0: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -0002c9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0002ca10: 2020 2020 7c7c 2033 3520 2d34 3920 2d33 || 35 -49 -3 │ │ │ │ -0002ca20: 3720 3332 202d 3438 2034 3220 2d31 3020 7 32 -48 42 -10 │ │ │ │ -0002ca30: 2d34 3920 3432 202d 3138 202d 3334 2032 -49 42 -18 -34 2 │ │ │ │ -0002ca40: 3520 2d32 3220 3332 202d 3335 2032 3420 5 -22 32 -35 24 │ │ │ │ -0002ca50: 3330 202d 3135 202d 3230 207c 0a7c 2020 30 -15 -20 |.| │ │ │ │ -0002ca60: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -0002ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002caa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0002cab0: 2020 2020 7c7c 202d 3239 202d 3430 2033 || -29 -40 3 │ │ │ │ -0002cac0: 3020 2d35 2033 3620 2d34 3120 2d32 3420 0 -5 36 -41 -24 │ │ │ │ -0002cad0: 3120 3334 202d 3135 2032 3020 2d33 3320 1 34 -15 20 -33 │ │ │ │ -0002cae0: 3333 202d 3439 202d 3134 202d 3230 202d 33 -49 -14 -20 - │ │ │ │ -0002caf0: 3438 2032 3120 3137 2030 207c 0a7c 2020 48 21 17 0 |.| │ │ │ │ -0002cb00: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -0002cb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002cb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0002cb50: 2020 2020 7c7c 202d 3238 2032 3720 3338 || -28 27 38 │ │ │ │ -0002cb60: 2031 3320 3336 2033 3820 3020 3335 2031 13 36 38 0 35 1 │ │ │ │ -0002cb70: 3220 3336 202d 3333 202d 3234 2034 2031 2 36 -33 -24 4 1 │ │ │ │ -0002cb80: 3320 2d33 3520 2d31 3120 2d33 3920 3334 3 -35 -11 -39 34 │ │ │ │ -0002cb90: 202d 3439 202d 3339 2032 327c 0a7c 2020 -49 -39 22|.| │ │ │ │ +0002c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002c600: 7c69 3130 203a 2063 6f6d 7073 4a20 3d20 |i10 : compsJ = │ │ │ │ +0002c610: 636f 6d70 734a 2f74 7269 6d3b 2020 2020 compsJ/trim; │ │ │ │ +0002c620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c640: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c650: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002c660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002c6a0: 7c69 3131 203a 2023 636f 6d70 734a 203d |i11 : #compsJ = │ │ │ │ +0002c6b0: 3d20 3220 2020 2020 2020 2020 2020 2020 = 2 │ │ │ │ +0002c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c6e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c6f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c730: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c740: 7c6f 3131 203d 2074 7275 6520 2020 2020 |o11 = true │ │ │ │ +0002c750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c780: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c790: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002c7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002c7e0: 7c69 3132 203a 2063 6f6d 7073 4a2f 6469 |i12 : compsJ/di │ │ │ │ +0002c7f0: 6d20 2020 2020 2020 2020 2020 2020 2020 m │ │ │ │ +0002c800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c820: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c830: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002c840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c870: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c880: 7c6f 3132 203d 207b 3131 2c20 387d 2020 |o12 = {11, 8} │ │ │ │ +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 2020 7c0a |. │ │ │ │ +0002c8d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002c8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c910: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c920: 7c6f 3132 203a 204c 6973 7420 2020 2020 |o12 : List │ │ │ │ +0002c930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002c9c0: 0a54 6865 7265 2061 7265 2032 2063 6f6d .There are 2 com │ │ │ │ +0002c9d0: 706f 6e65 6e74 732e 2020 5765 2061 7474 ponents. We att │ │ │ │ +0002c9e0: 656d 7074 2074 6f20 6669 6e64 2070 6f69 empt to find poi │ │ │ │ +0002c9f0: 6e74 7320 6f6e 2065 6163 6820 6f66 2074 nts on each of t │ │ │ │ +0002ca00: 6865 7365 2074 776f 0a63 6f6d 706f 6e65 hese two.compone │ │ │ │ +0002ca10: 6e74 732e 2057 6520 6172 6520 7375 6363 nts. We are succ │ │ │ │ +0002ca20: 6573 7366 756c 2e20 2054 6869 7320 696e essful. This in │ │ │ │ +0002ca30: 6469 6361 7465 7320 7468 6174 2074 6865 dicates that the │ │ │ │ +0002ca40: 2063 6f72 7265 7370 6f6e 6469 6e67 2076 corresponding v │ │ │ │ +0002ca50: 6172 6965 7469 6573 0a61 7265 2062 6f74 arieties.are bot │ │ │ │ +0002ca60: 6820 7261 7469 6f6e 616c 2e20 416c 736f h rational. Also │ │ │ │ +0002ca70: 2c20 6966 2077 6520 6361 6e20 6669 6e64 , if we can find │ │ │ │ +0002ca80: 206f 6e65 2070 6f69 6e74 2c20 7765 2063 one point, we c │ │ │ │ +0002ca90: 616e 2066 696e 6420 6173 206d 616e 7920 an find as many │ │ │ │ +0002caa0: 6173 2077 650a 7761 6e74 2e0a 0a2b 2d2d as we.want...+-- │ │ │ │ +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 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0002cb00: 3320 3a20 6e65 744c 6973 7420 7261 6e64 3 : netList rand │ │ │ │ +0002cb10: 6f6d 506f 696e 7473 4f6e 5261 7469 6f6e omPointsOnRation │ │ │ │ +0002cb20: 616c 5661 7269 6574 7928 636f 6d70 734a alVariety(compsJ │ │ │ │ +0002cb30: 5f30 2c20 3130 2920 2020 2020 2020 2020 _0, 10) │ │ │ │ +0002cb40: 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e260: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 2d2b 0a0a 4361 -----------+..Ca │ │ │ │ -0002e2c0: 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54 6869 veat.======..Thi │ │ │ │ -0002e2d0: 7320 726f 7574 696e 6520 6578 7065 6374 s routine expect │ │ │ │ -0002e2e0: 7320 7468 6520 696e 7075 7420 746f 2072 s the input to r │ │ │ │ -0002e2f0: 6570 7265 7365 6e74 2061 6e20 6972 7265 epresent an irre │ │ │ │ -0002e300: 6475 6369 626c 6520 7661 7269 6574 790a ducible variety. │ │ │ │ -0002e310: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -0002e320: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7261 ==.. * *note ra │ │ │ │ -0002e330: 6e64 6f6d 506f 696e 744f 6e52 6174 696f ndomPointOnRatio │ │ │ │ -0002e340: 6e61 6c56 6172 6965 7479 2849 6465 616c nalVariety(Ideal │ │ │ │ -0002e350: 293a 0a20 2020 2072 616e 646f 6d50 6f69 ):. randomPoi │ │ │ │ -0002e360: 6e74 4f6e 5261 7469 6f6e 616c 5661 7269 ntOnRationalVari │ │ │ │ -0002e370: 6574 795f 6c70 4964 6561 6c5f 7270 2c20 ety_lpIdeal_rp, │ │ │ │ -0002e380: 2d2d 2066 696e 6420 6120 7261 6e64 6f6d -- find a random │ │ │ │ -0002e390: 2070 6f69 6e74 206f 6e20 610a 2020 2020 point on a. │ │ │ │ -0002e3a0: 7661 7269 6574 7920 7468 6174 2063 616e variety that can │ │ │ │ -0002e3b0: 2062 6520 6465 7465 6374 6564 2074 6f20 be detected to │ │ │ │ -0002e3c0: 6265 2072 6174 696f 6e61 6c0a 0a57 6179 be rational..Way │ │ │ │ -0002e3d0: 7320 746f 2075 7365 2074 6869 7320 6d65 s to use this me │ │ │ │ -0002e3e0: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d thod:.========== │ │ │ │ -0002e3f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -0002e400: 2020 2a20 2a6e 6f74 6520 7261 6e64 6f6d * *note random │ │ │ │ -0002e410: 506f 696e 7473 4f6e 5261 7469 6f6e 616c PointsOnRational │ │ │ │ -0002e420: 5661 7269 6574 7928 4964 6561 6c2c 5a5a Variety(Ideal,ZZ │ │ │ │ -0002e430: 293a 0a20 2020 2072 616e 646f 6d50 6f69 ):. randomPoi │ │ │ │ -0002e440: 6e74 734f 6e52 6174 696f 6e61 6c56 6172 ntsOnRationalVar │ │ │ │ -0002e450: 6965 7479 5f6c 7049 6465 616c 5f63 6d5a iety_lpIdeal_cmZ │ │ │ │ -0002e460: 5a5f 7270 2c20 2d2d 2066 696e 6420 7261 Z_rp, -- find ra │ │ │ │ -0002e470: 6e64 6f6d 2070 6f69 6e74 7320 6f6e 2061 ndom points on a │ │ │ │ -0002e480: 0a20 2020 2076 6172 6965 7479 2074 6861 . variety tha │ │ │ │ -0002e490: 7420 6361 6e20 6265 2064 6574 6563 7465 t can be detecte │ │ │ │ -0002e4a0: 6420 746f 2062 6520 7261 7469 6f6e 616c d to be rational │ │ │ │ -0002e4b0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ -0002e4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -0002e510: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -0002e520: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -0002e530: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -0002e540: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ -0002e550: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -0002e560: 6163 6b61 6765 732f 0a47 726f 6562 6e65 ackages/.Groebne │ │ │ │ -0002e570: 7253 7472 6174 612e 6d32 3a39 3031 3a30 rStrata.m2:901:0 │ │ │ │ -0002e580: 2e0a 1f0a 4669 6c65 3a20 4772 6f65 626e ....File: Groebn │ │ │ │ -0002e590: 6572 5374 7261 7461 2e69 6e66 6f2c 204e erStrata.info, N │ │ │ │ -0002e5a0: 6f64 653a 2073 6d61 6c6c 6572 4d6f 6e6f ode: smallerMono │ │ │ │ -0002e5b0: 6d69 616c 732c 204e 6578 743a 2073 7461 mials, Next: sta │ │ │ │ -0002e5c0: 6e64 6172 644d 6f6e 6f6d 6961 6c73 2c20 ndardMonomials, │ │ │ │ -0002e5d0: 5072 6576 3a20 7261 6e64 6f6d 506f 696e Prev: randomPoin │ │ │ │ -0002e5e0: 7473 4f6e 5261 7469 6f6e 616c 5661 7269 tsOnRationalVari │ │ │ │ -0002e5f0: 6574 795f 6c70 4964 6561 6c5f 636d 5a5a ety_lpIdeal_cmZZ │ │ │ │ -0002e600: 5f72 702c 2055 703a 2054 6f70 0a0a 736d _rp, Up: Top..sm │ │ │ │ -0002e610: 616c 6c65 724d 6f6e 6f6d 6961 6c73 202d allerMonomials - │ │ │ │ -0002e620: 2d20 7265 7475 726e 7320 7468 6520 7374 - returns the st │ │ │ │ -0002e630: 616e 6461 7264 206d 6f6e 6f6d 6961 6c73 andard monomials │ │ │ │ -0002e640: 2073 6d61 6c6c 6572 2062 7574 206f 6620 smaller but of │ │ │ │ -0002e650: 7468 6520 7361 6d65 2064 6567 7265 6520 the same degree │ │ │ │ -0002e660: 6173 2067 6976 656e 206d 6f6e 6f6d 6961 as given monomia │ │ │ │ -0002e670: 6c28 7329 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a l(s).*********** │ │ │ │ -0002e680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -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 2a0a 0a20 202a ***********.. * │ │ │ │ -0002e6e0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0002e6f0: 204c 203d 2073 6d61 6c6c 6572 4d6f 6e6f L = smallerMono │ │ │ │ -0002e700: 6d69 616c 7320 4d0a 2020 2020 2020 2020 mials M. │ │ │ │ -0002e710: 4c20 3d20 736d 616c 6c65 724d 6f6e 6f6d L = smallerMonom │ │ │ │ -0002e720: 6961 6c73 284d 2c20 6d29 0a20 202a 2049 ials(M, m). * I │ │ │ │ -0002e730: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ -0002e740: 2c20 616e 202a 6e6f 7465 2069 6465 616c , an *note ideal │ │ │ │ -0002e750: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0002e760: 4964 6561 6c2c 2c20 244d 2420 7368 6f75 Ideal,, $M$ shou │ │ │ │ -0002e770: 6c64 2062 6520 6120 6d6f 6e6f 6d69 616c ld be a monomial │ │ │ │ -0002e780: 2069 6465 616c 0a20 2020 2020 2020 2028 ideal. ( │ │ │ │ -0002e790: 616e 2069 6465 616c 2067 656e 6572 6174 an ideal generat │ │ │ │ -0002e7a0: 6564 2062 7920 6d6f 6e6f 6d69 616c 7329 ed by monomials) │ │ │ │ -0002e7b0: 0a20 2020 2020 202a 206d 2c20 6120 2a6e . * m, a *n │ │ │ │ -0002e7c0: 6f74 6520 7269 6e67 2065 6c65 6d65 6e74 ote ring element │ │ │ │ -0002e7d0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0002e7e0: 5269 6e67 456c 656d 656e 742c 2c20 6f70 RingElement,, op │ │ │ │ -0002e7f0: 7469 6f6e 616c 2c0a 2020 2a20 4f75 7470 tional,. * Outp │ │ │ │ -0002e800: 7574 733a 0a20 2020 2020 202a 204c 2c20 uts:. * L, │ │ │ │ -0002e810: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ -0002e820: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ -0002e830: 2c2c 2061 206c 6973 7420 6f66 206c 6973 ,, a list of lis │ │ │ │ -0002e840: 7473 3a20 666f 7220 6561 6368 0a20 2020 ts: for each. │ │ │ │ -0002e850: 2020 2020 2067 656e 6572 6174 6f72 2024 generator $ │ │ │ │ -0002e860: 6d24 206f 6620 244d 242c 2074 6865 206c m$ of $M$, the l │ │ │ │ -0002e870: 6973 7420 6f66 2061 6c6c 206d 6f6e 6f6d ist of all monom │ │ │ │ -0002e880: 6961 6c73 206f 6620 7468 6520 7361 6d65 ials of the same │ │ │ │ -0002e890: 2064 6567 7265 6520 6173 0a20 2020 2020 degree as. │ │ │ │ -0002e8a0: 2020 2024 6d24 2c20 6e6f 7420 696e 2074 $m$, not in t │ │ │ │ -0002e8b0: 6865 206d 6f6e 6f6d 6961 6c20 6964 6561 he monomial idea │ │ │ │ -0002e8c0: 6c20 616e 6420 736d 616c 6c65 7220 7468 l and smaller th │ │ │ │ -0002e8d0: 616e 2074 6861 7420 6765 6e65 7261 746f an that generato │ │ │ │ -0002e8e0: 7220 696e 2074 6865 0a20 2020 2020 2020 r in the. │ │ │ │ -0002e8f0: 2074 6572 6d20 6f72 6465 7220 6f66 2074 term order of t │ │ │ │ -0002e900: 6865 2061 6d62 6965 6e74 2072 696e 672e he ambient ring. │ │ │ │ -0002e910: 2020 4966 2069 6e73 7465 6164 2024 6d24 If instead $m$ │ │ │ │ -0002e920: 2069 7320 6769 7665 6e2c 2074 6865 206c is given, the l │ │ │ │ -0002e930: 6973 7420 6f66 0a20 2020 2020 2020 2074 ist of. t │ │ │ │ -0002e940: 6865 2073 7461 6e64 6172 6420 6d6f 6e6f he standard mono │ │ │ │ -0002e950: 6d69 616c 7320 6f66 2074 6865 2073 616d mials of the sam │ │ │ │ -0002e960: 6520 6465 6772 6565 2c20 736d 616c 6c65 e degree, smalle │ │ │ │ -0002e970: 7220 7468 616e 2024 6d24 2c20 6973 0a20 r than $m$, is. │ │ │ │ -0002e980: 2020 2020 2020 2072 6574 7572 6e65 642e returned. │ │ │ │ -0002e990: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0002e9a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6e70 7574 =========..Input │ │ │ │ -0002e9b0: 7469 6e67 2061 6e20 6964 6561 6c20 244d ting an ideal $M │ │ │ │ -0002e9c0: 2420 7265 7475 726e 7320 7468 6520 736d $ returns the sm │ │ │ │ -0002e9d0: 616c 6c65 7220 6d6f 6e6f 6d69 616c 7320 aller monomials │ │ │ │ -0002e9e0: 6f66 2065 6163 6820 6f66 2074 6865 2067 of each of the g │ │ │ │ -0002e9f0: 6976 656e 0a67 656e 6572 6174 6f72 7320 iven.generators │ │ │ │ -0002ea00: 6f66 2074 6865 2069 6465 616c 2e0a 0a2b of the ideal...+ │ │ │ │ -0002ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ea40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002ea60: 6931 203a 2052 203d 205a 5a2f 3332 3030 i1 : R = ZZ/3200 │ │ │ │ -0002ea70: 335b 612e 2e64 5d3b 2020 2020 2020 2020 3[a..d]; │ │ │ │ -0002ea80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ea90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eaa0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ead0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002eb00: 6932 203a 204d 203d 2069 6465 616c 2028 i2 : M = ideal ( │ │ │ │ -0002eb10: 615e 322c 2062 5e32 2c20 612a 622a 6329 a^2, b^2, a*b*c) │ │ │ │ -0002eb20: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -0002eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002eba0: 6f32 203a 2049 6465 616c 206f 6620 5220 o2 : Ideal of R │ │ │ │ -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 207c 0a2b |.+ │ │ │ │ -0002ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ec00: 2d2d 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 2d2b 0a7c -------------+.| │ │ │ │ -0002ec40: 6933 203a 204c 3120 3d20 736d 616c 6c65 i3 : L1 = smalle │ │ │ │ -0002ec50: 724d 6f6e 6f6d 6961 6c73 204d 2020 2020 rMonomials M │ │ │ │ -0002ec60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002ec90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ecb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ecc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ecd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002ece0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ecf0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -0002ed00: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -0002ed10: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -0002ed20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002ed30: 6f33 203d 207b 7b61 2a62 2c20 612a 632c o3 = {{a*b, a*c, │ │ │ │ -0002ed40: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ -0002ed50: 2a64 2c20 632a 642c 2064 207d 2c20 7b61 *d, c*d, d }, {a │ │ │ │ -0002ed60: 2a63 2c20 622a 632c 2063 202c 2061 2a64 *c, b*c, c , a*d │ │ │ │ -0002ed70: 2c20 622a 642c 2063 2a64 2c20 207c 0a7c , b*d, c*d, |.| │ │ │ │ -0002ed80: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ -0002ed90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -0002edd0: 2020 2020 2020 3220 2020 2020 2020 3220 2 2 │ │ │ │ -0002ede0: 2020 2020 3220 2020 3320 2020 2020 2020 2 3 │ │ │ │ -0002edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee00: 2032 2020 2020 2020 3220 2020 2020 3220 2 2 2 │ │ │ │ -0002ee10: 2020 2020 3220 2020 3320 2020 207c 0a7c 2 3 |.| │ │ │ │ -0002ee20: 2020 2020 2064 207d 2c20 7b61 2a63 202c d }, {a*c , │ │ │ │ -0002ee30: 2062 2a63 202c 2063 202c 2061 2a62 2a64 b*c , c , a*b*d │ │ │ │ -0002ee40: 2c20 612a 632a 642c 2062 2a63 2a64 2c20 , a*c*d, b*c*d, │ │ │ │ -0002ee50: 6320 642c 2061 2a64 202c 2062 2a64 202c c d, a*d , b*d , │ │ │ │ -0002ee60: 2063 2a64 202c 2064 207d 7d20 207c 0a7c c*d , d }} |.| │ │ │ │ -0002ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eeb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002eec0: 6f33 203a 204c 6973 7420 2020 2020 2020 o3 : List │ │ │ │ -0002eed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 207c 0a2b |.+ │ │ │ │ +0002e260: 2020 2020 2020 2020 2020 207c 0a7c 3436 |.|46 │ │ │ │ +0002e270: 2031 3020 3020 7c20 2020 2020 2020 2020 10 0 | │ │ │ │ +0002e280: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e2b0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ +0002e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e2d0: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e300: 2020 2020 2020 2020 2020 207c 0a7c 3137 |.|17 │ │ │ │ +0002e310: 2038 2030 207c 2020 2020 2020 2020 2020 8 0 | │ │ │ │ +0002e320: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e350: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ +0002e360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e370: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3a0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ +0002e3b0: 3436 2032 3320 3020 7c20 2020 2020 2020 46 23 0 | │ │ │ │ +0002e3c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3f0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ +0002e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e410: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e440: 2020 2020 2020 2020 2020 207c 0a7c 3720 |.|7 │ │ │ │ +0002e450: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ +0002e460: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e490: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ +0002e4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e4b0: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e4e0: 2020 2020 2020 2020 2020 207c 0a7c 3720 |.|7 │ │ │ │ +0002e4f0: 2d32 3320 3139 2030 207c 2020 2020 2020 -23 19 0 | │ │ │ │ +0002e500: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e530: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ +0002e540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e550: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e580: 2020 2020 2020 2020 2020 207c 0a7c 3820 |.|8 │ │ │ │ +0002e590: 3437 202d 3333 2030 207c 2020 2020 2020 47 -33 0 | │ │ │ │ +0002e5a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5d0: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ +0002e5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e5f0: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e620: 2020 2020 2020 2020 2020 207c 0a7c 2035 |.| 5 │ │ │ │ +0002e630: 2034 3220 3020 7c20 2020 2020 2020 2020 42 0 | │ │ │ │ +0002e640: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e670: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ +0002e680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e690: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e6c0: 2020 2020 2020 2020 2020 207c 0a7c 3220 |.|2 │ │ │ │ +0002e6d0: 2d32 3920 3020 7c20 2020 2020 2020 2020 -29 0 | │ │ │ │ +0002e6e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e710: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ +0002e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e730: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e760: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361 -----------+..Ca │ │ │ │ +0002e7c0: 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54 6869 veat.======..Thi │ │ │ │ +0002e7d0: 7320 726f 7574 696e 6520 6578 7065 6374 s routine expect │ │ │ │ +0002e7e0: 7320 7468 6520 696e 7075 7420 746f 2072 s the input to r │ │ │ │ +0002e7f0: 6570 7265 7365 6e74 2061 6e20 6972 7265 epresent an irre │ │ │ │ +0002e800: 6475 6369 626c 6520 7661 7269 6574 790a ducible variety. │ │ │ │ +0002e810: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +0002e820: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7261 ==.. * *note ra │ │ │ │ +0002e830: 6e64 6f6d 506f 696e 744f 6e52 6174 696f ndomPointOnRatio │ │ │ │ +0002e840: 6e61 6c56 6172 6965 7479 2849 6465 616c nalVariety(Ideal │ │ │ │ +0002e850: 293a 0a20 2020 2072 616e 646f 6d50 6f69 ):. randomPoi │ │ │ │ +0002e860: 6e74 4f6e 5261 7469 6f6e 616c 5661 7269 ntOnRationalVari │ │ │ │ +0002e870: 6574 795f 6c70 4964 6561 6c5f 7270 2c20 ety_lpIdeal_rp, │ │ │ │ +0002e880: 2d2d 2066 696e 6420 6120 7261 6e64 6f6d -- find a random │ │ │ │ +0002e890: 2070 6f69 6e74 206f 6e20 610a 2020 2020 point on a. │ │ │ │ +0002e8a0: 7661 7269 6574 7920 7468 6174 2063 616e variety that can │ │ │ │ +0002e8b0: 2062 6520 6465 7465 6374 6564 2074 6f20 be detected to │ │ │ │ +0002e8c0: 6265 2072 6174 696f 6e61 6c0a 0a57 6179 be rational..Way │ │ │ │ +0002e8d0: 7320 746f 2075 7365 2074 6869 7320 6d65 s to use this me │ │ │ │ +0002e8e0: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d thod:.========== │ │ │ │ +0002e8f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0002e900: 2020 2a20 2a6e 6f74 6520 7261 6e64 6f6d * *note random │ │ │ │ +0002e910: 506f 696e 7473 4f6e 5261 7469 6f6e 616c PointsOnRational │ │ │ │ +0002e920: 5661 7269 6574 7928 4964 6561 6c2c 5a5a Variety(Ideal,ZZ │ │ │ │ +0002e930: 293a 0a20 2020 2072 616e 646f 6d50 6f69 ):. randomPoi │ │ │ │ +0002e940: 6e74 734f 6e52 6174 696f 6e61 6c56 6172 ntsOnRationalVar │ │ │ │ +0002e950: 6965 7479 5f6c 7049 6465 616c 5f63 6d5a iety_lpIdeal_cmZ │ │ │ │ +0002e960: 5a5f 7270 2c20 2d2d 2066 696e 6420 7261 Z_rp, -- find ra │ │ │ │ +0002e970: 6e64 6f6d 2070 6f69 6e74 7320 6f6e 2061 ndom points on a │ │ │ │ +0002e980: 0a20 2020 2076 6172 6965 7479 2074 6861 . variety tha │ │ │ │ +0002e990: 7420 6361 6e20 6265 2064 6574 6563 7465 t can be detecte │ │ │ │ +0002e9a0: 6420 746f 2062 6520 7261 7469 6f6e 616c d to be rational │ │ │ │ +0002e9b0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +0002e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ea00: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0002ea10: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0002ea20: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0002ea30: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0002ea40: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +0002ea50: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0002ea60: 6163 6b61 6765 732f 0a47 726f 6562 6e65 ackages/.Groebne │ │ │ │ +0002ea70: 7253 7472 6174 612e 6d32 3a39 3031 3a30 rStrata.m2:901:0 │ │ │ │ +0002ea80: 2e0a 1f0a 4669 6c65 3a20 4772 6f65 626e ....File: Groebn │ │ │ │ +0002ea90: 6572 5374 7261 7461 2e69 6e66 6f2c 204e erStrata.info, N │ │ │ │ +0002eaa0: 6f64 653a 2073 6d61 6c6c 6572 4d6f 6e6f ode: smallerMono │ │ │ │ +0002eab0: 6d69 616c 732c 204e 6578 743a 2073 7461 mials, Next: sta │ │ │ │ +0002eac0: 6e64 6172 644d 6f6e 6f6d 6961 6c73 2c20 ndardMonomials, │ │ │ │ +0002ead0: 5072 6576 3a20 7261 6e64 6f6d 506f 696e Prev: randomPoin │ │ │ │ +0002eae0: 7473 4f6e 5261 7469 6f6e 616c 5661 7269 tsOnRationalVari │ │ │ │ +0002eaf0: 6574 795f 6c70 4964 6561 6c5f 636d 5a5a ety_lpIdeal_cmZZ │ │ │ │ +0002eb00: 5f72 702c 2055 703a 2054 6f70 0a0a 736d _rp, Up: Top..sm │ │ │ │ +0002eb10: 616c 6c65 724d 6f6e 6f6d 6961 6c73 202d allerMonomials - │ │ │ │ +0002eb20: 2d20 7265 7475 726e 7320 7468 6520 7374 - returns the st │ │ │ │ +0002eb30: 616e 6461 7264 206d 6f6e 6f6d 6961 6c73 andard monomials │ │ │ │ +0002eb40: 2073 6d61 6c6c 6572 2062 7574 206f 6620 smaller but of │ │ │ │ +0002eb50: 7468 6520 7361 6d65 2064 6567 7265 6520 the same degree │ │ │ │ +0002eb60: 6173 2067 6976 656e 206d 6f6e 6f6d 6961 as given monomia │ │ │ │ +0002eb70: 6c28 7329 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a l(s).*********** │ │ │ │ +0002eb80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002eb90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002eba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ebb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ebc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002ebd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0002ebe0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0002ebf0: 204c 203d 2073 6d61 6c6c 6572 4d6f 6e6f L = smallerMono │ │ │ │ +0002ec00: 6d69 616c 7320 4d0a 2020 2020 2020 2020 mials M. │ │ │ │ +0002ec10: 4c20 3d20 736d 616c 6c65 724d 6f6e 6f6d L = smallerMonom │ │ │ │ +0002ec20: 6961 6c73 284d 2c20 6d29 0a20 202a 2049 ials(M, m). * I │ │ │ │ +0002ec30: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ +0002ec40: 2c20 616e 202a 6e6f 7465 2069 6465 616c , an *note ideal │ │ │ │ +0002ec50: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0002ec60: 4964 6561 6c2c 2c20 244d 2420 7368 6f75 Ideal,, $M$ shou │ │ │ │ +0002ec70: 6c64 2062 6520 6120 6d6f 6e6f 6d69 616c ld be a monomial │ │ │ │ +0002ec80: 2069 6465 616c 0a20 2020 2020 2020 2028 ideal. ( │ │ │ │ +0002ec90: 616e 2069 6465 616c 2067 656e 6572 6174 an ideal generat │ │ │ │ +0002eca0: 6564 2062 7920 6d6f 6e6f 6d69 616c 7329 ed by monomials) │ │ │ │ +0002ecb0: 0a20 2020 2020 202a 206d 2c20 6120 2a6e . * m, a *n │ │ │ │ +0002ecc0: 6f74 6520 7269 6e67 2065 6c65 6d65 6e74 ote ring element │ │ │ │ +0002ecd0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0002ece0: 5269 6e67 456c 656d 656e 742c 2c20 6f70 RingElement,, op │ │ │ │ +0002ecf0: 7469 6f6e 616c 2c0a 2020 2a20 4f75 7470 tional,. * Outp │ │ │ │ +0002ed00: 7574 733a 0a20 2020 2020 202a 204c 2c20 uts:. * L, │ │ │ │ +0002ed10: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ +0002ed20: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ +0002ed30: 2c2c 2061 206c 6973 7420 6f66 206c 6973 ,, a list of lis │ │ │ │ +0002ed40: 7473 3a20 666f 7220 6561 6368 0a20 2020 ts: for each. │ │ │ │ +0002ed50: 2020 2020 2067 656e 6572 6174 6f72 2024 generator $ │ │ │ │ +0002ed60: 6d24 206f 6620 244d 242c 2074 6865 206c m$ of $M$, the l │ │ │ │ +0002ed70: 6973 7420 6f66 2061 6c6c 206d 6f6e 6f6d ist of all monom │ │ │ │ +0002ed80: 6961 6c73 206f 6620 7468 6520 7361 6d65 ials of the same │ │ │ │ +0002ed90: 2064 6567 7265 6520 6173 0a20 2020 2020 degree as. │ │ │ │ +0002eda0: 2020 2024 6d24 2c20 6e6f 7420 696e 2074 $m$, not in t │ │ │ │ +0002edb0: 6865 206d 6f6e 6f6d 6961 6c20 6964 6561 he monomial idea │ │ │ │ +0002edc0: 6c20 616e 6420 736d 616c 6c65 7220 7468 l and smaller th │ │ │ │ +0002edd0: 616e 2074 6861 7420 6765 6e65 7261 746f an that generato │ │ │ │ +0002ede0: 7220 696e 2074 6865 0a20 2020 2020 2020 r in the. │ │ │ │ +0002edf0: 2074 6572 6d20 6f72 6465 7220 6f66 2074 term order of t │ │ │ │ +0002ee00: 6865 2061 6d62 6965 6e74 2072 696e 672e he ambient ring. │ │ │ │ +0002ee10: 2020 4966 2069 6e73 7465 6164 2024 6d24 If instead $m$ │ │ │ │ +0002ee20: 2069 7320 6769 7665 6e2c 2074 6865 206c is given, the l │ │ │ │ +0002ee30: 6973 7420 6f66 0a20 2020 2020 2020 2074 ist of. t │ │ │ │ +0002ee40: 6865 2073 7461 6e64 6172 6420 6d6f 6e6f he standard mono │ │ │ │ +0002ee50: 6d69 616c 7320 6f66 2074 6865 2073 616d mials of the sam │ │ │ │ +0002ee60: 6520 6465 6772 6565 2c20 736d 616c 6c65 e degree, smalle │ │ │ │ +0002ee70: 7220 7468 616e 2024 6d24 2c20 6973 0a20 r than $m$, is. │ │ │ │ +0002ee80: 2020 2020 2020 2072 6574 7572 6e65 642e returned. │ │ │ │ +0002ee90: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0002eea0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6e70 7574 =========..Input │ │ │ │ +0002eeb0: 7469 6e67 2061 6e20 6964 6561 6c20 244d ting an ideal $M │ │ │ │ +0002eec0: 2420 7265 7475 726e 7320 7468 6520 736d $ returns the sm │ │ │ │ +0002eed0: 616c 6c65 7220 6d6f 6e6f 6d69 616c 7320 aller monomials │ │ │ │ +0002eee0: 6f66 2065 6163 6820 6f66 2074 6865 2067 of each of the g │ │ │ │ +0002eef0: 6976 656e 0a67 656e 6572 6174 6f72 7320 iven.generators │ │ │ │ +0002ef00: 6f66 2074 6865 2069 6465 616c 2e0a 0a2b of the ideal...+ │ │ │ │ 0002ef10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ef20: 2d2d 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 2d2b 0a7c -------------+.| │ │ │ │ -0002ef60: 6934 203a 2073 6d61 6c6c 6572 4d6f 6e6f i4 : smallerMono │ │ │ │ -0002ef70: 6d69 616c 7328 4d2c 2062 5e32 2920 2020 mials(M, b^2) │ │ │ │ +0002ef60: 6931 203a 2052 203d 205a 5a2f 3332 3030 i1 : R = ZZ/3200 │ │ │ │ +0002ef70: 335b 612e 2e64 5d3b 2020 2020 2020 2020 3[a..d]; │ │ │ │ 0002ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002efa0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eff0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f010: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002f020: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0002efa0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002efb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002efc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002efd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002efe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0002f000: 6932 203a 204d 203d 2069 6465 616c 2028 i2 : M = ideal ( │ │ │ │ +0002f010: 615e 322c 2062 5e32 2c20 612a 622a 6329 a^2, b^2, a*b*c) │ │ │ │ +0002f020: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ 0002f030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f040: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002f050: 6f34 203d 207b 612a 632c 2062 2a63 2c20 o4 = {a*c, b*c, │ │ │ │ -0002f060: 6320 2c20 612a 642c 2062 2a64 2c20 632a c , a*d, b*d, c* │ │ │ │ -0002f070: 642c 2064 207d 2020 2020 2020 2020 2020 d, d } │ │ │ │ +0002f050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f060: 2020 2020 2020 2020 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 207c 0a7c |.| │ │ │ │ -0002f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f0a0: 6f32 203a 2049 6465 616c 206f 6620 5220 o2 : Ideal of R │ │ │ │ 0002f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f0e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002f0f0: 6f34 203a 204c 6973 7420 2020 2020 2020 o4 : List │ │ │ │ -0002f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f130: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0002f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0002f190: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ -0002f1a0: 3d0a 0a20 202a 202a 6e6f 7465 2074 6169 =.. * *note tai │ │ │ │ -0002f1b0: 6c4d 6f6e 6f6d 6961 6c73 3a20 7461 696c lMonomials: tail │ │ │ │ -0002f1c0: 4d6f 6e6f 6d69 616c 732c 202d 2d20 6669 Monomials, -- fi │ │ │ │ -0002f1d0: 6e64 2074 6169 6c20 6d6f 6e6f 6d69 616c nd tail monomial │ │ │ │ -0002f1e0: 730a 2020 2a20 2a6e 6f74 6520 7374 616e s. * *note stan │ │ │ │ -0002f1f0: 6461 7264 4d6f 6e6f 6d69 616c 733a 2073 dardMonomials: s │ │ │ │ -0002f200: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ -0002f210: 2c20 2d2d 2063 6f6d 7075 7465 7320 7374 , -- computes st │ │ │ │ -0002f220: 616e 6461 7264 206d 6f6e 6f6d 6961 6c73 andard monomials │ │ │ │ -0002f230: 0a0a 5761 7973 2074 6f20 7573 6520 736d ..Ways to use sm │ │ │ │ -0002f240: 616c 6c65 724d 6f6e 6f6d 6961 6c73 3a0a allerMonomials:. │ │ │ │ -0002f250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002f260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0002f270: 202a 2022 736d 616c 6c65 724d 6f6e 6f6d * "smallerMonom │ │ │ │ -0002f280: 6961 6c73 2849 6465 616c 2922 0a20 202a ials(Ideal)". * │ │ │ │ -0002f290: 2022 736d 616c 6c65 724d 6f6e 6f6d 6961 "smallerMonomia │ │ │ │ -0002f2a0: 6c73 2849 6465 616c 2c52 696e 6745 6c65 ls(Ideal,RingEle │ │ │ │ -0002f2b0: 6d65 6e74 2922 0a0a 466f 7220 7468 6520 ment)"..For the │ │ │ │ -0002f2c0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0002f2d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0002f2e0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0002f2f0: 736d 616c 6c65 724d 6f6e 6f6d 6961 6c73 smallerMonomials │ │ │ │ -0002f300: 3a20 736d 616c 6c65 724d 6f6e 6f6d 6961 : smallerMonomia │ │ │ │ -0002f310: 6c73 2c20 6973 2061 202a 6e6f 7465 206d ls, is a *note m │ │ │ │ -0002f320: 6574 686f 640a 6675 6e63 7469 6f6e 3a20 ethod.function: │ │ │ │ -0002f330: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -0002f340: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ -0002f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f360: 2d2d 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 2d0a ---------------. │ │ │ │ -0002f3a0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0002f3b0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0002f3c0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0002f3d0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -0002f3e0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -0002f3f0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -0002f400: 636b 6167 6573 2f0a 4772 6f65 626e 6572 ckages/.Groebner │ │ │ │ -0002f410: 5374 7261 7461 2e6d 323a 3534 323a 302e Strata.m2:542:0. │ │ │ │ -0002f420: 0a1f 0a46 696c 653a 2047 726f 6562 6e65 ...File: Groebne │ │ │ │ -0002f430: 7253 7472 6174 612e 696e 666f 2c20 4e6f rStrata.info, No │ │ │ │ -0002f440: 6465 3a20 7374 616e 6461 7264 4d6f 6e6f de: standardMono │ │ │ │ -0002f450: 6d69 616c 732c 204e 6578 743a 2074 6169 mials, Next: tai │ │ │ │ -0002f460: 6c4d 6f6e 6f6d 6961 6c73 2c20 5072 6576 lMonomials, Prev │ │ │ │ -0002f470: 3a20 736d 616c 6c65 724d 6f6e 6f6d 6961 : smallerMonomia │ │ │ │ -0002f480: 6c73 2c20 5570 3a20 546f 700a 0a73 7461 ls, Up: Top..sta │ │ │ │ -0002f490: 6e64 6172 644d 6f6e 6f6d 6961 6c73 202d ndardMonomials - │ │ │ │ -0002f4a0: 2d20 636f 6d70 7574 6573 2073 7461 6e64 - computes stand │ │ │ │ -0002f4b0: 6172 6420 6d6f 6e6f 6d69 616c 730a 2a2a ard monomials.** │ │ │ │ -0002f4c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002f4d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002f4e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -0002f4f0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -0002f500: 2020 2020 4c20 3d20 7374 616e 6461 7264 L = standard │ │ │ │ -0002f510: 4d6f 6e6f 6d69 616c 7320 4d0a 2020 2020 Monomials M. │ │ │ │ -0002f520: 2020 2020 4c20 3d20 7374 616e 6461 7264 L = standard │ │ │ │ -0002f530: 4d6f 6e6f 6d69 616c 7328 642c 204d 290a Monomials(d, M). │ │ │ │ -0002f540: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -0002f550: 2020 2a20 4d2c 2061 6e20 2a6e 6f74 6520 * M, an *note │ │ │ │ -0002f560: 6964 6561 6c3a 2028 4d61 6361 756c 6179 ideal: (Macaulay │ │ │ │ -0002f570: 3244 6f63 2949 6465 616c 2c2c 204d 2073 2Doc)Ideal,, M s │ │ │ │ -0002f580: 686f 756c 6420 6265 2061 206d 6f6e 6f6d hould be a monom │ │ │ │ -0002f590: 6961 6c20 6964 6561 6c0a 2020 2020 2020 ial ideal. │ │ │ │ -0002f5a0: 2a20 642c 2061 202a 6e6f 7465 206c 6973 * d, a *note lis │ │ │ │ -0002f5b0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -0002f5c0: 294c 6973 742c 2c20 6120 6465 6772 6565 )List,, a degree │ │ │ │ -0002f5d0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -0002f5e0: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465 * L, a *note │ │ │ │ -0002f5f0: 206c 6973 743a 2028 4d61 6361 756c 6179 list: (Macaulay │ │ │ │ -0002f600: 3244 6f63 294c 6973 742c 2c20 4c20 6973 2Doc)List,, L is │ │ │ │ -0002f610: 2061 206c 6973 7420 6f66 206c 6973 7473 a list of lists │ │ │ │ -0002f620: 206f 6620 7374 616e 6461 7264 0a20 2020 of standard. │ │ │ │ -0002f630: 2020 2020 206d 6f6e 6f6d 6961 6c73 2066 monomials f │ │ │ │ -0002f640: 6f72 2074 6865 2069 6465 616c 2024 4d24 or the ideal $M$ │ │ │ │ -0002f650: 2c20 6f6e 6520 666f 7220 6561 6368 2067 , one for each g │ │ │ │ -0002f660: 656e 6572 6174 6f72 206f 6620 244d 240a enerator of $M$. │ │ │ │ -0002f670: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ -0002f680: 3d3d 3d3d 3d3d 3d3d 0a0a 4120 6d6f 6e6f ========..A mono │ │ │ │ -0002f690: 6d69 616c 2024 6d24 2069 7320 7374 616e mial $m$ is stan │ │ │ │ -0002f6a0: 6461 7264 2077 6974 6820 7265 7370 6563 dard with respec │ │ │ │ -0002f6b0: 7420 746f 2061 206d 6f6e 6f6d 6961 6c20 t to a monomial │ │ │ │ -0002f6c0: 6964 6561 6c20 244d 2420 616e 6420 6120 ideal $M$ and a │ │ │ │ -0002f6d0: 6765 6e65 7261 746f 720a 2467 2420 6f66 generator.$g$ of │ │ │ │ -0002f6e0: 2024 4d24 2069 6620 246d 2420 6973 206f $M$ if $m$ is o │ │ │ │ -0002f6f0: 6620 7468 6520 7361 6d65 2064 6567 7265 f the same degre │ │ │ │ -0002f700: 6520 6173 2024 6724 2062 7574 2069 7320 e as $g$ but is │ │ │ │ -0002f710: 6e6f 7420 616e 2065 6c65 6d65 6e74 206f not an element o │ │ │ │ -0002f720: 6620 244d 242e 0a0a 496e 7075 7474 696e f $M$...Inputtin │ │ │ │ -0002f730: 6720 616e 2069 6465 616c 2024 4d24 2072 g an ideal $M$ r │ │ │ │ -0002f740: 6574 7572 6e73 2074 6865 2073 7461 6e64 eturns the stand │ │ │ │ -0002f750: 6172 6420 6d6f 6e6f 6d69 616c 7320 6f66 ard monomials of │ │ │ │ -0002f760: 2065 6163 6820 6f66 2074 6865 2067 6976 each of the giv │ │ │ │ -0002f770: 656e 0a67 656e 6572 6174 6f72 7320 6f66 en.generators of │ │ │ │ -0002f780: 2074 6865 2069 6465 616c 2e0a 0a2b 2d2d the ideal...+-- │ │ │ │ -0002f790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0002f7e0: 203a 2052 203d 205a 5a2f 3332 3030 335b : R = ZZ/32003[ │ │ │ │ -0002f7f0: 612e 2e64 5d3b 2020 2020 2020 2020 2020 a..d]; │ │ │ │ -0002f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f820: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002f830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f0e0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f100: 2d2d 2d2d 2d2d 2d2d 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 2d2b 0a7c -------------+.| │ │ │ │ +0002f140: 6933 203a 204c 3120 3d20 736d 616c 6c65 i3 : L1 = smalle │ │ │ │ +0002f150: 724d 6f6e 6f6d 6961 6c73 204d 2020 2020 rMonomials M │ │ │ │ +0002f160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f180: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002f190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f1d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f1f0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0002f200: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002f210: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002f220: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002f230: 6f33 203d 207b 7b61 2a62 2c20 612a 632c o3 = {{a*b, a*c, │ │ │ │ +0002f240: 2062 2a63 2c20 6320 2c20 612a 642c 2062 b*c, c , a*d, b │ │ │ │ +0002f250: 2a64 2c20 632a 642c 2064 207d 2c20 7b61 *d, c*d, d }, {a │ │ │ │ +0002f260: 2a63 2c20 622a 632c 2063 202c 2061 2a64 *c, b*c, c , a*d │ │ │ │ +0002f270: 2c20 622a 642c 2063 2a64 2c20 207c 0a7c , b*d, c*d, |.| │ │ │ │ +0002f280: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ +0002f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +0002f2d0: 2020 2020 2020 3220 2020 2020 2020 3220 2 2 │ │ │ │ +0002f2e0: 2020 2020 3220 2020 3320 2020 2020 2020 2 3 │ │ │ │ +0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f300: 2032 2020 2020 2020 3220 2020 2020 3220 2 2 2 │ │ │ │ +0002f310: 2020 2020 3220 2020 3320 2020 207c 0a7c 2 3 |.| │ │ │ │ +0002f320: 2020 2020 2064 207d 2c20 7b61 2a63 202c d }, {a*c , │ │ │ │ +0002f330: 2062 2a63 202c 2063 202c 2061 2a62 2a64 b*c , c , a*b*d │ │ │ │ +0002f340: 2c20 612a 632a 642c 2062 2a63 2a64 2c20 , a*c*d, b*c*d, │ │ │ │ +0002f350: 6320 642c 2061 2a64 202c 2062 2a64 202c c d, a*d , b*d , │ │ │ │ +0002f360: 2063 2a64 202c 2064 207d 7d20 207c 0a7c c*d , d }} |.| │ │ │ │ +0002f370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f3b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002f3c0: 6f33 203a 204c 6973 7420 2020 2020 2020 o3 : List │ │ │ │ +0002f3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 207c 0a2b |.+ │ │ │ │ +0002f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0002f460: 6934 203a 2073 6d61 6c6c 6572 4d6f 6e6f i4 : smallerMono │ │ │ │ +0002f470: 6d69 616c 7328 4d2c 2062 5e32 2920 2020 mials(M, b^2) │ │ │ │ +0002f480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f4a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002f4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f4f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f510: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002f520: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0002f530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002f550: 6f34 203d 207b 612a 632c 2062 2a63 2c20 o4 = {a*c, b*c, │ │ │ │ +0002f560: 6320 2c20 612a 642c 2062 2a64 2c20 632a c , a*d, b*d, c* │ │ │ │ +0002f570: 642c 2064 207d 2020 2020 2020 2020 2020 d, d } │ │ │ │ +0002f580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f590: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002f5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f5e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002f5f0: 6f34 203a 204c 6973 7420 2020 2020 2020 o4 : List │ │ │ │ +0002f600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f630: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002f640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0002f690: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +0002f6a0: 3d0a 0a20 202a 202a 6e6f 7465 2074 6169 =.. * *note tai │ │ │ │ +0002f6b0: 6c4d 6f6e 6f6d 6961 6c73 3a20 7461 696c lMonomials: tail │ │ │ │ +0002f6c0: 4d6f 6e6f 6d69 616c 732c 202d 2d20 6669 Monomials, -- fi │ │ │ │ +0002f6d0: 6e64 2074 6169 6c20 6d6f 6e6f 6d69 616c nd tail monomial │ │ │ │ +0002f6e0: 730a 2020 2a20 2a6e 6f74 6520 7374 616e s. * *note stan │ │ │ │ +0002f6f0: 6461 7264 4d6f 6e6f 6d69 616c 733a 2073 dardMonomials: s │ │ │ │ +0002f700: 7461 6e64 6172 644d 6f6e 6f6d 6961 6c73 tandardMonomials │ │ │ │ +0002f710: 2c20 2d2d 2063 6f6d 7075 7465 7320 7374 , -- computes st │ │ │ │ +0002f720: 616e 6461 7264 206d 6f6e 6f6d 6961 6c73 andard monomials │ │ │ │ +0002f730: 0a0a 5761 7973 2074 6f20 7573 6520 736d ..Ways to use sm │ │ │ │ +0002f740: 616c 6c65 724d 6f6e 6f6d 6961 6c73 3a0a allerMonomials:. │ │ │ │ +0002f750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002f760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +0002f770: 202a 2022 736d 616c 6c65 724d 6f6e 6f6d * "smallerMonom │ │ │ │ +0002f780: 6961 6c73 2849 6465 616c 2922 0a20 202a ials(Ideal)". * │ │ │ │ +0002f790: 2022 736d 616c 6c65 724d 6f6e 6f6d 6961 "smallerMonomia │ │ │ │ +0002f7a0: 6c73 2849 6465 616c 2c52 696e 6745 6c65 ls(Ideal,RingEle │ │ │ │ +0002f7b0: 6d65 6e74 2922 0a0a 466f 7220 7468 6520 ment)"..For the │ │ │ │ +0002f7c0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +0002f7d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +0002f7e0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +0002f7f0: 736d 616c 6c65 724d 6f6e 6f6d 6961 6c73 smallerMonomials │ │ │ │ +0002f800: 3a20 736d 616c 6c65 724d 6f6e 6f6d 6961 : smallerMonomia │ │ │ │ +0002f810: 6c73 2c20 6973 2061 202a 6e6f 7465 206d ls, is a *note m │ │ │ │ +0002f820: 6574 686f 640a 6675 6e63 7469 6f6e 3a20 ethod.function: │ │ │ │ +0002f830: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +0002f840: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ 0002f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -0002f880: 203a 204d 203d 2069 6465 616c 2028 615e : M = ideal (a^ │ │ │ │ -0002f890: 322c 2061 2a62 2c20 625e 332c 2061 2a63 2, a*b, b^3, a*c │ │ │ │ -0002f8a0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -0002f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f910: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0002f920: 203a 2049 6465 616c 206f 6620 5220 2020 : Ideal of R │ │ │ │ -0002f930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f960: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002f970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -0002f9c0: 203a 204c 3120 3d20 7374 616e 6461 7264 : L1 = standard │ │ │ │ -0002f9d0: 4d6f 6e6f 6d69 616c 7320 4d20 2020 2020 Monomials M │ │ │ │ -0002f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fa00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002fa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 207c 0a7c 2020 |.| │ │ │ │ -0002fa60: 2020 2020 2020 3220 2020 2020 2020 2032 2 2 │ │ │ │ -0002fa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fa80: 2020 3220 2020 2020 3220 2020 2020 2020 2 2 │ │ │ │ -0002fa90: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002faa0: 2020 2020 3220 2020 2020 207c 0a7c 6f33 2 |.|o3 │ │ │ │ -0002fab0: 203d 207b 7b62 202c 2062 2a63 2c20 6320 = {{b , b*c, c │ │ │ │ -0002fac0: 2c20 612a 642c 2062 2a64 2c20 632a 642c , a*d, b*d, c*d, │ │ │ │ -0002fad0: 2064 207d 2c20 7b62 202c 2062 2a63 2c20 d }, {b , b*c, │ │ │ │ -0002fae0: 6320 2c20 612a 642c 2062 2a64 2c20 632a c , a*d, b*d, c* │ │ │ │ -0002faf0: 642c 2064 207d 2c20 2020 207c 0a7c 2020 d, d }, |.| │ │ │ │ -0002fb00: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ -0002fb10: 2d2d 2d2d 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 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0002fb50: 2020 2020 2032 2020 2020 2020 3220 2020 2 2 │ │ │ │ -0002fb60: 3320 2020 3220 2020 2020 2020 2020 2020 3 2 │ │ │ │ -0002fb70: 3220 2020 2020 2032 2020 2020 2032 2020 2 2 2 │ │ │ │ -0002fb80: 2020 2032 2020 2033 2020 2020 2032 2020 2 3 2 │ │ │ │ -0002fb90: 2020 2020 2020 3220 2020 207c 0a7c 2020 2 |.| │ │ │ │ -0002fba0: 2020 207b 6220 632c 2062 2a63 202c 2063 {b c, b*c , c │ │ │ │ -0002fbb0: 202c 2062 2064 2c20 622a 632a 642c 2063 , b d, b*c*d, c │ │ │ │ -0002fbc0: 2064 2c20 612a 6420 2c20 622a 6420 2c20 d, a*d , b*d , │ │ │ │ -0002fbd0: 632a 6420 2c20 6420 7d2c 207b 6220 2c20 c*d , d }, {b , │ │ │ │ -0002fbe0: 622a 632c 2063 202c 2020 207c 0a7c 2020 b*c, c , |.| │ │ │ │ -0002fbf0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ -0002fc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0002fc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc50: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fc80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002fc90: 2020 2061 2a64 2c20 622a 642c 2063 2a64 a*d, b*d, c*d │ │ │ │ -0002fca0: 2c20 6420 7d7d 2020 2020 2020 2020 2020 , d }} │ │ │ │ -0002fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fcd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002fce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +0002f8a0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +0002f8b0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +0002f8c0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +0002f8d0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +0002f8e0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +0002f8f0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +0002f900: 636b 6167 6573 2f0a 4772 6f65 626e 6572 ckages/.Groebner │ │ │ │ +0002f910: 5374 7261 7461 2e6d 323a 3534 323a 302e Strata.m2:542:0. │ │ │ │ +0002f920: 0a1f 0a46 696c 653a 2047 726f 6562 6e65 ...File: Groebne │ │ │ │ +0002f930: 7253 7472 6174 612e 696e 666f 2c20 4e6f rStrata.info, No │ │ │ │ +0002f940: 6465 3a20 7374 616e 6461 7264 4d6f 6e6f de: standardMono │ │ │ │ +0002f950: 6d69 616c 732c 204e 6578 743a 2074 6169 mials, Next: tai │ │ │ │ +0002f960: 6c4d 6f6e 6f6d 6961 6c73 2c20 5072 6576 lMonomials, Prev │ │ │ │ +0002f970: 3a20 736d 616c 6c65 724d 6f6e 6f6d 6961 : smallerMonomia │ │ │ │ +0002f980: 6c73 2c20 5570 3a20 546f 700a 0a73 7461 ls, Up: Top..sta │ │ │ │ +0002f990: 6e64 6172 644d 6f6e 6f6d 6961 6c73 202d ndardMonomials - │ │ │ │ +0002f9a0: 2d20 636f 6d70 7574 6573 2073 7461 6e64 - computes stand │ │ │ │ +0002f9b0: 6172 6420 6d6f 6e6f 6d69 616c 730a 2a2a ard monomials.** │ │ │ │ +0002f9c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002f9d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002f9e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0002f9f0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0002fa00: 2020 2020 4c20 3d20 7374 616e 6461 7264 L = standard │ │ │ │ +0002fa10: 4d6f 6e6f 6d69 616c 7320 4d0a 2020 2020 Monomials M. │ │ │ │ +0002fa20: 2020 2020 4c20 3d20 7374 616e 6461 7264 L = standard │ │ │ │ +0002fa30: 4d6f 6e6f 6d69 616c 7328 642c 204d 290a Monomials(d, M). │ │ │ │ +0002fa40: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0002fa50: 2020 2a20 4d2c 2061 6e20 2a6e 6f74 6520 * M, an *note │ │ │ │ +0002fa60: 6964 6561 6c3a 2028 4d61 6361 756c 6179 ideal: (Macaulay │ │ │ │ +0002fa70: 3244 6f63 2949 6465 616c 2c2c 204d 2073 2Doc)Ideal,, M s │ │ │ │ +0002fa80: 686f 756c 6420 6265 2061 206d 6f6e 6f6d hould be a monom │ │ │ │ +0002fa90: 6961 6c20 6964 6561 6c0a 2020 2020 2020 ial ideal. │ │ │ │ +0002faa0: 2a20 642c 2061 202a 6e6f 7465 206c 6973 * d, a *note lis │ │ │ │ +0002fab0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +0002fac0: 294c 6973 742c 2c20 6120 6465 6772 6565 )List,, a degree │ │ │ │ +0002fad0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +0002fae0: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465 * L, a *note │ │ │ │ +0002faf0: 206c 6973 743a 2028 4d61 6361 756c 6179 list: (Macaulay │ │ │ │ +0002fb00: 3244 6f63 294c 6973 742c 2c20 4c20 6973 2Doc)List,, L is │ │ │ │ +0002fb10: 2061 206c 6973 7420 6f66 206c 6973 7473 a list of lists │ │ │ │ +0002fb20: 206f 6620 7374 616e 6461 7264 0a20 2020 of standard. │ │ │ │ +0002fb30: 2020 2020 206d 6f6e 6f6d 6961 6c73 2066 monomials f │ │ │ │ +0002fb40: 6f72 2074 6865 2069 6465 616c 2024 4d24 or the ideal $M$ │ │ │ │ +0002fb50: 2c20 6f6e 6520 666f 7220 6561 6368 2067 , one for each g │ │ │ │ +0002fb60: 656e 6572 6174 6f72 206f 6620 244d 240a enerator of $M$. │ │ │ │ +0002fb70: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ +0002fb80: 3d3d 3d3d 3d3d 3d3d 0a0a 4120 6d6f 6e6f ========..A mono │ │ │ │ +0002fb90: 6d69 616c 2024 6d24 2069 7320 7374 616e mial $m$ is stan │ │ │ │ +0002fba0: 6461 7264 2077 6974 6820 7265 7370 6563 dard with respec │ │ │ │ +0002fbb0: 7420 746f 2061 206d 6f6e 6f6d 6961 6c20 t to a monomial │ │ │ │ +0002fbc0: 6964 6561 6c20 244d 2420 616e 6420 6120 ideal $M$ and a │ │ │ │ +0002fbd0: 6765 6e65 7261 746f 720a 2467 2420 6f66 generator.$g$ of │ │ │ │ +0002fbe0: 2024 4d24 2069 6620 246d 2420 6973 206f $M$ if $m$ is o │ │ │ │ +0002fbf0: 6620 7468 6520 7361 6d65 2064 6567 7265 f the same degre │ │ │ │ +0002fc00: 6520 6173 2024 6724 2062 7574 2069 7320 e as $g$ but is │ │ │ │ +0002fc10: 6e6f 7420 616e 2065 6c65 6d65 6e74 206f not an element o │ │ │ │ +0002fc20: 6620 244d 242e 0a0a 496e 7075 7474 696e f $M$...Inputtin │ │ │ │ +0002fc30: 6720 616e 2069 6465 616c 2024 4d24 2072 g an ideal $M$ r │ │ │ │ +0002fc40: 6574 7572 6e73 2074 6865 2073 7461 6e64 eturns the stand │ │ │ │ +0002fc50: 6172 6420 6d6f 6e6f 6d69 616c 7320 6f66 ard monomials of │ │ │ │ +0002fc60: 2065 6163 6820 6f66 2074 6865 2067 6976 each of the giv │ │ │ │ +0002fc70: 656e 0a67 656e 6572 6174 6f72 7320 6f66 en.generators of │ │ │ │ +0002fc80: 2074 6865 2069 6465 616c 2e0a 0a2b 2d2d the ideal...+-- │ │ │ │ +0002fc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0002fce0: 203a 2052 203d 205a 5a2f 3332 3030 335b : R = ZZ/32003[ │ │ │ │ +0002fcf0: 612e 2e64 5d3b 2020 2020 2020 2020 2020 a..d]; │ │ │ │ 0002fd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fd20: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -0002fd30: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ -0002fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fd70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002fd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fd90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -0002fdd0: 203a 2073 7461 6e64 6172 644d 6f6e 6f6d : standardMonom │ │ │ │ -0002fde0: 6961 6c73 287b 337d 2c20 4d29 2020 2020 ials({3}, M) │ │ │ │ +0002fd20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002fd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0002fd80: 203a 204d 203d 2069 6465 616c 2028 615e : M = ideal (a^ │ │ │ │ +0002fd90: 322c 2061 2a62 2c20 625e 332c 2061 2a63 2, a*b, b^3, a*c │ │ │ │ +0002fda0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ +0002fdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fdc0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fe00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fe10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002fe20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fe10: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0002fe20: 203a 2049 6465 616c 206f 6620 5220 2020 : Ideal of R │ │ │ │ 0002fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002fe50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002fe60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002fe70: 2020 2020 2032 2020 2020 2020 3220 2020 2 2 │ │ │ │ -0002fe80: 3320 2020 3220 2020 2020 2020 2020 2020 3 2 │ │ │ │ -0002fe90: 3220 2020 2020 2032 2020 2020 2032 2020 2 2 2 │ │ │ │ -0002fea0: 2020 2032 2020 2033 2020 2020 2020 2020 2 3 │ │ │ │ -0002feb0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -0002fec0: 203d 207b 6220 632c 2062 2a63 202c 2063 = {b c, b*c , c │ │ │ │ -0002fed0: 202c 2062 2064 2c20 622a 632a 642c 2063 , b d, b*c*d, c │ │ │ │ -0002fee0: 2064 2c20 612a 6420 2c20 622a 6420 2c20 d, a*d , b*d , │ │ │ │ -0002fef0: 632a 6420 2c20 6420 7d20 2020 2020 2020 c*d , d } │ │ │ │ +0002fe60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002fe70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002feb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ +0002fec0: 203a 204c 3120 3d20 7374 616e 6461 7264 : L1 = standard │ │ │ │ +0002fed0: 4d6f 6e6f 6d69 616c 7320 4d20 2020 2020 Monomials M │ │ │ │ +0002fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002fef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ff00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0002ff10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 207c 0a7c 6f34 |.|o4 │ │ │ │ -0002ff60: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +0002ff50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002ff60: 2020 2020 2020 3220 2020 2020 2020 2032 2 2 │ │ │ │ 0002ff70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ff80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ff90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ffa0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 2d2b 0a0a 496e -----------+..In │ │ │ │ -00030000: 7075 7474 696e 6720 616e 2069 6e74 6567 putting an integ │ │ │ │ -00030010: 6572 2024 6424 2028 6f72 2064 6567 7265 er $d$ (or degre │ │ │ │ -00030020: 6520 2464 2429 2061 6e64 2061 6e20 6964 e $d$) and an id │ │ │ │ -00030030: 6561 6c20 6769 7665 7320 7468 6520 7374 eal gives the st │ │ │ │ -00030040: 616e 6461 7264 0a6d 6f6e 6f6d 6961 6c73 andard.monomials │ │ │ │ -00030050: 2066 6f72 2074 6865 2073 7065 6369 6669 for the specifi │ │ │ │ -00030060: 6564 2069 6465 616c 2069 6e20 6465 6772 ed ideal in degr │ │ │ │ -00030070: 6565 2024 6424 2e0a 0a2b 2d2d 2d2d 2d2d ee $d$...+------ │ │ │ │ -00030080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000300a0: 0a7c 6935 203a 2073 7461 6e64 6172 644d .|i5 : standardM │ │ │ │ -000300b0: 6f6e 6f6d 6961 6c73 2832 2c20 4d29 2020 onomials(2, M) │ │ │ │ -000300c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000300d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000300e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000300f0: 0a7c 2020 2020 2020 2032 2020 2020 2020 .| 2 │ │ │ │ -00030100: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00030110: 2020 2020 2032 207c 0a7c 6f35 203d 207b 2 |.|o5 = { │ │ │ │ -00030120: 6220 2c20 622a 632c 2063 202c 2061 2a64 b , b*c, c , a*d │ │ │ │ -00030130: 2c20 622a 642c 2063 2a64 2c20 6420 7d7c , b*d, c*d, d }| │ │ │ │ -00030140: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00030150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030160: 2020 2020 2020 207c 0a7c 6f35 203a 204c |.|o5 : L │ │ │ │ -00030170: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ -00030180: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00030190: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -000301a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000301b0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -000301c0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -000301d0: 202a 6e6f 7465 2074 6169 6c4d 6f6e 6f6d *note tailMonom │ │ │ │ -000301e0: 6961 6c73 3a20 7461 696c 4d6f 6e6f 6d69 ials: tailMonomi │ │ │ │ -000301f0: 616c 732c 202d 2d20 6669 6e64 2074 6169 als, -- find tai │ │ │ │ -00030200: 6c20 6d6f 6e6f 6d69 616c 730a 2020 2a20 l monomials. * │ │ │ │ -00030210: 2a6e 6f74 6520 736d 616c 6c65 724d 6f6e *note smallerMon │ │ │ │ -00030220: 6f6d 6961 6c73 3a20 736d 616c 6c65 724d omials: smallerM │ │ │ │ -00030230: 6f6e 6f6d 6961 6c73 2c20 2d2d 2072 6574 onomials, -- ret │ │ │ │ -00030240: 7572 6e73 2074 6865 2073 7461 6e64 6172 urns the standar │ │ │ │ -00030250: 6420 6d6f 6e6f 6d69 616c 730a 2020 2020 d monomials. │ │ │ │ -00030260: 736d 616c 6c65 7220 6275 7420 6f66 2074 smaller but of t │ │ │ │ -00030270: 6865 2073 616d 6520 6465 6772 6565 2061 he same degree a │ │ │ │ -00030280: 7320 6769 7665 6e20 6d6f 6e6f 6d69 616c s given monomial │ │ │ │ -00030290: 2873 290a 0a57 6179 7320 746f 2075 7365 (s)..Ways to use │ │ │ │ -000302a0: 2073 7461 6e64 6172 644d 6f6e 6f6d 6961 standardMonomia │ │ │ │ -000302b0: 6c73 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ls:.============ │ │ │ │ -000302c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000302d0: 3d3d 0a0a 2020 2a20 2273 7461 6e64 6172 ==.. * "standar │ │ │ │ -000302e0: 644d 6f6e 6f6d 6961 6c73 2849 6465 616c dMonomials(Ideal │ │ │ │ -000302f0: 2922 0a20 202a 2022 7374 616e 6461 7264 )". * "standard │ │ │ │ -00030300: 4d6f 6e6f 6d69 616c 7328 4c69 7374 2c49 Monomials(List,I │ │ │ │ -00030310: 6465 616c 2922 0a20 202a 2022 7374 616e deal)". * "stan │ │ │ │ -00030320: 6461 7264 4d6f 6e6f 6d69 616c 7328 5a5a dardMonomials(ZZ │ │ │ │ -00030330: 2c49 6465 616c 2922 0a0a 466f 7220 7468 ,Ideal)"..For th │ │ │ │ -00030340: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00030350: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00030360: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00030370: 6520 7374 616e 6461 7264 4d6f 6e6f 6d69 e standardMonomi │ │ │ │ -00030380: 616c 733a 2073 7461 6e64 6172 644d 6f6e als: standardMon │ │ │ │ -00030390: 6f6d 6961 6c73 2c20 6973 2061 202a 6e6f omials, is a *no │ │ │ │ -000303a0: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ -000303b0: 6f6e 3a20 284d 6163 6175 6c61 7932 446f on: (Macaulay2Do │ │ │ │ -000303c0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -000303d0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ -000303e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -00030430: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -00030440: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -00030450: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -00030460: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -00030470: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -00030480: 322f 7061 636b 6167 6573 2f0a 4772 6f65 2/packages/.Groe │ │ │ │ -00030490: 626e 6572 5374 7261 7461 2e6d 323a 3530 bnerStrata.m2:50 │ │ │ │ -000304a0: 363a 302e 0a1f 0a46 696c 653a 2047 726f 6:0....File: Gro │ │ │ │ -000304b0: 6562 6e65 7253 7472 6174 612e 696e 666f ebnerStrata.info │ │ │ │ -000304c0: 2c20 4e6f 6465 3a20 7461 696c 4d6f 6e6f , Node: tailMono │ │ │ │ -000304d0: 6d69 616c 732c 2050 7265 763a 2073 7461 mials, Prev: sta │ │ │ │ -000304e0: 6e64 6172 644d 6f6e 6f6d 6961 6c73 2c20 ndardMonomials, │ │ │ │ -000304f0: 5570 3a20 546f 700a 0a74 6169 6c4d 6f6e Up: Top..tailMon │ │ │ │ -00030500: 6f6d 6961 6c73 202d 2d20 6669 6e64 2074 omials -- find t │ │ │ │ -00030510: 6169 6c20 6d6f 6e6f 6d69 616c 730a 2a2a ail monomials.** │ │ │ │ -00030520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00030530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00030540: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ -00030550: 2020 2020 2020 2020 4c20 3d20 7461 696c L = tail │ │ │ │ -00030560: 4d6f 6e6f 6d69 616c 7320 4d0a 2020 2020 Monomials M. │ │ │ │ -00030570: 2020 2020 4c20 3d20 7461 696c 4d6f 6e6f L = tailMono │ │ │ │ -00030580: 6d69 616c 7328 4d2c 206d 290a 2020 2a20 mials(M, m). * │ │ │ │ -00030590: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -000305a0: 4d2c 2061 6e20 2a6e 6f74 6520 6964 6561 M, an *note idea │ │ │ │ -000305b0: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ -000305c0: 2949 6465 616c 2c2c 2024 4d24 2073 686f )Ideal,, $M$ sho │ │ │ │ -000305d0: 756c 6420 6265 2061 206d 6f6e 6f6d 6961 uld be a monomia │ │ │ │ -000305e0: 6c20 6964 6561 6c0a 2020 2020 2020 2020 l ideal. │ │ │ │ -000305f0: 2861 6e20 6964 6561 6c20 6765 6e65 7261 (an ideal genera │ │ │ │ -00030600: 7465 6420 6279 206d 6f6e 6f6d 6961 6c73 ted by monomials │ │ │ │ -00030610: 290a 2020 2020 2020 2a20 6d2c 2061 202a ). * m, a * │ │ │ │ -00030620: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ -00030630: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -00030640: 2952 696e 6745 6c65 6d65 6e74 2c2c 206f )RingElement,, o │ │ │ │ -00030650: 7074 696f 6e61 6c2c 206f 6e6c 790a 2020 ptional, only. │ │ │ │ -00030660: 2020 2020 2020 7265 7475 726e 2061 2073 return a s │ │ │ │ -00030670: 696e 676c 6520 6c69 7374 206f 6620 7468 ingle list of th │ │ │ │ -00030680: 6520 7461 696c 206d 6f6e 6f6d 6961 6c73 e tail monomials │ │ │ │ -00030690: 2066 6f72 2074 6869 7320 6d6f 6e6f 6d69 for this monomi │ │ │ │ -000306a0: 616c 0a20 202a 202a 6e6f 7465 204f 7074 al. * *note Opt │ │ │ │ -000306b0: 696f 6e61 6c20 696e 7075 7473 3a20 284d ional inputs: (M │ │ │ │ -000306c0: 6163 6175 6c61 7932 446f 6329 7573 696e acaulay2Doc)usin │ │ │ │ -000306d0: 6720 6675 6e63 7469 6f6e 7320 7769 7468 g functions with │ │ │ │ -000306e0: 206f 7074 696f 6e61 6c20 696e 7075 7473 optional inputs │ │ │ │ -000306f0: 2c3a 0a20 2020 2020 202a 2041 6c6c 5374 ,:. * AllSt │ │ │ │ -00030700: 616e 6461 7264 203d 3e20 6120 2a6e 6f74 andard => a *not │ │ │ │ -00030710: 6520 426f 6f6c 6561 6e20 7661 6c75 653a e Boolean value: │ │ │ │ -00030720: 2028 4d61 6361 756c 6179 3244 6f63 2942 (Macaulay2Doc)B │ │ │ │ -00030730: 6f6f 6c65 616e 2c2c 2064 6566 6175 6c74 oolean,, default │ │ │ │ -00030740: 0a20 2020 2020 2020 2076 616c 7565 2066 . value f │ │ │ │ -00030750: 616c 7365 2c20 7768 6963 6820 6d6f 6e6f alse, which mono │ │ │ │ -00030760: 6d69 616c 7320 7368 6f75 6c64 2062 6520 mials should be │ │ │ │ -00030770: 636f 6e73 6964 6572 6564 2074 6169 6c20 considered tail │ │ │ │ -00030780: 6d6f 6e6f 6d69 616c 7320 6f66 2061 0a20 monomials of a. │ │ │ │ -00030790: 2020 2020 2020 206d 6f6e 6f6d 6961 6c20 monomial │ │ │ │ -000307a0: 246d 243a 2065 6974 6865 7220 616c 6c20 $m$: either all │ │ │ │ -000307b0: 7374 616e 6461 7264 206d 6f6e 6f6d 6961 standard monomia │ │ │ │ -000307c0: 6c73 206f 6620 6120 6769 7665 6e20 6465 ls of a given de │ │ │ │ -000307d0: 6772 6565 2c20 6f72 2061 6c6c 0a20 2020 gree, or all. │ │ │ │ -000307e0: 2020 2020 206d 6f6e 6f6d 6961 6c73 2073 monomials s │ │ │ │ -000307f0: 6d61 6c6c 6572 2074 6861 6e20 246d 2420 maller than $m$ │ │ │ │ -00030800: 696e 2074 6865 2067 6976 656e 2074 6572 in the given ter │ │ │ │ -00030810: 6d20 6f72 6465 7220 2862 7574 2073 7469 m order (but sti │ │ │ │ -00030820: 6c6c 206f 6620 7468 650a 2020 2020 2020 ll of the. │ │ │ │ -00030830: 2020 7361 6d65 2064 6567 7265 6529 0a20 same degree). │ │ │ │ -00030840: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -00030850: 2020 2a20 4c2c 2061 202a 6e6f 7465 206c * L, a *note l │ │ │ │ -00030860: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ -00030870: 6f63 294c 6973 742c 2c20 6120 6c69 7374 oc)List,, a list │ │ │ │ -00030880: 206f 6620 6c69 7374 733a 2066 6f72 2065 of lists: for e │ │ │ │ -00030890: 6163 680a 2020 2020 2020 2020 6765 6e65 ach. gene │ │ │ │ -000308a0: 7261 746f 7220 246d 2420 6f66 2024 4d24 rator $m$ of $M$ │ │ │ │ -000308b0: 2c20 7468 6520 6c69 7374 206f 6620 616c , the list of al │ │ │ │ -000308c0: 6c20 7461 696c 206d 6f6e 6f6d 6961 6c73 l tail monomials │ │ │ │ -000308d0: 2049 6620 696e 7374 6561 6420 246d 2420 If instead $m$ │ │ │ │ -000308e0: 6973 0a20 2020 2020 2020 2067 6976 656e is. given │ │ │ │ -000308f0: 2c20 7468 6520 6c69 7374 206f 6620 7468 , the list of th │ │ │ │ -00030900: 6520 7461 696c 206d 6f6e 6f6d 6961 6c73 e tail monomials │ │ │ │ -00030910: 206f 6620 246d 2420 6973 2072 6574 7572 of $m$ is retur │ │ │ │ -00030920: 6e65 640a 0a44 6573 6372 6970 7469 6f6e ned..Description │ │ │ │ -00030930: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 496e .===========..In │ │ │ │ -00030940: 7075 7474 696e 6720 616e 2069 6465 616c putting an ideal │ │ │ │ -00030950: 2024 4d24 2067 656e 6572 6174 6564 2062 $M$ generated b │ │ │ │ -00030960: 7920 6d6f 6e6f 6d69 616c 7320 7265 7475 y monomials retu │ │ │ │ -00030970: 726e 7320 6120 6c69 7374 206f 6620 6c69 rns a list of li │ │ │ │ -00030980: 7374 7320 6f66 2074 6169 6c0a 6d6f 6e6f sts of tail.mono │ │ │ │ -00030990: 6d69 616c 7320 666f 7220 6561 6368 2067 mials for each g │ │ │ │ -000309a0: 656e 6572 6174 6f72 206f 6620 244d 2420 enerator of $M$ │ │ │ │ -000309b0: 2869 6e20 7468 6520 7361 6d65 206f 7264 (in the same ord │ │ │ │ -000309c0: 6572 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d er)...+--------- │ │ │ │ -000309d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 2b0a 7c69 3120 3a20 5220 3d20 ----+.|i1 : R = │ │ │ │ -00030a20: 5a5a 2f33 3230 3033 5b61 2e2e 645d 3b20 ZZ/32003[a..d]; │ │ │ │ -00030a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00030a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030ab0: 2d2d 2d2d 2b0a 7c69 3220 3a20 4d20 3d20 ----+.|i2 : M = │ │ │ │ -00030ac0: 6964 6561 6c20 2861 5e32 2c20 625e 322c ideal (a^2, b^2, │ │ │ │ -00030ad0: 2061 2a62 2a63 293b 2020 2020 2020 2020 a*b*c); │ │ │ │ -00030ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00030b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b50: 2020 2020 7c0a 7c6f 3220 3a20 4964 6561 |.|o2 : Idea │ │ │ │ -00030b60: 6c20 6f66 2052 2020 2020 2020 2020 2020 l of R │ │ │ │ -00030b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ba0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00030bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030bf0: 2d2d 2d2d 2b0a 7c69 3320 3a20 7461 696c ----+.|i3 : tail │ │ │ │ -00030c00: 4d6f 6e6f 6d69 616c 7320 4d20 2020 2020 Monomials M │ │ │ │ -00030c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00030c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00030ca0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00030cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030cc0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00030cd0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00030ce0: 2020 2020 7c0a 7c6f 3320 3d20 7b7b 612a |.|o3 = {{a* │ │ │ │ -00030cf0: 622c 2061 2a63 2c20 622a 632c 2063 202c b, a*c, b*c, c , │ │ │ │ -00030d00: 2061 2a64 2c20 622a 642c 2063 2a64 2c20 a*d, b*d, c*d, │ │ │ │ -00030d10: 6420 7d2c 207b 612a 632c 2062 2a63 2c20 d }, {a*c, b*c, │ │ │ │ -00030d20: 6320 2c20 612a 642c 2062 2a64 2c20 632a c , a*d, b*d, c* │ │ │ │ -00030d30: 642c 2020 7c0a 7c20 2020 2020 2d2d 2d2d d, |.| ---- │ │ │ │ -00030d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030d80: 2d2d 2d2d 7c0a 7c20 2020 2020 2032 2020 ----|.| 2 │ │ │ │ -00030d90: 2020 2020 2032 2020 2020 2032 2020 2033 2 2 3 │ │ │ │ -00030da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030db0: 2020 2020 2020 2020 3220 2020 2020 2032 2 2 │ │ │ │ -00030dc0: 2020 2020 2032 2020 2020 2032 2020 2033 2 2 3 │ │ │ │ -00030dd0: 2020 2020 7c0a 7c20 2020 2020 6420 7d2c |.| d }, │ │ │ │ -00030de0: 207b 612a 6320 2c20 622a 6320 2c20 6320 {a*c , b*c , c │ │ │ │ -00030df0: 2c20 612a 622a 642c 2061 2a63 2a64 2c20 , a*b*d, a*c*d, │ │ │ │ -00030e00: 622a 632a 642c 2063 2064 2c20 612a 6420 b*c*d, c d, a*d │ │ │ │ -00030e10: 2c20 622a 6420 2c20 632a 6420 2c20 6420 , b*d , c*d , d │ │ │ │ -00030e20: 7d7d 2020 7c0a 7c20 2020 2020 2020 2020 }} |.| │ │ │ │ -00030e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e70: 2020 2020 7c0a 7c6f 3320 3a20 4c69 7374 |.|o3 : List │ │ │ │ -00030e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ec0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002ff80: 2020 3220 2020 2020 3220 2020 2020 2020 2 2 │ │ │ │ +0002ff90: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002ffa0: 2020 2020 3220 2020 2020 207c 0a7c 6f33 2 |.|o3 │ │ │ │ +0002ffb0: 203d 207b 7b62 202c 2062 2a63 2c20 6320 = {{b , b*c, c │ │ │ │ +0002ffc0: 2c20 612a 642c 2062 2a64 2c20 632a 642c , a*d, b*d, c*d, │ │ │ │ +0002ffd0: 2064 207d 2c20 7b62 202c 2062 2a63 2c20 d }, {b , b*c, │ │ │ │ +0002ffe0: 6320 2c20 612a 642c 2062 2a64 2c20 632a c , a*d, b*d, c* │ │ │ │ +0002fff0: 642c 2064 207d 2c20 2020 207c 0a7c 2020 d, d }, |.| │ │ │ │ +00030000: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ +00030010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ +00030050: 2020 2020 2032 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00030060: 3320 2020 3220 2020 2020 2020 2020 2020 3 2 │ │ │ │ +00030070: 3220 2020 2020 2032 2020 2020 2032 2020 2 2 2 │ │ │ │ +00030080: 2020 2032 2020 2033 2020 2020 2032 2020 2 3 2 │ │ │ │ +00030090: 2020 2020 2020 3220 2020 207c 0a7c 2020 2 |.| │ │ │ │ +000300a0: 2020 207b 6220 632c 2062 2a63 202c 2063 {b c, b*c , c │ │ │ │ +000300b0: 202c 2062 2064 2c20 622a 632a 642c 2063 , b d, b*c*d, c │ │ │ │ +000300c0: 2064 2c20 612a 6420 2c20 622a 6420 2c20 d, a*d , b*d , │ │ │ │ +000300d0: 632a 6420 2c20 6420 7d2c 207b 6220 2c20 c*d , d }, {b , │ │ │ │ +000300e0: 622a 632c 2063 202c 2020 207c 0a7c 2020 b*c, c , |.| │ │ │ │ +000300f0: 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------- │ │ │ │ +00030100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ +00030140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030150: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00030160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030180: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00030190: 2020 2061 2a64 2c20 622a 642c 2063 2a64 a*d, b*d, c*d │ │ │ │ +000301a0: 2c20 6420 7d7d 2020 2020 2020 2020 2020 , d }} │ │ │ │ +000301b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000301c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000301d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000301e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000301f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030220: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +00030230: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +00030240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030270: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00030280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000302a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000302b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000302c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ +000302d0: 203a 2073 7461 6e64 6172 644d 6f6e 6f6d : standardMonom │ │ │ │ +000302e0: 6961 6c73 287b 337d 2c20 4d29 2020 2020 ials({3}, M) │ │ │ │ +000302f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030310: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00030320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030360: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00030370: 2020 2020 2032 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00030380: 3320 2020 3220 2020 2020 2020 2020 2020 3 2 │ │ │ │ +00030390: 3220 2020 2020 2032 2020 2020 2032 2020 2 2 2 │ │ │ │ +000303a0: 2020 2032 2020 2033 2020 2020 2020 2020 2 3 │ │ │ │ +000303b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +000303c0: 203d 207b 6220 632c 2062 2a63 202c 2063 = {b c, b*c , c │ │ │ │ +000303d0: 202c 2062 2064 2c20 622a 632a 642c 2063 , b d, b*c*d, c │ │ │ │ +000303e0: 2064 2c20 612a 6420 2c20 622a 6420 2c20 d, a*d , b*d , │ │ │ │ +000303f0: 632a 6420 2c20 6420 7d20 2020 2020 2020 c*d , d } │ │ │ │ +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 6f34 |.|o4 │ │ │ │ +00030460: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +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 0a2b 2d2d |.+-- │ │ │ │ +000304b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000304c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000304d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000304e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000304f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 496e -----------+..In │ │ │ │ +00030500: 7075 7474 696e 6720 616e 2069 6e74 6567 putting an integ │ │ │ │ +00030510: 6572 2024 6424 2028 6f72 2064 6567 7265 er $d$ (or degre │ │ │ │ +00030520: 6520 2464 2429 2061 6e64 2061 6e20 6964 e $d$) and an id │ │ │ │ +00030530: 6561 6c20 6769 7665 7320 7468 6520 7374 eal gives the st │ │ │ │ +00030540: 616e 6461 7264 0a6d 6f6e 6f6d 6961 6c73 andard.monomials │ │ │ │ +00030550: 2066 6f72 2074 6865 2073 7065 6369 6669 for the specifi │ │ │ │ +00030560: 6564 2069 6465 616c 2069 6e20 6465 6772 ed ideal in degr │ │ │ │ +00030570: 6565 2024 6424 2e0a 0a2b 2d2d 2d2d 2d2d ee $d$...+------ │ │ │ │ +00030580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +000305a0: 0a7c 6935 203a 2073 7461 6e64 6172 644d .|i5 : standardM │ │ │ │ +000305b0: 6f6e 6f6d 6961 6c73 2832 2c20 4d29 2020 onomials(2, M) │ │ │ │ +000305c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000305d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000305e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000305f0: 0a7c 2020 2020 2020 2032 2020 2020 2020 .| 2 │ │ │ │ +00030600: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00030610: 2020 2020 2032 207c 0a7c 6f35 203d 207b 2 |.|o5 = { │ │ │ │ +00030620: 6220 2c20 622a 632c 2063 202c 2061 2a64 b , b*c, c , a*d │ │ │ │ +00030630: 2c20 622a 642c 2063 2a64 2c20 6420 7d7c , b*d, c*d, d }| │ │ │ │ +00030640: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00030650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030660: 2020 2020 2020 207c 0a7c 6f35 203a 204c |.|o5 : L │ │ │ │ +00030670: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +00030680: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00030690: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000306a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000306b0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ +000306c0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ +000306d0: 202a 6e6f 7465 2074 6169 6c4d 6f6e 6f6d *note tailMonom │ │ │ │ +000306e0: 6961 6c73 3a20 7461 696c 4d6f 6e6f 6d69 ials: tailMonomi │ │ │ │ +000306f0: 616c 732c 202d 2d20 6669 6e64 2074 6169 als, -- find tai │ │ │ │ +00030700: 6c20 6d6f 6e6f 6d69 616c 730a 2020 2a20 l monomials. * │ │ │ │ +00030710: 2a6e 6f74 6520 736d 616c 6c65 724d 6f6e *note smallerMon │ │ │ │ +00030720: 6f6d 6961 6c73 3a20 736d 616c 6c65 724d omials: smallerM │ │ │ │ +00030730: 6f6e 6f6d 6961 6c73 2c20 2d2d 2072 6574 onomials, -- ret │ │ │ │ +00030740: 7572 6e73 2074 6865 2073 7461 6e64 6172 urns the standar │ │ │ │ +00030750: 6420 6d6f 6e6f 6d69 616c 730a 2020 2020 d monomials. │ │ │ │ +00030760: 736d 616c 6c65 7220 6275 7420 6f66 2074 smaller but of t │ │ │ │ +00030770: 6865 2073 616d 6520 6465 6772 6565 2061 he same degree a │ │ │ │ +00030780: 7320 6769 7665 6e20 6d6f 6e6f 6d69 616c s given monomial │ │ │ │ +00030790: 2873 290a 0a57 6179 7320 746f 2075 7365 (s)..Ways to use │ │ │ │ +000307a0: 2073 7461 6e64 6172 644d 6f6e 6f6d 6961 standardMonomia │ │ │ │ +000307b0: 6c73 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ls:.============ │ │ │ │ +000307c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000307d0: 3d3d 0a0a 2020 2a20 2273 7461 6e64 6172 ==.. * "standar │ │ │ │ +000307e0: 644d 6f6e 6f6d 6961 6c73 2849 6465 616c dMonomials(Ideal │ │ │ │ +000307f0: 2922 0a20 202a 2022 7374 616e 6461 7264 )". * "standard │ │ │ │ +00030800: 4d6f 6e6f 6d69 616c 7328 4c69 7374 2c49 Monomials(List,I │ │ │ │ +00030810: 6465 616c 2922 0a20 202a 2022 7374 616e deal)". * "stan │ │ │ │ +00030820: 6461 7264 4d6f 6e6f 6d69 616c 7328 5a5a dardMonomials(ZZ │ │ │ │ +00030830: 2c49 6465 616c 2922 0a0a 466f 7220 7468 ,Ideal)"..For th │ │ │ │ +00030840: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +00030850: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00030860: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +00030870: 6520 7374 616e 6461 7264 4d6f 6e6f 6d69 e standardMonomi │ │ │ │ +00030880: 616c 733a 2073 7461 6e64 6172 644d 6f6e als: standardMon │ │ │ │ +00030890: 6f6d 6961 6c73 2c20 6973 2061 202a 6e6f omials, is a *no │ │ │ │ +000308a0: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ +000308b0: 6f6e 3a20 284d 6163 6175 6c61 7932 446f on: (Macaulay2Do │ │ │ │ +000308c0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +000308d0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +000308e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000308f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030920: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +00030930: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +00030940: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +00030950: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +00030960: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +00030970: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +00030980: 322f 7061 636b 6167 6573 2f0a 4772 6f65 2/packages/.Groe │ │ │ │ +00030990: 626e 6572 5374 7261 7461 2e6d 323a 3530 bnerStrata.m2:50 │ │ │ │ +000309a0: 363a 302e 0a1f 0a46 696c 653a 2047 726f 6:0....File: Gro │ │ │ │ +000309b0: 6562 6e65 7253 7472 6174 612e 696e 666f ebnerStrata.info │ │ │ │ +000309c0: 2c20 4e6f 6465 3a20 7461 696c 4d6f 6e6f , Node: tailMono │ │ │ │ +000309d0: 6d69 616c 732c 2050 7265 763a 2073 7461 mials, Prev: sta │ │ │ │ +000309e0: 6e64 6172 644d 6f6e 6f6d 6961 6c73 2c20 ndardMonomials, │ │ │ │ +000309f0: 5570 3a20 546f 700a 0a74 6169 6c4d 6f6e Up: Top..tailMon │ │ │ │ +00030a00: 6f6d 6961 6c73 202d 2d20 6669 6e64 2074 omials -- find t │ │ │ │ +00030a10: 6169 6c20 6d6f 6e6f 6d69 616c 730a 2a2a ail monomials.** │ │ │ │ +00030a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00030a30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00030a40: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +00030a50: 2020 2020 2020 2020 4c20 3d20 7461 696c L = tail │ │ │ │ +00030a60: 4d6f 6e6f 6d69 616c 7320 4d0a 2020 2020 Monomials M. │ │ │ │ +00030a70: 2020 2020 4c20 3d20 7461 696c 4d6f 6e6f L = tailMono │ │ │ │ +00030a80: 6d69 616c 7328 4d2c 206d 290a 2020 2a20 mials(M, m). * │ │ │ │ +00030a90: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ +00030aa0: 4d2c 2061 6e20 2a6e 6f74 6520 6964 6561 M, an *note idea │ │ │ │ +00030ab0: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ +00030ac0: 2949 6465 616c 2c2c 2024 4d24 2073 686f )Ideal,, $M$ sho │ │ │ │ +00030ad0: 756c 6420 6265 2061 206d 6f6e 6f6d 6961 uld be a monomia │ │ │ │ +00030ae0: 6c20 6964 6561 6c0a 2020 2020 2020 2020 l ideal. │ │ │ │ +00030af0: 2861 6e20 6964 6561 6c20 6765 6e65 7261 (an ideal genera │ │ │ │ +00030b00: 7465 6420 6279 206d 6f6e 6f6d 6961 6c73 ted by monomials │ │ │ │ +00030b10: 290a 2020 2020 2020 2a20 6d2c 2061 202a ). * m, a * │ │ │ │ +00030b20: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ +00030b30: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +00030b40: 2952 696e 6745 6c65 6d65 6e74 2c2c 206f )RingElement,, o │ │ │ │ +00030b50: 7074 696f 6e61 6c2c 206f 6e6c 790a 2020 ptional, only. │ │ │ │ +00030b60: 2020 2020 2020 7265 7475 726e 2061 2073 return a s │ │ │ │ +00030b70: 696e 676c 6520 6c69 7374 206f 6620 7468 ingle list of th │ │ │ │ +00030b80: 6520 7461 696c 206d 6f6e 6f6d 6961 6c73 e tail monomials │ │ │ │ +00030b90: 2066 6f72 2074 6869 7320 6d6f 6e6f 6d69 for this monomi │ │ │ │ +00030ba0: 616c 0a20 202a 202a 6e6f 7465 204f 7074 al. * *note Opt │ │ │ │ +00030bb0: 696f 6e61 6c20 696e 7075 7473 3a20 284d ional inputs: (M │ │ │ │ +00030bc0: 6163 6175 6c61 7932 446f 6329 7573 696e acaulay2Doc)usin │ │ │ │ +00030bd0: 6720 6675 6e63 7469 6f6e 7320 7769 7468 g functions with │ │ │ │ +00030be0: 206f 7074 696f 6e61 6c20 696e 7075 7473 optional inputs │ │ │ │ +00030bf0: 2c3a 0a20 2020 2020 202a 2041 6c6c 5374 ,:. * AllSt │ │ │ │ +00030c00: 616e 6461 7264 203d 3e20 6120 2a6e 6f74 andard => a *not │ │ │ │ +00030c10: 6520 426f 6f6c 6561 6e20 7661 6c75 653a e Boolean value: │ │ │ │ +00030c20: 2028 4d61 6361 756c 6179 3244 6f63 2942 (Macaulay2Doc)B │ │ │ │ +00030c30: 6f6f 6c65 616e 2c2c 2064 6566 6175 6c74 oolean,, default │ │ │ │ +00030c40: 0a20 2020 2020 2020 2076 616c 7565 2066 . value f │ │ │ │ +00030c50: 616c 7365 2c20 7768 6963 6820 6d6f 6e6f alse, which mono │ │ │ │ +00030c60: 6d69 616c 7320 7368 6f75 6c64 2062 6520 mials should be │ │ │ │ +00030c70: 636f 6e73 6964 6572 6564 2074 6169 6c20 considered tail │ │ │ │ +00030c80: 6d6f 6e6f 6d69 616c 7320 6f66 2061 0a20 monomials of a. │ │ │ │ +00030c90: 2020 2020 2020 206d 6f6e 6f6d 6961 6c20 monomial │ │ │ │ +00030ca0: 246d 243a 2065 6974 6865 7220 616c 6c20 $m$: either all │ │ │ │ +00030cb0: 7374 616e 6461 7264 206d 6f6e 6f6d 6961 standard monomia │ │ │ │ +00030cc0: 6c73 206f 6620 6120 6769 7665 6e20 6465 ls of a given de │ │ │ │ +00030cd0: 6772 6565 2c20 6f72 2061 6c6c 0a20 2020 gree, or all. │ │ │ │ +00030ce0: 2020 2020 206d 6f6e 6f6d 6961 6c73 2073 monomials s │ │ │ │ +00030cf0: 6d61 6c6c 6572 2074 6861 6e20 246d 2420 maller than $m$ │ │ │ │ +00030d00: 696e 2074 6865 2067 6976 656e 2074 6572 in the given ter │ │ │ │ +00030d10: 6d20 6f72 6465 7220 2862 7574 2073 7469 m order (but sti │ │ │ │ +00030d20: 6c6c 206f 6620 7468 650a 2020 2020 2020 ll of the. │ │ │ │ +00030d30: 2020 7361 6d65 2064 6567 7265 6529 0a20 same degree). │ │ │ │ +00030d40: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +00030d50: 2020 2a20 4c2c 2061 202a 6e6f 7465 206c * L, a *note l │ │ │ │ +00030d60: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +00030d70: 6f63 294c 6973 742c 2c20 6120 6c69 7374 oc)List,, a list │ │ │ │ +00030d80: 206f 6620 6c69 7374 733a 2066 6f72 2065 of lists: for e │ │ │ │ +00030d90: 6163 680a 2020 2020 2020 2020 6765 6e65 ach. gene │ │ │ │ +00030da0: 7261 746f 7220 246d 2420 6f66 2024 4d24 rator $m$ of $M$ │ │ │ │ +00030db0: 2c20 7468 6520 6c69 7374 206f 6620 616c , the list of al │ │ │ │ +00030dc0: 6c20 7461 696c 206d 6f6e 6f6d 6961 6c73 l tail monomials │ │ │ │ +00030dd0: 2049 6620 696e 7374 6561 6420 246d 2420 If instead $m$ │ │ │ │ +00030de0: 6973 0a20 2020 2020 2020 2067 6976 656e is. given │ │ │ │ +00030df0: 2c20 7468 6520 6c69 7374 206f 6620 7468 , the list of th │ │ │ │ +00030e00: 6520 7461 696c 206d 6f6e 6f6d 6961 6c73 e tail monomials │ │ │ │ +00030e10: 206f 6620 246d 2420 6973 2072 6574 7572 of $m$ is retur │ │ │ │ +00030e20: 6e65 640a 0a44 6573 6372 6970 7469 6f6e ned..Description │ │ │ │ +00030e30: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 496e .===========..In │ │ │ │ +00030e40: 7075 7474 696e 6720 616e 2069 6465 616c putting an ideal │ │ │ │ +00030e50: 2024 4d24 2067 656e 6572 6174 6564 2062 $M$ generated b │ │ │ │ +00030e60: 7920 6d6f 6e6f 6d69 616c 7320 7265 7475 y monomials retu │ │ │ │ +00030e70: 726e 7320 6120 6c69 7374 206f 6620 6c69 rns a list of li │ │ │ │ +00030e80: 7374 7320 6f66 2074 6169 6c0a 6d6f 6e6f sts of tail.mono │ │ │ │ +00030e90: 6d69 616c 7320 666f 7220 6561 6368 2067 mials for each g │ │ │ │ +00030ea0: 656e 6572 6174 6f72 206f 6620 244d 2420 enerator of $M$ │ │ │ │ +00030eb0: 2869 6e20 7468 6520 7361 6d65 206f 7264 (in the same ord │ │ │ │ +00030ec0: 6572 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d er)...+--------- │ │ │ │ 00030ed0: 2d2d 2d2d 2d2d 2d2d 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 2b0a 7c69 3420 3a20 7461 696c ----+.|i4 : tail │ │ │ │ -00030f20: 4d6f 6e6f 6d69 616c 7328 4d2c 2041 6c6c Monomials(M, All │ │ │ │ -00030f30: 5374 616e 6461 7264 203d 3e20 7472 7565 Standard => true │ │ │ │ -00030f40: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00030f10: 2d2d 2d2d 2b0a 7c69 3120 3a20 5220 3d20 ----+.|i1 : R = │ │ │ │ +00030f20: 5a5a 2f33 3230 3033 5b61 2e2e 645d 3b20 ZZ/32003[a..d]; │ │ │ │ +00030f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00030f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030fb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00030fc0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ -00030fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030fe0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00030ff0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00031000: 2020 2020 7c0a 7c6f 3420 3d20 7b7b 612a |.|o4 = {{a* │ │ │ │ -00031010: 622c 2061 2a63 2c20 622a 632c 2063 202c b, a*c, b*c, c , │ │ │ │ -00031020: 2061 2a64 2c20 622a 642c 2063 2a64 2c20 a*d, b*d, c*d, │ │ │ │ -00031030: 6420 7d2c 207b 612a 622c 2061 2a63 2c20 d }, {a*b, a*c, │ │ │ │ -00031040: 622a 632c 2063 202c 2061 2a64 2c20 622a b*c, c , a*d, b* │ │ │ │ -00031050: 642c 2020 7c0a 7c20 2020 2020 2d2d 2d2d d, |.| ---- │ │ │ │ -00031060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ -000310b0: 2020 3220 2020 2020 2020 3220 2020 2020 2 2 │ │ │ │ -000310c0: 3220 2020 3320 2020 2020 2020 2020 2020 2 3 │ │ │ │ -000310d0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -000310e0: 2020 2020 3220 2020 2020 3220 2020 2020 2 2 │ │ │ │ -000310f0: 3220 2020 7c0a 7c20 2020 2020 632a 642c 2 |.| c*d, │ │ │ │ -00031100: 2064 207d 2c20 7b61 2a63 202c 2062 2a63 d }, {a*c , b*c │ │ │ │ -00031110: 202c 2063 202c 2061 2a62 2a64 2c20 612a , c , a*b*d, a* │ │ │ │ -00031120: 632a 642c 2062 2a63 2a64 2c20 6320 642c c*d, b*c*d, c d, │ │ │ │ -00031130: 2061 2a64 202c 2062 2a64 202c 2063 2a64 a*d , b*d , c*d │ │ │ │ -00031140: 202c 2020 7c0a 7c20 2020 2020 2d2d 2d2d , |.| ---- │ │ │ │ -00031150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031190: 2d2d 2d2d 7c0a 7c20 2020 2020 2033 2020 ----|.| 3 │ │ │ │ -000311a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030f60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00030f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00030fb0: 2d2d 2d2d 2b0a 7c69 3220 3a20 4d20 3d20 ----+.|i2 : M = │ │ │ │ +00030fc0: 6964 6561 6c20 2861 5e32 2c20 625e 322c ideal (a^2, b^2, │ │ │ │ +00030fd0: 2061 2a62 2a63 293b 2020 2020 2020 2020 a*b*c); │ │ │ │ +00030fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031000: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00031010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031050: 2020 2020 7c0a 7c6f 3220 3a20 4964 6561 |.|o2 : Idea │ │ │ │ +00031060: 6c20 6f66 2052 2020 2020 2020 2020 2020 l of R │ │ │ │ +00031070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000310a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000310b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000310c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000310d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000310e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000310f0: 2d2d 2d2d 2b0a 7c69 3320 3a20 7461 696c ----+.|i3 : tail │ │ │ │ +00031100: 4d6f 6e6f 6d69 616c 7320 4d20 2020 2020 Monomials M │ │ │ │ +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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00031150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031190: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000311a0: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ 000311b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311e0: 2020 2020 7c0a 7c20 2020 2020 6420 7d7d |.| d }} │ │ │ │ -000311f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031230: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00031240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031280: 2020 2020 7c0a 7c6f 3420 3a20 4c69 7374 |.|o4 : List │ │ │ │ -00031290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000311c0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000311d0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000311e0: 2020 2020 7c0a 7c6f 3320 3d20 7b7b 612a |.|o3 = {{a* │ │ │ │ +000311f0: 622c 2061 2a63 2c20 622a 632c 2063 202c b, a*c, b*c, c , │ │ │ │ +00031200: 2061 2a64 2c20 622a 642c 2063 2a64 2c20 a*d, b*d, c*d, │ │ │ │ +00031210: 6420 7d2c 207b 612a 632c 2062 2a63 2c20 d }, {a*c, b*c, │ │ │ │ +00031220: 6320 2c20 612a 642c 2062 2a64 2c20 632a c , a*d, b*d, c* │ │ │ │ +00031230: 642c 2020 7c0a 7c20 2020 2020 2d2d 2d2d d, |.| ---- │ │ │ │ +00031240: 2d2d 2d2d 2d2d 2d2d 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 7c0a 7c20 2020 2020 2032 2020 ----|.| 2 │ │ │ │ +00031290: 2020 2020 2032 2020 2020 2032 2020 2033 2 2 3 │ │ │ │ 000312a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000312b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000312c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000312d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -000312e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000312f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031320: 2d2d 2d2d 2b0a 7c69 3520 3a20 7461 696c ----+.|i5 : tail │ │ │ │ -00031330: 4d6f 6e6f 6d69 616c 7328 4d2c 2062 5e32 Monomials(M, b^2 │ │ │ │ -00031340: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +000312b0: 2020 2020 2020 2020 3220 2020 2020 2032 2 2 │ │ │ │ +000312c0: 2020 2020 2032 2020 2020 2032 2020 2033 2 2 3 │ │ │ │ +000312d0: 2020 2020 7c0a 7c20 2020 2020 6420 7d2c |.| d }, │ │ │ │ +000312e0: 207b 612a 6320 2c20 622a 6320 2c20 6320 {a*c , b*c , c │ │ │ │ +000312f0: 2c20 612a 622a 642c 2061 2a63 2a64 2c20 , a*b*d, a*c*d, │ │ │ │ +00031300: 622a 632a 642c 2063 2064 2c20 612a 6420 b*c*d, c d, a*d │ │ │ │ +00031310: 2c20 622a 6420 2c20 632a 6420 2c20 6420 , b*d , c*d , d │ │ │ │ +00031320: 7d7d 2020 7c0a 7c20 2020 2020 2020 2020 }} |.| │ │ │ │ +00031330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031370: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00031370: 2020 2020 7c0a 7c6f 3320 3a20 4c69 7374 |.|o3 : List │ │ │ │ 00031380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000313c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000313d0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -000313e0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -000313f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031410: 2020 2020 7c0a 7c6f 3520 3d20 7b61 2a63 |.|o5 = {a*c │ │ │ │ -00031420: 2c20 622a 632c 2063 202c 2061 2a64 2c20 , b*c, c , a*d, │ │ │ │ -00031430: 622a 642c 2063 2a64 2c20 6420 7d20 2020 b*d, c*d, d } │ │ │ │ -00031440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000313c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000313d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000313e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000313f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031410: 2d2d 2d2d 2b0a 7c69 3420 3a20 7461 696c ----+.|i4 : tail │ │ │ │ +00031420: 4d6f 6e6f 6d69 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2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00031510: 2d2d 2d2d 2d2d 2d2d 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 2b0a 7c69 3620 3a20 7461 696c ----+.|i6 : tail │ │ │ │ -00031560: 4d6f 6e6f 6d69 616c 7328 4d2c 2062 5e32 Monomials(M, b^2 │ │ │ │ -00031570: 2c20 416c 6c53 7461 6e64 6172 643d 3e74 , AllStandard=>t │ │ │ │ -00031580: 7275 6529 2020 2020 2020 2020 2020 2020 rue) │ │ │ │ -00031590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000315a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000315b0: 2020 2020 2020 2020 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 │ │ │ │ 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2c20 622a 642c 2063 2a64 2c20 a*d, b*d, c*d, │ │ │ │ +00031530: 6420 7d2c 207b 612a 622c 2061 2a63 2c20 d }, {a*b, a*c, │ │ │ │ +00031540: 622a 632c 2063 202c 2061 2a64 2c20 622a b*c, c , a*d, b* │ │ │ │ +00031550: 642c 2020 7c0a 7c20 2020 2020 2d2d 2d2d d, |.| ---- │ │ │ │ +00031560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000315a0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +000315b0: 2020 3220 2020 2020 2020 3220 2020 2020 2 2 │ │ │ │ +000315c0: 3220 2020 3320 2020 2020 2020 2020 2020 2 3 │ │ │ │ +000315d0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +000315e0: 2020 2020 3220 2020 2020 3220 2020 2020 2 2 │ │ │ │ +000315f0: 3220 2020 7c0a 7c20 2020 2020 632a 642c 2 |.| c*d, │ │ │ │ +00031600: 2064 207d 2c20 7b61 2a63 202c 2062 2a63 d }, {a*c , b*c │ │ │ │ +00031610: 202c 2063 202c 2061 2a62 2a64 2c20 612a , c , a*b*d, a* │ │ │ │ +00031620: 632a 642c 2062 2a63 2a64 2c20 6320 642c c*d, b*c*d, c d, │ │ │ │ +00031630: 2061 2a64 202c 2062 2a64 202c 2063 2a64 a*d , b*d , c*d │ │ │ │ +00031640: 202c 2020 7c0a 7c20 2020 2020 2d2d 2d2d , |.| ---- │ │ │ │ +00031650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031690: 2d2d 2d2d 7c0a 7c20 2020 2020 2033 2020 ----|.| 3 │ │ │ │ 000316a0: 2020 2020 2020 2020 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 7c0a 7c6f 3620 3a20 4c69 7374 |.|o6 : List │ │ │ │ +000316e0: 2020 2020 7c0a 7c20 2020 2020 6420 7d7d |.| d }} │ │ │ │ 000316f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031730: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00031740: 2d2d 2d2d 2d2d 2d2d 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 2b0a 0a53 6565 2061 6c73 6f0a ----+..See also. │ │ │ │ -00031790: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -000317a0: 6f74 6520 7374 616e 6461 7264 4d6f 6e6f ote standardMono │ │ │ │ -000317b0: 6d69 616c 733a 2073 7461 6e64 6172 644d mials: standardM │ │ │ │ -000317c0: 6f6e 6f6d 6961 6c73 2c20 2d2d 2063 6f6d onomials, -- com │ │ │ │ -000317d0: 7075 7465 7320 7374 616e 6461 7264 206d putes standard m │ │ │ │ -000317e0: 6f6e 6f6d 6961 6c73 0a20 202a 202a 6e6f onomials. * *no │ │ │ │ -000317f0: 7465 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 te smallerMonomi │ │ │ │ -00031800: 616c 733a 2073 6d61 6c6c 6572 4d6f 6e6f als: smallerMono │ │ │ │ -00031810: 6d69 616c 732c 202d 2d20 7265 7475 726e mials, -- return │ │ │ │ -00031820: 7320 7468 6520 7374 616e 6461 7264 206d s the standard m │ │ │ │ -00031830: 6f6e 6f6d 6961 6c73 0a20 2020 2073 6d61 onomials. sma │ │ │ │ -00031840: 6c6c 6572 2062 7574 206f 6620 7468 6520 ller but of the │ │ │ │ -00031850: 7361 6d65 2064 6567 7265 6520 6173 2067 same degree as g │ │ │ │ -00031860: 6976 656e 206d 6f6e 6f6d 6961 6c28 7329 iven monomial(s) │ │ │ │ -00031870: 0a0a 5761 7973 2074 6f20 7573 6520 7461 ..Ways to use ta │ │ │ │ -00031880: 696c 4d6f 6e6f 6d69 616c 733a 0a3d 3d3d ilMonomials:.=== │ │ │ │ -00031890: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000318a0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7461 =======.. * "ta │ │ │ │ -000318b0: 696c 4d6f 6e6f 6d69 616c 7328 4964 6561 ilMonomials(Idea │ │ │ │ -000318c0: 6c29 220a 2020 2a20 2274 6169 6c4d 6f6e l)". * "tailMon │ │ │ │ -000318d0: 6f6d 6961 6c73 2849 6465 616c 2c52 696e omials(Ideal,Rin │ │ │ │ -000318e0: 6745 6c65 6d65 6e74 2922 0a0a 466f 7220 gElement)"..For │ │ │ │ -000318f0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00031900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00031910: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00031920: 6f74 6520 7461 696c 4d6f 6e6f 6d69 616c ote tailMonomial │ │ │ │ -00031930: 733a 2074 6169 6c4d 6f6e 6f6d 6961 6c73 s: tailMonomials │ │ │ │ -00031940: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -00031950: 686f 6420 6675 6e63 7469 6f6e 2077 6974 hod function wit │ │ │ │ -00031960: 680a 6f70 7469 6f6e 733a 2028 4d61 6361 h.options: (Maca │ │ │ │ -00031970: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -00031980: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ -00031990: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ -000319a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000319b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000319c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000319d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000319e0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -000319f0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -00031a00: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -00031a10: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -00031a20: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -00031a30: 2e30 362b 6473 2f4d 322f 4d61 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000317d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000317e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000317f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031820: 2d2d 2d2d 2b0a 7c69 3520 3a20 7461 696c ----+.|i5 : tail │ │ │ │ +00031830: 4d6f 6e6f 6d69 616c 7328 4d2c 2062 5e32 Monomials(M, b^2 │ │ │ │ +00031840: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00031850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031870: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00031880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000318a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000318b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000318c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000318d0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +000318e0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +000318f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031910: 2020 2020 7c0a 7c6f 3520 3d20 7b61 2a63 |.|o5 = {a*c │ │ │ │ +00031920: 2c20 622a 632c 2063 202c 2061 2a64 2c20 , b*c, c , a*d, │ │ │ │ +00031930: 622a 642c 2063 2a64 2c20 6420 7d20 2020 b*d, c*d, d } │ │ │ │ +00031940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031960: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00031970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000319a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000319b0: 2020 2020 7c0a 7c6f 3520 3a20 4c69 7374 |.|o5 : List │ │ │ │ +000319c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000319d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000319e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000319f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031a00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00031a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031a50: 2d2d 2d2d 2b0a 7c69 3620 3a20 7461 696c ----+.|i6 : tail │ │ │ │ +00031a60: 4d6f 6e6f 6d69 616c 7328 4d2c 2062 5e32 Monomials(M, b^2 │ │ │ │ +00031a70: 2c20 416c 6c53 7461 6e64 6172 643d 3e74 , AllStandard=>t │ │ │ │ +00031a80: 7275 6529 2020 2020 2020 2020 2020 2020 rue) │ │ │ │ +00031a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031aa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00031ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031af0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00031b00: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00031b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031b20: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00031b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031b40: 2020 2020 7c0a 7c6f 3620 3d20 7b61 2a62 |.|o6 = {a*b │ │ │ │ +00031b50: 2c20 612a 632c 2062 2a63 2c20 6320 2c20 , a*c, b*c, c , │ │ │ │ +00031b60: 612a 642c 2062 2a64 2c20 632a 642c 2064 a*d, b*d, c*d, d │ │ │ │ +00031b70: 207d 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00031b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031b90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00031ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031be0: 2020 2020 7c0a 7c6f 3620 3a20 4c69 7374 |.|o6 : List │ │ │ │ +00031bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031c30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00031c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031c80: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a ----+..See also. │ │ │ │ +00031c90: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00031ca0: 6f74 6520 7374 616e 6461 7264 4d6f 6e6f ote standardMono │ │ │ │ +00031cb0: 6d69 616c 733a 2073 7461 6e64 6172 644d mials: standardM │ │ │ │ +00031cc0: 6f6e 6f6d 6961 6c73 2c20 2d2d 2063 6f6d onomials, -- com │ │ │ │ +00031cd0: 7075 7465 7320 7374 616e 6461 7264 206d putes standard m │ │ │ │ +00031ce0: 6f6e 6f6d 6961 6c73 0a20 202a 202a 6e6f onomials. * *no │ │ │ │ +00031cf0: 7465 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 te smallerMonomi │ │ │ │ +00031d00: 616c 733a 2073 6d61 6c6c 6572 4d6f 6e6f als: smallerMono │ │ │ │ +00031d10: 6d69 616c 732c 202d 2d20 7265 7475 726e mials, -- return │ │ │ │ +00031d20: 7320 7468 6520 7374 616e 6461 7264 206d s the standard m │ │ │ │ +00031d30: 6f6e 6f6d 6961 6c73 0a20 2020 2073 6d61 onomials. sma │ │ │ │ +00031d40: 6c6c 6572 2062 7574 206f 6620 7468 6520 ller but of the │ │ │ │ +00031d50: 7361 6d65 2064 6567 7265 6520 6173 2067 same degree as g │ │ │ │ +00031d60: 6976 656e 206d 6f6e 6f6d 6961 6c28 7329 iven monomial(s) │ │ │ │ +00031d70: 0a0a 5761 7973 2074 6f20 7573 6520 7461 ..Ways to use ta │ │ │ │ +00031d80: 696c 4d6f 6e6f 6d69 616c 733a 0a3d 3d3d ilMonomials:.=== │ │ │ │ +00031d90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00031da0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7461 =======.. * "ta │ │ │ │ +00031db0: 696c 4d6f 6e6f 6d69 616c 7328 4964 6561 ilMonomials(Idea │ │ │ │ +00031dc0: 6c29 220a 2020 2a20 2274 6169 6c4d 6f6e l)". * "tailMon │ │ │ │ +00031dd0: 6f6d 6961 6c73 2849 6465 616c 2c52 696e omials(Ideal,Rin │ │ │ │ +00031de0: 6745 6c65 6d65 6e74 2922 0a0a 466f 7220 gElement)"..For │ │ │ │ +00031df0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +00031e00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00031e10: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +00031e20: 6f74 6520 7461 696c 4d6f 6e6f 6d69 616c ote tailMonomial │ │ │ │ +00031e30: 733a 2074 6169 6c4d 6f6e 6f6d 6961 6c73 s: tailMonomials │ │ │ │ +00031e40: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +00031e50: 686f 6420 6675 6e63 7469 6f6e 2077 6974 hod function wit │ │ │ │ +00031e60: 680a 6f70 7469 6f6e 733a 2028 4d61 6361 h.options: (Maca │ │ │ │ +00031e70: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +00031e80: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ +00031e90: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ +00031ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00031ee0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +00031ef0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +00031f00: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +00031f10: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +00031f20: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +00031f30: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +00031f40: 6179 322f 7061 636b 6167 6573 2f0a 4772 ay2/packages/.Gr │ │ │ │ +00031f50: 6f65 626e 6572 5374 7261 7461 2e6d 323a oebnerStrata.m2: │ │ │ │ +00031f60: 3538 333a 302e 0a1f 0a54 6167 2054 6162 583:0....Tag Tab │ │ │ │ +00031f70: 6c65 3a0a 4e6f 6465 3a20 546f 707f 3235 le:.Node: Top.25 │ │ │ │ +00031f80: 370a 4e6f 6465 3a20 416c 6c53 7461 6e64 7.Node: AllStand │ │ │ │ +00031f90: 6172 647f 3531 3136 370a 4e6f 6465 3a20 ard.51167.Node: │ │ │ │ +00031fa0: 6669 6e64 5765 6967 6874 436f 6e73 7472 findWeightConstr │ │ │ │ +00031fb0: 6169 6e74 737f 3533 3131 380a 4e6f 6465 aints.53118.Node │ │ │ │ +00031fc0: 3a20 6669 6e64 5765 6967 6874 5665 6374 : findWeightVect │ │ │ │ +00031fd0: 6f72 7f36 3139 3238 0a4e 6f64 653a 2067 or.61928.Node: g │ │ │ │ +00031fe0: 726f 6562 6e65 7246 616d 696c 797f 3636 roebnerFamily.66 │ │ │ │ +00031ff0: 3633 380a 4e6f 6465 3a20 6772 6f65 626e 638.Node: groebn │ │ │ │ +00032000: 6572 5374 7261 7475 6d7f 3131 3036 3935 erStratum.110695 │ │ │ │ +00032010: 0a4e 6f64 653a 206c 696e 6561 7250 6172 .Node: linearPar │ │ │ │ +00032020: 747f 3131 3939 3631 0a4e 6f64 653a 204d t.119961.Node: M │ │ │ │ +00032030: 696e 696d 616c 697a 657f 3132 3230 3632 inimalize.122062 │ │ │ │ +00032040: 0a4e 6f64 653a 206e 6f6e 6d69 6e69 6d61 .Node: nonminima │ │ │ │ +00032050: 6c4d 6170 737f 3132 3336 3632 0a4e 6f64 lMaps.123662.Nod │ │ │ │ +00032060: 653a 2072 616e 646f 6d50 6f69 6e74 4f6e e: randomPointOn │ │ │ │ +00032070: 5261 7469 6f6e 616c 5661 7269 6574 795f RationalVariety_ │ │ │ │ +00032080: 6c70 4964 6561 6c5f 7270 7f31 3531 3033 lpIdeal_rp.15103 │ │ │ │ +00032090: 360a 4e6f 6465 3a20 7261 6e64 6f6d 506f 6.Node: randomPo │ │ │ │ +000320a0: 696e 7473 4f6e 5261 7469 6f6e 616c 5661 intsOnRationalVa │ │ │ │ +000320b0: 7269 6574 795f 6c70 4964 6561 6c5f 636d riety_lpIdeal_cm │ │ │ │ +000320c0: 5a5a 5f72 707f 3137 3533 3533 0a4e 6f64 ZZ_rp.175353.Nod │ │ │ │ +000320d0: 653a 2073 6d61 6c6c 6572 4d6f 6e6f 6d69 e: smallerMonomi │ │ │ │ +000320e0: 616c 737f 3139 3131 3036 0a4e 6f64 653a als.191106.Node: │ │ │ │ +000320f0: 2073 7461 6e64 6172 644d 6f6e 6f6d 6961 standardMonomia │ │ │ │ +00032100: 6c73 7f31 3934 3834 390a 4e6f 6465 3a20 ls.194849.Node: │ │ │ │ +00032110: 7461 696c 4d6f 6e6f 6d69 616c 737f 3139 tailMonomials.19 │ │ │ │ +00032120: 3930 3737 0a1f 0a45 6e64 2054 6167 2054 9077...End Tag T │ │ │ │ +00032130: 6162 6c65 0a able. │ │ ├── ./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 2e35 |.| -- 3.5 │ │ │ │ -00000d60: 3138 3437 7320 656c 6170 7365 6420 2020 1847s elapsed │ │ │ │ +00000d50: 2020 2020 2020 7c0a 7c20 2d2d 2031 2e37 |.| -- 1.7 │ │ │ │ +00000d60: 3837 3831 7320 656c 6170 7365 6420 2020 8781s 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 2033 2e30 3834 3535 7320 656c | -- 3.08455s el │ │ │ │ +00000ff0: 7c20 2d2d 2031 2e34 3633 3436 7320 656c | -- 1.46346s 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 │ │ │ │ @@ -577,62 +577,62 @@ │ │ │ │ 00002400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002430: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00002440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00002470: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00002470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002480: 2020 2020 7c0a 7c6f 3220 3d20 7b50 6f69 |.|o2 = {Poi │ │ │ │ -00002490: 6e74 7b31 2c20 342c 2038 7d2c 2050 6f69 nt{1, 4, 8}, Poi │ │ │ │ -000024a0: 6e74 7b31 2c20 302c 2031 367d 2c20 506f nt{1, 0, 16}, Po │ │ │ │ -000024b0: 696e 747b 312c 2030 2c20 317d 2c20 506f int{1, 0, 1}, Po │ │ │ │ -000024c0: 696e 747b 312c 2031 2c20 2d7d 2c20 2020 int{1, 1, -}, │ │ │ │ +00002490: 6e74 7b31 2c20 382c 2036 347d 2c20 506f nt{1, 8, 64}, Po │ │ │ │ +000024a0: 696e 747b 312c 2034 2c20 387d 2c20 506f int{1, 4, 8}, Po │ │ │ │ +000024b0: 696e 747b 312c 2030 2c20 3136 7d2c 2050 int{1, 0, 16}, P │ │ │ │ +000024c0: 6f69 6e74 7b31 2c20 302c 2031 7d2c 2020 oint{1, 0, 1}, │ │ │ │ 000024d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000024e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000024f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00002510: 2020 2020 2020 2020 2020 3820 2020 2020 8 │ │ │ │ +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 │ │ │ │ -000025a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00002580: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ +00002590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000025a0: 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 327d 2c20 506f 696e t{1, 0, 2}, Poin │ │ │ │ -000025e0: 747b 312c 2030 2c20 2d7d 2c20 506f 696e t{1, 0, -}, Poin │ │ │ │ -000025f0: 747b 312c 2032 2c20 317d 2c20 506f 696e t{1, 2, 1}, Poin │ │ │ │ -00002600: 747b 312c 2030 2c20 347d 2c20 506f 696e t{1, 0, 4}, Poin │ │ │ │ +000025d0: 747b 312c 2031 2c20 2d7d 2c20 506f 696e t{1, 1, -}, Poin │ │ │ │ +000025e0: 747b 312c 2030 2c20 327d 2c20 506f 696e t{1, 0, 2}, Poin │ │ │ │ +000025f0: 747b 312c 2030 2c20 2d7d 2c20 506f 696e t{1, 0, -}, Poin │ │ │ │ +00002600: 747b 312c 2032 2c20 317d 2c20 506f 696e t{1, 2, 1}, 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 3220 2020 2020 2020 2 │ │ │ │ -00002640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00002620: 2020 2020 2020 2020 3820 2020 2020 2020 8 │ │ │ │ +00002630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00002640: 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 2020 2031 ----|.| 1 │ │ │ │ -000026c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000026b0: 2d2d 2d2d 7c0a 7c20 2020 2020 2020 2020 ----|.| │ │ │ │ +000026c0: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ 000026d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 302c 202d |.| 0, - │ │ │ │ -00002710: 7d2c 2050 6f69 6e74 7b31 2c20 382c 2036 }, Point{1, 8, 6 │ │ │ │ -00002720: 347d 7d20 2020 2020 2020 2020 2020 2020 4}} │ │ │ │ +00002700: 2020 2020 7c0a 7c20 2020 2020 302c 2034 |.| 0, 4 │ │ │ │ +00002710: 7d2c 2050 6f69 6e74 7b31 2c20 302c 202d }, Point{1, 0, - │ │ │ │ +00002720: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ 00002730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00002750: 2020 2020 7c0a 7c20 2020 2020 2020 2034 |.| 4 │ │ │ │ -00002760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00002750: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00002760: 2020 2020 2020 2020 2020 2020 2020 2034 4 │ │ │ │ 00002770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000027a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 000027b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000027c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000027d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1620,26 +1620,26 @@ │ │ │ │ 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}, │ │ │ │ +000065a0: 506f 696e 747b 302c 2031 2c20 307d 2c20 Point{0, 1, 0}, │ │ │ │ 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}, │ │ │ │ +000065c0: 2050 6f69 6e74 7b30 2c20 312c 2031 7d2c Point{0, 1, 1}, │ │ │ │ +000065d0: 2050 6f69 6e74 7b31 2c20 312c 2030 7d2c Point{1, 1, 0}, │ │ │ │ 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}} │ │ │ │ +00006640: 6f69 6e74 7b31 2c20 322c 2030 7d2c 2050 oint{1, 2, 0}, P │ │ │ │ +00006650: 6f69 6e74 7b31 2c20 342c 2031 7d7d 2020 oint{1, 4, 1}} │ │ │ │ 00006660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 │ │ ├── ./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 3438 |.| -- .0000048 │ │ │ │ -0000faf0: 3339 7320 656c 6170 7365 6420 2020 2020 39s elapsed │ │ │ │ +0000fae0: 207c 0a7c 202d 2d20 2e30 3030 3030 3434 |.| -- .0000044 │ │ │ │ +0000faf0: 3531 7320 656c 6170 7365 6420 2020 2020 51s 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 3438 |.| -- .0000048 │ │ │ │ -0000fb40: 3739 7320 656c 6170 7365 6420 2020 2020 79s elapsed │ │ │ │ +0000fb30: 207c 0a7c 202d 2d20 2e30 3030 3030 3432 |.| -- .0000042 │ │ │ │ +0000fb40: 3834 7320 656c 6170 7365 6420 2020 2020 84s 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 3531 |.| -- .0000051 │ │ │ │ -0000fb90: 3373 2065 6c61 7073 6564 2020 2020 2020 3s elapsed │ │ │ │ +0000fb80: 207c 0a7c 202d 2d20 2e30 3030 3030 3434 |.| -- .0000044 │ │ │ │ +0000fb90: 3736 7320 656c 6170 7365 6420 2020 2020 76s 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 3531 |.| -- .0000051 │ │ │ │ -0000fbe0: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ +0000fbd0: 207c 0a7c 202d 2d20 2e30 3030 3030 3536 |.| -- .0000056 │ │ │ │ +0000fbe0: 3936 7320 656c 6170 7365 6420 2020 2020 96s 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 3339 |.| -- .0000039 │ │ │ │ -0000fc30: 3837 7320 656c 6170 7365 6420 2020 2020 87s elapsed │ │ │ │ +0000fc20: 207c 0a7c 202d 2d20 2e30 3030 3030 3535 |.| -- .0000055 │ │ │ │ +0000fc30: 3037 7320 656c 6170 7365 6420 2020 2020 07s elapsed │ │ │ │ 0000fc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000fc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000fca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5568,15 +5568,15 @@ │ │ │ │ 00015bf0: 2063 6f65 6666 6963 6965 6e74 2072 696e coefficient rin │ │ │ │ 00015c00: 6720 6f72 2069 6e68 6f6d 6f67 7c0a 7c43 g or inhomog|.|C │ │ │ │ 00015c10: 6f6e 7665 7274 696e 6720 746f 204e 6169 onverting to Nai │ │ │ │ 00015c20: 7665 2061 6c67 6f72 6974 686d 2e20 2020 ve algorithm. │ │ │ │ 00015c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00015c60: 2d2d 202e 3030 3030 3036 3039 3273 2065 -- .000006092s e │ │ │ │ +00015c60: 2d2d 202e 3030 3030 3035 3537 3373 2065 -- .000005573s e │ │ │ │ 00015c70: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00015c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00015cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5688,15 +5688,15 @@ │ │ │ │ 00016370: 2063 6f65 6666 6963 6965 6e74 2072 696e coefficient rin │ │ │ │ 00016380: 6720 6f72 2069 6e68 6f6d 6f67 7c0a 7c43 g or inhomog|.|C │ │ │ │ 00016390: 6f6e 7665 7274 696e 6720 746f 204e 6169 onverting to Nai │ │ │ │ 000163a0: 7665 2061 6c67 6f72 6974 686d 2e20 2020 ve algorithm. │ │ │ │ 000163b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000163c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000163d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000163e0: 2d2d 202e 3030 3030 3037 3735 3573 2065 -- .000007755s e │ │ │ │ +000163e0: 2d2d 202e 3030 3030 3035 3330 3773 2065 -- .000005307s e │ │ │ │ 000163f0: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00016400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016420: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00016430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/HomotopyLieAlgebra.info.gz │ │ │ ├── HomotopyLieAlgebra.info │ │ │ │ @@ -2585,470 +2585,470 @@ │ │ │ │ 0000a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a1c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000a1d0: 6f31 3520 3d20 7b28 7b54 2020 202c 2054 o15 = {({T , T │ │ │ │ 0000a1e0: 2020 207d 2c20 2d20 5420 2020 5420 2020 }, - T T │ │ │ │ -0000a1f0: 202d 2054 2020 2054 2020 2020 2b20 792a - T T + y* │ │ │ │ -0000a200: 5420 2020 202b 207a 2a54 2020 2029 2c20 T + z*T ), │ │ │ │ -0000a210: 287b 5420 2020 2c20 2020 2020 207c 0a7c ({T , |.| │ │ │ │ -0000a220: 2020 2020 2020 2020 2020 312c 3420 2020 1,4 │ │ │ │ -0000a230: 322c 3320 2020 2020 2031 2c32 2032 2c32 2,3 1,2 2,2 │ │ │ │ -0000a240: 2020 2020 312c 3420 322c 3320 2020 2020 1,4 2,3 │ │ │ │ -0000a250: 2033 2c32 2020 2020 2020 332c 3420 2020 3,2 3,4 │ │ │ │ -0000a260: 2020 2031 2c33 2020 2020 2020 207c 0a7c 1,3 |.| │ │ │ │ +0000a1f0: 202b 2079 2a54 2020 2029 2c20 287b 5420 + y*T ), ({T │ │ │ │ +0000a200: 2020 2c20 5420 2020 7d2c 2054 2020 2054 , T }, T T │ │ │ │ +0000a210: 2020 2020 2d20 2020 2020 2020 207c 0a7c - |.| │ │ │ │ +0000a220: 2020 2020 2020 2020 2020 312c 3520 2020 1,5 │ │ │ │ +0000a230: 322c 3520 2020 2020 2031 2c35 2032 2c35 2,5 1,5 2,5 │ │ │ │ +0000a240: 2020 2020 2020 332c 3820 2020 2020 2031 3,8 1 │ │ │ │ +0000a250: 2c34 2020 2032 2c31 2020 2020 312c 3420 ,4 2,1 1,4 │ │ │ │ +0000a260: 322c 3120 2020 2020 2020 2020 207c 0a7c 2,1 |.| │ │ │ │ 0000a270: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ 0000a280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -0000a2c0: 2020 2020 2020 5420 2020 7d2c 2054 2020 T }, T │ │ │ │ -0000a2d0: 2054 2020 2020 2d20 7a2a 5420 2020 202b T - z*T + │ │ │ │ -0000a2e0: 2079 2a54 2020 2029 2c20 287b 5420 2020 y*T ), ({T │ │ │ │ -0000a2f0: 2c20 5420 2020 7d2c 202d 2054 2020 2054 , T }, - T T │ │ │ │ -0000a300: 2020 2020 2d20 2020 2020 2020 207c 0a7c - |.| │ │ │ │ -0000a310: 2020 2020 2020 2032 2c34 2020 2020 312c 2,4 1, │ │ │ │ -0000a320: 3320 322c 3420 2020 2020 2033 2c35 2020 3 2,4 3,5 │ │ │ │ -0000a330: 2020 2020 332c 3720 2020 2020 2031 2c33 3,7 1,3 │ │ │ │ -0000a340: 2020 2032 2c31 2020 2020 2020 312c 3320 2,1 1,3 │ │ │ │ -0000a350: 322c 3120 2020 2020 2020 2020 207c 0a7c 2,1 |.| │ │ │ │ +0000a2c0: 2020 2020 2020 5420 2020 5420 2020 202b T T + │ │ │ │ +0000a2d0: 2078 2a54 2020 2020 292c 2028 7b54 2020 x*T ), ({T │ │ │ │ +0000a2e0: 202c 2054 2020 207d 2c20 2d20 5420 2020 , T }, - T │ │ │ │ +0000a2f0: 5420 2020 202d 2054 2020 2054 2020 2020 T - T T │ │ │ │ +0000a300: 2b20 782a 5420 2020 292c 2020 207c 0a7c + x*T ), |.| │ │ │ │ +0000a310: 2020 2020 2020 2031 2c31 2032 2c35 2020 1,1 2,5 │ │ │ │ +0000a320: 2020 2020 332c 3130 2020 2020 2020 312c 3,10 1, │ │ │ │ +0000a330: 3420 2020 322c 3220 2020 2020 2031 2c31 4 2,2 1,1 │ │ │ │ +0000a340: 2032 2c31 2020 2020 312c 3420 322c 3220 2,1 1,4 2,2 │ │ │ │ +0000a350: 2020 2020 2033 2c31 2020 2020 207c 0a7c 3,1 |.| │ │ │ │ 0000a360: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ 0000a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -0000a3b0: 2020 2020 2020 5420 2020 5420 2020 202d T T - │ │ │ │ -0000a3c0: 2054 2020 2054 2020 2020 2b20 7a2a 5420 T T + z*T │ │ │ │ -0000a3d0: 2020 202b 2078 2a54 2020 2029 2c20 287b + x*T ), ({ │ │ │ │ -0000a3e0: 5420 2020 2c20 5420 2020 7d2c 202d 2054 T , T }, - T │ │ │ │ -0000a3f0: 2020 2054 2020 2020 2b20 2020 207c 0a7c T + |.| │ │ │ │ -0000a400: 2020 2020 2020 2031 2c35 2032 2c32 2020 1,5 2,2 │ │ │ │ -0000a410: 2020 312c 3120 322c 3320 2020 2020 2033 1,1 2,3 3 │ │ │ │ -0000a420: 2c32 2020 2020 2020 332c 3420 2020 2020 ,2 3,4 │ │ │ │ -0000a430: 2031 2c31 2020 2032 2c32 2020 2020 2020 1,1 2,2 │ │ │ │ -0000a440: 312c 3120 322c 3220 2020 2020 207c 0a7c 1,1 2,2 |.| │ │ │ │ +0000a3b0: 2020 2020 2020 287b 5420 2020 2c20 5420 ({T , T │ │ │ │ +0000a3c0: 2020 7d2c 2054 2020 2054 2020 2020 2b20 }, T T + │ │ │ │ +0000a3d0: 5420 2020 5420 2020 202b 2054 2020 2054 T T + T T │ │ │ │ +0000a3e0: 2020 2020 2b20 792a 5420 2020 202d 207a + y*T - z │ │ │ │ +0000a3f0: 2a54 2020 2029 2c20 2020 2020 207c 0a7c *T ), |.| │ │ │ │ +0000a400: 2020 2020 2020 2020 2031 2c32 2020 2032 1,2 2 │ │ │ │ +0000a410: 2c31 2020 2020 312c 3220 322c 3120 2020 ,1 1,2 2,1 │ │ │ │ +0000a420: 2031 2c33 2032 2c33 2020 2020 312c 3420 1,3 2,3 1,4 │ │ │ │ +0000a430: 322c 3420 2020 2020 2033 2c34 2020 2020 2,4 3,4 │ │ │ │ +0000a440: 2020 332c 3720 2020 2020 2020 207c 0a7c 3,7 |.| │ │ │ │ 0000a450: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ 0000a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -0000a4a0: 2020 2020 2020 782a 5420 2020 292c 2028 x*T ), ( │ │ │ │ -0000a4b0: 7b54 2020 202c 2054 2020 207d 2c20 2d20 {T , T }, - │ │ │ │ -0000a4c0: 5420 2020 5420 2020 202d 2054 2020 2054 T T - T T │ │ │ │ -0000a4d0: 2020 2020 2b20 792a 5420 2020 292c 2028 + y*T ), ( │ │ │ │ -0000a4e0: 7b54 2020 202c 2054 2020 207d 2c7c 0a7c {T , T },|.| │ │ │ │ -0000a4f0: 2020 2020 2020 2020 2033 2c33 2020 2020 3,3 │ │ │ │ -0000a500: 2020 312c 3520 2020 322c 3420 2020 2020 1,5 2,4 │ │ │ │ -0000a510: 2031 2c32 2032 2c33 2020 2020 312c 3520 1,2 2,3 1,5 │ │ │ │ -0000a520: 322c 3420 2020 2020 2033 2c35 2020 2020 2,4 3,5 │ │ │ │ -0000a530: 2020 312c 3320 2020 322c 3520 207c 0a7c 1,3 2,5 |.| │ │ │ │ +0000a4a0: 2020 2020 2020 287b 5420 2020 2c20 5420 ({T , T │ │ │ │ +0000a4b0: 2020 7d2c 2054 2020 2054 2020 2020 2d20 }, T T - │ │ │ │ +0000a4c0: 5420 2020 5420 2020 202d 207a 2a54 2020 T T - z*T │ │ │ │ +0000a4d0: 2020 2b20 7a2a 5420 2020 292c 2028 7b54 + z*T ), ({T │ │ │ │ +0000a4e0: 2020 202c 2054 2020 207d 2c20 207c 0a7c , T }, |.| │ │ │ │ +0000a4f0: 2020 2020 2020 2020 2031 2c35 2020 2032 1,5 2 │ │ │ │ +0000a500: 2c35 2020 2020 312c 3420 322c 3420 2020 ,5 1,4 2,4 │ │ │ │ +0000a510: 2031 2c35 2032 2c35 2020 2020 2020 332c 1,5 2,5 3, │ │ │ │ +0000a520: 3720 2020 2020 2033 2c39 2020 2020 2020 7 3,9 │ │ │ │ +0000a530: 312c 3320 2020 322c 3320 2020 207c 0a7c 1,3 2,3 |.| │ │ │ │ 0000a540: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ 0000a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -0000a590: 2020 2020 2020 2d20 5420 2020 5420 2020 - T T │ │ │ │ -0000a5a0: 202b 2054 2020 2054 2020 2020 2d20 7a2a + T T - z* │ │ │ │ -0000a5b0: 5420 2020 202b 2078 2a54 2020 2029 2c20 T + x*T ), │ │ │ │ -0000a5c0: 287b 5420 2020 2c20 5420 2020 7d2c 202d ({T , T }, - │ │ │ │ -0000a5d0: 2054 2020 2054 2020 2020 2d20 207c 0a7c T T - |.| │ │ │ │ -0000a5e0: 2020 2020 2020 2020 2031 2c34 2032 2c33 1,4 2,3 │ │ │ │ -0000a5f0: 2020 2020 312c 3320 322c 3520 2020 2020 1,3 2,5 │ │ │ │ -0000a600: 2033 2c38 2020 2020 2020 332c 3920 2020 3,8 3,9 │ │ │ │ -0000a610: 2020 2031 2c32 2020 2032 2c32 2020 2020 1,2 2,2 │ │ │ │ -0000a620: 2020 312c 3220 322c 3220 2020 207c 0a7c 1,2 2,2 |.| │ │ │ │ +0000a590: 2020 2020 2020 5420 2020 5420 2020 202b T T + │ │ │ │ +0000a5a0: 2054 2020 2054 2020 2020 2b20 5420 2020 T T + T │ │ │ │ +0000a5b0: 5420 2020 202b 2079 2a54 2020 2020 2d20 T + y*T - │ │ │ │ +0000a5c0: 7a2a 5420 2020 292c 2028 7b54 2020 202c z*T ), ({T , │ │ │ │ +0000a5d0: 2054 2020 207d 2c20 2020 2020 207c 0a7c T }, |.| │ │ │ │ +0000a5e0: 2020 2020 2020 2031 2c32 2032 2c31 2020 1,2 2,1 │ │ │ │ +0000a5f0: 2020 312c 3320 322c 3320 2020 2031 2c34 1,3 2,3 1,4 │ │ │ │ +0000a600: 2032 2c34 2020 2020 2020 332c 3420 2020 2,4 3,4 │ │ │ │ +0000a610: 2020 2033 2c37 2020 2020 2020 312c 3320 3,7 1,3 │ │ │ │ +0000a620: 2020 322c 3120 2020 2020 2020 207c 0a7c 2,1 |.| │ │ │ │ 0000a630: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ 0000a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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│ -0000af70: 2020 2020 312c 3320 322c 3520 2020 2020 1,3 2,5 │ │ │ │ -0000af80: 2033 2c38 2020 2020 2020 2020 207c 0a7c 3,8 |.| │ │ │ │ +0000aef0: 2020 2020 2020 792a 5420 2020 292c 2028 y*T ), ( │ │ │ │ +0000af00: 7b54 2020 202c 2054 2020 207d 2c20 2d20 {T , T }, - │ │ │ │ +0000af10: 5420 2020 5420 2020 202d 2054 2020 2054 T T - T T │ │ │ │ +0000af20: 2020 2020 2d20 5420 2020 5420 2020 202b - T T + │ │ │ │ +0000af30: 207a 2a54 2020 2020 2b20 2020 207c 0a7c z*T + |.| │ │ │ │ +0000af40: 2020 2020 2020 2020 2033 2c37 2020 2020 3,7 │ │ │ │ +0000af50: 2020 312c 3320 2020 322c 3120 2020 2020 1,3 2,1 │ │ │ │ +0000af60: 2031 2c33 2032 2c31 2020 2020 312c 3520 1,3 2,1 1,5 │ │ │ │ +0000af70: 322c 3220 2020 2031 2c31 2032 2c33 2020 2,2 1,1 2,3 │ │ │ │ +0000af80: 2020 2020 332c 3220 2020 2020 207c 0a7c 3,2 |.| │ │ │ │ 0000af90: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ 0000afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000afb0: 2d2d 2d2d 2d2d 2d2d 2d2d 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- │ │ │ │ +0000b000: 5420 2020 5420 2020 202b 2078 2a54 2020 T T + x*T │ │ │ │ +0000b010: 2029 2c20 287b 5420 2020 2c20 5420 2020 ), ({T , T │ │ │ │ +0000b020: 7d2c 202d 2054 2020 2054 2020 207c 0a7c }, - T T |.| │ │ │ │ +0000b030: 2020 2020 2020 2020 2033 2c34 2020 2020 3,4 │ │ │ │ +0000b040: 2020 312c 3120 2020 322c 3220 2020 2020 1,1 2,2 │ │ │ │ +0000b050: 2031 2c31 2032 2c32 2020 2020 2020 332c 1,1 2,2 3, │ │ │ │ +0000b060: 3320 2020 2020 2031 2c35 2020 2032 2c34 3 1,5 2,4 │ │ │ │ +0000b070: 2020 2020 2020 312c 3220 322c 337c 0a7c 1,2 2,3|.| │ │ │ │ 0000b080: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ 0000b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -0000b0d0: 2020 2020 2020 287b 5420 2020 2c20 5420 ({T , T │ │ │ │ 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│ │ │ 0000b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -0000b3a0: 2020 2020 2020 5420 2020 5420 2020 202d T T - │ │ │ │ -0000b3b0: 2054 2020 2054 2020 2020 2b20 782a 5420 T T + x*T │ │ │ │ -0000b3c0: 2020 2029 2c20 287b 5420 2020 2c20 5420 ), ({T , T │ │ │ │ -0000b3d0: 2020 7d2c 202d 2054 2020 2054 2020 2020 }, - T T │ │ │ │ -0000b3e0: 2d20 5420 2020 5420 2020 202b 207c 0a7c - T T + |.| │ │ │ │ -0000b3f0: 2020 2020 2020 2031 2c34 2032 2c31 2020 1,4 2,1 │ │ │ │ -0000b400: 2020 312c 3120 322c 3520 2020 2020 2033 1,1 2,5 3 │ │ │ │ -0000b410: 2c31 3020 2020 2020 2031 2c34 2020 2032 ,10 1,4 2 │ │ │ │ -0000b420: 2c32 2020 2020 2020 312c 3120 322c 3120 ,2 1,1 2,1 │ │ │ │ -0000b430: 2020 2031 2c34 2032 2c32 2020 207c 0a7c 1,4 2,2 |.| │ │ │ │ +0000b3a0: 2020 2020 2020 2d20 5420 2020 5420 2020 - T T │ │ │ │ +0000b3b0: 202d 2054 2020 2054 2020 2020 2b20 792a - T T + y* │ │ │ │ +0000b3c0: 5420 2020 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---------------- │ │ │ │ 0000bde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ 0000bdf0: 2020 2020 2020 5420 2020 5420 2020 202b T T + │ │ │ │ 0000be00: 2054 2020 2054 2020 2020 2d20 7a2a 5420 T T - z*T │ │ │ │ -0000be10: 2020 202b 2078 2a54 2020 2029 7d20 2020 + x*T )} │ │ │ │ +0000be10: 2020 202b 2079 2a54 2020 2029 7d20 2020 + y*T )} │ │ │ │ 0000be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000be30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0000be40: 2020 2020 2020 2031 2c35 2032 2c33 2020 1,5 2,3 │ │ │ │ -0000be50: 2020 312c 3320 322c 3420 2020 2020 2033 1,3 2,4 3 │ │ │ │ -0000be60: 2c35 2020 2020 2020 332c 3620 2020 2020 ,5 3,6 │ │ │ │ +0000be40: 2020 2020 2020 2031 2c34 2032 2c31 2020 1,4 2,1 │ │ │ │ +0000be50: 2020 312c 3320 322c 3220 2020 2020 2033 1,3 2,2 3 │ │ │ │ +0000be60: 2c31 2020 2020 2020 332c 3320 2020 2020 ,1 3,3 │ │ │ │ 0000be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000be80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000beb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bed0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ ├── ./usr/share/info/HyperplaneArrangements.info.gz │ │ │ ├── HyperplaneArrangements.info │ │ │ │ @@ -7211,16 +7211,16 @@ │ │ │ │ 0001c2a0: 203a 2063 4127 2720 3d20 7472 696d 2063 : cA'' = trim c │ │ │ │ 0001c2b0: 6f6e 6528 412c 2078 2920 2020 2020 2020 one(A, x) │ │ │ │ 0001c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c2d0: 2020 2020 207c 0a7c 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: 7c0a 7c6f 3134 203d 207b 792c 2078 2c20 |.|o14 = {y, x, │ │ │ │ -0001c320: 7820 2d20 797d 2020 2020 2020 2020 2020 x - y} │ │ │ │ +0001c310: 7c0a 7c6f 3134 203d 207b 7820 2d20 792c |.|o14 = {x - y, │ │ │ │ +0001c320: 2079 2c20 787d 2020 2020 2020 2020 2020 y, x} │ │ │ │ 0001c330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c340: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0001c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c380: 2020 2020 2020 7c0a 7c6f 3134 203a 2048 |.|o14 : H │ │ │ │ 0001c390: 7970 6572 706c 616e 6520 4172 7261 6e67 yperplane Arrang │ │ │ │ @@ -9843,15 +9843,15 @@ │ │ │ │ 00026720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026730: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00026740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026780: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ -00026790: 2028 7b78 202d 207a 2c20 7a2c 2078 7d2c ({x - z, z, x}, │ │ │ │ +00026790: 2028 7b7a 2c20 782c 2078 202d 207a 7d2c ({z, x, x - z}, │ │ │ │ 000267a0: 207b 312c 2031 2c20 317d 2920 2020 2020 {1, 1, 1}) │ │ │ │ 000267b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000267e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9902,16 +9902,16 @@ │ │ │ │ 00026ad0: 6d65 6e74 2c20 6469 6666 6572 656e 7420 ment, different │ │ │ │ 00026ae0: 6d75 6c74 6970 6c69 6369 7469 6573 7c0a multiplicities|. │ │ │ │ 00026af0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00026b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00026b40: 6f35 203d 207b 7820 2d20 7a2c 207a 2c20 o5 = {x - z, z, │ │ │ │ -00026b50: 787d 2020 2020 2020 2020 2020 2020 2020 x} │ │ │ │ +00026b40: 6f35 203d 207b 7a2c 2078 2c20 7820 2d20 o5 = {z, x, x - │ │ │ │ +00026b50: 7a7d 2020 2020 2020 2020 2020 2020 2020 z} │ │ │ │ 00026b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00026b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9959,16 +9959,16 @@ │ │ │ │ 00026e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00026e70: 6938 203a 2028 4127 272c 6d27 2729 203d i8 : (A'',m'') = │ │ │ │ 00026e80: 2065 756c 6572 5265 7374 7269 6374 696f eulerRestrictio │ │ │ │ 00026e90: 6e28 412c 6d2c 3329 2020 7c0a 7c20 2020 n(A,m,3) |.| │ │ │ │ 00026ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026ec0: 2020 2020 2020 207c 0a7c 6f38 203d 2028 |.|o8 = ( │ │ │ │ -00026ed0: 7b79 202d 207a 2c20 7a2c 2079 7d2c 207b {y - z, z, y}, { │ │ │ │ -00026ee0: 312c 2032 2c20 337d 2920 2020 2020 2020 1, 2, 3}) │ │ │ │ +00026ed0: 7b7a 2c20 792c 2079 202d 207a 7d2c 207b {z, y, y - z}, { │ │ │ │ +00026ee0: 322c 2033 2c20 317d 2920 2020 2020 2020 2, 3, 1}) │ │ │ │ 00026ef0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00026f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026f20: 207c 0a7c 6f38 203a 2053 6571 7565 6e63 |.|o8 : Sequenc │ │ │ │ 00026f30: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ 00026f40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00026f50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ @@ -10058,22 +10058,22 @@ │ │ │ │ 00027490: 2c31 2c31 2c31 2c31 2c31 2c31 2c31 2c31 ,1,1,1,1,1,1,1,1 │ │ │ │ 000274a0: 7d2c 3029 2020 2020 2020 2020 2020 2020 },0) │ │ │ │ 000274b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000274c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027500: 7c0a 7c6f 3133 203d 2028 7b78 202c 2078 |.|o13 = ({x , x │ │ │ │ -00027510: 2020 2b20 7820 2c20 7820 2c20 7820 2c20 + x , x , x , │ │ │ │ -00027520: 7820 202b 2078 2020 2b20 7820 2c20 7820 x + x + x , x │ │ │ │ -00027530: 202b 2078 207d 2c20 7b31 2c20 312c 2031 + x }, {1, 1, 1 │ │ │ │ +00027500: 7c0a 7c6f 3133 203d 2028 7b78 2020 2b20 |.|o13 = ({x + │ │ │ │ +00027510: 7820 2c20 7820 2c20 7820 2c20 7820 202b x , x , x , x + │ │ │ │ +00027520: 2078 2020 2b20 7820 2c20 7820 202b 2078 x + x , x + x │ │ │ │ +00027530: 202c 2078 207d 2c20 7b31 2c20 312c 2031 , x }, {1, 1, 1 │ │ │ │ 00027540: 2c20 312c 2031 2c20 317d 297c 0a7c 2020 , 1, 1, 1})|.| │ │ │ │ -00027550: 2020 2020 2020 2032 2020 2033 2020 2020 2 3 │ │ │ │ -00027560: 3420 2020 3320 2020 3420 2020 3220 2020 4 3 4 2 │ │ │ │ -00027570: 2033 2020 2020 3420 2020 3220 2020 2034 3 4 2 4 │ │ │ │ +00027550: 2020 2020 2020 2033 2020 2020 3420 2020 3 4 │ │ │ │ +00027560: 3320 2020 3420 2020 3220 2020 2033 2020 3 4 2 3 │ │ │ │ +00027570: 2020 3420 2020 3220 2020 2034 2020 2032 4 2 4 2 │ │ │ │ 00027580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027590: 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ 000275e0: 207c 0a7c 6f31 3320 3a20 5365 7175 656e |.|o13 : Sequen │ │ │ │ @@ -18275,16 +18275,16 @@ │ │ │ │ 00047620: 3320 3a20 4127 203d 2074 7269 6d20 4120 3 : A' = trim A │ │ │ │ 00047630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047650: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00047660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047690: 2020 2020 2020 7c0a 7c6f 3320 3d20 7b78 |.|o3 = {x │ │ │ │ -000476a0: 202b 2079 2c20 792c 2078 7d20 2020 2020 + y, y, x} │ │ │ │ +00047690: 2020 2020 2020 7c0a 7c6f 3320 3d20 7b79 |.|o3 = {y │ │ │ │ +000476a0: 2c20 782c 2078 202b 2079 7d20 2020 2020 , x, x + y} │ │ │ │ 000476b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000476c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000476d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000476e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000476f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047710: 7c0a 7c6f 3320 3a20 4879 7065 7270 6c61 |.|o3 : Hyperpla │ │ │ │ @@ -19192,20 +19192,20 @@ │ │ │ │ 0004af70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004af80: 2020 2020 2020 7c0a 7c20 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: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004afd0: 2020 2020 2020 7c0a 7c6f 3620 3d20 7b78 |.|o6 = {x │ │ │ │ -0004afe0: 202c 2078 202c 2078 2020 2b20 7820 7d20 , x , x + x } │ │ │ │ +0004afe0: 2020 2b20 7820 2c20 7820 2c20 7820 7d20 + x , x , x } │ │ │ │ 0004aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b020: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0004b030: 3220 2020 3120 2020 3120 2020 2032 2020 2 1 1 2 │ │ │ │ +0004b030: 3120 2020 2032 2020 2032 2020 2031 2020 1 2 2 1 │ │ │ │ 0004b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b070: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0004b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b0a0: 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 2d2b 0a7c 6934 203a 2074 696d -----+.|i4 : tim │ │ │ │ 000118c0: 6520 5227 203d 2069 6e74 6567 7261 6c43 e R' = integralC │ │ │ │ 000118d0: 6c6f 7375 7265 2052 2020 2020 2020 2020 losure R │ │ │ │ 000118e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000118f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011900: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00011910: 2030 2e33 3837 3839 3673 2028 6370 7529 0.387896s (cpu) │ │ │ │ -00011920: 3b20 302e 3330 3335 3935 7320 2874 6872 ; 0.303595s (thr │ │ │ │ +00011910: 2030 2e33 3836 3136 3573 2028 6370 7529 0.386165s (cpu) │ │ │ │ +00011920: 3b20 302e 3330 3939 3235 7320 2874 6872 ; 0.309925s (thr │ │ │ │ 00011930: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00011940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011950: 2020 2020 207c 0a7c 2020 2020 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 2d2b 0a7c 6931 3020 3a20 7469 -----+.|i10 : ti │ │ │ │ 00013760: 6d65 2052 2720 3d20 696e 7465 6772 616c me R' = integral │ │ │ │ 00013770: 436c 6f73 7572 6528 522c 2053 7472 6174 Closure(R, Strat │ │ │ │ 00013780: 6567 7920 3d3e 2052 6164 6963 616c 2920 egy => Radical) │ │ │ │ 00013790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000137a0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -000137b0: 2030 2e35 3038 3332 3973 2028 6370 7529 0.508329s (cpu) │ │ │ │ -000137c0: 3b20 302e 3334 3232 3931 7320 2874 6872 ; 0.342291s (thr │ │ │ │ +000137b0: 2030 2e33 3630 3538 3673 2028 6370 7529 0.360586s (cpu) │ │ │ │ +000137c0: 3b20 302e 3238 3738 3039 7320 2874 6872 ; 0.287809s (thr │ │ │ │ 000137d0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 000137e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000137f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00013800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013820: 2020 2020 2020 2020 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 2d2b 0a7c 6931 3620 3a20 7469 -----+.|i16 : ti │ │ │ │ 00015600: 6d65 2052 2720 3d20 696e 7465 6772 616c me R' = integral │ │ │ │ 00015610: 436c 6f73 7572 6528 522c 2053 7472 6174 Closure(R, Strat │ │ │ │ 00015620: 6567 7920 3d3e 2041 6c6c 436f 6469 6d65 egy => AllCodime │ │ │ │ 00015630: 6e73 696f 6e73 2920 2020 2020 2020 2020 nsions) │ │ │ │ 00015640: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00015650: 2030 2e35 3037 3230 3273 2028 6370 7529 0.507202s (cpu) │ │ │ │ -00015660: 3b20 302e 3334 3933 3038 7320 2874 6872 ; 0.349308s (thr │ │ │ │ +00015650: 2030 2e34 3838 3132 3273 2028 6370 7529 0.488122s (cpu) │ │ │ │ +00015660: 3b20 302e 3333 3238 3532 7320 2874 6872 ; 0.332852s (thr │ │ │ │ 00015670: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00015680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015690: 2020 2020 207c 0a7c 2020 2020 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 2d2b 0a7c 6932 3120 3a20 7469 -----+.|i21 : ti │ │ │ │ 000171d0: 6d65 2052 2720 3d20 696e 7465 6772 616c me R' = integral │ │ │ │ 000171e0: 436c 6f73 7572 6528 522c 2053 7472 6174 Closure(R, Strat │ │ │ │ 000171f0: 6567 7920 3d3e 2053 696d 706c 6966 7946 egy => SimplifyF │ │ │ │ 00017200: 7261 6374 696f 6e73 2920 2020 2020 2020 ractions) │ │ │ │ 00017210: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00017220: 2030 2e34 3030 3234 3973 2028 6370 7529 0.400249s (cpu) │ │ │ │ -00017230: 3b20 302e 3331 3736 3838 7320 2874 6872 ; 0.317688s (thr │ │ │ │ +00017220: 2030 2e33 3839 3931 3973 2028 6370 7529 0.389919s (cpu) │ │ │ │ +00017230: 3b20 302e 3331 3538 3133 7320 2874 6872 ; 0.315813s (thr │ │ │ │ 00017240: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00017250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017260: 2020 2020 207c 0a7c 2020 2020 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 │ │ │ │ @@ -6361,17 +6361,17 @@ │ │ │ │ 00018d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018d90: 2d2d 2d2d 2d2b 0a7c 6932 3620 3a20 7469 -----+.|i26 : ti │ │ │ │ 00018da0: 6d65 2052 2720 3d20 696e 7465 6772 616c me R' = integral │ │ │ │ 00018db0: 436c 6f73 7572 6520 2852 2c20 5374 7261 Closure (R, Stra │ │ │ │ 00018dc0: 7465 6779 203d 3e20 5261 6469 6361 6c43 tegy => RadicalC │ │ │ │ 00018dd0: 6f64 696d 3129 2020 2020 2020 2020 2020 odim1) │ │ │ │ 00018de0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00018df0: 2030 2e37 3936 3235 7320 2863 7075 293b 0.79625s (cpu); │ │ │ │ -00018e00: 2030 2e35 3530 3638 3573 2028 7468 7265 0.550685s (thre │ │ │ │ -00018e10: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +00018df0: 2030 2e37 3732 3737 3373 2028 6370 7529 0.772773s (cpu) │ │ │ │ +00018e00: 3b20 302e 3534 3035 3737 7320 2874 6872 ; 0.540577s (thr │ │ │ │ +00018e10: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00018e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00018e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e80: 2020 2020 207c 0a7c 6f32 3620 3d20 5227 |.|o26 = R' │ │ │ │ @@ -6806,16 +6806,16 @@ │ │ │ │ 0001a950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a960: 2d2d 2d2d 2d2b 0a7c 6933 3120 3a20 7469 -----+.|i31 : ti │ │ │ │ 0001a970: 6d65 2052 2720 3d20 696e 7465 6772 616c me R' = integral │ │ │ │ 0001a980: 436c 6f73 7572 6520 2852 2c20 5374 7261 Closure (R, Stra │ │ │ │ 0001a990: 7465 6779 203d 3e20 5661 7363 6f6e 6365 tegy => Vasconce │ │ │ │ 0001a9a0: 6c6f 7329 2020 2020 2020 2020 2020 2020 los) │ │ │ │ 0001a9b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001a9c0: 2030 2e35 3038 3537 3973 2028 6370 7529 0.508579s (cpu) │ │ │ │ -0001a9d0: 3b20 302e 3333 3632 3933 7320 2874 6872 ; 0.336293s (thr │ │ │ │ +0001a9c0: 2030 2e34 3737 3431 3573 2028 6370 7529 0.477415s (cpu) │ │ │ │ +0001a9d0: 3b20 302e 3332 3637 3239 7320 2874 6872 ; 0.326729s (thr │ │ │ │ 0001a9e0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 0001a9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa30: 2020 2020 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: 2d2b 0a7c 6933 3620 3a20 7469 6d65 2052 -+.|i36 : time R │ │ │ │ 0001c3a0: 2720 3d20 696e 7465 6772 616c 436c 6f73 ' = integralClos │ │ │ │ 0001c3b0: 7572 6520 5220 2020 2020 2020 2020 2020 ure R │ │ │ │ 0001c3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001c3d0: 2d2d 2075 7365 6420 302e 3034 3238 3334 -- used 0.042834 │ │ │ │ -0001c3e0: 3573 2028 6370 7529 3b20 302e 3034 3238 5s (cpu); 0.0428 │ │ │ │ -0001c3f0: 3231 3273 2028 7468 7265 6164 293b 2030 212s (thread); 0 │ │ │ │ -0001c400: 7320 2867 6329 207c 0a7c 2020 2020 2020 s (gc) |.| │ │ │ │ +0001c3d0: 2d2d 2075 7365 6420 302e 3034 3636 3934 -- used 0.046694 │ │ │ │ +0001c3e0: 3273 2028 6370 7529 3b20 302e 3034 3636 2s (cpu); 0.0466 │ │ │ │ +0001c3f0: 3936 7320 2874 6872 6561 6429 3b20 3073 96s (thread); 0s │ │ │ │ +0001c400: 2028 6763 2920 207c 0a7c 2020 2020 2020 (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 7c0a 7c6f 3336 203d 2052 2720 2020 |.|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 207c 0a7c |.| │ │ │ │ @@ -7412,18 +7412,18 @@ │ │ │ │ 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 2d2b 0a7c 6934 3120 3a20 7469 -----+.|i41 : ti │ │ │ │ 0001cf70: 6d65 2052 2720 3d20 696e 7465 6772 616c me R' = integral │ │ │ │ 0001cf80: 436c 6f73 7572 6528 522c 2053 7472 6174 Closure(R, Strat │ │ │ │ 0001cf90: 6567 7920 3d3e 2052 6164 6963 616c 2920 egy => Radical) │ │ │ │ -0001cfa0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3136 |.| -- used 0.16 │ │ │ │ -0001cfb0: 3735 3233 7320 2863 7075 293b 2030 2e30 7523s (cpu); 0.0 │ │ │ │ -0001cfc0: 3832 3832 3439 7320 2874 6872 6561 6429 828249s (thread) │ │ │ │ -0001cfd0: 3b20 3073 2028 6763 2920 207c 0a7c 2020 ; 0s (gc) |.| │ │ │ │ +0001cfa0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3135 |.| -- used 0.15 │ │ │ │ +0001cfb0: 3137 3373 2028 6370 7529 3b20 302e 3037 173s (cpu); 0.07 │ │ │ │ +0001cfc0: 3134 3037 3573 2028 7468 7265 6164 293b 14075s (thread); │ │ │ │ +0001cfd0: 2030 7320 2867 6329 2020 207c 0a7c 2020 0s (gc) |.| │ │ │ │ 0001cfe0: 2020 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 7c0a 7c6f 3431 203d 2052 |.|o41 = R │ │ │ │ 0001d020: 2720 2020 2020 2020 2020 2020 2020 2020 ' │ │ │ │ 0001d030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7554,17 +7554,17 @@ │ │ │ │ 0001d810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0001d840: 6934 3620 3a20 7469 6d65 2052 2720 3d20 i46 : time R' = │ │ │ │ 0001d850: 696e 7465 6772 616c 436c 6f73 7572 6528 integralClosure( │ │ │ │ 0001d860: 522c 2053 7472 6174 6567 7920 3d3e 2041 R, Strategy => A │ │ │ │ 0001d870: 6c6c 436f 6469 6d65 6e73 696f 6e73 297c llCodimensions)| │ │ │ │ -0001d880: 0a7c 202d 2d20 7573 6564 2030 2e30 3633 .| -- used 0.063 │ │ │ │ -0001d890: 3938 3731 7320 2863 7075 293b 2030 2e30 9871s (cpu); 0.0 │ │ │ │ -0001d8a0: 3633 3938 3639 7320 2874 6872 6561 6429 639869s (thread) │ │ │ │ +0001d880: 0a7c 202d 2d20 7573 6564 2030 2e30 3636 .| -- used 0.066 │ │ │ │ +0001d890: 3834 3933 7320 2863 7075 293b 2030 2e30 8493s (cpu); 0.0 │ │ │ │ +0001d8a0: 3636 3834 3631 7320 2874 6872 6561 6429 668461s (thread) │ │ │ │ 0001d8b0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0001d8c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d900: 2020 207c 0a7c 6f34 3620 3d20 5227 2020 |.|o46 = R' │ │ │ │ 0001d910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7697,18 +7697,18 @@ │ │ │ │ 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 2d2b ---------------+ │ │ │ │ 0001e130: 0a7c 6935 3120 3a20 7469 6d65 2052 2720 .|i51 : time R' │ │ │ │ 0001e140: 3d20 696e 7465 6772 616c 436c 6f73 7572 = integralClosur │ │ │ │ 0001e150: 6520 2852 2c20 5374 7261 7465 6779 203d e (R, Strategy = │ │ │ │ 0001e160: 3e20 5261 6469 6361 6c43 6f64 696d 3129 > RadicalCodim1) │ │ │ │ -0001e170: 7c0a 7c20 2d2d 2075 7365 6420 302e 3034 |.| -- used 0.04 │ │ │ │ -0001e180: 3334 3330 3673 2028 6370 7529 3b20 302e 34306s (cpu); 0. │ │ │ │ -0001e190: 3034 3334 3331 3573 2028 7468 7265 6164 0434315s (thread │ │ │ │ -0001e1a0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +0001e170: 7c0a 7c20 2d2d 2075 7365 6420 302e 3135 |.| -- used 0.15 │ │ │ │ +0001e180: 3135 3532 7320 2863 7075 293b 2030 2e30 1552s (cpu); 0.0 │ │ │ │ +0001e190: 3735 3332 3833 7320 2874 6872 6561 6429 753283s (thread) │ │ │ │ +0001e1a0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0001e1b0: 207c 0a7c 2020 2020 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 7c0a 7c6f 3531 203d 2052 2720 2020 |.|o51 = R' │ │ │ │ 0001e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7841,18 +7841,18 @@ │ │ │ │ 0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 0001ea30: 3620 3a20 7469 6d65 2052 2720 3d20 696e 6 : time R' = in │ │ │ │ 0001ea40: 7465 6772 616c 436c 6f73 7572 6520 2852 tegralClosure (R │ │ │ │ 0001ea50: 2c20 5374 7261 7465 6779 203d 3e20 5661 , Strategy => Va │ │ │ │ 0001ea60: 7363 6f6e 6365 6c6f 7329 7c0a 7c20 2d2d sconcelos)|.| -- │ │ │ │ -0001ea70: 2075 7365 6420 302e 3137 3538 3033 7320 used 0.175803s │ │ │ │ -0001ea80: 2863 7075 293b 2030 2e30 3839 3831 3639 (cpu); 0.0898169 │ │ │ │ -0001ea90: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -0001eaa0: 6763 2920 2020 2020 207c 0a7c 2020 2020 gc) |.| │ │ │ │ +0001ea70: 2075 7365 6420 302e 3034 3537 3738 3273 used 0.0457782s │ │ │ │ +0001ea80: 2028 6370 7529 3b20 302e 3034 3537 3738 (cpu); 0.045778 │ │ │ │ +0001ea90: 3173 2028 7468 7265 6164 293b 2030 7320 1s (thread); 0s │ │ │ │ +0001eaa0: 2867 6329 2020 2020 207c 0a7c 2020 2020 (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 7c0a 7c6f 3536 203d |.|o56 = │ │ │ │ 0001eaf0: 2052 2720 2020 2020 2020 2020 2020 2020 R' │ │ │ │ 0001eb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8362,16 +8362,16 @@ │ │ │ │ 00020a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020aa0: 2b0a 7c69 3637 203a 2074 696d 6520 5227 +.|i67 : time R' │ │ │ │ 00020ab0: 203d 2069 6e74 6567 7261 6c43 6c6f 7375 = integralClosu │ │ │ │ 00020ac0: 7265 2852 2c20 5374 7261 7465 6779 203d re(R, Strategy = │ │ │ │ 00020ad0: 3e20 5261 6469 6361 6c29 2020 2020 2020 > Radical) │ │ │ │ 00020ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020af0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3036 |.| -- used 0.06 │ │ │ │ -00020b00: 3232 3733 3473 2028 6370 7529 3b20 302e 22734s (cpu); 0. │ │ │ │ -00020b10: 3036 3232 3734 3573 2028 7468 7265 6164 0622745s (thread │ │ │ │ +00020b00: 3539 3738 3573 2028 6370 7529 3b20 302e 59785s (cpu); 0. │ │ │ │ +00020b10: 3036 3539 3739 3273 2028 7468 7265 6164 0659792s (thread │ │ │ │ 00020b20: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00020b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b40: 7c0a 7c20 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 2020 2020 2020 │ │ │ │ @@ -8856,17 +8856,17 @@ │ │ │ │ 00022970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022990: 2b0a 7c69 3738 203a 2074 696d 6520 5227 +.|i78 : time R' │ │ │ │ 000229a0: 203d 2069 6e74 6567 7261 6c43 6c6f 7375 = integralClosu │ │ │ │ 000229b0: 7265 2852 2c20 5374 7261 7465 6779 203d re(R, Strategy = │ │ │ │ 000229c0: 3e20 5261 6469 6361 6c29 2020 2020 2020 > Radical) │ │ │ │ 000229d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229e0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3530 |.| -- used 0.50 │ │ │ │ -000229f0: 3035 3533 7320 2863 7075 293b 2030 2e33 0553s (cpu); 0.3 │ │ │ │ -00022a00: 3333 3431 3273 2028 7468 7265 6164 293b 33412s (thread); │ │ │ │ +000229e0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3437 |.| -- used 0.47 │ │ │ │ +000229f0: 3437 3933 7320 2863 7075 293b 2030 2e33 4793s (cpu); 0.3 │ │ │ │ +00022a00: 3135 3938 3373 2028 7468 7265 6164 293b 15983s (thread); │ │ │ │ 00022a10: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00022a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8989,18 +8989,18 @@ │ │ │ │ 000231c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000231e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3832 ----------+.|i82 │ │ │ │ 000231f0: 203a 2074 696d 6520 5227 203d 2069 6e74 : time R' = int │ │ │ │ 00023200: 6567 7261 6c43 6c6f 7375 7265 2852 2c20 egralClosure(R, │ │ │ │ 00023210: 5374 7261 7465 6779 203d 3e20 416c 6c43 Strategy => AllC │ │ │ │ 00023220: 6f64 696d 656e 7369 6f6e 7329 7c0a 7c20 odimensions)|.| │ │ │ │ -00023230: 2d2d 2075 7365 6420 302e 3534 3331 3833 -- used 0.543183 │ │ │ │ -00023240: 7320 2863 7075 293b 2030 2e33 3835 3438 s (cpu); 0.38548 │ │ │ │ -00023250: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -00023260: 6763 2920 2020 2020 2020 2020 2020 7c0a gc) |. │ │ │ │ +00023230: 2d2d 2075 7365 6420 302e 3539 3436 3538 -- used 0.594658 │ │ │ │ +00023240: 7320 2863 7075 293b 2030 2e33 3635 3337 s (cpu); 0.36537 │ │ │ │ +00023250: 3673 2028 7468 7265 6164 293b 2030 7320 6s (thread); 0s │ │ │ │ +00023260: 2867 6329 2020 2020 2020 2020 2020 7c0a (gc) |. │ │ │ │ 00023270: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00023280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232b0: 7c0a 7c6f 3832 203d 2052 2720 2020 2020 |.|o82 = R' │ │ │ │ 000232c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9122,40 +9122,40 @@ │ │ │ │ 00023a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3836 ----------+.|i86 │ │ │ │ 00023a20: 203a 2074 696d 6520 5227 203d 2069 6e74 : time R' = int │ │ │ │ 00023a30: 6567 7261 6c43 6c6f 7375 7265 2028 522c egralClosure (R, │ │ │ │ 00023a40: 2053 7472 6174 6567 7920 3d3e 2052 6164 Strategy => Rad │ │ │ │ 00023a50: 6963 616c 436f 6469 6d31 2c20 5665 7262 icalCodim1, Verb │ │ │ │ 00023a60: 6f73 6974 7920 3d3e 2020 7c0a 7c20 5b6a osity => |.| [j │ │ │ │ 00023a70: 6163 6f62 6961 6e20 7469 6d65 202e 3030 acobian time .00 │ │ │ │ -00023a80: 3035 3332 3335 3820 7365 6320 236d 696e 0532358 sec #min │ │ │ │ +00023a80: 3035 3538 3835 3320 7365 6320 236d 696e 0558853 sec #min │ │ │ │ 00023a90: 6f72 7320 345d 2020 2020 2020 2020 2020 ors 4] │ │ │ │ 00023aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ab0: 2020 2020 2020 2020 2020 7c0a 7c69 6e74 |.|int │ │ │ │ 00023ac0: 6567 7261 6c20 636c 6f73 7572 6520 6e76 egral closure nv │ │ │ │ 00023ad0: 6172 7320 3420 6e75 6d67 656e 7320 3120 ars 4 numgens 1 │ │ │ │ 00023ae0: 6973 2053 3220 636f 6469 6d20 3120 636f is S2 codim 1 co │ │ │ │ 00023af0: 6469 6d4a 2032 2020 2020 2020 2020 2020 dimJ 2 │ │ │ │ 00023b00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00023b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b50: 2020 2020 2020 2020 2020 7c0a 7c20 5b73 |.| [s │ │ │ │ -00023b60: 7465 7020 303a 2020 2074 696d 6520 2e32 tep 0: time .2 │ │ │ │ -00023b70: 3238 3435 3520 7365 6320 2023 6672 6163 28455 sec #frac │ │ │ │ +00023b60: 7465 7020 303a 2020 2074 696d 6520 2e31 tep 0: time .1 │ │ │ │ +00023b70: 3036 3138 3120 7365 6320 2023 6672 6163 06181 sec #frac │ │ │ │ 00023b80: 7469 6f6e 7320 365d 2020 2020 2020 2020 tions 6] │ │ │ │ 00023b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ba0: 2020 2020 2020 2020 2020 7c0a 7c20 5b73 |.| [s │ │ │ │ 00023bb0: 7465 7020 313a 2020 2074 696d 6520 2e32 tep 1: time .2 │ │ │ │ -00023bc0: 3436 3733 3320 7365 6320 2023 6672 6163 46733 sec #frac │ │ │ │ +00023bc0: 3435 3934 3420 7365 6320 2023 6672 6163 45944 sec #frac │ │ │ │ 00023bd0: 7469 6f6e 7320 365d 2020 2020 2020 2020 tions 6] │ │ │ │ 00023be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023bf0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00023c00: 2075 7365 6420 302e 3437 3838 3733 7320 used 0.478873s │ │ │ │ -00023c10: 2863 7075 293b 2030 2e33 3034 3735 3373 (cpu); 0.304753s │ │ │ │ +00023c00: 2075 7365 6420 302e 3335 3539 3331 7320 used 0.355931s │ │ │ │ +00023c10: 2863 7075 293b 2030 2e32 3637 3039 3573 (cpu); 0.267095s │ │ │ │ 00023c20: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 00023c30: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ 00023c40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00023c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9301,40 +9301,40 @@ │ │ │ │ 00024540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00024550: 6939 3020 3a20 7469 6d65 2052 2720 3d20 i90 : time R' = │ │ │ │ 00024560: 696e 7465 6772 616c 436c 6f73 7572 6520 integralClosure │ │ │ │ 00024570: 2852 2c20 5374 7261 7465 6779 203d 3e20 (R, Strategy => │ │ │ │ 00024580: 5661 7363 6f6e 6365 6c6f 732c 2056 6572 Vasconcelos, Ver │ │ │ │ 00024590: 626f 7369 7479 203d 3e20 3129 7c0a 7c20 bosity => 1)|.| │ │ │ │ 000245a0: 5b6a 6163 6f62 6961 6e20 7469 6d65 202e [jacobian time . │ │ │ │ -000245b0: 3030 3035 3330 3039 3420 7365 6320 236d 000530094 sec #m │ │ │ │ +000245b0: 3030 3035 3234 3533 3620 7365 6320 236d 000524536 sec #m │ │ │ │ 000245c0: 696e 6f72 7320 345d 2020 2020 2020 2020 inors 4] │ │ │ │ 000245d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000245e0: 2020 2020 2020 2020 2020 207c 0a7c 696e |.|in │ │ │ │ 000245f0: 7465 6772 616c 2063 6c6f 7375 7265 206e tegral closure n │ │ │ │ 00024600: 7661 7273 2034 206e 756d 6765 6e73 2031 vars 4 numgens 1 │ │ │ │ 00024610: 2069 7320 5332 2063 6f64 696d 2031 2063 is S2 codim 1 c │ │ │ │ 00024620: 6f64 696d 4a20 3220 2020 2020 2020 2020 odimJ 2 │ │ │ │ 00024630: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 │ │ │ │ 00024680: 2020 2020 2020 2020 207c 0a7c 205b 7374 |.| [st │ │ │ │ 00024690: 6570 2030 3a20 2020 7469 6d65 202e 3231 ep 0: time .21 │ │ │ │ -000246a0: 3034 3432 2073 6563 2020 2366 7261 6374 0442 sec #fract │ │ │ │ +000246a0: 3336 3839 2073 6563 2020 2366 7261 6374 3689 sec #fract │ │ │ │ 000246b0: 696f 6e73 2036 5d20 2020 2020 2020 2020 ions 6] │ │ │ │ 000246c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000246d0: 2020 2020 2020 2020 7c0a 7c20 5b73 7465 |.| [ste │ │ │ │ -000246e0: 7020 313a 2020 2074 696d 6520 2e32 3639 p 1: time .269 │ │ │ │ -000246f0: 3137 3420 7365 6320 2023 6672 6163 7469 174 sec #fracti │ │ │ │ +000246e0: 7020 313a 2020 2074 696d 6520 2e32 3534 p 1: time .254 │ │ │ │ +000246f0: 3231 3520 7365 6320 2023 6672 6163 7469 215 sec #fracti │ │ │ │ 00024700: 6f6e 7320 365d 2020 2020 2020 2020 2020 ons 6] │ │ │ │ 00024710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024720: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00024730: 6564 2030 2e34 3833 3332 3973 2028 6370 ed 0.483329s (cp │ │ │ │ -00024740: 7529 3b20 302e 3239 3833 3838 7320 2874 u); 0.298388s (t │ │ │ │ +00024730: 6564 2030 2e34 3731 3834 3473 2028 6370 ed 0.471844s (cp │ │ │ │ +00024740: 7529 3b20 302e 3330 3437 3233 7320 2874 u); 0.304723s (t │ │ │ │ 00024750: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00024760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024770: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00024780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000247a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000247b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9478,42 +9478,42 @@ │ │ │ │ 00025050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025060: 2b0a 7c69 3934 203a 2074 696d 6520 5227 +.|i94 : time R' │ │ │ │ 00025070: 203d 2069 6e74 6567 7261 6c43 6c6f 7375 = integralClosu │ │ │ │ 00025080: 7265 2028 522c 2053 7472 6174 6567 7920 re (R, Strategy │ │ │ │ 00025090: 3d3e 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│ │ │ 00025170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251a0: 7c0a 7c20 5b73 7465 7020 303a 2020 2074 |.| [step 0: t │ │ │ │ -000251b0: 696d 6520 2e32 3731 3131 3120 7365 6320 ime .271111 sec │ │ │ │ +000251b0: 696d 6520 2e32 3537 3932 3920 7365 6320 ime .257929 sec │ │ │ │ 000251c0: 2023 6672 6163 7469 6f6e 7320 365d 2020 #fractions 6] │ │ │ │ 000251d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000251f0: 7c0a 7c20 5b73 7465 7020 313a 2020 2074 |.| [step 1: t │ │ │ │ -00025200: 696d 6520 2e38 3734 3839 3720 7365 6320 ime .874897 sec │ │ │ │ +00025200: 696d 6520 2e36 3739 3134 3820 7365 6320 ime .679148 sec │ │ │ │ 00025210: 2023 6672 6163 7469 6f6e 7320 365d 2020 #fractions 6] │ │ │ │ 00025220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025240: 7c0a 7c20 2d2d 2075 7365 6420 312e 3135 |.| -- used 1.15 │ │ │ │ -00025250: 3032 3773 2028 6370 7529 3b20 302e 3733 027s (cpu); 0.73 │ │ │ │ -00025260: 3232 3739 7320 2874 6872 6561 6429 3b20 2279s (thread); │ │ │ │ -00025270: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +00025240: 7c0a 7c20 2d2d 2075 7365 6420 302e 3934 |.| -- used 0.94 │ │ │ │ +00025250: 3130 3031 7320 2863 7075 293b 2030 2e36 1001s (cpu); 0.6 │ │ │ │ +00025260: 3030 3838 3373 2028 7468 7265 6164 293b 00883s (thread); │ │ │ │ +00025270: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 00025280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025290: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000252a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000252e0: 7c0a 7c6f 3934 203d 2052 2720 2020 2020 |.|o94 = R' │ │ │ │ @@ -10324,15 +10324,15 @@ │ │ │ │ 00028530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00028540: 6932 203a 2074 696d 6520 5227 203d 2069 i2 : time R' = i │ │ │ │ 00028550: 6e74 6567 7261 6c43 6c6f 7375 7265 2852 ntegralClosure(R │ │ │ │ 00028560: 2c20 5665 7262 6f73 6974 7920 3d3e 2032 , Verbosity => 2 │ │ │ │ 00028570: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00028580: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028590: 205b 6a61 636f 6269 616e 2074 696d 6520 [jacobian time │ │ │ │ -000285a0: 2e30 3030 3532 3232 3739 2073 6563 2023 .000522279 sec # │ │ │ │ +000285a0: 2e30 3030 3532 3833 3834 2073 6563 2023 .000528384 sec # │ │ │ │ 000285b0: 6d69 6e6f 7273 2033 5d20 2020 2020 2020 minors 3] │ │ │ │ 000285c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000285d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000285e0: 696e 7465 6772 616c 2063 6c6f 7375 7265 integral closure │ │ │ │ 000285f0: 206e 7661 7273 2033 206e 756d 6765 6e73 nvars 3 numgens │ │ │ │ 00028600: 2031 2069 7320 5332 2063 6f64 696d 2031 1 is S2 codim 1 │ │ │ │ 00028610: 2063 6f64 696d 4a20 3220 2020 2020 2020 codimJ 2 │ │ │ │ @@ -10345,180 +10345,180 @@ │ │ │ │ 00028680: 205b 7374 6570 2030 3a20 2020 2020 2020 [step 0: │ │ │ │ 00028690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000286a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000286b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000286c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000286d0: 2020 2020 2020 7261 6469 6361 6c20 2875 radical (u │ │ │ │ 000286e0: 7365 206d 696e 7072 696d 6573 2920 2e30 se minprimes) .0 │ │ │ │ -000286f0: 3032 3333 3333 3920 7365 636f 6e64 7320 0233339 seconds │ │ │ │ +000286f0: 3032 3437 3530 3420 7365 636f 6e64 7320 0247504 seconds │ │ │ │ 00028700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028710: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028720: 2020 2020 2020 6964 6c69 7a65 7231 3a20 idlizer1: │ │ │ │ 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│ │ │ 000287e0: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ 000287f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028800: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028810: 2020 7469 6d65 202e 3033 3731 3532 3720 time .0371527 │ │ │ │ +00028810: 2020 7469 6d65 202e 3034 3233 3535 3120 time .0423551 │ │ │ │ 00028820: 7365 6320 2023 6672 6163 7469 6f6e 7320 sec #fractions │ │ │ │ 00028830: 345d 2020 2020 2020 2020 2020 2020 2020 4] │ │ │ │ 00028840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028850: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028860: 205b 7374 6570 2031 3a20 2020 2020 2020 [step 1: │ │ │ │ 00028870: 2020 2020 2020 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 207c 0a7c |.| │ │ │ │ 000288b0: 2020 2020 2020 7261 6469 6361 6c20 2875 radical (u │ │ │ │ 000288c0: 7365 206d 696e 7072 696d 6573 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0a7c |.| │ │ │ │ 00028a40: 205b 7374 6570 2032 3a20 2020 2020 2020 [step 2: │ │ │ │ 00028a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028a90: 2020 2020 2020 7261 6469 6361 6c20 2875 radical (u │ │ │ │ 00028aa0: 7365 206d 696e 7072 696d 6573 2920 2e30 se minprimes) .0 │ │ │ │ -00028ab0: 3034 3133 3835 3220 7365 636f 6e64 7320 0413852 seconds │ │ │ │ +00028ab0: 3032 3732 3638 3920 7365 636f 6e64 7320 0272689 seconds │ │ │ │ 00028ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ad0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028ae0: 2020 2020 2020 6964 6c69 7a65 7231 3a20 idlizer1: │ │ │ │ -00028af0: 202e 3031 3639 3531 3220 7365 636f 6e64 .0169512 second │ │ │ │ -00028b00: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ +00028af0: 202e 3031 3138 3331 2073 6563 6f6e 6473 .011831 seconds │ │ │ │ +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 207c 0a7c |.| │ │ │ │ 00028b30: 2020 2020 2020 6964 6c69 7a65 7232 3a20 idlizer2: │ │ │ │ -00028b40: 202e 3031 3433 3337 3220 7365 636f 6e64 .0143372 second │ │ │ │ +00028b40: 202e 3031 3131 3334 3620 7365 636f 6e64 .0111346 second │ │ │ │ 00028b50: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ 00028b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028b80: 2020 2020 2020 6d69 6e70 7265 733a 2020 minpres: │ │ │ │ -00028b90: 202e 3031 3337 3532 3520 7365 636f 6e64 .0137525 second │ │ │ │ +00028b90: 202e 3031 3034 3832 3520 7365 636f 6e64 .0104825 second │ │ │ │ 00028ba0: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ 00028bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028bd0: 2020 7469 6d65 202e 3230 3334 3936 2073 time .203496 s │ │ │ │ +00028bd0: 2020 7469 6d65 202e 3134 3338 3733 2073 time .143873 s │ │ │ │ 00028be0: 6563 2020 2366 7261 6374 696f 6e73 2035 ec #fractions 5 │ │ │ │ 00028bf0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028c20: 205b 7374 6570 2033 3a20 2020 2020 2020 [step 3: │ │ │ │ 00028c30: 2020 2020 2020 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 207c 0a7c |.| │ │ │ │ 00028c70: 2020 2020 2020 7261 6469 6361 6c20 2875 radical (u │ │ │ │ 00028c80: 7365 206d 696e 7072 696d 6573 2920 2e30 se minprimes) .0 │ │ │ │ -00028c90: 3032 3739 3936 3220 7365 636f 6e64 7320 0279962 seconds │ │ │ │ +00028c90: 3033 3036 3732 3820 7365 636f 6e64 7320 0306728 seconds │ │ │ │ 00028ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028cb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028cc0: 2020 2020 2020 6964 6c69 7a65 7231 3a20 idlizer1: │ │ │ │ -00028cd0: 202e 3031 3431 3639 3220 7365 636f 6e64 .0141692 second │ │ │ │ +00028cd0: 202e 3031 3334 3136 3720 7365 636f 6e64 .0134167 second │ │ │ │ 00028ce0: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ 00028cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028d10: 2020 2020 2020 6964 6c69 7a65 7232 3a20 idlizer2: │ │ │ │ -00028d20: 202e 3031 3636 3130 3520 7365 636f 6e64 .0166105 second │ │ │ │ +00028d20: 202e 3031 3432 3836 3120 7365 636f 6e64 .0142861 second │ │ │ │ 00028d30: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ 00028d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028d50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028d60: 2020 2020 2020 6d69 6e70 7265 733a 2020 minpres: │ │ │ │ -00028d70: 202e 3031 3838 3830 3520 7365 636f 6e64 .0188805 second │ │ │ │ -00028d80: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ +00028d70: 202e 3031 3838 3136 2073 6563 6f6e 6473 .018816 seconds │ │ │ │ +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 207c 0a7c |.| │ │ │ │ -00028db0: 2020 7469 6d65 202e 3036 3830 3638 3820 time .0680688 │ │ │ │ +00028db0: 2020 7469 6d65 202e 3036 3431 3531 3720 time .0641517 │ │ │ │ 00028dc0: 7365 6320 2023 6672 6163 7469 6f6e 7320 sec #fractions │ │ │ │ 00028dd0: 355d 2020 2020 2020 2020 2020 2020 2020 5] │ │ │ │ 00028de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028df0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028e00: 205b 7374 6570 2034 3a20 2020 2020 2020 [step 4: │ │ │ │ 00028e10: 2020 2020 2020 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 207c 0a7c |.| │ │ │ │ 00028e50: 2020 2020 2020 7261 6469 6361 6c20 2875 radical (u │ │ │ │ 00028e60: 7365 206d 696e 7072 696d 6573 2920 2e30 se minprimes) .0 │ │ │ │ -00028e70: 3032 3535 3536 2073 6563 6f6e 6473 2020 025556 seconds │ │ │ │ +00028e70: 3033 3131 3738 3520 7365 636f 6e64 7320 0311785 seconds │ │ │ │ 00028e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028e90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028ea0: 2020 2020 2020 6964 6c69 7a65 7231 3a20 idlizer1: │ │ │ │ -00028eb0: 202e 3030 3933 3538 3936 2073 6563 6f6e .00935896 secon │ │ │ │ +00028eb0: 202e 3030 3938 3232 3531 2073 6563 6f6e .00982251 secon │ │ │ │ 00028ec0: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ 00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028ef0: 2020 2020 2020 6964 6c69 7a65 7232 3a20 idlizer2: │ │ │ │ -00028f00: 202e 3031 3831 3139 3220 7365 636f 6e64 .0181192 second │ │ │ │ +00028f00: 202e 3031 3834 3630 3820 7365 636f 6e64 .0184608 second │ │ │ │ 00028f10: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ 00028f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028f40: 2020 2020 2020 6d69 6e70 7265 733a 2020 minpres: │ │ │ │ -00028f50: 202e 3031 3139 3331 3420 7365 636f 6e64 .0119314 second │ │ │ │ +00028f50: 202e 3031 3139 3730 3220 7365 636f 6e64 .0119702 second │ │ │ │ 00028f60: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ 00028f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028f90: 2020 7469 6d65 202e 3035 3535 3839 3920 time .0555899 │ │ │ │ +00028f90: 2020 7469 6d65 202e 3035 3736 3631 3420 time .0576614 │ │ │ │ 00028fa0: 7365 6320 2023 6672 6163 7469 6f6e 7320 sec #fractions │ │ │ │ 00028fb0: 355d 2020 2020 2020 2020 2020 2020 2020 5] │ │ │ │ 00028fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00028fe0: 205b 7374 6570 2035 3a20 2020 2020 2020 [step 5: │ │ │ │ 00028ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029020: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00029030: 2020 2020 2020 7261 6469 6361 6c20 2875 radical (u │ │ │ │ 00029040: 7365 206d 696e 7072 696d 6573 2920 2e30 se minprimes) .0 │ │ │ │ -00029050: 3032 3631 3835 3120 7365 636f 6e64 7320 0261851 seconds │ │ │ │ +00029050: 3032 3539 3420 7365 636f 6e64 7320 2020 02594 seconds │ │ │ │ 00029060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029070: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00029080: 2020 2020 2020 6964 6c69 7a65 7231 3a20 idlizer1: │ │ │ │ -00029090: 202e 3030 3834 3938 3934 2073 6563 6f6e .00849894 secon │ │ │ │ +00029090: 202e 3030 3930 3035 3737 2073 6563 6f6e .00900577 secon │ │ │ │ 000290a0: 6473 2020 2020 2020 2020 2020 2020 2020 ds │ │ │ │ 000290b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000290c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000290d0: 2020 7469 6d65 202e 3031 3834 3539 3720 time .0184597 │ │ │ │ +000290d0: 2020 7469 6d65 202e 3031 3931 3235 3520 time .0191255 │ │ │ │ 000290e0: 7365 6320 2023 6672 6163 7469 6f6e 7320 sec #fractions │ │ │ │ 000290f0: 355d 2020 2020 2020 2020 2020 2020 2020 5] │ │ │ │ 00029100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029110: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00029120: 202d 2d20 7573 6564 2030 2e34 3335 3036 -- used 0.43506 │ │ │ │ -00029130: 3273 2028 6370 7529 3b20 302e 3336 3131 2s (cpu); 0.3611 │ │ │ │ -00029140: 3731 7320 2874 6872 6561 6429 3b20 3073 71s (thread); 0s │ │ │ │ +00029120: 202d 2d20 7573 6564 2030 2e33 3832 3834 -- used 0.38284 │ │ │ │ +00029130: 3773 2028 6370 7529 3b20 302e 3331 3431 7s (cpu); 0.3141 │ │ │ │ +00029140: 3635 7320 2874 6872 6561 6429 3b20 3073 65s (thread); 0s │ │ │ │ 00029150: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00029160: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00029170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00029190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000291a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000291b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -11017,16 +11017,16 @@ │ │ │ │ 0002b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ 0002b0a0: 3a20 7469 6d65 2069 6e74 6567 7261 6c43 : time integralC │ │ │ │ 0002b0b0: 6c6f 7375 7265 204a 2020 2020 2020 2020 losure J │ │ │ │ 0002b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b0e0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0002b0f0: 2075 7365 6420 312e 3030 3334 3873 2028 used 1.00348s ( │ │ │ │ -0002b100: 6370 7529 3b20 302e 3733 3939 3234 7320 cpu); 0.739924s │ │ │ │ +0002b0f0: 2075 7365 6420 312e 3135 3731 3973 2028 used 1.15719s ( │ │ │ │ +0002b100: 6370 7529 3b20 302e 3639 3737 3138 7320 cpu); 0.697718s │ │ │ │ 0002b110: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0002b120: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0002b130: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -11072,16 +11072,16 @@ │ │ │ │ 0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ 0002b410: 3a20 7469 6d65 2069 6e74 6567 7261 6c43 : time integralC │ │ │ │ 0002b420: 6c6f 7375 7265 284a 2c20 5374 7261 7465 losure(J, Strate │ │ │ │ 0002b430: 6779 3d3e 7b52 6164 6963 616c 436f 6469 gy=>{RadicalCodi │ │ │ │ 0002b440: 6d31 7d29 2020 2020 2020 2020 2020 2020 m1}) │ │ │ │ 0002b450: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0002b460: 2075 7365 6420 302e 3733 3336 3734 7320 used 0.733674s │ │ │ │ -0002b470: 2863 7075 293b 2030 2e35 3636 3935 7320 (cpu); 0.56695s │ │ │ │ +0002b460: 2075 7365 6420 312e 3136 3333 3473 2028 used 1.16334s ( │ │ │ │ +0002b470: 6370 7529 3b20 302e 3534 3734 3735 7320 cpu); 0.547475s │ │ │ │ 0002b480: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0002b490: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0002b4a0: 2020 2020 2020 2020 2020 7c0a 7c20 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 2020 │ │ │ │ 0002b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/InvariantRing.info.gz │ │ │ ├── InvariantRing.info │ │ │ │ @@ -3576,16 +3576,16 @@ │ │ │ │ 0000df70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000df90: 2d2d 2d2d 2b0a 7c69 3520 3a20 656c 6170 ----+.|i5 : elap │ │ │ │ 0000dfa0: 7365 6454 696d 6520 6571 7569 7661 7269 sedTime equivari │ │ │ │ 0000dfb0: 616e 7448 696c 6265 7274 5365 7269 6573 antHilbertSeries │ │ │ │ 0000dfc0: 2854 2c20 4f72 6465 7220 3d3e 2035 2920 (T, Order => 5) │ │ │ │ 0000dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0000dfe0: 2020 2020 7c0a 7c20 2d2d 202e 3030 3733 |.| -- .0073 │ │ │ │ -0000dff0: 3935 3732 7320 656c 6170 7365 6420 2020 9572s elapsed │ │ │ │ +0000dfe0: 2020 2020 7c0a 7c20 2d2d 202e 3030 3239 |.| -- .0029 │ │ │ │ +0000dff0: 3937 3931 7320 656c 6170 7365 6420 2020 9791s elapsed │ │ │ │ 0000e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e030: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0000e040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -3710,15 +3710,15 @@ │ │ │ │ 0000e7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ 0000e7f0: 3a20 656c 6170 7365 6454 696d 6520 6571 : elapsedTime eq │ │ │ │ 0000e800: 7569 7661 7269 616e 7448 696c 6265 7274 uivariantHilbert │ │ │ │ 0000e810: 5365 7269 6573 2854 2c20 4f72 6465 7220 Series(T, Order │ │ │ │ 0000e820: 3d3e 2035 293b 2020 2020 2020 2020 2020 => 5); │ │ │ │ 0000e830: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0000e840: 202e 3030 3036 3736 3836 3673 2065 6c61 .000676866s ela │ │ │ │ +0000e840: 202e 3030 3036 3836 3737 3573 2065 6c61 .000686775s ela │ │ │ │ 0000e850: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 0000e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e880: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0000e890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000e8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -6915,20 +6915,20 @@ │ │ │ │ 0001b020: 203a 2068 6972 6f6e 616b 6144 6563 6f6d : hironakaDecom │ │ │ │ 0001b030: 706f 7369 7469 6f6e 2043 3420 2020 2020 position C4 │ │ │ │ 0001b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b050: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b080: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001b090: 2020 2020 3220 2020 2032 2020 2032 2032 2 2 2 2 │ │ │ │ -0001b0a0: 2020 2020 2020 2020 3320 2020 2020 2020 3 │ │ │ │ -0001b0b0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0001b090: 2020 2020 3220 2020 2032 2020 2020 2033 2 2 3 │ │ │ │ +0001b0a0: 2020 2020 2020 2033 2020 2020 2020 2020 3 │ │ │ │ +0001b0b0: 3420 2020 2034 2020 2020 2020 2020 2020 4 4 │ │ │ │ 0001b0c0: 7c0a 7c6f 3220 3d20 287b 7820 202b 2079 |.|o2 = ({x + y │ │ │ │ -0001b0d0: 202c 2078 2079 207d 2c20 7b31 2c20 7820 , x y }, {1, x │ │ │ │ -0001b0e0: 7920 2d20 782a 7920 7d29 2020 2020 2020 y - x*y }) │ │ │ │ +0001b0d0: 202c 202d 2078 2079 202b 2078 2a79 207d , - x y + x*y } │ │ │ │ +0001b0e0: 2c20 7b31 2c20 7820 202b 2079 207d 2920 , {1, x + y }) │ │ │ │ 0001b0f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b120: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0001b130: 7c6f 3220 3a20 5365 7175 656e 6365 2020 |o2 : Sequence │ │ │ │ 0001b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7513,32 +7513,32 @@ │ │ │ │ 0001d580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d5a0: 2b0a 7c69 3420 3a20 7469 6d65 2050 313d +.|i4 : time P1= │ │ │ │ 0001d5b0: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ 0001d5c0: 7320 4334 7843 3220 2020 2020 2020 2020 s C4xC2 │ │ │ │ 0001d5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d5f0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3438 |.| -- used 0.48 │ │ │ │ -0001d600: 3735 3833 7320 2863 7075 293b 2030 2e33 7583s (cpu); 0.3 │ │ │ │ -0001d610: 3234 3536 3373 2028 7468 7265 6164 293b 24563s (thread); │ │ │ │ +0001d5f0: 7c0a 7c20 2d2d 2075 7365 6420 302e 3434 |.| -- used 0.44 │ │ │ │ +0001d600: 3232 3836 7320 2863 7075 293b 2030 2e33 2286s (cpu); 0.3 │ │ │ │ +0001d610: 3131 3132 3173 2028 7468 7265 6164 293b 11121s (thread); │ │ │ │ 0001d620: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001d630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d640: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d690: 7c0a 7c20 2020 2020 2020 3220 2020 3220 |.| 2 2 │ │ │ │ -0001d6a0: 2020 2032 2020 2034 2020 2020 3420 2020 2 4 4 │ │ │ │ +0001d6a0: 2020 2032 2020 2033 2020 2020 2020 2033 2 3 3 │ │ │ │ 0001d6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d6e0: 7c0a 7c6f 3420 3d20 7b7a 202c 2078 2020 |.|o4 = {z , x │ │ │ │ -0001d6f0: 2b20 7920 2c20 7820 202b 2079 207d 2020 + y , x + y } │ │ │ │ -0001d700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001d6f0: 2b20 7920 2c20 7820 7920 2d20 782a 7920 + y , x y - x*y │ │ │ │ +0001d700: 7d20 2020 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: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7553,17 +7553,17 @@ │ │ │ │ 0001d800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d820: 2b0a 7c69 3520 3a20 7469 6d65 2050 323d +.|i5 : time P2= │ │ │ │ 0001d830: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ 0001d840: 7328 4334 7843 322c 4461 6465 3d3e 7472 s(C4xC2,Dade=>tr │ │ │ │ 0001d850: 7565 2920 2020 2020 2020 2020 2020 2020 ue) │ │ │ │ 0001d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d870: 7c0a 7c20 2d2d 2075 7365 6420 302e 3434 |.| -- used 0.44 │ │ │ │ -0001d880: 3431 3339 7320 2863 7075 293b 2030 2e33 4139s (cpu); 0.3 │ │ │ │ -0001d890: 3735 3334 3573 2028 7468 7265 6164 293b 75345s (thread); │ │ │ │ +0001d870: 7c0a 7c20 2d2d 2075 7365 6420 302e 3337 |.| -- used 0.37 │ │ │ │ +0001d880: 3439 3036 7320 2863 7075 293b 2030 2e33 4906s (cpu); 0.3 │ │ │ │ +0001d890: 3034 3633 3673 2028 7468 7265 6164 293b 04636s (thread); │ │ │ │ 0001d8a0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7872,31 +7872,31 @@ │ │ │ │ 0001ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ec00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ec10: 2b0a 7c69 3620 3a20 7469 6d65 2073 6563 +.|i6 : time sec │ │ │ │ 0001ec20: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ 0001ec30: 2850 312c 4334 7843 3229 2020 2020 2020 (P1,C4xC2) │ │ │ │ 0001ec40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3133 |.| -- used 0.13 │ │ │ │ -0001ec70: 3738 3034 7320 2863 7075 293b 2030 2e30 7804s (cpu); 0.0 │ │ │ │ -0001ec80: 3635 3834 3533 7320 2874 6872 6561 6429 658453s (thread) │ │ │ │ +0001ec60: 7c0a 7c20 2d2d 2075 7365 6420 302e 3131 |.| -- used 0.11 │ │ │ │ +0001ec70: 3535 3633 7320 2863 7075 293b 2030 2e30 5563s (cpu); 0.0 │ │ │ │ +0001ec80: 3437 3236 3736 7320 2874 6872 6561 6429 472676s (thread) │ │ │ │ 0001ec90: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 0001eca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ecb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ecc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ece0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ed00: 7c0a 7c20 2020 2020 2020 2020 2033 2020 |.| 3 │ │ │ │ -0001ed10: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ +0001ed00: 7c0a 7c20 2020 2020 2020 2020 2034 2020 |.| 4 │ │ │ │ +0001ed10: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 0001ed20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ed30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ed40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ed50: 7c0a 7c6f 3620 3d20 7b31 2c20 7820 7920 |.|o6 = {1, x y │ │ │ │ -0001ed60: 2d20 782a 7920 7d20 2020 2020 2020 2020 - x*y } │ │ │ │ +0001ed50: 7c0a 7c6f 3620 3d20 7b31 2c20 7820 202b |.|o6 = {1, x + │ │ │ │ +0001ed60: 2079 207d 2020 2020 2020 2020 2020 2020 y } │ │ │ │ 0001ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ed90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eda0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001edb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001edc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7912,18 +7912,18 @@ │ │ │ │ 0001ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ee90: 2b0a 7c69 3720 3a20 7469 6d65 2073 6563 +.|i7 : time sec │ │ │ │ 0001eea0: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ 0001eeb0: 2850 322c 4334 7843 3229 2020 2020 2020 (P2,C4xC2) │ │ │ │ 0001eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eee0: 7c0a 7c20 2d2d 2075 7365 6420 312e 3431 |.| -- used 1.41 │ │ │ │ -0001eef0: 3830 3273 2028 6370 7529 3b20 312e 3133 802s (cpu); 1.13 │ │ │ │ -0001ef00: 3239 3173 2028 7468 7265 6164 293b 2030 291s (thread); 0 │ │ │ │ -0001ef10: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ +0001eee0: 7c0a 7c20 2d2d 2075 7365 6420 312e 3136 |.| -- used 1.16 │ │ │ │ +0001eef0: 3432 3773 2028 6370 7529 3b20 302e 3839 427s (cpu); 0.89 │ │ │ │ +0001ef00: 3233 3138 7320 2874 6872 6561 6429 3b20 2318s (thread); │ │ │ │ +0001ef10: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ 0001ef20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef80: 7c0a 7c20 2020 2020 2020 2020 2032 2020 |.| 2 │ │ │ │ @@ -9181,15 +9181,15 @@ │ │ │ │ 00023dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 00023de0: 6934 203a 2065 6c61 7073 6564 5469 6d65 i4 : elapsedTime │ │ │ │ 00023df0: 2069 6e76 6172 6961 6e74 7320 5334 2020 invariants S4 │ │ │ │ 00023e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023e30: 202d 2d20 2e34 3232 3230 3373 2065 6c61 -- .422203s ela │ │ │ │ +00023e30: 202d 2d20 2e32 3737 3131 3873 2065 6c61 -- .277118s ela │ │ │ │ 00023e40: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 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 207c 0a7c |.| │ │ │ │ 00023e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9271,16 +9271,16 @@ │ │ │ │ 00024360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024370: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00024380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000243a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000243b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000243c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000243d0: 202d 2d20 2e33 3339 3436 3973 2065 6c61 -- .339469s ela │ │ │ │ -000243e0: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ +000243d0: 202d 2d20 2e32 3530 3033 7320 656c 6170 -- .25003s elap │ │ │ │ +000243e0: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 000243f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024410: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00024420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9811,15 +9811,15 @@ │ │ │ │ 00026520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026540: 2d2d 2d2b 0a7c 6934 203a 2065 6c61 7073 ---+.|i4 : elaps │ │ │ │ 00026550: 6564 5469 6d65 2069 6e76 6172 6961 6e74 edTime invariant │ │ │ │ 00026560: 7320 5334 2020 2020 2020 2020 2020 2020 s S4 │ │ │ │ 00026570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026590: 2020 207c 0a7c 202d 2d20 2e37 3335 3731 |.| -- .73571 │ │ │ │ +00026590: 2020 207c 0a7c 202d 2d20 2e33 3736 3133 |.| -- .37613 │ │ │ │ 000265a0: 3473 2065 6c61 7073 6564 2020 2020 2020 4s elapsed │ │ │ │ 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 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000265f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9876,16 +9876,16 @@ │ │ │ │ 00026930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026950: 2d2d 2d2b 0a7c 6935 203a 2065 6c61 7073 ---+.|i5 : elaps │ │ │ │ 00026960: 6564 5469 6d65 2069 6e76 6172 6961 6e74 edTime invariant │ │ │ │ 00026970: 7328 5334 2c20 5374 7261 7465 6779 203d s(S4, Strategy = │ │ │ │ 00026980: 3e20 224c 696e 6561 7241 6c67 6562 7261 > "LinearAlgebra │ │ │ │ 00026990: 2229 2020 2020 2020 2020 2020 2020 2020 ") │ │ │ │ -000269a0: 2020 207c 0a7c 202d 2d20 2e31 3236 3434 |.| -- .12644 │ │ │ │ -000269b0: 3973 2065 6c61 7073 6564 2020 2020 2020 9s elapsed │ │ │ │ +000269a0: 2020 207c 0a7c 202d 2d20 2e30 3636 3035 |.| -- .06605 │ │ │ │ +000269b0: 3634 7320 656c 6170 7365 6420 2020 2020 64s elapsed │ │ │ │ 000269c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000269d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000269e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000269f0: 2020 207c 0a7c 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 2020 2020 2020 2020 2020 │ │ │ │ @@ -15923,4877 +15923,4831 @@ │ │ │ │ 0003e320: 6e6f 7465 2068 736f 7020 616c 676f 7269 note hsop algori │ │ │ │ 0003e330: 7468 6d73 3a0a 6873 6f70 2061 6c67 6f72 thms:.hsop algor │ │ │ │ 0003e340: 6974 686d 732c 2066 6f72 2061 2064 6973 ithms, for a dis │ │ │ │ 0003e350: 6375 7373 696f 6e20 636f 6d70 6172 696e cussion comparin │ │ │ │ 0003e360: 6720 7468 6520 7477 6f20 616c 676f 7269 g the two algori │ │ │ │ 0003e370: 7468 6d73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d thms...+-------- │ │ │ │ 0003e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e3a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2041 -------+.|i1 : A │ │ │ │ -0003e3b0: 3d6d 6174 7269 787b 7b30 2c31 2c30 7d2c =matrix{{0,1,0}, │ │ │ │ -0003e3c0: 7b30 2c30 2c31 7d2c 7b31 2c30 2c30 7d7d {0,0,1},{1,0,0}} │ │ │ │ -0003e3d0: 3b20 2020 2020 2020 207c 0a7c 2020 2020 ; |.| │ │ │ │ +0003e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003e3a0: 0a7c 6931 203a 2041 3d6d 6174 7269 787b .|i1 : A=matrix{ │ │ │ │ +0003e3b0: 7b30 2c31 2c30 7d2c 7b30 2c30 2c31 7d2c {0,1,0},{0,0,1}, │ │ │ │ +0003e3c0: 7b31 2c30 2c30 7d7d 3b7c 0a7c 2020 2020 {1,0,0}};|.| │ │ │ │ +0003e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e400: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0003e410: 2020 2020 2020 2020 2020 2020 3320 2020 3 │ │ │ │ -0003e420: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ -0003e430: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003e440: 6f31 203a 204d 6174 7269 7820 5a5a 2020 o1 : Matrix ZZ │ │ │ │ -0003e450: 3c2d 2d20 5a5a 2020 2020 2020 2020 2020 <-- ZZ │ │ │ │ -0003e460: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e470: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0003e480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e4a0: 2d2b 0a7c 6932 203a 2042 3d6d 6174 7269 -+.|i2 : B=matri │ │ │ │ -0003e4b0: 787b 7b30 2c31 2c30 7d2c 7b31 2c30 2c30 x{{0,1,0},{1,0,0 │ │ │ │ -0003e4c0: 7d2c 7b30 2c30 2c31 7d7d 3b20 2020 2020 },{0,0,1}}; │ │ │ │ -0003e4d0: 2020 207c 0a7c 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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0003e510: 2020 2020 2020 3320 2020 2020 2020 3320 3 3 │ │ │ │ -0003e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e530: 2020 2020 2020 207c 0a7c 6f32 203a 204d |.|o2 : M │ │ │ │ -0003e540: 6174 7269 7820 5a5a 2020 3c2d 2d20 5a5a atrix ZZ <-- ZZ │ │ │ │ -0003e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e560: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0003e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -0003e5a0: 203a 2053 333d 6669 6e69 7465 4163 7469 : S3=finiteActi │ │ │ │ -0003e5b0: 6f6e 287b 412c 427d 2c51 515b 782c 792c on({A,B},QQ[x,y, │ │ │ │ -0003e5c0: 7a5d 2920 2020 2020 2020 2020 207c 0a7c z]) |.| │ │ │ │ -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 207c | │ │ │ │ -0003e600: 0a7c 6f33 203d 2051 515b 782e 2e7a 5d20 .|o3 = QQ[x..z] │ │ │ │ -0003e610: 3c2d 203c 7c20 3020 3120 3020 7c2c 207c <- <| 0 1 0 |, | │ │ │ │ -0003e620: 2030 2031 2030 207c 3e20 2020 2020 2020 0 1 0 |> │ │ │ │ -0003e630: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003e640: 2020 2020 2020 7c20 3020 3020 3120 7c20 | 0 0 1 | │ │ │ │ -0003e650: 207c 2031 2030 2030 207c 2020 2020 2020 | 1 0 0 | │ │ │ │ -0003e660: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003e670: 2020 2020 2020 2020 7c20 3120 3020 3020 | 1 0 0 │ │ │ │ -0003e680: 7c20 207c 2030 2030 2031 207c 2020 2020 | | 0 0 1 | │ │ │ │ -0003e690: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0003e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e6c0: 2020 2020 2020 207c 0a7c 6f33 203a 2046 |.|o3 : F │ │ │ │ -0003e6d0: 696e 6974 6547 726f 7570 4163 7469 6f6e initeGroupAction │ │ │ │ -0003e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e6f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0003e700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -0003e730: 203a 2070 7269 6d61 7279 496e 7661 7269 : primaryInvari │ │ │ │ -0003e740: 616e 7473 2053 3320 2020 2020 2020 2020 ants S3 │ │ │ │ -0003e750: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e3f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003e400: 2020 2020 3320 2020 2020 2020 3320 2020 3 3 │ │ │ │ +0003e410: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003e420: 6f31 203a 204d 6174 7269 7820 5a5a 2020 o1 : Matrix ZZ │ │ │ │ +0003e430: 3c2d 2d20 5a5a 2020 2020 2020 2020 2020 <-- ZZ │ │ │ │ +0003e440: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0003e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e470: 2d2b 0a7c 6932 203a 2042 3d6d 6174 7269 -+.|i2 : B=matri │ │ │ │ +0003e480: 787b 7b30 2c31 2c30 7d2c 7b31 2c30 2c30 x{{0,1,0},{1,0,0 │ │ │ │ +0003e490: 7d2c 7b30 2c30 2c31 7d7d 3b7c 0a7c 2020 },{0,0,1}};|.| │ │ │ │ +0003e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e4c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003e4d0: 2020 2020 2020 3320 2020 2020 2020 3320 3 3 │ │ │ │ +0003e4e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e4f0: 0a7c 6f32 203a 204d 6174 7269 7820 5a5a .|o2 : Matrix ZZ │ │ │ │ +0003e500: 2020 3c2d 2d20 5a5a 2020 2020 2020 2020 <-- ZZ │ │ │ │ +0003e510: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e540: 2d2d 2d2b 0a7c 6933 203a 2053 333d 6669 ---+.|i3 : S3=fi │ │ │ │ +0003e550: 6e69 7465 4163 7469 6f6e 287b 412c 427d niteAction({A,B} │ │ │ │ +0003e560: 2c51 515b 782c 792c 7a5d 2920 207c 0a7c ,QQ[x,y,z]) |.| │ │ │ │ +0003e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e590: 2020 2020 2020 207c 0a7c 6f33 203d 2051 |.|o3 = Q │ │ │ │ +0003e5a0: 515b 782e 2e7a 5d20 3c2d 203c 7c20 3020 Q[x..z] <- <| 0 │ │ │ │ +0003e5b0: 3120 3020 7c2c 207c 2030 2031 2030 207c 1 0 |, | 0 1 0 | │ │ │ │ +0003e5c0: 3e7c 0a7c 2020 2020 2020 2020 2020 2020 >|.| │ │ │ │ +0003e5d0: 2020 2020 2020 7c20 3020 3020 3120 7c20 | 0 0 1 | │ │ │ │ +0003e5e0: 207c 2031 2030 2030 207c 207c 0a7c 2020 | 1 0 0 | |.| │ │ │ │ +0003e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e600: 7c20 3120 3020 3020 7c20 207c 2030 2030 | 1 0 0 | | 0 0 │ │ │ │ +0003e610: 2031 207c 207c 0a7c 2020 2020 2020 2020 1 | |.| │ │ │ │ +0003e620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e630: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e640: 0a7c 6f33 203a 2046 696e 6974 6547 726f .|o3 : FiniteGro │ │ │ │ +0003e650: 7570 4163 7469 6f6e 2020 2020 2020 2020 upAction │ │ │ │ +0003e660: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e690: 2d2d 2d2b 0a7c 6934 203a 2070 7269 6d61 ---+.|i4 : prima │ │ │ │ +0003e6a0: 7279 496e 7661 7269 616e 7473 2053 3320 ryInvariants S3 │ │ │ │ +0003e6b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e6e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0003e6f0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +0003e700: 2032 2020 2020 3220 2020 2020 2020 2020 2 2 │ │ │ │ +0003e710: 207c 0a7c 6f34 203d 207b 7820 2b20 7920 |.|o4 = {x + y │ │ │ │ +0003e720: 2b20 7a2c 2078 2020 2b20 7920 202b 207a + z, x + y + z │ │ │ │ +0003e730: 202c 2078 2a79 2a7a 7d20 207c 0a7c 2020 , x*y*z} |.| │ │ │ │ +0003e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e760: 2020 2020 207c 0a7c 6f34 203a 204c 6973 |.|o4 : Lis │ │ │ │ +0003e770: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 0003e780: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e790: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e7b0: 2020 2020 2033 2020 2020 3320 2020 2033 3 3 3 │ │ │ │ -0003e7c0: 207c 0a7c 6f34 203d 207b 7820 2b20 7920 |.|o4 = {x + y │ │ │ │ -0003e7d0: 2b20 7a2c 2078 2a79 202b 2078 2a7a 202b + z, x*y + x*z + │ │ │ │ -0003e7e0: 2079 2a7a 2c20 7820 202b 2079 2020 2b20 y*z, x + y + │ │ │ │ -0003e7f0: 7a20 7d7c 0a7c 2020 2020 2020 2020 2020 z }|.| │ │ │ │ -0003e800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e820: 2020 2020 207c 0a7c 6f34 203a 204c 6973 |.|o4 : Lis │ │ │ │ -0003e830: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -0003e840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e850: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0003e790: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003e7b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 2020 2020 ---------+.. │ │ │ │ +0003e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e7d0: 2020 2020 2020 2020 2020 2020 2020 2053 S │ │ │ │ +0003e7e0: 330a 4265 6c6f 772c 2074 6865 2069 6e76 3.Below, the inv │ │ │ │ +0003e7f0: 6172 6961 6e74 2072 696e 6720 5151 5b78 ariant ring QQ[x │ │ │ │ +0003e800: 2c79 2c7a 5d20 2020 6973 2063 616c 6375 ,y,z] is calcu │ │ │ │ +0003e810: 6c61 7465 6420 7769 7468 204b 2062 6569 lated with K bei │ │ │ │ +0003e820: 6e67 2074 6865 2066 6965 6c64 2077 6974 ng the field wit │ │ │ │ +0003e830: 680a 3130 3120 656c 656d 656e 7473 2e0a h.101 elements.. │ │ │ │ +0003e840: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e880: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 2020 2020 ---------+.. │ │ │ │ -0003e890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e8a0: 2020 2020 2020 2020 2020 2020 2020 2053 S │ │ │ │ -0003e8b0: 330a 4265 6c6f 772c 2074 6865 2069 6e76 3.Below, the inv │ │ │ │ -0003e8c0: 6172 6961 6e74 2072 696e 6720 5151 5b78 ariant ring QQ[x │ │ │ │ -0003e8d0: 2c79 2c7a 5d20 2020 6973 2063 616c 6375 ,y,z] is calcu │ │ │ │ -0003e8e0: 6c61 7465 6420 7769 7468 204b 2062 6569 lated with K bei │ │ │ │ -0003e8f0: 6e67 2074 6865 2066 6965 6c64 2077 6974 ng the field wit │ │ │ │ -0003e900: 680a 3130 3120 656c 656d 656e 7473 2e0a h.101 elements.. │ │ │ │ -0003e910: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0003e920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0003e960: 0a7c 6935 203a 204b 3d47 4628 3130 3129 .|i5 : K=GF(101) │ │ │ │ -0003e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003e890: 0a7c 6935 203a 204b 3d47 4628 3130 3129 .|i5 : K=GF(101) │ │ │ │ +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 2020 2020 207c | │ │ │ │ +0003e8e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e930: 0a7c 6f35 203d 204b 2020 2020 2020 2020 .|o5 = K │ │ │ │ +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 2020 2020 207c | │ │ │ │ +0003e980: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e9a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e9b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003e9c0: 2020 2020 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 207c | │ │ │ │ -0003ea00: 0a7c 6f35 203d 204b 2020 2020 2020 2020 .|o5 = K │ │ │ │ -0003ea10: 2020 2020 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 207c | │ │ │ │ -0003ea50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003ea60: 2020 2020 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 207c | │ │ │ │ -0003eaa0: 0a7c 6f35 203a 2047 616c 6f69 7346 6965 .|o5 : GaloisFie │ │ │ │ -0003eab0: 6c64 2020 2020 2020 2020 2020 2020 2020 ld │ │ │ │ -0003eac0: 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 2020 2020 207c | │ │ │ │ +0003e9d0: 0a7c 6f35 203a 2047 616c 6f69 7346 6965 .|o5 : GaloisFie │ │ │ │ +0003e9e0: 6c64 2020 2020 2020 2020 2020 2020 2020 ld │ │ │ │ +0003e9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ea00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ea10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003ea20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ea40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ea50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ea60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003ea70: 0a7c 6936 203a 2053 333d 6669 6e69 7465 .|i6 : S3=finite │ │ │ │ +0003ea80: 4163 7469 6f6e 287b 412c 427d 2c4b 5b78 Action({A,B},K[x │ │ │ │ +0003ea90: 2c79 2c7a 5d29 2020 2020 2020 2020 2020 ,y,z]) │ │ │ │ +0003eaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eab0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003eac0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003ead0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eae0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003eaf0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0003eb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003eb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003eb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003eb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0003eb40: 0a7c 6936 203a 2053 333d 6669 6e69 7465 .|i6 : S3=finite │ │ │ │ -0003eb50: 4163 7469 6f6e 287b 412c 427d 2c4b 5b78 Action({A,B},K[x │ │ │ │ -0003eb60: 2c79 2c7a 5d29 2020 2020 2020 2020 2020 ,y,z]) │ │ │ │ -0003eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003eb80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003eb90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ebd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003ebe0: 0a7c 6f36 203d 204b 5b78 2e2e 7a5d 203c .|o6 = K[x..z] < │ │ │ │ -0003ebf0: 2d20 3c7c 2030 2031 2030 207c 2c20 7c20 - <| 0 1 0 |, | │ │ │ │ -0003ec00: 3020 3120 3020 7c3e 2020 2020 2020 2020 0 1 0 |> │ │ │ │ +0003eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eb00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003eb10: 0a7c 6f36 203d 204b 5b78 2e2e 7a5d 203c .|o6 = K[x..z] < │ │ │ │ +0003eb20: 2d20 3c7c 2030 2031 2030 207c 2c20 7c20 - <| 0 1 0 |, | │ │ │ │ +0003eb30: 3020 3120 3020 7c3e 2020 2020 2020 2020 0 1 0 |> │ │ │ │ +0003eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eb50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003eb60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003eb70: 2020 207c 2030 2030 2031 207c 2020 7c20 | 0 0 1 | | │ │ │ │ +0003eb80: 3120 3020 3020 7c20 2020 2020 2020 2020 1 0 0 | │ │ │ │ +0003eb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003eba0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003ebb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003ebc0: 2020 207c 2031 2030 2030 207c 2020 7c20 | 1 0 0 | | │ │ │ │ +0003ebd0: 3020 3020 3120 7c20 2020 2020 2020 2020 0 0 1 | │ │ │ │ +0003ebe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ebf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003ec00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ec20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003ec30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003ec40: 2020 207c 2030 2030 2031 207c 2020 7c20 | 0 0 1 | | │ │ │ │ -0003ec50: 3120 3020 3020 7c20 2020 2020 2020 2020 1 0 0 | │ │ │ │ -0003ec60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ec70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003ec80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003ec90: 2020 207c 2031 2030 2030 207c 2020 7c20 | 1 0 0 | | │ │ │ │ -0003eca0: 3020 3020 3120 7c20 2020 2020 2020 2020 0 0 1 | │ │ │ │ -0003ecb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ecc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003ecd0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003ece0: 2020 2020 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 207c | │ │ │ │ -0003ed20: 0a7c 6f36 203a 2046 696e 6974 6547 726f .|o6 : FiniteGro │ │ │ │ -0003ed30: 7570 4163 7469 6f6e 2020 2020 2020 2020 upAction │ │ │ │ -0003ed40: 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 2020 2020 207c | │ │ │ │ +0003ec50: 0a7c 6f36 203a 2046 696e 6974 6547 726f .|o6 : FiniteGro │ │ │ │ +0003ec60: 7570 4163 7469 6f6e 2020 2020 2020 2020 upAction │ │ │ │ +0003ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ec80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ec90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003eca0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0003ecf0: 0a7c 6937 203a 2070 7269 6d61 7279 496e .|i7 : primaryIn │ │ │ │ +0003ed00: 7661 7269 616e 7473 2853 332c 4461 6465 variants(S3,Dade │ │ │ │ +0003ed10: 3d3e 7472 7565 2920 2020 2020 2020 2020 =>true) │ │ │ │ +0003ed20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003ed30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003ed40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ed60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003ed70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0003ed80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ed90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003eda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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│ +0003edc0: 3420 2020 2020 2020 2035 2020 2020 2020 4 5 │ │ │ │ +0003edd0: 3620 2020 2020 2035 2020 2020 2020 207c 6 5 | │ │ │ │ +0003ede0: 0a7c 6f37 203d 207b 2d20 3435 7820 202d .|o7 = {- 45x - │ │ │ │ +0003edf0: 2032 3878 2079 202d 2034 3578 2079 2020 28x y - 45x y │ │ │ │ +0003ee00: 2d20 3432 7820 7920 202d 2034 3578 2079 - 42x y - 45x y │ │ │ │ +0003ee10: 2020 2d20 3238 782a 7920 202d 2034 3579 - 28x*y - 45y │ │ │ │ +0003ee20: 2020 2d20 3238 7820 7a20 2b20 2020 207c - 28x z + | │ │ │ │ +0003ee30: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ +0003ee40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ee60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0003ee80: 0a7c 2020 2020 2020 2020 3420 2020 2020 .| 4 │ │ │ │ +0003ee90: 2020 2020 3320 3220 2020 2020 2020 3220 3 2 2 │ │ │ │ +0003eea0: 3320 2020 2020 2020 2020 3420 2020 2020 3 4 │ │ │ │ +0003eeb0: 2020 3520 2020 2020 2020 3420 3220 2020 5 4 2 │ │ │ │ +0003eec0: 2020 2033 2020 2032 2020 2020 2020 207c 3 2 | │ │ │ │ +0003eed0: 0a7c 2020 2020 2031 3178 2079 2a7a 202b .| 11x y*z + │ │ │ │ +0003eee0: 2032 3378 2079 207a 202b 2032 3378 2079 23x y z + 23x y │ │ │ │ +0003eef0: 207a 202b 2031 3178 2a79 207a 202d 2032 z + 11x*y z - 2 │ │ │ │ +0003ef00: 3879 207a 202d 2034 3578 207a 2020 2b20 8y z - 45x z + │ │ │ │ +0003ef10: 3233 7820 792a 7a20 202b 2020 2020 207c 23x y*z + | │ │ │ │ +0003ef20: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ 0003ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0003ef50: 0a7c 2020 2020 2020 2020 3420 2020 2020 .| 4 │ │ │ │ -0003ef60: 2020 2020 3320 3220 2020 2020 2020 3220 3 2 2 │ │ │ │ -0003ef70: 3320 2020 2020 2020 2020 3420 2020 2020 3 4 │ │ │ │ -0003ef80: 2020 3520 2020 2020 2020 3420 3220 2020 5 4 2 │ │ │ │ -0003ef90: 2020 2033 2020 2032 2020 2020 2020 207c 3 2 | │ │ │ │ -0003efa0: 0a7c 2020 2020 2031 3178 2079 2a7a 202b .| 11x y*z + │ │ │ │ -0003efb0: 2032 3378 2079 207a 202b 2032 3378 2079 23x y z + 23x y │ │ │ │ -0003efc0: 207a 202b 2031 3178 2a79 207a 202d 2032 z + 11x*y z - 2 │ │ │ │ -0003efd0: 3879 207a 202d 2034 3578 207a 2020 2b20 8y z - 45x z + │ │ │ │ -0003efe0: 3233 7820 792a 7a20 202b 2020 2020 207c 23x y*z + | │ │ │ │ -0003eff0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0003f000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ef40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ef50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003ef60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0003ef70: 0a7c 2020 2020 2020 2020 3220 3220 3220 .| 2 2 2 │ │ │ │ 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2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0003f150: 0a7c 2020 2020 2020 2020 3420 3220 2020 .| 4 2 │ │ │ │ +0003f160: 2020 2033 2033 2020 2020 2020 3220 3420 3 3 2 4 │ │ │ │ +0003f170: 2020 2020 2020 2035 2020 2020 2036 2020 5 6 │ │ │ │ +0003f180: 2020 2020 3520 2020 2020 2020 3420 2020 5 4 │ │ │ │ +0003f190: 2020 2020 2020 3320 3220 2020 2020 207c 3 2 | │ │ │ │ +0003f1a0: 0a7c 2020 2020 2031 3378 2079 2020 2d20 .| 13x y - │ │ │ │ +0003f1b0: 3231 7820 7920 202b 2031 3378 2079 2020 21x y + 13x y │ │ │ │ +0003f1c0: 2b20 3332 782a 7920 202b 2039 7920 202b + 32x*y + 9y + │ │ │ │ +0003f1d0: 2033 3278 207a 202b 2034 3878 2079 2a7a 32x z + 48x y*z │ │ │ │ +0003f1e0: 202b 2031 3478 2079 207a 202b 2020 207c + 14x y z + | │ │ │ │ +0003f1f0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ 0003f200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0003f220: 0a7c 2020 2020 2020 2020 3420 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---------------- │ │ │ │ +0003f310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0003f330: 0a7c 2020 2020 2020 2020 3420 3220 2020 .| 4 2 │ │ │ │ +0003f340: 2020 2033 2033 2020 2020 2020 3220 2020 3 3 2 │ │ │ │ +0003f350: 3320 2020 2020 2020 2032 2033 2020 2020 3 2 3 │ │ │ │ +0003f360: 2020 3320 3320 2020 2020 2032 2034 2020 3 3 2 4 │ │ │ │ +0003f370: 2020 2020 2020 2020 3420 2020 2020 207c 4 | │ │ │ │ +0003f380: 0a7c 2020 2020 2031 3379 207a 2020 2d20 .| 13y z - │ │ │ │ +0003f390: 3231 7820 7a20 202b 2031 3478 2079 2a7a 21x z + 14x y*z │ │ │ │ +0003f3a0: 2020 2b20 3134 782a 7920 7a20 202d 2032 + 14x*y z - 2 │ │ │ │ +0003f3b0: 3179 207a 2020 2b20 3133 7820 7a20 202b 1y z + 13x z + │ │ │ │ +0003f3c0: 2034 3878 2a79 2a7a 2020 2b20 2020 207c 48x*y*z + | │ │ │ │ +0003f3d0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ 0003f3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0003f400: 0a7c 2020 2020 2020 2020 3420 3220 2020 .| 4 2 │ │ │ │ -0003f410: 2020 2033 2033 2020 2020 2020 3220 2020 3 3 2 │ │ │ │ -0003f420: 3320 2020 2020 2020 2032 2033 2020 2020 3 2 3 │ │ │ │ -0003f430: 2020 3320 3320 2020 2020 2032 2034 2020 3 3 2 4 │ │ │ │ -0003f440: 2020 2020 2020 2020 3420 2020 2020 207c 4 | │ │ │ │ -0003f450: 0a7c 2020 2020 2031 3379 207a 2020 2d20 .| 13y z - │ │ │ │ -0003f460: 3231 7820 7a20 202b 2031 3478 2079 2a7a 21x z + 14x y*z │ │ │ │ -0003f470: 2020 2b20 3134 782a 7920 7a20 202d 2032 + 14x*y z - 2 │ │ │ │ -0003f480: 3179 207a 2020 2b20 3133 7820 7a20 202b 1y z + 13x z + │ │ │ │ -0003f490: 2034 3878 2a79 2a7a 2020 2b20 2020 207c 48x*y*z + | │ │ │ │ -0003f4a0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0003f4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0003f420: 0a7c 2020 2020 2020 2020 3220 3420 2020 .| 2 4 │ │ │ │ +0003f430: 2020 2020 2035 2020 2020 2020 2020 3520 5 5 │ │ │ │ +0003f440: 2020 2020 3620 2020 2020 3620 2020 2020 6 6 │ │ │ │ +0003f450: 2035 2020 2020 2020 2034 2032 2020 2020 5 4 2 │ │ │ │ +0003f460: 2020 3220 3420 2020 2020 2020 2035 207c 2 4 5 | │ │ │ │ +0003f470: 0a7c 2020 2020 2031 3379 207a 2020 2b20 .| 13y z + │ │ │ │ +0003f480: 3332 782a 7a20 202b 2033 3279 2a7a 2020 32x*z + 32y*z │ │ │ │ +0003f490: 2b20 397a 202c 2031 3378 2020 2d20 3433 + 9z , 13x - 43 │ │ │ │ +0003f4a0: 7820 7920 2b20 3435 7820 7920 202b 2034 x y + 45x y + 4 │ │ │ │ +0003f4b0: 3578 2079 2020 2d20 3433 782a 7920 207c 5x y - 43x*y | │ │ │ │ +0003f4c0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ 0003f4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0003f4f0: 0a7c 2020 2020 2020 2020 3220 3420 2020 .| 2 4 │ │ │ │ -0003f500: 2020 2020 2035 2020 2020 2020 2020 3520 5 5 │ │ │ │ -0003f510: 2020 2020 3620 2020 2020 3620 2020 2020 6 6 │ │ │ │ -0003f520: 2035 2020 2020 2020 2034 2032 2020 2020 5 4 2 │ │ │ │ -0003f530: 2020 3220 3420 2020 2020 2020 2035 207c 2 4 5 | │ │ │ │ -0003f540: 0a7c 2020 2020 2031 3379 207a 2020 2b20 .| 13y z + │ │ │ │ -0003f550: 3332 782a 7a20 202b 2033 3279 2a7a 2020 32x*z + 32y*z │ │ │ │ -0003f560: 2b20 397a 202c 2031 3378 2020 2d20 3433 + 9z , 13x - 43 │ │ │ │ -0003f570: 7820 7920 2b20 3435 7820 7920 202b 2034 x y + 45x y + 4 │ │ │ │ -0003f580: 3578 2079 2020 2d20 3433 782a 7920 207c 5x y - 43x*y | │ │ │ │ -0003f590: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0003f5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0003f510: 0a7c 2020 2020 2020 2020 2020 3620 2020 .| 6 │ │ │ │ +0003f520: 2020 2035 2020 2020 2020 2034 2020 2020 5 4 │ │ │ │ +0003f530: 2020 2020 2033 2032 2020 2020 2020 2032 3 2 2 │ │ │ │ +0003f540: 2033 2020 2020 2020 2020 2034 2020 2020 3 4 │ │ │ │ +0003f550: 2020 2035 2020 2020 2020 2034 2032 207c 5 4 2 | │ │ │ │ +0003f560: 0a7c 2020 2020 202b 2031 3379 2020 2d20 .| + 13y - │ │ │ │ +0003f570: 3433 7820 7a20 2d20 3233 7820 792a 7a20 43x z - 23x y*z │ │ │ │ +0003f580: 2d20 3339 7820 7920 7a20 2d20 3339 7820 - 39x y z - 39x │ │ │ │ +0003f590: 7920 7a20 2d20 3233 782a 7920 7a20 2d20 y z - 23x*y z - │ │ │ │ +0003f5a0: 3433 7920 7a20 2b20 3435 7820 7a20 207c 43y z + 45x z | │ │ │ │ +0003f5b0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ 0003f5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003f5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0003f5e0: 0a7c 2020 2020 2020 2020 2020 3620 2020 .| 6 │ │ │ │ -0003f5f0: 2020 2035 2020 2020 2020 2034 2020 2020 5 4 │ │ │ │ -0003f600: 2020 2020 2033 2032 2020 2020 2020 2032 3 2 2 │ │ │ │ -0003f610: 2033 2020 2020 2020 2020 2034 2020 2020 3 4 │ │ │ │ -0003f620: 2020 2035 2020 2020 2020 2034 2032 207c 5 4 2 | │ │ │ │ -0003f630: 0a7c 2020 2020 202b 2031 3379 2020 2d20 .| + 13y - │ │ │ │ -0003f640: 3433 7820 7a20 2d20 3233 7820 792a 7a20 43x z - 23x y*z │ │ │ │ -0003f650: 2d20 3339 7820 7920 7a20 2d20 3339 7820 - 39x y z - 39x │ │ │ │ -0003f660: 7920 7a20 2d20 3233 782a 7920 7a20 2d20 y z - 23x*y z - │ │ │ │ -0003f670: 3433 7920 7a20 2b20 3435 7820 7a20 207c 43y z + 45x z | │ │ │ │ -0003f680: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ 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2020 2020 2020 2020 207c | │ │ │ │ -0003f860: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f8a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003f8b0: 0a7c 6f37 203a 204c 6973 7420 2020 2020 .|o7 : List │ │ │ │ -0003f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0003f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +0003f6f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003f700: 3420 2020 2020 2032 2034 2020 2020 2020 4 2 4 │ │ │ │ +0003f710: 2020 3520 2020 2020 2020 2035 2020 2020 5 5 │ │ │ │ +0003f720: 2020 3620 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +0003f730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003f740: 0a7c 2020 2020 202d 2032 3378 2a79 2a7a .| - 23x*y*z │ │ │ │ +0003f750: 2020 2b20 3435 7920 7a20 202d 2034 3378 + 45y z - 43x │ │ │ │ +0003f760: 2a7a 2020 2d20 3433 792a 7a20 202b 2031 *z - 43y*z + 1 │ │ │ │ +0003f770: 337a 207d 2020 2020 2020 2020 2020 2020 3z } │ │ │ │ +0003f780: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003f790: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003f7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f7d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003f7e0: 0a7c 6f37 203a 204c 6973 7420 2020 2020 .|o7 : List │ │ │ │ +0003f7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003f820: 2020 2020 2020 2020 2020 2020 2020 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-0003fdd0: 0a74 6f20 636f 6e74 696e 7565 2077 6974 .to continue wit │ │ │ │ -0003fde0: 6820 7468 6520 636f 6d70 7574 6174 696f h the computatio │ │ │ │ -0003fdf0: 6e20 696e 2074 6869 7320 6361 7365 2e20 n in this case. │ │ │ │ -0003fe00: 5365 6520 2a6e 6f74 6520 6873 6f70 2061 See *note hsop a │ │ │ │ -0003fe10: 6c67 6f72 6974 686d 733a 2068 736f 700a lgorithms: hsop. │ │ │ │ -0003fe20: 616c 676f 7269 7468 6d73 2c20 666f 7220 algorithms, for │ │ │ │ -0003fe30: 6120 6469 7363 7573 7369 6f6e 206f 6e20 a discussion on │ │ │ │ -0003fe40: 7468 6520 4461 6465 2061 6c67 6f72 6974 the Dade algorit │ │ │ │ -0003fe50: 686d 2e0a 0a57 6179 7320 746f 2075 7365 hm...Ways to use │ │ │ │ -0003fe60: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ -0003fe70: 7473 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ts:.============ │ │ │ │ -0003fe80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0003fe90: 3d3d 0a0a 2020 2a20 2270 7269 6d61 7279 ==.. * "primary │ │ │ │ -0003fea0: 496e 7661 7269 616e 7473 2846 696e 6974 Invariants(Finit │ │ │ │ -0003feb0: 6547 726f 7570 4163 7469 6f6e 2922 0a0a eGroupAction)".. │ │ │ │ -0003fec0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0003fed0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0003fee0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0003fef0: 7420 2a6e 6f74 6520 7072 696d 6172 7949 t *note primaryI │ │ │ │ -0003ff00: 6e76 6172 6961 6e74 733a 2070 7269 6d61 nvariants: prima │ │ │ │ -0003ff10: 7279 496e 7661 7269 616e 7473 2c20 6973 ryInvariants, is │ │ │ │ -0003ff20: 2061 202a 6e6f 7465 206d 6574 686f 640a a *note method. │ │ │ │ -0003ff30: 6675 6e63 7469 6f6e 2077 6974 6820 6f70 function with op │ │ │ │ -0003ff40: 7469 6f6e 733a 2028 4d61 6361 756c 6179 tions: (Macaulay │ │ │ │ -0003ff50: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -0003ff60: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ -0003ff70: 0a0a 2d2d 2d2d 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0a1f 0a46 696c .m2:175:0....Fil │ │ │ │ -00040050: 653a 2049 6e76 6172 6961 6e74 5269 6e67 e: InvariantRing │ │ │ │ -00040060: 2e69 6e66 6f2c 204e 6f64 653a 2070 7269 .info, Node: pri │ │ │ │ -00040070: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ -00040080: 705f 7064 5f70 645f 7064 5f63 6d44 6164 p_pd_pd_pd_cmDad │ │ │ │ -00040090: 653d 3e5f 7064 5f70 645f 7064 5f72 702c e=>_pd_pd_pd_rp, │ │ │ │ -000400a0: 204e 6578 743a 2070 7269 6d61 7279 496e Next: primaryIn │ │ │ │ -000400b0: 7661 7269 616e 7473 5f6c 705f 7064 5f70 variants_lp_pd_p │ │ │ │ -000400c0: 645f 7064 5f63 6d44 6567 7265 6556 6563 d_pd_cmDegreeVec │ │ │ │ -000400d0: 746f 723d 3e5f 7064 5f70 645f 7064 5f72 tor=>_pd_pd_pd_r │ │ │ │ -000400e0: 702c 2050 7265 763a 2070 7269 6d61 7279 p, Prev: primary │ │ │ │ -000400f0: 496e 7661 7269 616e 7473 2c20 5570 3a20 Invariants, Up: │ │ │ │ -00040100: 546f 700a 0a70 7269 6d61 7279 496e 7661 Top..primaryInva │ │ │ │ -00040110: 7269 616e 7473 282e 2e2e 2c44 6164 653d 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6d61 7279 en *note primary │ │ │ │ -00040470: 496e 7661 7269 616e 7473 3a20 7072 696d Invariants: prim │ │ │ │ -00040480: 6172 7949 6e76 6172 6961 6e74 732c 2077 aryInvariants, w │ │ │ │ -00040490: 696c 6c20 7573 650a 7468 6520 4461 6465 ill use.the Dade │ │ │ │ -000404a0: 2061 6c67 6f72 6974 686d 2074 6f20 6361 algorithm to ca │ │ │ │ -000404b0: 6c63 756c 6174 6520 6120 686f 6d6f 6765 lculate a homoge │ │ │ │ -000404c0: 6e65 6f75 7320 7379 7374 656d 206f 6620 neous system of │ │ │ │ -000404d0: 7061 7261 6d65 7465 7273 2028 6873 6f70 parameters (hsop │ │ │ │ -000404e0: 2920 666f 720a 7468 6520 696e 7661 7269 ) for.the invari │ │ │ │ -000404f0: 616e 7420 7269 6e67 206f 6620 6120 6669 ant ring of a fi │ │ │ │ -00040500: 6e69 7465 2067 726f 7570 2e0a 0a54 6865 nite group...The │ │ │ │ -00040510: 2065 7861 6d70 6c65 2062 656c 6f77 2063 example below c │ │ │ │ -00040520: 6f6d 7075 7465 7320 7468 6520 696e 7661 omputes the inva │ │ │ │ -00040530: 7269 616e 7420 7269 6e67 206f 6620 5333 riant ring of S3 │ │ │ │ -00040540: 2061 6374 696e 6720 6f6e 2051 515b 782c acting on QQ[x, │ │ │ │ -00040550: 792c 7a5d 2062 790a 7065 726d 7574 6174 y,z] by.permutat │ │ │ │ -00040560: 696f 6e73 206f 6e20 7468 6520 7661 7269 ions on the vari │ │ │ │ -00040570: 6162 6c65 732e 202a 6e6f 7465 2044 6164 ables. *note Dad │ │ │ │ -00040580: 653a 0a70 7269 6d61 7279 496e 7661 7269 e:.primaryInvari │ │ │ │ -00040590: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ -000405a0: 5f63 6d44 6164 653d 3e5f 7064 5f70 645f _cmDade=>_pd_pd_ │ │ │ │ -000405b0: 7064 5f72 702c 2069 7320 7365 7420 746f pd_rp, is set to │ │ │ │ -000405c0: 202a 6e6f 7465 2074 7275 653a 0a28 4d61 *note true:.(Ma │ │ │ │ -000405d0: 6361 756c 6179 3244 6f63 2974 7275 652c caulay2Doc)true, │ │ │ │ -000405e0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -000405f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00040600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ 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User │ │ │ │ +0003fa60: 7320 7368 6f75 6c64 2065 6e74 6572 2074 s should enter t │ │ │ │ +0003fa70: 6865 2066 696e 6974 6520 6669 656c 6420 he finite field │ │ │ │ +0003fa80: 6173 2061 202a 6e6f 7465 2047 616c 6f69 as a *note Galoi │ │ │ │ +0003fa90: 7346 6965 6c64 3a0a 2020 2020 2020 2020 sField:. │ │ │ │ +0003faa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fad0: 2020 2020 2020 2020 2020 2020 0a28 4d61 .(Ma │ │ │ │ +0003fae0: 6361 756c 6179 3244 6f63 2947 616c 6f69 caulay2Doc)Galoi │ │ │ │ +0003faf0: 7346 6965 6c64 2c20 6f72 2061 2071 756f sField, or a quo │ │ │ │ +0003fb00: 7469 656e 7420 6669 656c 6420 6f66 2074 tient field of t │ │ │ │ +0003fb10: 6865 2066 6f72 6d20 2a6e 6f74 6520 5a5a he form *note ZZ │ │ │ │ +0003fb20: 3a0a 2020 2020 2020 2020 2020 2020 2020 :. │ │ │ │ +0003fb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003fb60: 2020 2020 2020 2020 200a 284d 6163 6175 .(Macau │ │ │ │ +0003fb70: 6c61 7932 446f 6329 5a5a 2c2f 7020 616e lay2Doc)ZZ,/p an │ │ │ │ +0003fb80: 6420 6172 6520 6164 7669 7365 6420 746f d are advised to │ │ │ │ +0003fb90: 2065 6e73 7572 6520 7468 6174 2074 6865 ensure that the │ │ │ │ +0003fba0: 2067 726f 756e 6420 6669 656c 6420 6861 ground field ha │ │ │ │ +0003fbb0: 730a 2020 2020 2020 2020 2020 2020 2020 s. │ │ │ │ +0003fbc0: 2020 2020 2020 2020 2020 2020 2020 6e2d n- │ │ │ │ +0003fbd0: 310a 6361 7264 696e 616c 6974 7920 6772 1.cardinality gr │ │ │ │ +0003fbe0: 6561 7465 7220 7468 616e 207c 477c 2020 eater than |G| │ │ │ │ +0003fbf0: 202c 2077 6865 7265 206e 2069 7320 7468 , where n is th │ │ │ │ +0003fc00: 6520 6e75 6d62 6572 206f 6620 7661 7269 e number of vari │ │ │ │ +0003fc10: 6162 6c65 7320 696e 2074 6865 0a70 6f6c ables in the.pol │ │ │ │ +0003fc20: 796e 6f6d 6961 6c20 7269 6e67 202e 2055 ynomial ring . U │ │ │ │ +0003fc30: 7369 6e67 2061 2067 726f 756e 6420 6669 sing a ground fi │ │ │ │ +0003fc40: 656c 6420 736d 616c 6c65 7220 7468 616e eld smaller than │ │ │ │ +0003fc50: 2074 6869 7320 7275 6e73 2074 6865 2072 this runs the r │ │ │ │ +0003fc60: 6973 6b20 6f66 2074 6865 0a61 6c67 6f72 isk of the.algor │ │ │ │ +0003fc70: 6974 686d 2067 6574 7469 6e67 2073 7475 ithm getting stu │ │ │ │ +0003fc80: 636b 2069 6e20 616e 2069 6e66 696e 6974 ck in an infinit │ │ │ │ +0003fc90: 6520 6c6f 6f70 3b20 2a6e 6f74 6520 7072 e loop; *note pr │ │ │ │ +0003fca0: 696d 6172 7949 6e76 6172 6961 6e74 733a imaryInvariants: │ │ │ │ +0003fcb0: 0a70 7269 6d61 7279 496e 7661 7269 616e .primaryInvarian │ │ │ │ +0003fcc0: 7473 2c20 6469 7370 6c61 7973 2061 2077 ts, displays a w │ │ │ │ +0003fcd0: 6172 6e69 6e67 206d 6573 7361 6765 2061 arning message a │ │ │ │ +0003fce0: 736b 696e 6720 7468 6520 7573 6572 2077 sking the user w │ │ │ │ +0003fcf0: 6865 7468 6572 2074 6865 7920 7769 7368 hether they wish │ │ │ │ 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0a1f 0a46 696c .m2:175:0....Fil │ │ │ │ +0003ff80: 653a 2049 6e76 6172 6961 6e74 5269 6e67 e: InvariantRing │ │ │ │ +0003ff90: 2e69 6e66 6f2c 204e 6f64 653a 2070 7269 .info, Node: pri │ │ │ │ +0003ffa0: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ +0003ffb0: 705f 7064 5f70 645f 7064 5f63 6d44 6164 p_pd_pd_pd_cmDad │ │ │ │ +0003ffc0: 653d 3e5f 7064 5f70 645f 7064 5f72 702c e=>_pd_pd_pd_rp, │ │ │ │ +0003ffd0: 204e 6578 743a 2070 7269 6d61 7279 496e Next: primaryIn │ │ │ │ +0003ffe0: 7661 7269 616e 7473 5f6c 705f 7064 5f70 variants_lp_pd_p │ │ │ │ +0003fff0: 645f 7064 5f63 6d44 6567 7265 6556 6563 d_pd_cmDegreeVec │ │ │ │ +00040000: 746f 723d 3e5f 7064 5f70 645f 7064 5f72 tor=>_pd_pd_pd_r │ │ │ │ +00040010: 702c 2050 7265 763a 2070 7269 6d61 7279 p, Prev: primary │ │ │ │ +00040020: 496e 7661 7269 616e 7473 2c20 5570 3a20 Invariants, Up: │ │ │ │ +00040030: 546f 700a 0a70 7269 6d61 7279 496e 7661 Top..primaryInva │ │ │ │ +00040040: 7269 616e 7473 282e 2e2e 2c44 6164 653d riants(...,Dade= │ │ │ │ +00040050: 3e2e 2e2e 2920 2d2d 2061 6e20 6f70 7469 >...) -- an opti │ │ │ │ +00040060: 6f6e 616c 2061 7267 756d 656e 7420 666f onal argument fo │ │ │ │ +00040070: 7220 7072 696d 6172 7949 6e76 6172 6961 r primaryInvaria │ │ │ │ +00040080: 6e74 7320 6465 7465 726d 696e 696e 6720 nts determining │ │ │ │ +00040090: 7768 6574 6865 7220 746f 2075 7365 2074 whether to use t │ │ │ │ +000400a0: 6865 2044 6164 6520 616c 676f 7269 7468 he Dade algorith │ │ │ │ +000400b0: 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a m.************** │ │ │ │ +000400c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000400d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000400e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000400f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00040100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00040110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 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6973 7469 6e67 206f 6620 6120 686f 6d6f isting of a homo │ │ │ │ +00040200: 6765 6e65 6f75 7320 7379 7374 656d 0a20 geneous system. │ │ │ │ +00040210: 2020 2020 2020 206f 6620 7061 7261 6d65 of parame │ │ │ │ +00040220: 7465 7273 2028 6873 6f70 2920 666f 7220 ters (hsop) for │ │ │ │ +00040230: 7468 6520 696e 7661 7269 616e 7420 7269 the invariant ri │ │ │ │ +00040240: 6e67 206f 6620 7468 6520 6772 6f75 7020 ng of the group │ │ │ │ +00040250: 6163 7469 6f6e 0a0a 4465 7363 7269 7074 action..Descript │ │ │ │ +00040260: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +00040270: 0a2a 6e6f 7465 2044 6164 653a 2070 7269 .*note Dade: pri │ │ │ │ +00040280: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ +00040290: 705f 7064 5f70 645f 7064 5f63 6d44 6164 p_pd_pd_pd_cmDad │ │ │ │ +000402a0: 653d 3e5f 7064 5f70 645f 7064 5f72 702c e=>_pd_pd_pd_rp, │ │ │ │ +000402b0: 2074 616b 6573 202a 6e6f 7465 0a42 6f6f takes *note.Boo │ │ │ │ +000402c0: 6c65 616e 3a20 284d 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6d61 7279 en *note primary │ │ │ │ +000403a0: 496e 7661 7269 616e 7473 3a20 7072 696d Invariants: prim │ │ │ │ +000403b0: 6172 7949 6e76 6172 6961 6e74 732c 2077 aryInvariants, w │ │ │ │ +000403c0: 696c 6c20 7573 650a 7468 6520 4461 6465 ill use.the Dade │ │ │ │ +000403d0: 2061 6c67 6f72 6974 686d 2074 6f20 6361 algorithm to ca │ │ │ │ +000403e0: 6c63 756c 6174 6520 6120 686f 6d6f 6765 lculate a homoge │ │ │ │ +000403f0: 6e65 6f75 7320 7379 7374 656d 206f 6620 neous system of │ │ │ │ +00040400: 7061 7261 6d65 7465 7273 2028 6873 6f70 parameters (hsop │ │ │ │ +00040410: 2920 666f 720a 7468 6520 696e 7661 7269 ) for.the invari │ │ │ │ +00040420: 616e 7420 7269 6e67 206f 6620 6120 6669 ant ring of a fi │ │ │ │ +00040430: 6e69 7465 2067 726f 7570 2e0a 0a54 6865 nite group...The │ │ │ │ +00040440: 2065 7861 6d70 6c65 2062 656c 6f77 2063 example below c │ │ │ │ +00040450: 6f6d 7075 7465 7320 7468 6520 696e 7661 omputes the inva │ │ │ │ +00040460: 7269 616e 7420 7269 6e67 206f 6620 5333 riant 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2020 │ │ │ │ +00041db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041dc0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00041dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00041e10: 2d2b 0a0a 436f 6d70 6172 6520 7468 6973 -+..Compare this │ │ │ │ +00041e20: 2072 6573 756c 7420 746f 2074 6865 2068 result to the h │ │ │ │ +00041e30: 736f 7020 6f75 7470 7574 2077 6865 6e20 sop output when │ │ │ │ +00041e40: 4461 6465 2069 7320 6c65 6674 2074 6f20 Dade is left to │ │ │ │ +00041e50: 6974 7320 6465 6661 756c 7420 7661 6c75 its default valu │ │ │ │ +00041e60: 650a 2a6e 6f74 6520 6661 6c73 653a 2028 e.*note false: ( │ │ │ │ +00041e70: 4d61 6361 756c 6179 3244 6f63 2966 616c Macaulay2Doc)fal │ │ │ │ +00041e80: 7365 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d se,...+--------- │ │ │ │ +00041e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041ee0: 2d2b 0a0a 436f 6d70 6172 6520 7468 6973 -+..Compare this │ │ │ │ -00041ef0: 2072 6573 756c 7420 746f 2074 6865 2068 result to the h │ │ │ │ -00041f00: 736f 7020 6f75 7470 7574 2077 6865 6e20 sop output when │ │ │ │ -00041f10: 4461 6465 2069 7320 6c65 6674 2074 6f20 Dade is left to │ │ │ │ -00041f20: 6974 7320 6465 6661 756c 7420 7661 6c75 its default valu │ │ │ │ -00041f30: 650a 2a6e 6f74 6520 6661 6c73 653a 2028 e.*note false: ( │ │ │ │ -00041f40: 4d61 6361 756c 6179 3244 6f63 2966 616c Macaulay2Doc)fal │ │ │ │ -00041f50: 7365 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d se,...+--------- │ │ │ │ -00041f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00041f80: 3520 3a20 7072 696d 6172 7949 6e76 6172 5 : primaryInvar │ │ │ │ -00041f90: 6961 6e74 7328 5333 2920 2020 2020 2020 iants(S3) │ │ │ │ -00041fa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00041fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00041eb0: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7072 ------+.|i5 : pr │ │ │ │ +00041ec0: 696d 6172 7949 6e76 6172 6961 6e74 7328 imaryInvariants( │ │ │ │ +00041ed0: 5333 2920 2020 2020 2020 2020 2020 2020 S3) │ │ │ │ +00041ee0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00041ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041f10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00041f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041f40: 3320 2020 2033 2020 2020 3320 7c0a 7c6f 3 3 3 |.|o │ │ │ │ +00041f50: 3520 3d20 7b78 202b 2079 202b 207a 2c20 5 = {x + y + z, │ │ │ │ +00041f60: 782a 7920 2b20 782a 7a20 2b20 792a 7a2c x*y + x*z + y*z, │ │ │ │ +00041f70: 2078 2020 2b20 7920 202b 207a 207d 7c0a x + y + z }|. │ │ │ │ +00041f80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00041f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041fb0: 7c0a 7c6f 3520 3a20 4c69 7374 2020 2020 |.|o5 : List │ │ │ │ +00041fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041fe0: 2032 2020 2020 3220 2020 2032 2020 2020 2 2 2 │ │ │ │ -00041ff0: 2020 2020 7c0a 7c6f 3520 3d20 7b78 202b |.|o5 = {x + │ │ │ │ -00042000: 2079 202b 207a 2c20 7820 202b 2079 2020 y + z, x + y │ │ │ │ -00042010: 2b20 7a20 2c20 782a 792a 7a7d 7c0a 7c20 + z , x*y*z}|.| │ │ │ │ +00041fe0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00041ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042010: 2d2d 2d2d 2b0a 0a20 2020 2020 2020 2020 ----+.. │ │ │ │ 00042020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042040: 2020 2020 7c0a 7c6f 3520 3a20 4c69 7374 |.|o5 : List │ │ │ │ -00042050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042060: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00042070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042090: 2d2d 2d2d 2b0a 0a20 2020 2020 2020 2020 ----+.. │ │ │ │ -000420a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000420b0: 2020 2020 2020 2020 2020 5333 0a42 656c S3.Bel │ │ │ │ -000420c0: 6f77 2c20 7468 6520 696e 7661 7269 616e ow, the invarian │ │ │ │ -000420d0: 7420 7269 6e67 2051 515b 782c 792c 7a5d t ring QQ[x,y,z] │ │ │ │ -000420e0: 2020 2069 7320 6361 6c63 756c 6174 6564 is calculated │ │ │ │ -000420f0: 2077 6974 6820 4b20 6265 696e 6720 7468 with K being th │ │ │ │ -00042100: 6520 6669 656c 6420 7769 7468 0a31 3031 e field with.101 │ │ │ │ -00042110: 2065 6c65 6d65 6e74 732e 0a0a 2b2d 2d2d elements...+--- │ │ │ │ -00042120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042160: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -00042170: 3a20 4b3d 4746 2831 3031 2920 2020 2020 : K=GF(101) │ │ │ │ -00042180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042030: 2020 2020 2020 2020 2020 5333 0a42 656c S3.Bel │ │ │ │ +00042040: 6f77 2c20 7468 6520 696e 7661 7269 616e ow, the invarian │ │ │ │ +00042050: 7420 7269 6e67 2051 515b 782c 792c 7a5d t ring QQ[x,y,z] │ │ │ │ +00042060: 2020 2069 7320 6361 6c63 756c 6174 6564 is calculated │ │ │ │ +00042070: 2077 6974 6820 4b20 6265 696e 6720 7468 with K being th │ │ │ │ +00042080: 6520 6669 656c 6420 7769 7468 0a31 3031 e field with.101 │ │ │ │ +00042090: 2065 6c65 6d65 6e74 732e 0a0a 2b2d 2d2d elements...+--- │ │ │ │ +000420a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000420b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000420c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000420d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000420e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ +000420f0: 3a20 4b3d 4746 2831 3031 2920 2020 2020 : K=GF(101) │ │ │ │ +00042100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042130: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00042140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042180: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +00042190: 3d20 4b20 2020 2020 2020 2020 2020 2020 = K │ │ │ │ 000421a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000421b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000421b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000421c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000421d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000421d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 000421e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000421f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042200: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -00042210: 3d20 4b20 2020 2020 2020 2020 2020 2020 = K │ │ │ │ -00042220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042220: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +00042230: 3a20 4761 6c6f 6973 4669 656c 6420 2020 : GaloisField │ │ │ │ 00042240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042250: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00042250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00042260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000422a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -000422b0: 3a20 4761 6c6f 6973 4669 656c 6420 2020 : GaloisField │ │ │ │ -000422c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000422d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000422e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000422f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00042300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042340: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ -00042350: 3a20 5333 3d66 696e 6974 6541 6374 696f : S3=finiteActio │ │ │ │ -00042360: 6e28 7b41 2c42 7d2c 4b5b 782c 792c 7a5d n({A,B},K[x,y,z] │ │ │ │ -00042370: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00042380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042390: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00042270: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00042280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000422a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000422b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000422c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ +000422d0: 3a20 5333 3d66 696e 6974 6541 6374 696f : S3=finiteActio │ │ │ │ +000422e0: 6e28 7b41 2c42 7d2c 4b5b 782c 792c 7a5d n({A,B},K[x,y,z] │ │ │ │ +000422f0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00042300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042310: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00042320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042360: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ +00042370: 3d20 4b5b 782e 2e7a 5d20 3c2d 203c 7c20 = K[x..z] <- <| │ │ │ │ +00042380: 3020 3120 3020 7c2c 207c 2030 2031 2030 0 1 0 |, | 0 1 0 │ │ │ │ +00042390: 207c 3e20 2020 2020 2020 2020 2020 2020 |> │ │ │ │ 000423a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000423b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000423c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000423d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000423e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -000423f0: 3d20 4b5b 782e 2e7a 5d20 3c2d 203c 7c20 = K[x..z] <- <| │ │ │ │ -00042400: 3020 3120 3020 7c2c 207c 2030 2031 2030 0 1 0 |, | 0 1 0 │ │ │ │ -00042410: 207c 3e20 2020 2020 2020 2020 2020 2020 |> │ │ │ │ -00042420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042430: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00042440: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ -00042450: 3020 3020 3120 7c20 207c 2031 2030 2030 0 0 1 | | 1 0 0 │ │ │ │ -00042460: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000423b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000423c0: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ +000423d0: 3020 3020 3120 7c20 207c 2031 2030 2030 0 0 1 | | 1 0 0 │ │ │ │ +000423e0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000423f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042400: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00042410: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ +00042420: 3120 3020 3020 7c20 207c 2030 2030 2031 1 0 0 | | 0 0 1 │ │ │ │ +00042430: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00042440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042450: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00042460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00042470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042480: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00042490: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ -000424a0: 3120 3020 3020 7c20 207c 2030 2030 2031 1 0 0 | | 0 0 1 │ │ │ │ -000424b0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000424c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000424d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00042480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00042490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000424a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ +000424b0: 3a20 4669 6e69 7465 4772 6f75 7041 6374 : FiniteGroupAct │ │ │ │ +000424c0: 696f 6e20 2020 2020 2020 2020 2020 2020 ion │ │ │ │ +000424d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000424e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000424f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042520: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -00042530: 3a20 4669 6e69 7465 4772 6f75 7041 6374 : FiniteGroupAct │ │ │ │ -00042540: 696f 6e20 2020 2020 2020 2020 2020 2020 ion │ │ │ │ -00042550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042570: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00042580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00042590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000425a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000425b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000425c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 ----------+.|i8 │ │ │ │ -000425d0: 3a20 7072 696d 6172 7949 6e76 6172 6961 : primaryInvaria │ │ │ │ -000425e0: 6e74 7328 5333 2c44 6164 653d 3e74 7275 nts(S3,Dade=>tru │ │ │ │ -000425f0: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ -00042600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042610: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00042620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042660: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00042670: 2020 2020 2020 3620 2020 2020 2035 2020 6 5 │ │ │ │ -00042680: 2020 2020 2034 2032 2020 2020 2020 3320 4 2 3 │ │ │ │ -00042690: 3320 2020 2020 2032 2034 2020 2020 2020 3 2 4 │ │ │ │ -000426a0: 2020 3520 2020 2020 2036 2020 2020 2020 5 6 │ │ │ │ -000426b0: 3520 2020 2020 2020 2020 7c0a 7c6f 3820 5 |.|o8 │ │ │ │ -000426c0: 3d20 7b32 3278 2020 2d20 3230 7820 7920 = {22x - 20x y │ │ │ │ -000426d0: 2d20 3139 7820 7920 202d 2035 3078 2079 - 19x y - 50x y │ │ │ │ -000426e0: 2020 2d20 3139 7820 7920 202d 2032 3078 - 19x y - 20x │ │ │ │ -000426f0: 2a79 2020 2b20 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----------|.| │ │ │ │ +000428c0: 2020 2020 2032 2034 2020 2020 2020 2020 2 4 │ │ │ │ +000428d0: 2020 3420 2020 2020 2032 2034 2020 2020 4 2 4 │ │ │ │ +000428e0: 2020 2020 3520 2020 2020 2020 2035 2020 5 5 │ │ │ │ +000428f0: 2020 2020 3620 2020 2036 2020 2020 2020 6 6 │ │ │ │ +00042900: 3520 2020 2020 2034 2032 7c0a 7c20 2020 5 4 2|.| │ │ │ │ +00042910: 2020 3139 7820 7a20 202b 2031 3178 2a79 19x z + 11x*y │ │ │ │ +00042920: 2a7a 2020 2d20 3139 7920 7a20 202d 2032 *z - 19y z - 2 │ │ │ │ +00042930: 3078 2a7a 2020 2d20 3230 792a 7a20 202b 0x*z - 20y*z + │ │ │ │ +00042940: 2032 327a 202c 2036 7820 202b 2033 3778 22z , 6x + 37x │ │ │ │ +00042950: 2079 202b 2032 7820 7920 7c0a 7c20 2020 y + 2x y |.| │ │ │ │ +00042960: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ +00042970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000429a0: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +000429b0: 2020 2020 2020 2033 2033 2020 2020 2032 3 3 2 │ │ │ │ +000429c0: 2034 2020 2020 2020 2020 3520 2020 2020 4 5 │ │ │ │ +000429d0: 3620 2020 2020 2035 2020 2020 2020 2034 6 5 4 │ │ │ │ +000429e0: 2020 2020 2020 2020 2033 2032 2020 2020 3 2 │ │ │ │ +000429f0: 2020 2032 2033 2020 2020 7c0a 7c20 2020 2 3 |.| │ │ │ │ +00042a00: 2020 2b20 3234 7820 7920 202b 2032 7820 + 24x y + 2x │ │ │ │ +00042a10: 7920 202b 2033 3778 2a79 2020 2b20 3679 y + 37x*y + 6y │ │ │ │ +00042a20: 2020 2b20 3337 7820 7a20 2b20 3133 7820 + 37x z + 13x │ │ │ │ +00042a30: 792a 7a20 2b20 3239 7820 7920 7a20 2b20 y*z + 29x y z + │ │ │ │ +00042a40: 3239 7820 7920 7a20 2b20 7c0a 7c20 2020 29x y z + |.| │ │ │ │ +00042a50: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ +00042a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042a90: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +00042aa0: 2020 2020 2020 2034 2020 2020 2020 2035 4 5 │ │ │ │ +00042ab0: 2020 2020 2020 3420 3220 2020 2020 2033 4 2 3 │ │ │ │ +00042ac0: 2020 2032 2020 2020 2020 3220 3220 3220 2 2 2 2 │ │ │ │ +00042ad0: 2020 2020 2020 2033 2032 2020 2020 2034 3 2 4 │ │ │ │ +00042ae0: 2032 2020 2020 2020 2020 7c0a 7c20 2020 2 |.| │ │ │ │ +00042af0: 2020 3133 782a 7920 7a20 2b20 3337 7920 13x*y z + 37y │ │ │ │ +00042b00: 7a20 2b20 3278 207a 2020 2b20 3239 7820 z + 2x z + 29x │ │ │ │ +00042b10: 792a 7a20 202b 2031 3578 2079 207a 2020 y*z + 15x y z │ │ │ │ +00042b20: 2b20 3239 782a 7920 7a20 202b 2032 7920 + 29x*y z + 2y │ │ │ │ +00042b30: 7a20 202b 2020 2020 2020 7c0a 7c20 2020 z + |.| │ │ │ │ +00042b40: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- │ │ │ │ +00042b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00042b80: 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 ----------|.| │ │ │ │ +00042b90: 2020 2020 2033 2033 2020 2020 2020 3220 3 3 2 │ │ │ │ +00042ba0: 2020 3320 2020 2020 2020 2032 2033 2020 3 2 3 │ │ │ │ +00042bb0: 2020 2020 3320 3320 2020 2020 3220 3420 3 3 2 4 │ │ │ │ +00042bc0: 2020 2020 2020 2020 2034 2020 2020 2032 4 2 │ │ │ │ +00042bd0: 2034 2020 2020 2020 2020 7c0a 7c20 2020 4 |.| │ │ │ │ +00042be0: 2020 3234 7820 7a20 202b 2032 3978 2079 24x z + 29x y │ │ │ │ +00042bf0: 2a7a 2020 2b20 3239 782a 7920 7a20 202b *z + 29x*y z + │ │ │ │ +00042c00: 2032 3479 207a 2020 2b20 3278 207a 2020 24y z + 2x z │ │ │ │ +00042c10: 2b20 3133 782a 792a 7a20 202b 2032 7920 + 13x*y*z + 2y │ │ │ │ +00042c20: 7a20 202b 2020 2020 2020 7c0a 7c20 2020 z + |.| │ │ │ │ +00042c30: 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d -------------- 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207a 2020 2d20 3432 782a 7a20 202d 3y z - 42x*z - │ │ │ │ +00042fd0: 2034 3279 2a7a 2020 2b20 3231 7a20 7d20 42y*z + 21z } │ │ │ │ +00042fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00042ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043030: 2020 2020 2020 2020 2020 7c0a 7c6f 3820 |.|o8 │ │ │ │ +00043040: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +00043050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000430a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000430b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3820 |.|o8 │ │ │ │ -000430c0: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ -000430d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000430e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000430f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043100: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00043110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043150: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 6f72 ----------+..For │ │ │ │ -00043160: 206d 6f72 6520 696e 666f 726d 6174 696f more informatio │ │ │ │ -00043170: 6e20 6162 6f75 7420 7468 6520 616c 676f n about the algo │ │ │ │ -00043180: 7269 7468 6d73 2075 7365 6420 746f 2063 rithms used to c │ │ │ │ -00043190: 616c 6375 6c61 7465 2061 2068 736f 7020 alculate a hsop │ │ │ │ -000431a0: 696e 0a70 7269 6d61 7279 496e 7661 7269 in.primaryInvari │ │ │ │ -000431b0: 616e 7473 2c20 7365 6520 2a6e 6f74 6520 ants, see *note │ │ │ │ -000431c0: 6873 6f70 2061 6c67 6f72 6974 686d 733a hsop algorithms: │ │ │ │ -000431d0: 2068 736f 7020 616c 676f 7269 7468 6d73 hsop algorithms │ │ │ │ -000431e0: 2c2e 0a0a 4361 7665 6174 0a3d 3d3d 3d3d ,...Caveat.===== │ │ │ │ -000431f0: 3d0a 0a20 2020 2020 2020 2020 2020 2020 =.. │ │ │ │ -00043200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043080: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00043090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000430a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000430b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000430c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000430d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 6f72 ----------+..For │ │ │ │ +000430e0: 206d 6f72 6520 696e 666f 726d 6174 696f more informatio │ │ │ │ +000430f0: 6e20 6162 6f75 7420 7468 6520 616c 676f n about the algo │ │ │ │ +00043100: 7269 7468 6d73 2075 7365 6420 746f 2063 rithms used to c │ │ │ │ +00043110: 616c 6375 6c61 7465 2061 2068 736f 7020 alculate a hsop │ │ │ │ +00043120: 696e 0a70 7269 6d61 7279 496e 7661 7269 in.primaryInvari │ │ │ │ +00043130: 616e 7473 2c20 7365 6520 2a6e 6f74 6520 ants, see *note │ │ │ │ +00043140: 6873 6f70 2061 6c67 6f72 6974 686d 733a hsop algorithms: │ │ │ │ +00043150: 2068 736f 7020 616c 676f 7269 7468 6d73 hsop algorithms │ │ │ │ +00043160: 2c2e 0a0a 4361 7665 6174 0a3d 3d3d 3d3d ,...Caveat.===== │ │ │ │ +00043170: 3d0a 0a20 2020 2020 2020 2020 2020 2020 =.. │ │ │ │ +00043180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000431a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000431b0: 2020 2020 2020 2020 2020 2020 2020 0a43 .C │ │ │ │ +000431c0: 7572 7265 6e74 6c79 2075 7365 7273 2063 urrently users c │ │ │ │ +000431d0: 616e 206f 6e6c 7920 7573 6520 2a6e 6f74 an only use *not │ │ │ │ +000431e0: 6520 7072 696d 6172 7949 6e76 6172 6961 e primaryInvaria │ │ │ │ +000431f0: 6e74 733a 2070 7269 6d61 7279 496e 7661 nts: primaryInva │ │ │ │ +00043200: 7269 616e 7473 2c20 746f 0a20 2020 2020 riants, to. │ │ │ │ 00043210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043230: 2020 2020 2020 2020 2020 2020 2020 0a43 .C │ │ │ │ -00043240: 7572 7265 6e74 6c79 2075 7365 7273 2063 urrently users c │ │ │ │ -00043250: 616e 206f 6e6c 7920 7573 6520 2a6e 6f74 an only use *not │ │ │ │ -00043260: 6520 7072 696d 6172 7949 6e76 6172 6961 e primaryInvaria │ │ │ │ -00043270: 6e74 733a 2070 7269 6d61 7279 496e 7661 nts: primaryInva │ │ │ │ -00043280: 7269 616e 7473 2c20 746f 0a20 2020 2020 riants, to. │ │ │ │ -00043290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000432a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043250: 2020 2020 2020 2020 0a63 616c 6375 6c61 .calcula │ │ │ │ +00043260: 7465 2061 2068 736f 7020 666f 7220 7468 te a hsop for th │ │ │ │ +00043270: 6520 696e 7661 7269 616e 7420 7269 6e67 e invariant ring │ │ │ │ +00043280: 206f 7665 7220 6120 6669 6e69 7465 2066 over a finite f │ │ │ │ +00043290: 6965 6c64 2062 7920 7573 696e 6720 7468 ield by using th │ │ │ │ +000432a0: 6520 4461 6465 0a20 2020 2020 2020 2020 e Dade. │ │ │ │ 000432b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000432c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000432d0: 2020 2020 2020 2020 0a63 616c 6375 6c61 .calcula │ │ │ │ -000432e0: 7465 2061 2068 736f 7020 666f 7220 7468 te a hsop for th │ │ │ │ -000432f0: 6520 696e 7661 7269 616e 7420 7269 6e67 e invariant ring │ │ │ │ -00043300: 206f 7665 7220 6120 6669 6e69 7465 2066 over a finite f │ │ │ │ -00043310: 6965 6c64 2062 7920 7573 696e 6720 7468 ield by using th │ │ │ │ -00043320: 6520 4461 6465 0a20 2020 2020 2020 2020 e Dade. │ │ │ │ -00043330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000432d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000432e0: 2020 2020 2020 2020 2020 2020 200a 616c .al │ │ │ │ +000432f0: 676f 7269 7468 6d2e 2055 7365 7273 2073 gorithm. Users s │ │ │ │ +00043300: 686f 756c 6420 656e 7465 7220 7468 6520 hould enter the │ │ │ │ +00043310: 6669 6e69 7465 2066 6965 6c64 2061 7320 finite field as │ │ │ │ +00043320: 6120 2a6e 6f74 6520 4761 6c6f 6973 4669 a *note GaloisFi │ │ │ │ +00043330: 656c 643a 0a20 2020 2020 2020 2020 2020 eld:. │ │ │ │ 00043340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043360: 2020 2020 2020 2020 2020 2020 200a 616c .al │ │ │ │ -00043370: 676f 7269 7468 6d2e 2055 7365 7273 2073 gorithm. Users s │ │ │ │ -00043380: 686f 756c 6420 656e 7465 7220 7468 6520 hould enter the │ │ │ │ -00043390: 6669 6e69 7465 2066 6965 6c64 2061 7320 finite field as │ │ │ │ -000433a0: 6120 2a6e 6f74 6520 4761 6c6f 6973 4669 a *note GaloisFi │ │ │ │ -000433b0: 656c 643a 0a20 2020 2020 2020 2020 2020 eld:. │ │ │ │ +00043360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043370: 2020 2020 2020 2020 200a 284d 6163 6175 .(Macau │ │ │ │ +00043380: 6c61 7932 446f 6329 4761 6c6f 6973 4669 lay2Doc)GaloisFi │ │ │ │ +00043390: 656c 642c 206f 7220 6120 7175 6f74 6965 eld, or a quotie │ │ │ │ +000433a0: 6e74 2066 6965 6c64 206f 6620 7468 6520 nt field of the │ │ │ │ +000433b0: 666f 726d 202a 6e6f 7465 205a 5a3a 0a20 form *note ZZ:. │ │ │ │ 000433c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000433d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000433e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000433f0: 2020 2020 2020 2020 200a 284d 6163 6175 .(Macau │ │ │ │ -00043400: 6c61 7932 446f 6329 4761 6c6f 6973 4669 lay2Doc)GaloisFi │ │ │ │ -00043410: 656c 642c 206f 7220 6120 7175 6f74 6965 eld, or a quotie │ │ │ │ -00043420: 6e74 2066 6965 6c64 206f 6620 7468 6520 nt field of the │ │ │ │ -00043430: 666f 726d 202a 6e6f 7465 205a 5a3a 0a20 form *note ZZ:. │ │ │ │ -00043440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000433f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043400: 2020 2020 2020 0a28 4d61 6361 756c 6179 .(Macaulay │ │ │ │ +00043410: 3244 6f63 295a 5a2c 2f70 2061 6e64 2061 2Doc)ZZ,/p and a │ │ │ │ +00043420: 7265 2061 6476 6973 6564 2074 6f20 656e re advised to en │ │ │ │ +00043430: 7375 7265 2074 6861 7420 7468 6520 6772 sure that the gr │ │ │ │ +00043440: 6f75 6e64 2066 6965 6c64 2068 6173 0a20 ound field has. │ │ │ │ 00043450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043480: 2020 2020 2020 0a28 4d61 6361 756c 6179 .(Macaulay │ │ │ │ -00043490: 3244 6f63 295a 5a2c 2f70 2061 6e64 2061 2Doc)ZZ,/p and a │ │ │ │ -000434a0: 7265 2061 6476 6973 6564 2074 6f20 656e re advised to en │ │ │ │ -000434b0: 7375 7265 2074 6861 7420 7468 6520 6772 sure that the gr │ │ │ │ -000434c0: 6f75 6e64 2066 6965 6c64 2068 6173 0a20 ound field has. │ │ │ │ -000434d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000434e0: 2020 2020 2020 2020 2020 206e 2d31 0a63 n-1.c │ │ │ │ -000434f0: 6172 6469 6e61 6c69 7479 2067 7265 6174 ardinality great │ │ │ │ -00043500: 6572 2074 6861 6e20 7c47 7c20 2020 2c20 er than |G| , │ │ │ │ -00043510: 7768 6572 6520 6e20 6973 2074 6865 206e where n is the n │ │ │ │ -00043520: 756d 6265 7220 6f66 2076 6172 6961 626c umber of variabl │ │ │ │ -00043530: 6573 2069 6e20 7468 650a 706f 6c79 6e6f es in the.polyno │ │ │ │ -00043540: 6d69 616c 2072 696e 6720 522e 2055 7369 mial ring R. Usi │ │ │ │ -00043550: 6e67 2061 2067 726f 756e 6420 6669 656c ng a ground fiel │ │ │ │ -00043560: 6420 736d 616c 6c65 7220 7468 616e 2074 d smaller than t │ │ │ │ -00043570: 6869 7320 7275 6e73 2074 6865 2072 6973 his runs the ris │ │ │ │ -00043580: 6b20 6f66 2074 6865 0a61 6c67 6f72 6974 k of the.algorit │ │ │ │ -00043590: 686d 2067 6574 7469 6e67 2073 7475 636b hm getting stuck │ │ │ │ -000435a0: 2069 6e20 616e 2069 6e66 696e 6974 6520 in an infinite │ │ │ │ -000435b0: 6c6f 6f70 3b20 2a6e 6f74 6520 7072 696d loop; *note prim │ │ │ │ -000435c0: 6172 7949 6e76 6172 6961 6e74 733a 0a70 aryInvariants:.p │ │ │ │ -000435d0: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ -000435e0: 2c20 6469 7370 6c61 7973 2061 2077 6172 , displays a war │ │ │ │ -000435f0: 6e69 6e67 206d 6573 7361 6765 2061 736b ning message ask │ │ │ │ -00043600: 696e 6720 7468 6520 7573 6572 2077 6865 ing the user whe │ │ │ │ -00043610: 7468 6572 2074 6865 7920 7769 7368 0a74 ther they wish.t │ │ │ │ -00043620: 6f20 636f 6e74 696e 7565 2077 6974 6820 o continue with │ │ │ │ -00043630: 7468 6520 636f 6d70 7574 6174 696f 6e20 the computation │ │ │ │ -00043640: 696e 2074 6869 7320 6361 7365 2e20 5365 in this case. Se │ │ │ │ -00043650: 6520 2a6e 6f74 6520 6873 6f70 2061 6c67 e *note hsop alg │ │ │ │ -00043660: 6f72 6974 686d 733a 2068 736f 700a 616c orithms: hsop.al │ │ │ │ -00043670: 676f 7269 7468 6d73 2c20 666f 7220 6120 gorithms, for a │ │ │ │ -00043680: 6469 7363 7573 7369 6f6e 206f 6e20 7468 discussion on th │ │ │ │ -00043690: 6520 4461 6465 2061 6c67 6f72 6974 686d e Dade algorithm │ │ │ │ -000436a0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -000436b0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -000436c0: 6873 6f70 2061 6c67 6f72 6974 686d 733a hsop algorithms: │ │ │ │ -000436d0: 2068 736f 7020 616c 676f 7269 7468 6d73 hsop algorithms │ │ │ │ -000436e0: 2c20 2d2d 2061 6e20 6f76 6572 7669 6577 , -- an overview │ │ │ │ -000436f0: 206f 6620 7468 6520 616c 676f 7269 7468 of the algorith │ │ │ │ -00043700: 6d73 0a20 2020 2075 7365 6420 696e 2070 ms. used in p │ │ │ │ -00043710: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ -00043720: 0a20 202a 202a 6e6f 7465 2070 7269 6d61 . * *note prima │ │ │ │ -00043730: 7279 496e 7661 7269 616e 7473 3a20 7072 ryInvariants: pr │ │ │ │ -00043740: 696d 6172 7949 6e76 6172 6961 6e74 732c imaryInvariants, │ │ │ │ -00043750: 202d 2d20 636f 6d70 7574 6573 2061 206c -- computes a l │ │ │ │ -00043760: 6973 7420 6f66 2070 7269 6d61 7279 0a20 ist of primary. │ │ │ │ -00043770: 2020 2069 6e76 6172 6961 6e74 7320 666f invariants fo │ │ │ │ -00043780: 7220 7468 6520 696e 7661 7269 616e 7420 r the invariant │ │ │ │ -00043790: 7269 6e67 206f 6620 6120 6669 6e69 7465 ring of a finite │ │ │ │ -000437a0: 2067 726f 7570 0a0a 4675 6e63 7469 6f6e group..Function │ │ │ │ -000437b0: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ -000437c0: 6172 6775 6d65 6e74 206e 616d 6564 2044 argument named D │ │ │ │ -000437d0: 6164 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ade:.=========== │ │ │ │ -000437e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000437f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00043800: 3d0a 0a20 202a 202a 6e6f 7465 2070 7269 =.. * *note pri │ │ │ │ -00043810: 6d61 7279 496e 7661 7269 616e 7473 282e maryInvariants(. │ │ │ │ -00043820: 2e2e 2c44 6164 653d 3e2e 2e2e 293a 0a20 ..,Dade=>...):. │ │ │ │ -00043830: 2020 2070 7269 6d61 7279 496e 7661 7269 primaryInvari │ │ │ │ -00043840: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ -00043850: 5f63 6d44 6164 653d 3e5f 7064 5f70 645f _cmDade=>_pd_pd_ │ │ │ │ -00043860: 7064 5f72 702c 202d 2d20 616e 206f 7074 pd_rp, -- an opt │ │ │ │ -00043870: 696f 6e61 6c20 6172 6775 6d65 6e74 0a20 ional argument. │ │ │ │ -00043880: 2020 2066 6f72 2070 7269 6d61 7279 496e for primaryIn │ │ │ │ -00043890: 7661 7269 616e 7473 2064 6574 6572 6d69 variants determi │ │ │ │ -000438a0: 6e69 6e67 2077 6865 7468 6572 2074 6f20 ning whether to │ │ │ │ -000438b0: 7573 6520 7468 6520 4461 6465 2061 6c67 use the Dade alg │ │ │ │ -000438c0: 6f72 6974 686d 0a0a 4675 7274 6865 7220 orithm..Further │ │ │ │ -000438d0: 696e 666f 726d 6174 696f 6e0a 3d3d 3d3d information.==== │ │ │ │ -000438e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -000438f0: 0a20 202a 2044 6566 6175 6c74 2076 616c . * Default val │ │ │ │ -00043900: 7565 3a20 2a6e 6f74 6520 6661 6c73 653a ue: *note false: │ │ │ │ -00043910: 2028 4d61 6361 756c 6179 3244 6f63 2966 (Macaulay2Doc)f │ │ │ │ -00043920: 616c 7365 2c0a 2020 2a20 4675 6e63 7469 alse,. * Functi │ │ │ │ -00043930: 6f6e 3a20 2a6e 6f74 6520 7072 696d 6172 on: *note primar │ │ │ │ -00043940: 7949 6e76 6172 6961 6e74 733a 2070 7269 yInvariants: pri │ │ │ │ -00043950: 6d61 7279 496e 7661 7269 616e 7473 2c20 maryInvariants, │ │ │ │ -00043960: 2d2d 2063 6f6d 7075 7465 7320 6120 6c69 -- computes a li │ │ │ │ -00043970: 7374 206f 660a 2020 2020 7072 696d 6172 st of. primar │ │ │ │ -00043980: 7920 696e 7661 7269 616e 7473 2066 6f72 y invariants for │ │ │ │ -00043990: 2074 6865 2069 6e76 6172 6961 6e74 2072 the invariant r │ │ │ │ -000439a0: 696e 6720 6f66 2061 2066 696e 6974 6520 ing of a finite │ │ │ │ -000439b0: 6772 6f75 700a 2020 2a20 4f70 7469 6f6e group. * Option │ │ │ │ -000439c0: 206b 6579 3a20 2a6e 6f74 6520 4461 6465 key: *note Dade │ │ │ │ -000439d0: 3a20 7072 696d 6172 7949 6e76 6172 6961 : primaryInvaria │ │ │ │ -000439e0: 6e74 735f 6c70 5f70 645f 7064 5f70 645f nts_lp_pd_pd_pd_ │ │ │ │ -000439f0: 636d 4461 6465 3d3e 5f70 645f 7064 5f70 cmDade=>_pd_pd_p │ │ │ │ -00043a00: 645f 7270 2c0a 2020 2020 2d2d 2061 6e20 d_rp,. -- an │ │ │ │ -00043a10: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ -00043a20: 7420 666f 7220 7072 696d 6172 7949 6e76 t for primaryInv │ │ │ │ -00043a30: 6172 6961 6e74 7320 6465 7465 726d 696e ariants determin │ │ │ │ -00043a40: 696e 6720 7768 6574 6865 7220 746f 2075 ing whether to u │ │ │ │ -00043a50: 7365 0a20 2020 2074 6865 2044 6164 6520 se. the Dade │ │ │ │ -00043a60: 616c 676f 7269 7468 6d0a 2d2d 2d2d 2d2d algorithm.------ │ │ │ │ -00043a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00043ab0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00043ac0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00043ad0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -00043ae0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -00043af0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -00043b00: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -00043b10: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -00043b20: 2f0a 496e 7661 7269 616e 7452 696e 672f /.InvariantRing/ │ │ │ │ -00043b30: 4861 7765 7344 6f63 2e6d 323a 3234 343a HawesDoc.m2:244: │ │ │ │ -00043b40: 302e 0a1f 0a46 696c 653a 2049 6e76 6172 0....File: Invar │ │ │ │ -00043b50: 6961 6e74 5269 6e67 2e69 6e66 6f2c 204e iantRing.info, N │ │ │ │ -00043b60: 6f64 653a 2070 7269 6d61 7279 496e 7661 ode: primaryInva │ │ │ │ -00043b70: 7269 616e 7473 5f6c 705f 7064 5f70 645f riants_lp_pd_pd_ │ │ │ │ -00043b80: 7064 5f63 6d44 6567 7265 6556 6563 746f pd_cmDegreeVecto │ │ │ │ -00043b90: 723d 3e5f 7064 5f70 645f 7064 5f72 702c r=>_pd_pd_pd_rp, │ │ │ │ -00043ba0: 204e 6578 743a 2072 616e 6b5f 6c70 4469 Next: rank_lpDi │ │ │ │ -00043bb0: 6167 6f6e 616c 4163 7469 6f6e 5f72 702c agonalAction_rp, │ │ │ │ -00043bc0: 2050 7265 763a 2070 7269 6d61 7279 496e Prev: primaryIn │ │ │ │ -00043bd0: 7661 7269 616e 7473 5f6c 705f 7064 5f70 variants_lp_pd_p │ │ │ │ -00043be0: 645f 7064 5f63 6d44 6164 653d 3e5f 7064 d_pd_cmDade=>_pd │ │ │ │ -00043bf0: 5f70 645f 7064 5f72 702c 2055 703a 2054 _pd_pd_rp, Up: T │ │ │ │ -00043c00: 6f70 0a0a 7072 696d 6172 7949 6e76 6172 op..primaryInvar │ │ │ │ -00043c10: 6961 6e74 7328 2e2e 2e2c 4465 6772 6565 iants(...,Degree │ │ │ │ -00043c20: 5665 6374 6f72 3d3e 2e2e 2e29 202d 2d20 Vector=>...) -- │ │ │ │ -00043c30: 616e 206f 7074 696f 6e61 6c20 6172 6775 an optional argu │ │ │ │ -00043c40: 6d65 6e74 2066 6f72 2070 7269 6d61 7279 ment for primary │ │ │ │ -00043c50: 496e 7661 7269 616e 7473 2074 6861 7420 Invariants that │ │ │ │ -00043c60: 6669 6e64 7320 696e 7661 7269 616e 7473 finds invariants │ │ │ │ -00043c70: 206f 6620 6365 7274 6169 6e20 6465 6772 of certain degr │ │ │ │ -00043c80: 6565 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ees.************ │ │ │ │ -00043c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00043ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00043cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00043cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00043cd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00043ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00043cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00043d00: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -00043d10: 0a20 2020 2020 2020 2070 7269 6d61 7279 . primary │ │ │ │ -00043d20: 496e 7661 7269 616e 7473 2047 0a20 202a Invariants G. * │ │ │ │ -00043d30: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00043d40: 2047 2c20 616e 2069 6e73 7461 6e63 6520 G, an instance │ │ │ │ -00043d50: 6f66 2074 6865 2074 7970 6520 2a6e 6f74 of the type *not │ │ │ │ -00043d60: 6520 4669 6e69 7465 4772 6f75 7041 6374 e FiniteGroupAct │ │ │ │ -00043d70: 696f 6e3a 2046 696e 6974 6547 726f 7570 ion: FiniteGroup │ │ │ │ -00043d80: 4163 7469 6f6e 2c0a 2020 2a20 4f75 7470 Action,. * Outp │ │ │ │ -00043d90: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ -00043da0: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ -00043db0: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ -00043dc0: 2063 6f6e 7369 7374 696e 6720 6f66 2061 consisting of a │ │ │ │ -00043dd0: 2068 6f6d 6f67 656e 656f 7573 2073 7973 homogeneous sys │ │ │ │ -00043de0: 7465 6d0a 2020 2020 2020 2020 6f66 2070 tem. of p │ │ │ │ -00043df0: 6172 616d 6574 6572 7320 2868 736f 7029 arameters (hsop) │ │ │ │ -00043e00: 2066 6f72 2074 6865 2069 6e76 6172 6961 for the invaria │ │ │ │ -00043e10: 6e74 2072 696e 6720 6f66 2074 6865 2067 nt ring of the g │ │ │ │ -00043e20: 726f 7570 2061 6374 696f 6e0a 0a44 6573 roup action..Des │ │ │ │ -00043e30: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00043e40: 3d3d 3d3d 0a0a 2020 2020 2020 2020 2020 ====.. │ │ │ │ -00043e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043460: 2020 2020 2020 2020 2020 206e 2d31 0a63 n-1.c │ │ │ │ +00043470: 6172 6469 6e61 6c69 7479 2067 7265 6174 ardinality great │ │ │ │ +00043480: 6572 2074 6861 6e20 7c47 7c20 2020 2c20 er than |G| , │ │ │ │ +00043490: 7768 6572 6520 6e20 6973 2074 6865 206e where n is the n │ │ │ │ +000434a0: 756d 6265 7220 6f66 2076 6172 6961 626c umber of variabl │ │ │ │ +000434b0: 6573 2069 6e20 7468 650a 706f 6c79 6e6f es in the.polyno │ │ │ │ +000434c0: 6d69 616c 2072 696e 6720 522e 2055 7369 mial ring R. Usi │ │ │ │ +000434d0: 6e67 2061 2067 726f 756e 6420 6669 656c ng a ground fiel │ │ │ │ +000434e0: 6420 736d 616c 6c65 7220 7468 616e 2074 d smaller than t │ │ │ │ +000434f0: 6869 7320 7275 6e73 2074 6865 2072 6973 his runs the ris │ │ │ │ +00043500: 6b20 6f66 2074 6865 0a61 6c67 6f72 6974 k of the.algorit │ │ │ │ +00043510: 686d 2067 6574 7469 6e67 2073 7475 636b hm getting stuck │ │ │ │ +00043520: 2069 6e20 616e 2069 6e66 696e 6974 6520 in an infinite │ │ │ │ +00043530: 6c6f 6f70 3b20 2a6e 6f74 6520 7072 696d loop; *note prim │ │ │ │ +00043540: 6172 7949 6e76 6172 6961 6e74 733a 0a70 aryInvariants:.p │ │ │ │ +00043550: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +00043560: 2c20 6469 7370 6c61 7973 2061 2077 6172 , displays a war │ │ │ │ +00043570: 6e69 6e67 206d 6573 7361 6765 2061 736b ning message ask │ │ │ │ +00043580: 696e 6720 7468 6520 7573 6572 2077 6865 ing the user whe │ │ │ │ +00043590: 7468 6572 2074 6865 7920 7769 7368 0a74 ther they wish.t │ │ │ │ +000435a0: 6f20 636f 6e74 696e 7565 2077 6974 6820 o continue with │ │ │ │ +000435b0: 7468 6520 636f 6d70 7574 6174 696f 6e20 the computation │ │ │ │ +000435c0: 696e 2074 6869 7320 6361 7365 2e20 5365 in this case. Se │ │ │ │ +000435d0: 6520 2a6e 6f74 6520 6873 6f70 2061 6c67 e *note hsop alg │ │ │ │ +000435e0: 6f72 6974 686d 733a 2068 736f 700a 616c orithms: hsop.al │ │ │ │ +000435f0: 676f 7269 7468 6d73 2c20 666f 7220 6120 gorithms, for a │ │ │ │ +00043600: 6469 7363 7573 7369 6f6e 206f 6e20 7468 discussion on th │ │ │ │ +00043610: 6520 4461 6465 2061 6c67 6f72 6974 686d e Dade algorithm │ │ │ │ +00043620: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ +00043630: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +00043640: 6873 6f70 2061 6c67 6f72 6974 686d 733a hsop algorithms: │ │ │ │ +00043650: 2068 736f 7020 616c 676f 7269 7468 6d73 hsop algorithms │ │ │ │ +00043660: 2c20 2d2d 2061 6e20 6f76 6572 7669 6577 , -- an overview │ │ │ │ +00043670: 206f 6620 7468 6520 616c 676f 7269 7468 of the algorith │ │ │ │ +00043680: 6d73 0a20 2020 2075 7365 6420 696e 2070 ms. used in p │ │ │ │ +00043690: 7269 6d61 7279 496e 7661 7269 616e 7473 rimaryInvariants │ │ │ │ +000436a0: 0a20 202a 202a 6e6f 7465 2070 7269 6d61 . * *note prima │ │ │ │ +000436b0: 7279 496e 7661 7269 616e 7473 3a20 7072 ryInvariants: pr │ │ │ │ +000436c0: 696d 6172 7949 6e76 6172 6961 6e74 732c imaryInvariants, │ │ │ │ +000436d0: 202d 2d20 636f 6d70 7574 6573 2061 206c -- computes a l │ │ │ │ +000436e0: 6973 7420 6f66 2070 7269 6d61 7279 0a20 ist of primary. │ │ │ │ +000436f0: 2020 2069 6e76 6172 6961 6e74 7320 666f invariants fo │ │ │ │ +00043700: 7220 7468 6520 696e 7661 7269 616e 7420 r the invariant │ │ │ │ +00043710: 7269 6e67 206f 6620 6120 6669 6e69 7465 ring of a finite │ │ │ │ +00043720: 2067 726f 7570 0a0a 4675 6e63 7469 6f6e group..Function │ │ │ │ +00043730: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ +00043740: 6172 6775 6d65 6e74 206e 616d 6564 2044 argument named D │ │ │ │ +00043750: 6164 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ade:.=========== │ │ │ │ +00043760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00043770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00043780: 3d0a 0a20 202a 202a 6e6f 7465 2070 7269 =.. * *note pri │ │ │ │ +00043790: 6d61 7279 496e 7661 7269 616e 7473 282e maryInvariants(. │ │ │ │ +000437a0: 2e2e 2c44 6164 653d 3e2e 2e2e 293a 0a20 ..,Dade=>...):. │ │ │ │ +000437b0: 2020 2070 7269 6d61 7279 496e 7661 7269 primaryInvari │ │ │ │ +000437c0: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ +000437d0: 5f63 6d44 6164 653d 3e5f 7064 5f70 645f _cmDade=>_pd_pd_ │ │ │ │ +000437e0: 7064 5f72 702c 202d 2d20 616e 206f 7074 pd_rp, -- an opt │ │ │ │ +000437f0: 696f 6e61 6c20 6172 6775 6d65 6e74 0a20 ional argument. │ │ │ │ +00043800: 2020 2066 6f72 2070 7269 6d61 7279 496e for primaryIn │ │ │ │ +00043810: 7661 7269 616e 7473 2064 6574 6572 6d69 variants determi │ │ │ │ +00043820: 6e69 6e67 2077 6865 7468 6572 2074 6f20 ning whether to │ │ │ │ +00043830: 7573 6520 7468 6520 4461 6465 2061 6c67 use the Dade alg │ │ │ │ +00043840: 6f72 6974 686d 0a0a 4675 7274 6865 7220 orithm..Further │ │ │ │ +00043850: 696e 666f 726d 6174 696f 6e0a 3d3d 3d3d information.==== │ │ │ │ +00043860: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00043870: 0a20 202a 2044 6566 6175 6c74 2076 616c . * Default val │ │ │ │ +00043880: 7565 3a20 2a6e 6f74 6520 6661 6c73 653a ue: *note false: │ │ │ │ +00043890: 2028 4d61 6361 756c 6179 3244 6f63 2966 (Macaulay2Doc)f │ │ │ │ +000438a0: 616c 7365 2c0a 2020 2a20 4675 6e63 7469 alse,. * Functi │ │ │ │ +000438b0: 6f6e 3a20 2a6e 6f74 6520 7072 696d 6172 on: *note primar │ │ │ │ +000438c0: 7949 6e76 6172 6961 6e74 733a 2070 7269 yInvariants: pri │ │ │ │ +000438d0: 6d61 7279 496e 7661 7269 616e 7473 2c20 maryInvariants, │ │ │ │ +000438e0: 2d2d 2063 6f6d 7075 7465 7320 6120 6c69 -- computes a li │ │ │ │ +000438f0: 7374 206f 660a 2020 2020 7072 696d 6172 st of. primar │ │ │ │ +00043900: 7920 696e 7661 7269 616e 7473 2066 6f72 y invariants for │ │ │ │ +00043910: 2074 6865 2069 6e76 6172 6961 6e74 2072 the invariant r │ │ │ │ +00043920: 696e 6720 6f66 2061 2066 696e 6974 6520 ing of a finite │ │ │ │ +00043930: 6772 6f75 700a 2020 2a20 4f70 7469 6f6e group. * Option │ │ │ │ +00043940: 206b 6579 3a20 2a6e 6f74 6520 4461 6465 key: *note Dade │ │ │ │ +00043950: 3a20 7072 696d 6172 7949 6e76 6172 6961 : primaryInvaria │ │ │ │ +00043960: 6e74 735f 6c70 5f70 645f 7064 5f70 645f nts_lp_pd_pd_pd_ │ │ │ │ +00043970: 636d 4461 6465 3d3e 5f70 645f 7064 5f70 cmDade=>_pd_pd_p │ │ │ │ +00043980: 645f 7270 2c0a 2020 2020 2d2d 2061 6e20 d_rp,. -- an │ │ │ │ +00043990: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ +000439a0: 7420 666f 7220 7072 696d 6172 7949 6e76 t for primaryInv │ │ │ │ +000439b0: 6172 6961 6e74 7320 6465 7465 726d 696e ariants determin │ │ │ │ +000439c0: 696e 6720 7768 6574 6865 7220 746f 2075 ing whether to u │ │ │ │ +000439d0: 7365 0a20 2020 2074 6865 2044 6164 6520 se. the Dade │ │ │ │ +000439e0: 616c 676f 7269 7468 6d0a 2d2d 2d2d 2d2d algorithm.------ │ │ │ │ +000439f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00043a30: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00043a40: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00043a50: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00043a60: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00043a70: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00043a80: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ +00043a90: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00043aa0: 2f0a 496e 7661 7269 616e 7452 696e 672f /.InvariantRing/ │ │ │ │ +00043ab0: 4861 7765 7344 6f63 2e6d 323a 3234 343a HawesDoc.m2:244: │ │ │ │ +00043ac0: 302e 0a1f 0a46 696c 653a 2049 6e76 6172 0....File: Invar │ │ │ │ +00043ad0: 6961 6e74 5269 6e67 2e69 6e66 6f2c 204e iantRing.info, N │ │ │ │ +00043ae0: 6f64 653a 2070 7269 6d61 7279 496e 7661 ode: primaryInva │ │ │ │ +00043af0: 7269 616e 7473 5f6c 705f 7064 5f70 645f riants_lp_pd_pd_ │ │ │ │ +00043b00: 7064 5f63 6d44 6567 7265 6556 6563 746f pd_cmDegreeVecto │ │ │ │ +00043b10: 723d 3e5f 7064 5f70 645f 7064 5f72 702c r=>_pd_pd_pd_rp, │ │ │ │ +00043b20: 204e 6578 743a 2072 616e 6b5f 6c70 4469 Next: rank_lpDi │ │ │ │ +00043b30: 6167 6f6e 616c 4163 7469 6f6e 5f72 702c agonalAction_rp, │ │ │ │ +00043b40: 2050 7265 763a 2070 7269 6d61 7279 496e Prev: primaryIn │ │ │ │ +00043b50: 7661 7269 616e 7473 5f6c 705f 7064 5f70 variants_lp_pd_p │ │ │ │ +00043b60: 645f 7064 5f63 6d44 6164 653d 3e5f 7064 d_pd_cmDade=>_pd │ │ │ │ +00043b70: 5f70 645f 7064 5f72 702c 2055 703a 2054 _pd_pd_rp, Up: T │ │ │ │ +00043b80: 6f70 0a0a 7072 696d 6172 7949 6e76 6172 op..primaryInvar │ │ │ │ +00043b90: 6961 6e74 7328 2e2e 2e2c 4465 6772 6565 iants(...,Degree │ │ │ │ +00043ba0: 5665 6374 6f72 3d3e 2e2e 2e29 202d 2d20 Vector=>...) -- │ │ │ │ +00043bb0: 616e 206f 7074 696f 6e61 6c20 6172 6775 an optional argu │ │ │ │ +00043bc0: 6d65 6e74 2066 6f72 2070 7269 6d61 7279 ment for primary │ │ │ │ +00043bd0: 496e 7661 7269 616e 7473 2074 6861 7420 Invariants that │ │ │ │ +00043be0: 6669 6e64 7320 696e 7661 7269 616e 7473 finds invariants │ │ │ │ +00043bf0: 206f 6620 6365 7274 6169 6e20 6465 6772 of certain degr │ │ │ │ +00043c00: 6565 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ees.************ │ │ │ │ +00043c10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00043c20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00043c30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00043c40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00043c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00043c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00043c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00043c80: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00043c90: 0a20 2020 2020 2020 2070 7269 6d61 7279 . primary │ │ │ │ +00043ca0: 496e 7661 7269 616e 7473 2047 0a20 202a Invariants G. * │ │ │ │ +00043cb0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00043cc0: 2047 2c20 616e 2069 6e73 7461 6e63 6520 G, an instance │ │ │ │ +00043cd0: 6f66 2074 6865 2074 7970 6520 2a6e 6f74 of the type *not │ │ │ │ +00043ce0: 6520 4669 6e69 7465 4772 6f75 7041 6374 e FiniteGroupAct │ │ │ │ +00043cf0: 696f 6e3a 2046 696e 6974 6547 726f 7570 ion: FiniteGroup │ │ │ │ +00043d00: 4163 7469 6f6e 2c0a 2020 2a20 4f75 7470 Action,. * Outp │ │ │ │ +00043d10: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ +00043d20: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ +00043d30: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ +00043d40: 2063 6f6e 7369 7374 696e 6720 6f66 2061 consisting of a │ │ │ │ +00043d50: 2068 6f6d 6f67 656e 656f 7573 2073 7973 homogeneous sys │ │ │ │ +00043d60: 7465 6d0a 2020 2020 2020 2020 6f66 2070 tem. of p │ │ │ │ +00043d70: 6172 616d 6574 6572 7320 2868 736f 7029 arameters (hsop) │ │ │ │ +00043d80: 2066 6f72 2074 6865 2069 6e76 6172 6961 for the invaria │ │ │ │ +00043d90: 6e74 2072 696e 6720 6f66 2074 6865 2067 nt ring of the g │ │ │ │ +00043da0: 726f 7570 2061 6374 696f 6e0a 0a44 6573 roup action..Des │ │ │ │ +00043db0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +00043dc0: 3d3d 3d3d 0a0a 2020 2020 2020 2020 2020 ====.. │ │ │ │ +00043dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043e10: 0a42 7920 6465 6661 756c 742c 202a 6e6f .By default, *no │ │ │ │ +00043e20: 7465 2070 7269 6d61 7279 496e 7661 7269 te primaryInvari │ │ │ │ +00043e30: 616e 7473 3a20 7072 696d 6172 7949 6e76 ants: primaryInv │ │ │ │ +00043e40: 6172 6961 6e74 732c 2075 7365 7320 616e ariants, uses an │ │ │ │ +00043e50: 206f 7074 696d 6973 696e 670a 2020 2020 optimising. │ │ │ │ 00043e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043e90: 0a42 7920 6465 6661 756c 742c 202a 6e6f .By default, *no │ │ │ │ -00043ea0: 7465 2070 7269 6d61 7279 496e 7661 7269 te primaryInvari │ │ │ │ -00043eb0: 616e 7473 3a20 7072 696d 6172 7949 6e76 ants: primaryInv │ │ │ │ -00043ec0: 6172 6961 6e74 732c 2075 7365 7320 616e ariants, uses an │ │ │ │ -00043ed0: 206f 7074 696d 6973 696e 670a 2020 2020 optimising. │ │ │ │ +00043e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043ea0: 2020 2020 2020 0a20 2020 2020 2020 2020 . │ │ │ │ +00043eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043f20: 2020 2020 2020 0a20 2020 2020 2020 2020 . │ │ │ │ -00043f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043ef0: 2020 2020 0a61 6c67 6f72 6974 686d 2077 .algorithm w │ │ │ │ +00043f00: 6869 6368 2074 6573 7473 2066 6f72 2074 hich tests for t │ │ │ │ +00043f10: 6865 2065 7869 7374 656e 6365 206f 6620 he existence of │ │ │ │ +00043f20: 6120 686f 6d6f 6765 6e65 6f75 7320 7379 a homogeneous sy │ │ │ │ +00043f30: 7374 656d 206f 6620 7061 7261 6d65 7465 stem of paramete │ │ │ │ +00043f40: 7273 0a20 2020 2020 2020 2020 2020 2020 rs. │ │ │ │ 00043f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00043f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043f70: 2020 2020 0a61 6c67 6f72 6974 686d 2077 .algorithm w │ │ │ │ -00043f80: 6869 6368 2074 6573 7473 2066 6f72 2074 hich tests for t │ │ │ │ -00043f90: 6865 2065 7869 7374 656e 6365 206f 6620 he existence of │ │ │ │ -00043fa0: 6120 686f 6d6f 6765 6e65 6f75 7320 7379 a homogeneous sy │ │ │ │ -00043fb0: 7374 656d 206f 6620 7061 7261 6d65 7465 stem of paramete │ │ │ │ -00043fc0: 7273 0a20 2020 2020 2020 2020 2020 2020 rs. │ │ │ │ -00043fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00043ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044010: 0a20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -00044020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043f90: 0a20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +00043fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00043fd0: 2020 2020 2020 2020 2020 2020 2020 200a . │ │ │ │ +00043fe0: 2868 736f 7029 2028 6620 2c2e 2e2e 2c66 (hsop) (f ,...,f │ │ │ │ +00043ff0: 2029 2077 6974 6820 706f 7369 7469 7665 ) with positive │ │ │ │ +00044000: 2064 6567 7265 6573 2063 6f72 7265 7370 degrees corresp │ │ │ │ +00044010: 6f6e 6469 6e67 2074 6f20 2864 202c 2e2e onding to (d ,.. │ │ │ │ +00044020: 2e2c 6420 2920 696e 202a 6e6f 7465 0a20 .,d ) in *note. │ │ │ │ +00044030: 2020 2020 2020 2020 3120 2020 2020 206e 1 n │ │ │ │ 00044040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044050: 2020 2020 2020 2020 2020 2020 2020 200a . │ │ │ │ -00044060: 2868 736f 7029 2028 6620 2c2e 2e2e 2c66 (hsop) (f ,...,f │ │ │ │ -00044070: 2029 2077 6974 6820 706f 7369 7469 7665 ) with positive │ │ │ │ -00044080: 2064 6567 7265 6573 2063 6f72 7265 7370 degrees corresp │ │ │ │ -00044090: 6f6e 6469 6e67 2074 6f20 2864 202c 2e2e onding to (d ,.. │ │ │ │ -000440a0: 2e2c 6420 2920 696e 202a 6e6f 7465 0a20 .,d ) in *note. │ │ │ │ -000440b0: 2020 2020 2020 2020 3120 2020 2020 206e 1 n │ │ │ │ -000440c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000440d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000440e0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -000440f0: 2020 6e0a 2020 2020 2020 2020 2020 2020 n. │ │ │ │ -00044100: 2020 2020 2020 2020 206e 0a5a 5a3a 2028 n.ZZ: ( │ │ │ │ -00044110: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -00044120: 202e 2049 6620 6974 2069 7320 6b6e 6f77 . If it is know │ │ │ │ -00044130: 6e20 7468 6174 2061 2068 736f 7020 6578 n that a hsop ex │ │ │ │ -00044140: 6973 7473 2066 6f72 2061 2063 6572 7461 ists for a certa │ │ │ │ -00044150: 696e 0a63 6f6c 6c65 6374 696f 6e20 6f66 in.collection of │ │ │ │ -00044160: 2064 6567 7265 6573 2c20 7468 6973 2063 degrees, this c │ │ │ │ -00044170: 616e 2062 6520 6173 7369 676e 6564 2c20 an be assigned, │ │ │ │ -00044180: 6173 2061 202a 6e6f 7465 204c 6973 743a as a *note List: │ │ │ │ -00044190: 0a28 4d61 6361 756c 6179 3244 6f63 294c .(Macaulay2Doc)L │ │ │ │ -000441a0: 6973 742c 2c20 746f 2074 6865 206f 7074 ist,, to the opt │ │ │ │ -000441b0: 696f 6e61 6c20 6172 6775 6d65 6e74 202a ional argument * │ │ │ │ -000441c0: 6e6f 7465 2044 6567 7265 6556 6563 746f note DegreeVecto │ │ │ │ -000441d0: 723a 0a70 7269 6d61 7279 496e 7661 7269 r:.primaryInvari │ │ │ │ -000441e0: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ -000441f0: 5f63 6d44 6567 7265 6556 6563 746f 723d _cmDegreeVector= │ │ │ │ -00044200: 3e5f 7064 5f70 645f 7064 5f72 702c 2e20 >_pd_pd_pd_rp,. │ │ │ │ -00044210: 2a6e 6f74 650a 7072 696d 6172 7949 6e76 *note.primaryInv │ │ │ │ -00044220: 6172 6961 6e74 733a 2070 7269 6d61 7279 ariants: primary │ │ │ │ -00044230: 496e 7661 7269 616e 7473 2c20 7769 6c6c Invariants, will │ │ │ │ -00044240: 2074 6865 6e20 6f75 7470 7574 2061 2068 then output a h │ │ │ │ -00044250: 736f 7020 636f 7272 6573 706f 6e64 696e sop correspondin │ │ │ │ -00044260: 6720 746f 0a74 6869 7320 6c69 7374 206f g to.this list o │ │ │ │ -00044270: 6620 6465 6772 6565 732e 2049 6620 686f f degrees. If ho │ │ │ │ -00044280: 7765 7665 7220 6e6f 2073 7563 6820 6873 wever no such hs │ │ │ │ -00044290: 6f70 2065 7869 7374 732c 202a 6e6f 7465 op exists, *note │ │ │ │ -000442a0: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ -000442b0: 7473 3a0a 7072 696d 6172 7949 6e76 6172 ts:.primaryInvar │ │ │ │ -000442c0: 6961 6e74 732c 206f 7574 7075 7473 2061 iants, outputs a │ │ │ │ -000442d0: 6e20 6572 726f 7220 6d65 7373 6167 652e n error message. │ │ │ │ -000442e0: 0a0a 4e6f 7465 2074 6861 7420 7468 6520 ..Note that the │ │ │ │ -000442f0: 2a6e 6f74 6520 4c69 7374 3a20 284d 6163 *note List: (Mac │ │ │ │ -00044300: 6175 6c61 7932 446f 6329 4c69 7374 2c20 aulay2Doc)List, │ │ │ │ -00044310: 6173 7369 676e 6564 2074 6f20 2a6e 6f74 assigned to *not │ │ │ │ -00044320: 6520 4465 6772 6565 5665 6374 6f72 3a0a e DegreeVector:. │ │ │ │ -00044330: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ -00044340: 735f 6c70 5f70 645f 7064 5f70 645f 636d s_lp_pd_pd_pd_cm │ │ │ │ -00044350: 4465 6772 6565 5665 6374 6f72 3d3e 5f70 DegreeVector=>_p │ │ │ │ -00044360: 645f 7064 5f70 645f 7270 2c20 6973 2069 d_pd_pd_rp, is i │ │ │ │ -00044370: 676e 6f72 6564 2069 6620 2a6e 6f74 650a gnored if *note. │ │ │ │ -00044380: 4461 6465 3a20 7072 696d 6172 7949 6e76 Dade: primaryInv │ │ │ │ -00044390: 6172 6961 6e74 735f 6c70 5f70 645f 7064 ariants_lp_pd_pd │ │ │ │ -000443a0: 5f70 645f 636d 4461 6465 3d3e 5f70 645f _pd_cmDade=>_pd_ │ │ │ │ -000443b0: 7064 5f70 645f 7270 2c20 6973 2073 6574 pd_pd_rp, is set │ │ │ │ -000443c0: 2074 6f20 2a6e 6f74 6520 7472 7565 3a0a to *note true:. │ │ │ │ -000443d0: 284d 6163 6175 6c61 7932 446f 6329 7472 (Macaulay2Doc)tr │ │ │ │ -000443e0: 7565 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ue,...+--------- │ │ │ │ -000443f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00044430: 7c69 3120 3a20 413d 6d61 7472 6978 7b7b |i1 : A=matrix{{ │ │ │ │ -00044440: 302c 312c 307d 2c7b 302c 302c 317d 2c7b 0,1,0},{0,0,1},{ │ │ │ │ -00044450: 312c 302c 307d 7d3b 2020 2020 2020 2020 1,0,0}}; │ │ │ │ -00044460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044470: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00044480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000444a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000444b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000444c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -000444d0: 2020 2033 2020 2020 2020 2033 2020 2020 3 3 │ │ │ │ +00044050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044060: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +00044070: 2020 6e0a 2020 2020 2020 2020 2020 2020 n. │ │ │ │ +00044080: 2020 2020 2020 2020 206e 0a5a 5a3a 2028 n.ZZ: ( │ │ │ │ +00044090: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ +000440a0: 202e 2049 6620 6974 2069 7320 6b6e 6f77 . If it is know │ │ │ │ +000440b0: 6e20 7468 6174 2061 2068 736f 7020 6578 n that a hsop ex │ │ │ │ +000440c0: 6973 7473 2066 6f72 2061 2063 6572 7461 ists for a certa │ │ │ │ +000440d0: 696e 0a63 6f6c 6c65 6374 696f 6e20 6f66 in.collection of │ │ │ │ +000440e0: 2064 6567 7265 6573 2c20 7468 6973 2063 degrees, this c │ │ │ │ +000440f0: 616e 2062 6520 6173 7369 676e 6564 2c20 an be assigned, │ │ │ │ +00044100: 6173 2061 202a 6e6f 7465 204c 6973 743a as a *note List: │ │ │ │ +00044110: 0a28 4d61 6361 756c 6179 3244 6f63 294c .(Macaulay2Doc)L │ │ │ │ +00044120: 6973 742c 2c20 746f 2074 6865 206f 7074 ist,, to the opt │ │ │ │ +00044130: 696f 6e61 6c20 6172 6775 6d65 6e74 202a ional argument * │ │ │ │ +00044140: 6e6f 7465 2044 6567 7265 6556 6563 746f note DegreeVecto │ │ │ │ +00044150: 723a 0a70 7269 6d61 7279 496e 7661 7269 r:.primaryInvari │ │ │ │ +00044160: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ +00044170: 5f63 6d44 6567 7265 6556 6563 746f 723d _cmDegreeVector= │ │ │ │ +00044180: 3e5f 7064 5f70 645f 7064 5f72 702c 2e20 >_pd_pd_pd_rp,. │ │ │ │ +00044190: 2a6e 6f74 650a 7072 696d 6172 7949 6e76 *note.primaryInv │ │ │ │ +000441a0: 6172 6961 6e74 733a 2070 7269 6d61 7279 ariants: primary │ │ │ │ +000441b0: 496e 7661 7269 616e 7473 2c20 7769 6c6c Invariants, will │ │ │ │ +000441c0: 2074 6865 6e20 6f75 7470 7574 2061 2068 then output a h │ │ │ │ +000441d0: 736f 7020 636f 7272 6573 706f 6e64 696e sop correspondin │ │ │ │ +000441e0: 6720 746f 0a74 6869 7320 6c69 7374 206f g to.this list o │ │ │ │ +000441f0: 6620 6465 6772 6565 732e 2049 6620 686f f degrees. If ho │ │ │ │ +00044200: 7765 7665 7220 6e6f 2073 7563 6820 6873 wever no such hs │ │ │ │ +00044210: 6f70 2065 7869 7374 732c 202a 6e6f 7465 op exists, *note │ │ │ │ +00044220: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +00044230: 7473 3a0a 7072 696d 6172 7949 6e76 6172 ts:.primaryInvar │ │ │ │ +00044240: 6961 6e74 732c 206f 7574 7075 7473 2061 iants, outputs a │ │ │ │ +00044250: 6e20 6572 726f 7220 6d65 7373 6167 652e n error message. │ │ │ │ +00044260: 0a0a 4e6f 7465 2074 6861 7420 7468 6520 ..Note that the │ │ │ │ +00044270: 2a6e 6f74 6520 4c69 7374 3a20 284d 6163 *note List: (Mac │ │ │ │ +00044280: 6175 6c61 7932 446f 6329 4c69 7374 2c20 aulay2Doc)List, │ │ │ │ +00044290: 6173 7369 676e 6564 2074 6f20 2a6e 6f74 assigned to *not │ │ │ │ +000442a0: 6520 4465 6772 6565 5665 6374 6f72 3a0a e DegreeVector:. │ │ │ │ +000442b0: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ +000442c0: 735f 6c70 5f70 645f 7064 5f70 645f 636d s_lp_pd_pd_pd_cm │ │ │ │ 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+00044490: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 423d ------+.|i2 : B= │ │ │ │ +000444a0: 6d61 7472 6978 7b7b 302c 312c 307d 2c7b matrix{{0,1,0},{ │ │ │ │ +000444b0: 312c 302c 307d 2c7b 302c 302c 317d 7d3b 1,0,0},{0,0,1}}; │ │ │ │ +000444c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000444d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000444e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000444f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044500: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00044510: 3120 3a20 4d61 7472 6978 205a 5a20 203c 1 : Matrix ZZ < │ │ │ │ -00044520: 2d2d 205a 5a20 2020 2020 2020 2020 2020 -- ZZ │ │ │ │ -00044530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044550: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -00044560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000444f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 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2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00044730: 6934 203a 2070 7269 6d61 7279 496e 7661 i4 : primaryInva │ │ │ │ +00044740: 7269 616e 7473 2853 332c 4465 6772 6565 riants(S3,Degree │ │ │ │ +00044750: 5665 6374 6f72 3d3e 7b33 2c33 2c34 7d29 Vector=>{3,3,4}) │ │ │ │ +00044760: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00044770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000447a0: 2020 2020 2020 7c0a 7c6f 3320 3d20 5151 |.|o3 = QQ │ │ │ │ -000447b0: 5b78 2e2e 7a5d 203c 2d20 3c7c 2030 2031 [x..z] <- <| 0 1 │ │ │ │ -000447c0: 2030 207c 2c20 7c20 3020 3120 3020 7c3e 0 |, | 0 1 0 |> │ │ │ │ -000447d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000447e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000447f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00044800: 2020 2020 207c 2030 2030 2031 207c 2020 | 0 0 1 | │ │ │ │ -00044810: 7c20 3120 3020 3020 7c20 2020 2020 2020 | 1 0 0 | │ │ │ │ -00044820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044830: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00044840: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00044850: 2031 2030 2030 207c 2020 7c20 3020 3020 1 0 0 | | 0 0 │ │ │ │ -00044860: 3120 7c20 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -00044870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044880: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00044890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000448a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044790: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000447a0: 2020 2020 3320 2020 2033 2020 2020 3320 3 3 3 │ │ │ │ +000447b0: 2020 3220 3220 2020 2032 2032 2020 2020 2 2 2 2 │ │ │ │ +000447c0: 3220 3220 2020 7c0a 7c6f 3420 3d20 7b78 2 2 |.|o4 = {x │ │ │ │ +000447d0: 2a79 2a7a 2c20 7820 202b 2079 2020 2b20 *y*z, x + y + │ │ │ │ +000447e0: 7a20 2c20 7820 7920 202b 2078 207a 2020 z , x y + x z │ │ │ │ +000447f0: 2b20 7920 7a20 7d20 207c 0a7c 2020 2020 + y z } |.| │ │ │ │ +00044800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044820: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00044830: 3420 3a20 4c69 7374 2020 2020 2020 2020 4 : List │ │ │ │ +00044840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044850: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00044860: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00044870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044890: 2d2d 2b0a 0a43 6176 6561 740a 3d3d 3d3d --+..Caveat.==== │ │ │ │ +000448a0: 3d3d 0a0a 2020 2020 2020 2020 2020 2020 ==.. │ │ │ │ 000448b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000448c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000448d0: 7c6f 3320 3a20 4669 6e69 7465 4772 6f75 |o3 : FiniteGrou │ │ │ │ -000448e0: 7041 6374 696f 6e20 2020 2020 2020 2020 pAction │ │ │ │ -000448f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044910: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00044920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044960: 2d2d 2b0a 7c69 3420 3a20 7072 696d 6172 --+.|i4 : primar │ │ │ │ -00044970: 7949 6e76 6172 6961 6e74 7328 5333 2c44 yInvariants(S3,D │ │ │ │ -00044980: 6567 7265 6556 6563 746f 723d 3e7b 332c egreeVector=>{3, │ │ │ │ -00044990: 332c 347d 2920 2020 2020 2020 2020 2020 3,4}) │ │ │ │ -000449a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000449b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000449c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000449d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000448c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000448d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000448e0: 2020 2020 2020 2020 2020 2020 2020 200a . │ │ │ │ +000448f0: 4375 7272 656e 746c 7920 7573 6572 7320 Currently users │ │ │ │ +00044900: 6361 6e20 6f6e 6c79 2075 7365 202a 6e6f can only use *no │ │ │ │ +00044910: 7465 2070 7269 6d61 7279 496e 7661 7269 te primaryInvari │ │ │ │ +00044920: 616e 7473 3a20 7072 696d 6172 7949 6e76 ants: primaryInv │ │ │ │ +00044930: 6172 6961 6e74 732c 2074 6f0a 2020 2020 ariants, to. │ │ │ │ +00044940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044980: 2020 2020 2020 2020 200a 6361 6c63 756c .calcul │ │ │ │ +00044990: 6174 6520 6120 6873 6f70 2066 6f72 2074 ate a hsop for t │ │ │ │ +000449a0: 6865 2069 6e76 6172 6961 6e74 2072 696e he invariant rin │ │ │ │ +000449b0: 6720 6f76 6572 2061 2066 696e 6974 6520 g over a finite │ │ │ │ +000449c0: 6669 656c 6420 6279 2075 7369 6e67 2074 field by using t │ │ │ │ +000449d0: 6865 2044 6164 650a 2020 2020 2020 2020 he Dade. │ │ │ │ 000449e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000449f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00044a00: 3320 2020 2033 2020 2020 3320 2020 3220 3 3 3 2 │ │ │ │ -00044a10: 2020 2020 2020 3220 2020 2032 2020 2020 2 2 │ │ │ │ -00044a20: 2032 2020 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ -00044a30: 3220 2020 3420 2020 2034 2020 2020 3420 2 4 4 4 │ │ │ │ -00044a40: 7c0a 7c6f 3420 3d20 7b78 2020 2b20 7920 |.|o4 = {x + y │ │ │ │ -00044a50: 202b 207a 202c 2078 2079 202b 2078 2a79 + z , x y + x*y │ │ │ │ -00044a60: 2020 2b20 7820 7a20 2b20 7920 7a20 2b20 + x z + y z + │ │ │ │ -00044a70: 782a 7a20 202b 2079 2a7a 202c 2078 2020 x*z + y*z , x │ │ │ │ -00044a80: 2b20 7920 202b 207a 207d 7c0a 7c20 2020 + y + z }|.| │ │ │ │ +000449f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044a10: 2020 2020 2020 2020 2020 2020 2020 0a61 .a │ │ │ │ +00044a20: 6c67 6f72 6974 686d 2e20 5573 6572 7320 lgorithm. Users │ │ │ │ +00044a30: 7368 6f75 6c64 2065 6e74 6572 2074 6865 should enter the │ │ │ │ +00044a40: 2066 696e 6974 6520 6669 656c 6420 6173 finite field as │ │ │ │ +00044a50: 2061 202a 6e6f 7465 2047 616c 6f69 7346 a *note GaloisF │ │ │ │ +00044a60: 6965 6c64 3a0a 2020 2020 2020 2020 2020 ield:. │ │ │ │ +00044a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044ad0: 2020 2020 7c0a 7c6f 3420 3a20 4c69 7374 |.|o4 : List │ │ │ │ -00044ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044aa0: 2020 2020 2020 2020 2020 0a28 4d61 6361 .(Maca │ │ │ │ +00044ab0: 756c 6179 3244 6f63 2947 616c 6f69 7346 ulay2Doc)GaloisF │ │ │ │ +00044ac0: 6965 6c64 2c20 6f72 2061 2071 756f 7469 ield, or a quoti │ │ │ │ +00044ad0: 656e 7420 6669 656c 6420 6f66 2074 6865 ent field of the │ │ │ │ +00044ae0: 2066 6f72 6d20 2a6e 6f74 6520 5a5a 3a0a form *note ZZ:. │ │ │ │ 00044af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00044b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044b10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00044b20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00044b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00044b60: 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 6176 6561 --------+..Cavea │ │ │ │ -00044b70: 740a 3d3d 3d3d 3d3d 0a0a 2020 2020 2020 t.======.. │ │ │ │ +00044b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00044b30: 2020 2020 2020 200a 284d 6163 6175 6c61 .(Macaula │ │ │ │ +00044b40: 7932 446f 6329 5a5a 2c2f 7020 616e 6420 y2Doc)ZZ,/p and │ │ │ │ +00044b50: 6172 6520 6164 7669 7365 6420 746f 2065 are advised to e │ │ │ │ +00044b60: 6e73 7572 6520 7468 6174 2074 6865 2067 nsure that the g │ │ │ │ +00044b70: 726f 756e 6420 6669 656c 6420 6861 730a round field has. │ │ │ │ 00044b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044bc0: 2020 2020 200a 4375 7272 656e 746c 7920 .Currently │ │ │ │ -00044bd0: 7573 6572 7320 6361 6e20 6f6e 6c79 2075 users can only u │ │ │ │ -00044be0: 7365 202a 6e6f 7465 2070 7269 6d61 7279 se *note primary │ │ │ │ -00044bf0: 496e 7661 7269 616e 7473 3a20 7072 696d Invariants: prim │ │ │ │ -00044c00: 6172 7949 6e76 6172 6961 6e74 732c 2074 aryInvariants, t │ │ │ │ -00044c10: 6f0a 2020 2020 2020 2020 2020 2020 2020 o. │ │ │ │ -00044c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044c50: 2020 2020 2020 2020 2020 2020 2020 200a . │ │ │ │ -00044c60: 6361 6c63 756c 6174 6520 6120 6873 6f70 calculate a hsop │ │ │ │ -00044c70: 2066 6f72 2074 6865 2069 6e76 6172 6961 for the invaria │ │ │ │ -00044c80: 6e74 2072 696e 6720 6f76 6572 2061 2066 nt ring over a f │ │ │ │ -00044c90: 696e 6974 6520 6669 656c 6420 6279 2075 inite field by u │ │ │ │ -00044ca0: 7369 6e67 2074 6865 2044 6164 650a 2020 sing the Dade. │ │ │ │ -00044cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044cf0: 2020 2020 0a61 6c67 6f72 6974 686d 2e20 .algorithm. │ │ │ │ -00044d00: 5573 6572 7320 7368 6f75 6c64 2065 6e74 Users should ent │ │ │ │ -00044d10: 6572 2074 6865 2066 696e 6974 6520 6669 er the finite fi │ │ │ │ -00044d20: 656c 6420 6173 2061 202a 6e6f 7465 2047 eld as a *note G │ │ │ │ -00044d30: 616c 6f69 7346 6965 6c64 3a0a 2020 2020 aloisField:. │ │ │ │ -00044d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044d80: 0a28 4d61 6361 756c 6179 3244 6f63 2947 .(Macaulay2Doc)G │ │ │ │ -00044d90: 616c 6f69 7346 6965 6c64 2c20 6f72 2061 aloisField, or a │ │ │ │ -00044da0: 2071 756f 7469 656e 7420 6669 656c 6420 quotient field │ │ │ │ -00044db0: 6f66 2074 6865 2066 6f72 6d20 2a6e 6f74 of the form *not │ │ │ │ -00044dc0: 6520 5a5a 3a0a 2020 2020 2020 2020 2020 e ZZ:. │ │ │ │ -00044dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044e00: 2020 2020 2020 2020 2020 2020 200a 284d .(M │ │ │ │ -00044e10: 6163 6175 6c61 7932 446f 6329 5a5a 2c2f acaulay2Doc)ZZ,/ │ │ │ │ -00044e20: 7020 616e 6420 6172 6520 6164 7669 7365 p and are advise │ │ │ │ -00044e30: 6420 746f 2065 6e73 7572 6520 7468 6174 d to ensure that │ │ │ │ -00044e40: 2074 6865 2067 726f 756e 6420 6669 656c the ground fiel │ │ │ │ -00044e50: 6420 6861 730a 2020 2020 2020 2020 2020 d has. │ │ │ │ -00044e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00044e70: 2020 6e2d 310a 6361 7264 696e 616c 6974 n-1.cardinalit │ │ │ │ -00044e80: 7920 6772 6561 7465 7220 7468 616e 207c y greater than | │ │ │ │ -00044e90: 477c 2020 202c 2077 6865 7265 206e 2069 G| , where n i │ │ │ │ -00044ea0: 7320 7468 6520 6e75 6d62 6572 206f 6620 s the number of │ │ │ │ -00044eb0: 7661 7269 6162 6c65 7320 696e 2074 6865 variables in the │ │ │ │ -00044ec0: 0a70 6f6c 796e 6f6d 6961 6c20 7269 6e67 .polynomial ring │ │ │ │ -00044ed0: 2052 2e20 5573 696e 6720 6120 6772 6f75 R. Using a grou │ │ │ │ -00044ee0: 6e64 2066 6965 6c64 2073 6d61 6c6c 6572 nd field smaller │ │ │ │ -00044ef0: 2074 6861 6e20 7468 6973 2072 756e 7320 than this runs │ │ │ │ -00044f00: 7468 6520 7269 736b 206f 6620 7468 650a the risk of the. │ │ │ │ -00044f10: 616c 676f 7269 7468 6d20 6765 7474 696e algorithm gettin │ │ │ │ -00044f20: 6720 7374 7563 6b20 696e 2061 6e20 696e g stuck in an in │ │ │ │ -00044f30: 6669 6e69 7465 206c 6f6f 703b 202a 6e6f finite loop; *no │ │ │ │ -00044f40: 7465 2070 7269 6d61 7279 496e 7661 7269 te primaryInvari │ │ │ │ -00044f50: 616e 7473 3a0a 7072 696d 6172 7949 6e76 ants:.primaryInv │ │ │ │ -00044f60: 6172 6961 6e74 732c 2064 6973 706c 6179 ariants, display │ │ │ │ -00044f70: 7320 6120 7761 726e 696e 6720 6d65 7373 s a warning mess │ │ │ │ -00044f80: 6167 6520 6173 6b69 6e67 2074 6865 2075 age asking the u │ │ │ │ -00044f90: 7365 7220 7768 6574 6865 7220 7468 6579 ser whether they │ │ │ │ -00044fa0: 2077 6973 680a 746f 2063 6f6e 7469 6e75 wish.to continu │ │ │ │ -00044fb0: 6520 7769 7468 2074 6865 2063 6f6d 7075 e with the compu │ │ │ │ -00044fc0: 7461 7469 6f6e 2069 6e20 7468 6973 2063 tation in this c │ │ │ │ -00044fd0: 6173 652e 2053 6565 202a 6e6f 7465 2068 ase. See *note h │ │ │ │ -00044fe0: 736f 7020 616c 676f 7269 7468 6d73 3a20 sop algorithms: │ │ │ │ -00044ff0: 6873 6f70 0a61 6c67 6f72 6974 686d 732c hsop.algorithms, │ │ │ │ -00045000: 2066 6f72 2061 2064 6973 6375 7373 696f for a discussio │ │ │ │ -00045010: 6e20 6f6e 2074 6865 2044 6164 6520 616c n on the Dade al │ │ │ │ -00045020: 676f 7269 7468 6d2e 0a0a 5365 6520 616c gorithm...See al │ │ │ │ -00045030: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00045040: 202a 6e6f 7465 2070 7269 6d61 7279 496e *note primaryIn │ │ │ │ -00045050: 7661 7269 616e 7473 3a20 7072 696d 6172 variants: primar │ │ │ │ -00045060: 7949 6e76 6172 6961 6e74 732c 202d 2d20 yInvariants, -- │ │ │ │ -00045070: 636f 6d70 7574 6573 2061 206c 6973 7420 computes a list │ │ │ │ -00045080: 6f66 2070 7269 6d61 7279 0a20 2020 2069 of primary. i │ │ │ │ -00045090: 6e76 6172 6961 6e74 7320 666f 7220 7468 nvariants for th │ │ │ │ -000450a0: 6520 696e 7661 7269 616e 7420 7269 6e67 e invariant ring │ │ │ │ -000450b0: 206f 6620 6120 6669 6e69 7465 2067 726f of a finite gro │ │ │ │ -000450c0: 7570 0a0a 4675 6e63 7469 6f6e 7320 7769 up..Functions wi │ │ │ │ -000450d0: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ -000450e0: 6d65 6e74 206e 616d 6564 2044 6567 7265 ment named Degre │ │ │ │ -000450f0: 6556 6563 746f 723a 0a3d 3d3d 3d3d 3d3d eVector:.======= │ │ │ │ -00045100: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00045110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00045120: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -00045130: 202a 2022 6869 726f 6e61 6b61 4465 636f * "hironakaDeco │ │ │ │ -00045140: 6d70 6f73 6974 696f 6e28 2e2e 2e2c 4465 mposition(...,De │ │ │ │ -00045150: 6772 6565 5665 6374 6f72 3d3e 2e2e 2e29 greeVector=>...) │ │ │ │ -00045160: 220a 2020 2a20 2a6e 6f74 6520 7072 696d ". * *note prim │ │ │ │ -00045170: 6172 7949 6e76 6172 6961 6e74 7328 2e2e aryInvariants(.. │ │ │ │ -00045180: 2e2c 4465 6772 6565 5665 6374 6f72 3d3e .,DegreeVector=> │ │ │ │ -00045190: 2e2e 2e29 3a0a 2020 2020 7072 696d 6172 ...):. primar │ │ │ │ -000451a0: 7949 6e76 6172 6961 6e74 735f 6c70 5f70 yInvariants_lp_p │ │ │ │ -000451b0: 645f 7064 5f70 645f 636d 4465 6772 6565 d_pd_pd_cmDegree │ │ │ │ -000451c0: 5665 6374 6f72 3d3e 5f70 645f 7064 5f70 Vector=>_pd_pd_p │ │ │ │ -000451d0: 645f 7270 2c20 2d2d 2061 6e20 6f70 7469 d_rp, -- an opti │ │ │ │ -000451e0: 6f6e 616c 0a20 2020 2061 7267 756d 656e onal. argumen │ │ │ │ -000451f0: 7420 666f 7220 7072 696d 6172 7949 6e76 t for primaryInv │ │ │ │ -00045200: 6172 6961 6e74 7320 7468 6174 2066 696e ariants that fin │ │ │ │ -00045210: 6473 2069 6e76 6172 6961 6e74 7320 6f66 ds invariants of │ │ │ │ -00045220: 2063 6572 7461 696e 2064 6567 7265 6573 certain degrees │ │ │ │ -00045230: 0a0a 4675 7274 6865 7220 696e 666f 726d ..Further inform │ │ │ │ -00045240: 6174 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ation.========== │ │ │ │ -00045250: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2044 =========.. * D │ │ │ │ -00045260: 6566 6175 6c74 2076 616c 7565 3a20 300a efault value: 0. │ │ │ │ -00045270: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e * Function: *n │ │ │ │ -00045280: 6f74 6520 7072 696d 6172 7949 6e76 6172 ote primaryInvar │ │ │ │ -00045290: 6961 6e74 733a 2070 7269 6d61 7279 496e iants: primaryIn │ │ │ │ -000452a0: 7661 7269 616e 7473 2c20 2d2d 2063 6f6d variants, -- com │ │ │ │ -000452b0: 7075 7465 7320 6120 6c69 7374 206f 660a putes a list of. │ │ │ │ -000452c0: 2020 2020 7072 696d 6172 7920 696e 7661 primary inva │ │ │ │ -000452d0: 7269 616e 7473 2066 6f72 2074 6865 2069 riants for the i │ │ │ │ -000452e0: 6e76 6172 6961 6e74 2072 696e 6720 6f66 nvariant ring of │ │ │ │ -000452f0: 2061 2066 696e 6974 6520 6772 6f75 700a a finite group. │ │ │ │ -00045300: 2020 2a20 4f70 7469 6f6e 206b 6579 3a20 * Option key: │ │ │ │ -00045310: 2a6e 6f74 6520 4465 6772 6565 5665 6374 *note DegreeVect │ │ │ │ -00045320: 6f72 3a0a 2020 2020 7072 696d 6172 7949 or:. primaryI │ │ │ │ -00045330: 6e76 6172 6961 6e74 735f 6c70 5f70 645f nvariants_lp_pd_ │ │ │ │ -00045340: 7064 5f70 645f 636d 4465 6772 6565 5665 pd_pd_cmDegreeVe │ │ │ │ -00045350: 6374 6f72 3d3e 5f70 645f 7064 5f70 645f ctor=>_pd_pd_pd_ │ │ │ │ -00045360: 7270 2c20 2d2d 2061 6e20 6f70 7469 6f6e rp, -- an option │ │ │ │ -00045370: 616c 0a20 2020 2061 7267 756d 656e 7420 al. argument │ │ │ │ -00045380: 666f 7220 7072 696d 6172 7949 6e76 6172 for primaryInvar │ │ │ │ -00045390: 6961 6e74 7320 7468 6174 2066 696e 6473 iants that finds │ │ │ │ -000453a0: 2069 6e76 6172 6961 6e74 7320 6f66 2063 invariants of c │ │ │ │ -000453b0: 6572 7461 696e 2064 6567 7265 6573 0a2d ertain degrees.- │ │ │ │ -000453c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000453d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000453e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000453f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -00045410: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -00045420: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -00045430: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -00045440: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -00045450: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -00045460: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00045470: 6b61 6765 732f 0a49 6e76 6172 6961 6e74 kages/.Invariant │ │ │ │ -00045480: 5269 6e67 2f48 6177 6573 446f 632e 6d32 Ring/HawesDoc.m2 │ │ │ │ -00045490: 3a32 3938 3a30 2e0a 1f0a 4669 6c65 3a20 :298:0....File: │ │ │ │ -000454a0: 496e 7661 7269 616e 7452 696e 672e 696e InvariantRing.in │ │ │ │ -000454b0: 666f 2c20 4e6f 6465 3a20 7261 6e6b 5f6c fo, Node: rank_l │ │ │ │ -000454c0: 7044 6961 676f 6e61 6c41 6374 696f 6e5f pDiagonalAction_ │ │ │ │ -000454d0: 7270 2c20 4e65 7874 3a20 7265 6c61 7469 rp, Next: relati │ │ │ │ -000454e0: 6f6e 735f 6c70 4669 6e69 7465 4772 6f75 ons_lpFiniteGrou │ │ │ │ -000454f0: 7041 6374 696f 6e5f 7270 2c20 5072 6576 pAction_rp, Prev │ │ │ │ -00045500: 3a20 7072 696d 6172 7949 6e76 6172 6961 : primaryInvaria │ │ │ │ -00045510: 6e74 735f 6c70 5f70 645f 7064 5f70 645f nts_lp_pd_pd_pd_ │ │ │ │ -00045520: 636d 4465 6772 6565 5665 6374 6f72 3d3e cmDegreeVector=> │ │ │ │ -00045530: 5f70 645f 7064 5f70 645f 7270 2c20 5570 _pd_pd_pd_rp, Up │ │ │ │ -00045540: 3a20 546f 700a 0a72 616e 6b28 4469 6167 : Top..rank(Diag │ │ │ │ -00045550: 6f6e 616c 4163 7469 6f6e 2920 2d2d 206f onalAction) -- o │ │ │ │ -00045560: 6620 6120 6469 6167 6f6e 616c 2061 6374 f a diagonal act │ │ │ │ -00045570: 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ion.************ │ │ │ │ -00045580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00045590: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000455a0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20 .. * Function: │ │ │ │ -000455b0: 2a6e 6f74 6520 7261 6e6b 3a20 284d 6163 *note rank: (Mac │ │ │ │ -000455c0: 6175 6c61 7932 446f 6329 7261 6e6b 2c0a aulay2Doc)rank,. │ │ │ │ -000455d0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -000455e0: 2020 2020 7261 6e6b 2044 0a20 202a 2049 rank D. * I │ │ │ │ -000455f0: 6e70 7574 733a 0a20 2020 2020 202a 2044 nputs:. * D │ │ │ │ -00045600: 2c20 616e 2069 6e73 7461 6e63 6520 6f66 , an instance of │ │ │ │ -00045610: 2074 6865 2074 7970 6520 2a6e 6f74 6520 the type *note │ │ │ │ -00045620: 4469 6167 6f6e 616c 4163 7469 6f6e 3a20 DiagonalAction: │ │ │ │ -00045630: 4469 6167 6f6e 616c 4163 7469 6f6e 2c0a DiagonalAction,. │ │ │ │ -00045640: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00045650: 2020 202a 2061 6e20 2a6e 6f74 6520 696e * an *note in │ │ │ │ -00045660: 7465 6765 723a 2028 4d61 6361 756c 6179 teger: (Macaulay │ │ │ │ -00045670: 3244 6f63 295a 5a2c 2c20 7468 6520 7261 2Doc)ZZ,, the ra │ │ │ │ -00045680: 6e6b 206f 6620 7468 6520 746f 7275 7320 nk of the torus │ │ │ │ -00045690: 6661 6374 6f72 206f 6620 610a 2020 2020 factor of a. │ │ │ │ -000456a0: 2020 2020 6469 6167 6f6e 616c 2061 6374 diagonal act │ │ │ │ -000456b0: 696f 6e0a 0a44 6573 6372 6970 7469 6f6e ion..Description │ │ │ │ -000456c0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -000456d0: 6973 2066 756e 6374 696f 6e20 6973 2070 is function is p │ │ │ │ -000456e0: 726f 7669 6465 6420 6279 2074 6865 2070 rovided by the p │ │ │ │ -000456f0: 6163 6b61 6765 202a 6e6f 7465 2049 6e76 ackage *note Inv │ │ │ │ -00045700: 6172 6961 6e74 5269 6e67 3a20 546f 702c ariantRing: Top, │ │ │ │ -00045710: 2e20 0a0a 5573 6520 7468 6973 2066 756e . ..Use this fun │ │ │ │ -00045720: 6374 696f 6e20 746f 2072 6563 6f76 6572 ction to recover │ │ │ │ -00045730: 2074 6865 2072 616e 6b20 6f66 2074 6865 the rank of the │ │ │ │ -00045740: 2074 6f72 7573 2066 6163 746f 7220 6f66 torus factor of │ │ │ │ -00045750: 2061 2064 6961 676f 6e61 6c20 6163 7469 a diagonal acti │ │ │ │ -00045760: 6f6e 2e0a 0a54 6865 2066 6f6c 6c6f 7769 on...The followi │ │ │ │ -00045770: 6e67 2065 7861 6d70 6c65 2064 6566 696e ng example defin │ │ │ │ -00045780: 6573 2061 6e20 6163 7469 6f6e 206f 6620 es an action of │ │ │ │ -00045790: 6120 7477 6f2d 6469 6d65 6e73 696f 6e61 a two-dimensiona │ │ │ │ -000457a0: 6c20 746f 7275 7320 6f6e 2061 0a70 6f6c l torus on a.pol │ │ │ │ -000457b0: 796e 6f6d 6961 6c20 7269 6e67 2069 6e20 ynomial ring in │ │ │ │ -000457c0: 666f 7572 2076 6172 6961 626c 6573 2e0a four variables.. │ │ │ │ -000457d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -000457e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000457f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ -00045800: 2052 203d 2051 515b 785f 312e 2e78 5f34 R = QQ[x_1..x_4 │ │ │ │ -00045810: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00045820: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00044b90: 2020 2020 2020 2020 2020 2020 6e2d 310a n-1. │ │ │ │ +00044ba0: 6361 7264 696e 616c 6974 7920 6772 6561 cardinality grea │ │ │ │ +00044bb0: 7465 7220 7468 616e 207c 477c 2020 202c ter than |G| , │ │ │ │ +00044bc0: 2077 6865 7265 206e 2069 7320 7468 6520 where n is the │ │ │ │ +00044bd0: 6e75 6d62 6572 206f 6620 7661 7269 6162 number of variab │ │ │ │ +00044be0: 6c65 7320 696e 2074 6865 0a70 6f6c 796e les in the.polyn │ │ │ │ +00044bf0: 6f6d 6961 6c20 7269 6e67 2052 2e20 5573 omial ring R. Us │ │ │ │ +00044c00: 696e 6720 6120 6772 6f75 6e64 2066 6965 ing a ground fie │ │ │ │ +00044c10: 6c64 2073 6d61 6c6c 6572 2074 6861 6e20 ld smaller than │ │ │ │ +00044c20: 7468 6973 2072 756e 7320 7468 6520 7269 this runs the ri │ │ │ │ +00044c30: 736b 206f 6620 7468 650a 616c 676f 7269 sk of the.algori │ │ │ │ +00044c40: 7468 6d20 6765 7474 696e 6720 7374 7563 thm getting stuc │ │ │ │ +00044c50: 6b20 696e 2061 6e20 696e 6669 6e69 7465 k in an infinite │ │ │ │ +00044c60: 206c 6f6f 703b 202a 6e6f 7465 2070 7269 loop; *note pri │ │ │ │ +00044c70: 6d61 7279 496e 7661 7269 616e 7473 3a0a maryInvariants:. │ │ │ │ +00044c80: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ +00044c90: 732c 2064 6973 706c 6179 7320 6120 7761 s, displays a wa │ │ │ │ +00044ca0: 726e 696e 6720 6d65 7373 6167 6520 6173 rning message as │ │ │ │ +00044cb0: 6b69 6e67 2074 6865 2075 7365 7220 7768 king the user wh │ │ │ │ +00044cc0: 6574 6865 7220 7468 6579 2077 6973 680a ether they wish. │ │ │ │ +00044cd0: 746f 2063 6f6e 7469 6e75 6520 7769 7468 to continue with │ │ │ │ +00044ce0: 2074 6865 2063 6f6d 7075 7461 7469 6f6e the computation │ │ │ │ +00044cf0: 2069 6e20 7468 6973 2063 6173 652e 2053 in this case. S │ │ │ │ +00044d00: 6565 202a 6e6f 7465 2068 736f 7020 616c ee *note hsop al │ │ │ │ +00044d10: 676f 7269 7468 6d73 3a20 6873 6f70 0a61 gorithms: hsop.a │ │ │ │ +00044d20: 6c67 6f72 6974 686d 732c 2066 6f72 2061 lgorithms, for a │ │ │ │ +00044d30: 2064 6973 6375 7373 696f 6e20 6f6e 2074 discussion on t │ │ │ │ +00044d40: 6865 2044 6164 6520 616c 676f 7269 7468 he Dade algorith │ │ │ │ +00044d50: 6d2e 0a0a 5365 6520 616c 736f 0a3d 3d3d m...See also.=== │ │ │ │ +00044d60: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ +00044d70: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +00044d80: 7473 3a20 7072 696d 6172 7949 6e76 6172 ts: primaryInvar │ │ │ │ +00044d90: 6961 6e74 732c 202d 2d20 636f 6d70 7574 iants, -- comput │ │ │ │ +00044da0: 6573 2061 206c 6973 7420 6f66 2070 7269 es a list of pri │ │ │ │ +00044db0: 6d61 7279 0a20 2020 2069 6e76 6172 6961 mary. invaria │ │ │ │ +00044dc0: 6e74 7320 666f 7220 7468 6520 696e 7661 nts for the inva │ │ │ │ +00044dd0: 7269 616e 7420 7269 6e67 206f 6620 6120 riant ring of a │ │ │ │ +00044de0: 6669 6e69 7465 2067 726f 7570 0a0a 4675 finite group..Fu │ │ │ │ +00044df0: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +00044e00: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ +00044e10: 616d 6564 2044 6567 7265 6556 6563 746f amed DegreeVecto │ │ │ │ +00044e20: 723a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r:.============= │ │ │ │ +00044e30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00044e40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00044e50: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6869 =======.. * "hi │ │ │ │ +00044e60: 726f 6e61 6b61 4465 636f 6d70 6f73 6974 ronakaDecomposit │ │ │ │ +00044e70: 696f 6e28 2e2e 2e2c 4465 6772 6565 5665 ion(...,DegreeVe │ │ │ │ +00044e80: 6374 6f72 3d3e 2e2e 2e29 220a 2020 2a20 ctor=>...)". * │ │ │ │ +00044e90: 2a6e 6f74 6520 7072 696d 6172 7949 6e76 *note primaryInv │ │ │ │ +00044ea0: 6172 6961 6e74 7328 2e2e 2e2c 4465 6772 ariants(...,Degr │ │ │ │ +00044eb0: 6565 5665 6374 6f72 3d3e 2e2e 2e29 3a0a eeVector=>...):. │ │ │ │ +00044ec0: 2020 2020 7072 696d 6172 7949 6e76 6172 primaryInvar │ │ │ │ +00044ed0: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ +00044ee0: 645f 636d 4465 6772 6565 5665 6374 6f72 d_cmDegreeVector │ │ │ │ +00044ef0: 3d3e 5f70 645f 7064 5f70 645f 7270 2c20 =>_pd_pd_pd_rp, │ │ │ │ +00044f00: 2d2d 2061 6e20 6f70 7469 6f6e 616c 0a20 -- an optional. │ │ │ │ +00044f10: 2020 2061 7267 756d 656e 7420 666f 7220 argument for │ │ │ │ +00044f20: 7072 696d 6172 7949 6e76 6172 6961 6e74 primaryInvariant │ │ │ │ +00044f30: 7320 7468 6174 2066 696e 6473 2069 6e76 s that finds inv │ │ │ │ +00044f40: 6172 6961 6e74 7320 6f66 2063 6572 7461 ariants of certa │ │ │ │ +00044f50: 696e 2064 6567 7265 6573 0a0a 4675 7274 in degrees..Furt │ │ │ │ +00044f60: 6865 7220 696e 666f 726d 6174 696f 6e0a her information. │ │ │ │ +00044f70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00044f80: 3d3d 3d0a 0a20 202a 2044 6566 6175 6c74 ===.. * Default │ │ │ │ +00044f90: 2076 616c 7565 3a20 300a 2020 2a20 4675 value: 0. * Fu │ │ │ │ +00044fa0: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7072 nction: *note pr │ │ │ │ +00044fb0: 696d 6172 7949 6e76 6172 6961 6e74 733a imaryInvariants: │ │ │ │ +00044fc0: 2070 7269 6d61 7279 496e 7661 7269 616e primaryInvarian │ │ │ │ +00044fd0: 7473 2c20 2d2d 2063 6f6d 7075 7465 7320 ts, -- computes │ │ │ │ +00044fe0: 6120 6c69 7374 206f 660a 2020 2020 7072 a list of. pr │ │ │ │ +00044ff0: 696d 6172 7920 696e 7661 7269 616e 7473 imary invariants │ │ │ │ +00045000: 2066 6f72 2074 6865 2069 6e76 6172 6961 for the invaria │ │ │ │ +00045010: 6e74 2072 696e 6720 6f66 2061 2066 696e nt ring of a fin │ │ │ │ +00045020: 6974 6520 6772 6f75 700a 2020 2a20 4f70 ite group. * Op │ │ │ │ +00045030: 7469 6f6e 206b 6579 3a20 2a6e 6f74 6520 tion key: *note │ │ │ │ +00045040: 4465 6772 6565 5665 6374 6f72 3a0a 2020 DegreeVector:. │ │ │ │ +00045050: 2020 7072 696d 6172 7949 6e76 6172 6961 primaryInvaria │ │ │ │ +00045060: 6e74 735f 6c70 5f70 645f 7064 5f70 645f nts_lp_pd_pd_pd_ │ │ │ │ +00045070: 636d 4465 6772 6565 5665 6374 6f72 3d3e cmDegreeVector=> │ │ │ │ +00045080: 5f70 645f 7064 5f70 645f 7270 2c20 2d2d _pd_pd_pd_rp, -- │ │ │ │ +00045090: 2061 6e20 6f70 7469 6f6e 616c 0a20 2020 an optional. │ │ │ │ +000450a0: 2061 7267 756d 656e 7420 666f 7220 7072 argument for pr │ │ │ │ +000450b0: 696d 6172 7949 6e76 6172 6961 6e74 7320 imaryInvariants │ │ │ │ +000450c0: 7468 6174 2066 696e 6473 2069 6e76 6172 that finds invar │ │ │ │ +000450d0: 6961 6e74 7320 6f66 2063 6572 7461 696e iants of certain │ │ │ │ +000450e0: 2064 6567 7265 6573 0a2d 2d2d 2d2d 2d2d degrees.------- │ │ │ │ +000450f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045130: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00045140: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00045150: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00045160: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00045170: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00045180: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00045190: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +000451a0: 0a49 6e76 6172 6961 6e74 5269 6e67 2f48 .InvariantRing/H │ │ │ │ +000451b0: 6177 6573 446f 632e 6d32 3a32 3938 3a30 awesDoc.m2:298:0 │ │ │ │ +000451c0: 2e0a 1f0a 4669 6c65 3a20 496e 7661 7269 ....File: Invari │ │ │ │ +000451d0: 616e 7452 696e 672e 696e 666f 2c20 4e6f antRing.info, No │ │ │ │ +000451e0: 6465 3a20 7261 6e6b 5f6c 7044 6961 676f de: rank_lpDiago │ │ │ │ +000451f0: 6e61 6c41 6374 696f 6e5f 7270 2c20 4e65 nalAction_rp, Ne │ │ │ │ +00045200: 7874 3a20 7265 6c61 7469 6f6e 735f 6c70 xt: relations_lp │ │ │ │ +00045210: 4669 6e69 7465 4772 6f75 7041 6374 696f FiniteGroupActio │ │ │ │ +00045220: 6e5f 7270 2c20 5072 6576 3a20 7072 696d n_rp, Prev: prim │ │ │ │ +00045230: 6172 7949 6e76 6172 6961 6e74 735f 6c70 aryInvariants_lp │ │ │ │ +00045240: 5f70 645f 7064 5f70 645f 636d 4465 6772 _pd_pd_pd_cmDegr │ │ │ │ +00045250: 6565 5665 6374 6f72 3d3e 5f70 645f 7064 eeVector=>_pd_pd │ │ │ │ +00045260: 5f70 645f 7270 2c20 5570 3a20 546f 700a _pd_rp, Up: Top. │ │ │ │ +00045270: 0a72 616e 6b28 4469 6167 6f6e 616c 4163 .rank(DiagonalAc │ │ │ │ +00045280: 7469 6f6e 2920 2d2d 206f 6620 6120 6469 tion) -- of a di │ │ │ │ +00045290: 6167 6f6e 616c 2061 6374 696f 6e0a 2a2a agonal action.** │ │ │ │ +000452a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000452b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000452c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +000452d0: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520 Function: *note │ │ │ │ +000452e0: 7261 6e6b 3a20 284d 6163 6175 6c61 7932 rank: (Macaulay2 │ │ │ │ +000452f0: 446f 6329 7261 6e6b 2c0a 2020 2a20 5573 Doc)rank,. * Us │ │ │ │ +00045300: 6167 653a 200a 2020 2020 2020 2020 7261 age: . ra │ │ │ │ +00045310: 6e6b 2044 0a20 202a 2049 6e70 7574 733a nk D. * Inputs: │ │ │ │ +00045320: 0a20 2020 2020 202a 2044 2c20 616e 2069 . * D, an i │ │ │ │ +00045330: 6e73 7461 6e63 6520 6f66 2074 6865 2074 nstance of the t │ │ │ │ +00045340: 7970 6520 2a6e 6f74 6520 4469 6167 6f6e ype *note Diagon │ │ │ │ +00045350: 616c 4163 7469 6f6e 3a20 4469 6167 6f6e alAction: Diagon │ │ │ │ +00045360: 616c 4163 7469 6f6e 2c0a 2020 2a20 4f75 alAction,. * Ou │ │ │ │ +00045370: 7470 7574 733a 0a20 2020 2020 202a 2061 tputs:. * a │ │ │ │ +00045380: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ +00045390: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ +000453a0: 5a2c 2c20 7468 6520 7261 6e6b 206f 6620 Z,, the rank of │ │ │ │ +000453b0: 7468 6520 746f 7275 7320 6661 6374 6f72 the torus factor │ │ │ │ +000453c0: 206f 6620 610a 2020 2020 2020 2020 6469 of a. di │ │ │ │ +000453d0: 6167 6f6e 616c 2061 6374 696f 6e0a 0a44 agonal action..D │ │ │ │ +000453e0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +000453f0: 3d3d 3d3d 3d3d 0a0a 5468 6973 2066 756e ======..This fun │ │ │ │ +00045400: 6374 696f 6e20 6973 2070 726f 7669 6465 ction is provide │ │ │ │ +00045410: 6420 6279 2074 6865 2070 6163 6b61 6765 d by the package │ │ │ │ +00045420: 202a 6e6f 7465 2049 6e76 6172 6961 6e74 *note Invariant │ │ │ │ +00045430: 5269 6e67 3a20 546f 702c 2e20 0a0a 5573 Ring: Top,. ..Us │ │ │ │ +00045440: 6520 7468 6973 2066 756e 6374 696f 6e20 e this function │ │ │ │ +00045450: 746f 2072 6563 6f76 6572 2074 6865 2072 to recover the r │ │ │ │ +00045460: 616e 6b20 6f66 2074 6865 2074 6f72 7573 ank of the torus │ │ │ │ +00045470: 2066 6163 746f 7220 6f66 2061 2064 6961 factor of a dia │ │ │ │ +00045480: 676f 6e61 6c20 6163 7469 6f6e 2e0a 0a54 gonal action...T │ │ │ │ +00045490: 6865 2066 6f6c 6c6f 7769 6e67 2065 7861 he following exa │ │ │ │ +000454a0: 6d70 6c65 2064 6566 696e 6573 2061 6e20 mple defines an │ │ │ │ +000454b0: 6163 7469 6f6e 206f 6620 6120 7477 6f2d action of a two- │ │ │ │ +000454c0: 6469 6d65 6e73 696f 6e61 6c20 746f 7275 dimensional toru │ │ │ │ +000454d0: 7320 6f6e 2061 0a70 6f6c 796e 6f6d 6961 s on a.polynomia │ │ │ │ +000454e0: 6c20 7269 6e67 2069 6e20 666f 7572 2076 l ring in four v │ │ │ │ +000454f0: 6172 6961 626c 6573 2e0a 0a2b 2d2d 2d2d ariables...+---- │ │ │ │ +00045500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045520: 2d2d 2d2b 0a7c 6931 203a 2052 203d 2051 ---+.|i1 : R = Q │ │ │ │ +00045530: 515b 785f 312e 2e78 5f34 5d20 2020 2020 Q[x_1..x_4] │ │ │ │ +00045540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00045550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045570: 2020 2020 2020 207c 0a7c 6f31 203d 2052 |.|o1 = R │ │ │ │ +00045580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000455a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000455b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000455c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000455d0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ +000455e0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +000455f0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00045600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00045620: 0a7c 6932 203a 2057 203d 206d 6174 7269 .|i2 : W = matri │ │ │ │ +00045630: 787b 7b30 2c31 2c2d 312c 317d 2c7b 312c x{{0,1,-1,1},{1, │ │ │ │ +00045640: 302c 2d31 2c2d 317d 7d7c 0a7c 2020 2020 0,-1,-1}}|.| │ │ │ │ +00045650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045670: 2020 207c 0a7c 6f32 203d 207c 2030 2031 |.|o2 = | 0 1 │ │ │ │ +00045680: 202d 3120 3120 207c 2020 2020 2020 2020 -1 1 | │ │ │ │ +00045690: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000456a0: 2020 2020 207c 2031 2030 202d 3120 2d31 | 1 0 -1 -1 │ │ │ │ +000456b0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000456c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000456d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000456e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000456f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00045700: 2020 3220 2020 2020 2020 3420 2020 2020 2 4 │ │ │ │ +00045710: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +00045720: 203a 204d 6174 7269 7820 5a5a 2020 3c2d : Matrix ZZ <- │ │ │ │ +00045730: 2d20 5a5a 2020 2020 2020 2020 2020 2020 - ZZ │ │ │ │ +00045740: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00045750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00045770: 0a7c 6933 203a 2054 203d 2064 6961 676f .|i3 : T = diago │ │ │ │ +00045780: 6e61 6c41 6374 696f 6e28 572c 2052 2920 nalAction(W, R) │ │ │ │ +00045790: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000457a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000457b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000457c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000457d0: 2020 202a 2032 2020 2020 2020 2020 2020 * 2 │ │ │ │ +000457e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000457f0: 6f33 203d 2052 203c 2d20 2851 5120 2920 o3 = R <- (QQ ) │ │ │ │ +00045800: 2076 6961 2020 2020 2020 2020 2020 2020 via │ │ │ │ +00045810: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00045820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045840: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00045850: 6f31 203d 2052 2020 2020 2020 2020 2020 o1 = R │ │ │ │ -00045860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045870: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00045840: 207c 0a7c 2020 2020 207c 2030 2031 202d |.| | 0 1 - │ │ │ │ +00045850: 3120 3120 207c 2020 2020 2020 2020 2020 1 1 | │ │ │ │ +00045860: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00045870: 2020 207c 2031 2030 202d 3120 2d31 207c | 1 0 -1 -1 | │ │ │ │ 00045880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000458a0: 207c 0a7c 6f31 203a 2050 6f6c 796e 6f6d |.|o1 : Polynom │ │ │ │ -000458b0: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ -000458c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000458d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000458e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000458f0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2057 203d -----+.|i2 : W = │ │ │ │ -00045900: 206d 6174 7269 787b 7b30 2c31 2c2d 312c matrix{{0,1,-1, │ │ │ │ -00045910: 317d 2c7b 312c 302c 2d31 2c2d 317d 7d7c 1},{1,0,-1,-1}}| │ │ │ │ -00045920: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00045930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045940: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ -00045950: 207c 2030 2031 202d 3120 3120 207c 2020 | 0 1 -1 1 | │ │ │ │ -00045960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045970: 2020 207c 0a7c 2020 2020 207c 2031 2030 |.| | 1 0 │ │ │ │ -00045980: 202d 3120 2d31 207c 2020 2020 2020 2020 -1 -1 | │ │ │ │ -00045990: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000459a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000459b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000459c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000459d0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -000459e0: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -000459f0: 207c 0a7c 6f32 203a 204d 6174 7269 7820 |.|o2 : Matrix │ │ │ │ -00045a00: 5a5a 2020 3c2d 2d20 5a5a 2020 2020 2020 ZZ <-- ZZ │ │ │ │ -00045a10: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00045a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045a40: 2d2d 2d2d 2d2b 0a7c 6933 203a 2054 203d -----+.|i3 : T = │ │ │ │ -00045a50: 2064 6961 676f 6e61 6c41 6374 696f 6e28 diagonalAction( │ │ │ │ -00045a60: 572c 2052 2920 2020 2020 2020 2020 207c W, R) | │ │ │ │ -00045a70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00045a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045a90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00045aa0: 2020 2020 2020 2020 202a 2032 2020 2020 * 2 │ │ │ │ -00045ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045ac0: 2020 207c 0a7c 6f33 203d 2052 203c 2d20 |.|o3 = R <- │ │ │ │ -00045ad0: 2851 5120 2920 2076 6961 2020 2020 2020 (QQ ) via │ │ │ │ -00045ae0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00045af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045b10: 2020 2020 2020 207c 0a7c 2020 2020 207c |.| | │ │ │ │ -00045b20: 2030 2031 202d 3120 3120 207c 2020 2020 0 1 -1 1 | │ │ │ │ -00045b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045b40: 207c 0a7c 2020 2020 207c 2031 2030 202d |.| | 1 0 - │ │ │ │ -00045b50: 3120 2d31 207c 2020 2020 2020 2020 2020 1 -1 | │ │ │ │ -00045b60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00045b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045b90: 2020 2020 207c 0a7c 6f33 203a 2044 6961 |.|o3 : Dia │ │ │ │ -00045ba0: 676f 6e61 6c41 6374 696f 6e20 2020 2020 gonalAction │ │ │ │ -00045bb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00045bc0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00045bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045be0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -00045bf0: 2072 616e 6b20 5420 2020 2020 2020 2020 rank T │ │ │ │ -00045c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045c10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00045c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045c30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00045c40: 6f34 203d 2032 2020 2020 2020 2020 2020 o4 = 2 │ │ │ │ -00045c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045c60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00045c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045c90: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -00045ca0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -00045cb0: 2044 6961 676f 6e61 6c41 6374 696f 6e3a DiagonalAction: │ │ │ │ -00045cc0: 2044 6961 676f 6e61 6c41 6374 696f 6e2c DiagonalAction, │ │ │ │ -00045cd0: 202d 2d20 7468 6520 636c 6173 7320 6f66 -- the class of │ │ │ │ -00045ce0: 2061 6c6c 2064 6961 676f 6e61 6c20 6163 all diagonal ac │ │ │ │ -00045cf0: 7469 6f6e 730a 2020 2a20 2a6e 6f74 6520 tions. * *note │ │ │ │ -00045d00: 6469 6167 6f6e 616c 4163 7469 6f6e 3a20 diagonalAction: │ │ │ │ -00045d10: 6469 6167 6f6e 616c 4163 7469 6f6e 2c20 diagonalAction, │ │ │ │ -00045d20: 2d2d 2064 6961 676f 6e61 6c20 6772 6f75 -- diagonal grou │ │ │ │ -00045d30: 7020 6163 7469 6f6e 2076 6961 2077 6569 p action via wei │ │ │ │ -00045d40: 6768 7473 0a0a 5761 7973 2074 6f20 7573 ghts..Ways to us │ │ │ │ -00045d50: 6520 7468 6973 206d 6574 686f 643a 0a3d e this method:.= │ │ │ │ -00045d60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00045d70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -00045d80: 7465 2072 616e 6b28 4469 6167 6f6e 616c te rank(Diagonal │ │ │ │ -00045d90: 4163 7469 6f6e 293a 2072 616e 6b5f 6c70 Action): rank_lp │ │ │ │ -00045da0: 4469 6167 6f6e 616c 4163 7469 6f6e 5f72 DiagonalAction_r │ │ │ │ -00045db0: 702c 202d 2d20 6f66 2061 2064 6961 676f p, -- of a diago │ │ │ │ -00045dc0: 6e61 6c0a 2020 2020 6163 7469 6f6e 0a2d nal. action.- │ │ │ │ -00045dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -00045e20: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -00045e30: 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2e20 4561 6368 2065 6c65 6d65 6e74 0a69 . Each element.i │ │ │ │ -00046190: 7320 7265 7072 6573 656e 7465 6420 6279 s represented by │ │ │ │ -000461a0: 2061 2077 6f72 6420 6f66 206d 696e 696d a word of minim │ │ │ │ -000461b0: 616c 206c 656e 6774 6820 696e 2074 6865 al length in the │ │ │ │ -000461c0: 2043 6f78 7465 7220 6765 6e65 7261 746f Coxter generato │ │ │ │ -000461d0: 7273 2e20 416e 6420 6561 6368 0a72 656c rs. And each.rel │ │ │ │ -000461e0: 6174 696f 6e20 6973 2072 6570 7265 7365 ation is represe │ │ │ │ -000461f0: 6e74 6564 2062 7920 6120 6c69 7374 206f nted by a list o │ │ │ │ -00046200: 6620 7477 6f20 776f 7264 7320 7468 6174 f two words that │ │ │ │ -00046210: 2065 7175 6174 6573 2074 6865 2067 726f equates the gro │ │ │ │ -00046220: 7570 2065 6c65 6d65 6e74 0a72 6570 7265 up element.repre │ │ │ │ -00046230: 7365 6e74 6564 2062 7920 7468 6f73 6520 sented by those │ │ │ │ -00046240: 7477 6f20 776f 7264 732e 0a0a 5468 6520 two words...The │ │ │ │ -00046250: 666f 6c6c 6f77 696e 6720 6578 616d 706c following exampl │ │ │ │ -00046260: 6520 6465 6669 6e65 7320 7468 6520 7065 e defines the pe │ │ │ │ -00046270: 726d 7574 6174 696f 6e20 6163 7469 6f6e rmutation action │ │ │ │ -00046280: 206f 6620 6120 7379 6d6d 6574 7269 6320 of a symmetric │ │ │ │ -00046290: 6772 6f75 7020 6f6e 0a74 6872 6565 2065 group on.three e │ │ │ │ -000462a0: 6c65 6d65 6e74 7320 7573 696e 6720 7468 lements using th │ │ │ │ -000462b0: 7265 6520 7472 616e 7370 6f73 6974 696f ree transpositio │ │ │ │ -000462c0: 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ns...+---------- │ │ │ │ -000462d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000462e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000462f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046310: 2d2d 2d2b 0a7c 6931 203a 2052 203d 2051 ---+.|i1 : R = Q │ │ │ │ -00046320: 515b 785f 312e 2e78 5f33 5d20 2020 2020 Q[x_1..x_3] │ │ │ │ -00046330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045890: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000458a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000458b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000458c0: 0a7c 6f33 203a 2044 6961 676f 6e61 6c41 .|o3 : DiagonalA │ │ │ │ +000458d0: 6374 696f 6e20 2020 2020 2020 2020 2020 ction │ │ │ │ +000458e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000458f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045910: 2d2d 2d2b 0a7c 6934 203a 2072 616e 6b20 ---+.|i4 : rank │ │ │ │ +00045920: 5420 2020 2020 2020 2020 2020 2020 2020 T │ │ │ │ +00045930: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00045940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045960: 2020 2020 2020 207c 0a7c 6f34 203d 2032 |.|o4 = 2 │ │ │ │ +00045970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045990: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +000459a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000459b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 -----------+..Se │ │ │ │ +000459c0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +000459d0: 0a20 202a 202a 6e6f 7465 2044 6961 676f . * *note Diago │ │ │ │ +000459e0: 6e61 6c41 6374 696f 6e3a 2044 6961 676f nalAction: Diago │ │ │ │ +000459f0: 6e61 6c41 6374 696f 6e2c 202d 2d20 7468 nalAction, -- th │ │ │ │ +00045a00: 6520 636c 6173 7320 6f66 2061 6c6c 2064 e class of all d │ │ │ │ +00045a10: 6961 676f 6e61 6c20 6163 7469 6f6e 730a iagonal actions. │ │ │ │ +00045a20: 2020 2a20 2a6e 6f74 6520 6469 6167 6f6e * *note diagon │ │ │ │ +00045a30: 616c 4163 7469 6f6e 3a20 6469 6167 6f6e alAction: diagon │ │ │ │ +00045a40: 616c 4163 7469 6f6e 2c20 2d2d 2064 6961 alAction, -- dia │ │ │ │ +00045a50: 676f 6e61 6c20 6772 6f75 7020 6163 7469 gonal group acti │ │ │ │ +00045a60: 6f6e 2076 6961 2077 6569 6768 7473 0a0a on via weights.. │ │ │ │ +00045a70: 5761 7973 2074 6f20 7573 6520 7468 6973 Ways to use this │ │ │ │ +00045a80: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d method:.======= │ │ │ │ +00045a90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00045aa0: 3d0a 0a20 202a 202a 6e6f 7465 2072 616e =.. * *note ran │ │ │ │ +00045ab0: 6b28 4469 6167 6f6e 616c 4163 7469 6f6e k(DiagonalAction │ │ │ │ +00045ac0: 293a 2072 616e 6b5f 6c70 4469 6167 6f6e ): rank_lpDiagon │ │ │ │ +00045ad0: 616c 4163 7469 6f6e 5f72 702c 202d 2d20 alAction_rp, -- │ │ │ │ +00045ae0: 6f66 2061 2064 6961 676f 6e61 6c0a 2020 of a diagonal. │ │ │ │ +00045af0: 2020 6163 7469 6f6e 0a2d 2d2d 2d2d 2d2d action.------- │ │ │ │ +00045b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045b40: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00045b50: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00045b60: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00045b70: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00045b80: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00045b90: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00045ba0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00045bb0: 0a49 6e76 6172 6961 6e74 5269 6e67 2f41 .InvariantRing/A │ │ │ │ +00045bc0: 6265 6c69 616e 4772 6f75 7073 446f 632e belianGroupsDoc. │ │ │ │ +00045bd0: 6d32 3a33 3932 3a30 2e0a 1f0a 4669 6c65 m2:392:0....File │ │ │ │ +00045be0: 3a20 496e 7661 7269 616e 7452 696e 672e : InvariantRing. │ │ │ │ +00045bf0: 696e 666f 2c20 4e6f 6465 3a20 7265 6c61 info, Node: rela │ │ │ │ +00045c00: 7469 6f6e 735f 6c70 4669 6e69 7465 4772 tions_lpFiniteGr │ │ │ │ +00045c10: 6f75 7041 6374 696f 6e5f 7270 2c20 4e65 oupAction_rp, Ne │ │ │ │ +00045c20: 7874 3a20 7265 796e 6f6c 6473 4f70 6572 xt: reynoldsOper │ │ │ │ +00045c30: 6174 6f72 2c20 5072 6576 3a20 7261 6e6b ator, Prev: rank │ │ │ │ +00045c40: 5f6c 7044 6961 676f 6e61 6c41 6374 696f _lpDiagonalActio │ │ │ │ +00045c50: 6e5f 7270 2c20 5570 3a20 546f 700a 0a72 n_rp, Up: Top..r │ │ │ │ +00045c60: 656c 6174 696f 6e73 2846 696e 6974 6547 elations(FiniteG │ │ │ │ +00045c70: 726f 7570 4163 7469 6f6e 2920 2d2d 2072 roupAction) -- r │ │ │ │ +00045c80: 656c 6174 696f 6e73 206f 6620 6120 6669 elations of a fi │ │ │ │ +00045c90: 6e69 7465 2067 726f 7570 0a2a 2a2a 2a2a nite group.***** │ │ │ │ +00045ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00045cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00045cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00045cd0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 ******.. * Func │ │ │ │ +00045ce0: 7469 6f6e 3a20 2a6e 6f74 6520 7265 6c61 tion: *note rela │ │ │ │ +00045cf0: 7469 6f6e 733a 2028 4d61 6361 756c 6179 tions: (Macaulay │ │ │ │ +00045d00: 3244 6f63 2972 656c 6174 696f 6e73 2c0a 2Doc)relations,. │ │ │ │ +00045d10: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00045d20: 2020 2020 7265 6c61 7469 6f6e 7320 470a relations G. │ │ │ │ +00045d30: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +00045d40: 2020 2a20 472c 2061 6e20 696e 7374 616e * G, an instan │ │ │ │ +00045d50: 6365 206f 6620 7468 6520 7479 7065 202a ce of the type * │ │ │ │ +00045d60: 6e6f 7465 2046 696e 6974 6547 726f 7570 note FiniteGroup │ │ │ │ +00045d70: 4163 7469 6f6e 3a20 4669 6e69 7465 4772 Action: FiniteGr │ │ │ │ +00045d80: 6f75 7041 6374 696f 6e2c 2c0a 2020 2020 oupAction,,. │ │ │ │ +00045d90: 2020 2020 7468 6520 6163 7469 6f6e 206f the action o │ │ │ │ +00045da0: 6620 6120 6669 6e69 7465 2067 726f 7570 f a finite group │ │ │ │ +00045db0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00045dc0: 2020 2020 2a20 6120 2a6e 6f74 6520 6c69 * a *note li │ │ │ │ +00045dd0: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ +00045de0: 6329 4c69 7374 2c2c 2061 206c 6973 7420 c)List,, a list │ │ │ │ +00045df0: 6f66 2072 656c 6174 696f 6e73 206f 6620 of relations of │ │ │ │ +00045e00: 7468 6520 6772 6f75 700a 0a44 6573 6372 the group..Descr │ │ │ │ +00045e10: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00045e20: 3d3d 0a0a 5468 6973 2066 756e 6374 696f ==..This functio │ │ │ │ +00045e30: 6e20 6973 2070 726f 7669 6465 6420 6279 n is provided by │ │ │ │ +00045e40: 2074 6865 2070 6163 6b61 6765 202a 6e6f the package *no │ │ │ │ +00045e50: 7465 2049 6e76 6172 6961 6e74 5269 6e67 te InvariantRing │ │ │ │ +00045e60: 3a20 546f 702c 2e20 0a0a 5573 6520 7468 : Top,. ..Use th │ │ │ │ +00045e70: 6973 2066 756e 6374 696f 6e20 746f 2067 is function to g │ │ │ │ +00045e80: 6574 2074 6865 2072 656c 6174 696f 6e73 et the relations │ │ │ │ +00045e90: 2061 6d6f 6e67 2065 6c65 6d65 6e74 7320 among elements │ │ │ │ +00045ea0: 6f66 2061 2067 726f 7570 2e20 4561 6368 of a group. Each │ │ │ │ +00045eb0: 2065 6c65 6d65 6e74 0a69 7320 7265 7072 element.is repr │ │ │ │ +00045ec0: 6573 656e 7465 6420 6279 2061 2077 6f72 esented by a wor │ │ │ │ +00045ed0: 6420 6f66 206d 696e 696d 616c 206c 656e d of minimal len │ │ │ │ +00045ee0: 6774 6820 696e 2074 6865 2043 6f78 7465 gth in the Coxte │ │ │ │ +00045ef0: 7220 6765 6e65 7261 746f 7273 2e20 416e r generators. An │ │ │ │ +00045f00: 6420 6561 6368 0a72 656c 6174 696f 6e20 d each.relation │ │ │ │ +00045f10: 6973 2072 6570 7265 7365 6e74 6564 2062 is represented b │ │ │ │ +00045f20: 7920 6120 6c69 7374 206f 6620 7477 6f20 y a list of two │ │ │ │ +00045f30: 776f 7264 7320 7468 6174 2065 7175 6174 words that equat │ │ │ │ +00045f40: 6573 2074 6865 2067 726f 7570 2065 6c65 es the group ele │ │ │ │ +00045f50: 6d65 6e74 0a72 6570 7265 7365 6e74 6564 ment.represented │ │ │ │ +00045f60: 2062 7920 7468 6f73 6520 7477 6f20 776f by those two wo │ │ │ │ +00045f70: 7264 732e 0a0a 5468 6520 666f 6c6c 6f77 rds...The follow │ │ │ │ +00045f80: 696e 6720 6578 616d 706c 6520 6465 6669 ing example defi │ │ │ │ +00045f90: 6e65 7320 7468 6520 7065 726d 7574 6174 nes the permutat │ │ │ │ +00045fa0: 696f 6e20 6163 7469 6f6e 206f 6620 6120 ion action of a │ │ │ │ +00045fb0: 7379 6d6d 6574 7269 6320 6772 6f75 7020 symmetric group │ │ │ │ +00045fc0: 6f6e 0a74 6872 6565 2065 6c65 6d65 6e74 on.three element │ │ │ │ +00045fd0: 7320 7573 696e 6720 7468 7265 6520 7472 s using three tr │ │ │ │ +00045fe0: 616e 7370 6f73 6974 696f 6e73 2e0a 0a2b anspositions...+ │ │ │ │ +00045ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00046040: 6931 203a 2052 203d 2051 515b 785f 312e i1 : R = QQ[x_1. │ │ │ │ +00046050: 2e78 5f33 5d20 2020 2020 2020 2020 2020 .x_3] │ │ │ │ +00046060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046080: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000460a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000460b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000460c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000460d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000460e0: 6f31 203d 2052 2020 2020 2020 2020 2020 o1 = R │ │ │ │ +000460f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046120: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046170: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046180: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ +00046190: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +000461a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000461b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000461c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000461d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000461e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000461f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00046220: 6932 203a 204c 203d 207b 6d61 7472 6978 i2 : L = {matrix │ │ │ │ +00046230: 207b 7b30 2c31 2c30 7d2c 7b31 2c30 2c30 {{0,1,0},{1,0,0 │ │ │ │ +00046240: 7d2c 7b30 2c30 2c31 7d7d 2c20 6d61 7472 },{0,0,1}}, matr │ │ │ │ +00046250: 6978 207b 7b30 2c30 2c31 7d2c 7b30 2c31 ix {{0,0,1},{0,1 │ │ │ │ +00046260: 2c30 7d2c 7b31 2c30 2c30 7d7d 2c7c 0a7c ,0},{1,0,0}},|.| │ │ │ │ +00046270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000462a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000462b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000462c0: 6f32 203d 207b 7c20 3020 3120 3020 7c2c o2 = {| 0 1 0 |, │ │ │ │ +000462d0: 207c 2030 2030 2031 207c 2c20 7c20 3120 | 0 0 1 |, | 1 │ │ │ │ +000462e0: 3020 3020 7c7d 2020 2020 2020 2020 2020 0 0 |} │ │ │ │ +000462f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046300: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046310: 2020 2020 2020 7c20 3120 3020 3020 7c20 | 1 0 0 | │ │ │ │ +00046320: 207c 2030 2031 2030 207c 2020 7c20 3020 | 0 1 0 | | 0 │ │ │ │ +00046330: 3020 3120 7c20 2020 2020 2020 2020 2020 0 1 | │ │ │ │ 00046340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046360: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00046370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046350: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046360: 2020 2020 2020 7c20 3020 3020 3120 7c20 | 0 0 1 | │ │ │ │ +00046370: 207c 2031 2030 2030 207c 2020 7c20 3020 | 1 0 0 | | 0 │ │ │ │ +00046380: 3120 3020 7c20 2020 2020 2020 2020 2020 1 0 | │ │ │ │ 00046390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000463a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000463b0: 2020 207c 0a7c 6f31 203d 2052 2020 2020 |.|o1 = R │ │ │ │ +000463a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000463b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000463c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000463d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000463e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000463f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046400: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000463f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046400: 6f32 203a 204c 6973 7420 2020 2020 2020 o2 : List │ │ │ │ 00046410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046450: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ -00046460: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ -00046470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000464a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -000464b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000464c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000464d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000464e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000464f0: 2d2d 2d2b 0a7c 6932 203a 204c 203d 207b ---+.|i2 : L = { │ │ │ │ -00046500: 6d61 7472 6978 207b 7b30 2c31 2c30 7d2c matrix {{0,1,0}, │ │ │ │ -00046510: 7b31 2c30 2c30 7d2c 7b30 2c30 2c31 7d7d {1,0,0},{0,0,1}} │ │ │ │ -00046520: 2c20 6d61 7472 6978 207b 7b30 2c30 2c31 , matrix {{0,0,1 │ │ │ │ -00046530: 7d2c 7b30 2c31 2c30 7d2c 7b31 2c30 2c30 },{0,1,0},{1,0,0 │ │ │ │ -00046540: 7d7d 2c7c 0a7c 2020 2020 2020 2020 2020 }},|.| │ │ │ │ -00046550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046440: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +000464a0: 6d61 7472 6978 207b 7b31 2c30 2c30 7d2c matrix {{1,0,0}, │ │ │ │ +000464b0: 7b30 2c30 2c31 7d2c 7b30 2c31 2c30 7d7d {0,0,1},{0,1,0}} │ │ │ │ +000464c0: 207d 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +000464d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000464e0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000464f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00046540: 6933 203a 2047 203d 2066 696e 6974 6541 i3 : G = finiteA │ │ │ │ +00046550: 6374 696f 6e28 4c2c 2052 2920 2020 2020 ction(L, R) │ │ │ │ 00046560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046590: 2020 207c 0a7c 6f32 203d 207b 7c20 3020 |.|o2 = {| 0 │ │ │ │ -000465a0: 3120 3020 7c2c 207c 2030 2030 2031 207c 1 0 |, | 0 0 1 | │ │ │ │ -000465b0: 2c20 7c20 3120 3020 3020 7c7d 2020 2020 , | 1 0 0 |} │ │ │ │ +00046580: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000465a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000465b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000465c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000465d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000465e0: 2020 207c 0a7c 2020 2020 2020 7c20 3120 |.| | 1 │ │ │ │ -000465f0: 3020 3020 7c20 207c 2030 2031 2030 207c 0 0 | | 0 1 0 | │ │ │ │ -00046600: 2020 7c20 3020 3020 3120 7c20 2020 2020 | 0 0 1 | │ │ │ │ +000465d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000465e0: 6f33 203d 2052 203c 2d20 3c7c 2030 2031 o3 = R <- <| 0 1 │ │ │ │ +000465f0: 2030 207c 2c20 7c20 3020 3020 3120 7c2c 0 |, | 0 0 1 |, │ │ │ │ +00046600: 207c 2031 2030 2030 207c 3e20 2020 2020 | 1 0 0 |> │ │ │ │ 00046610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046630: 2020 207c 0a7c 2020 2020 2020 7c20 3020 |.| | 0 │ │ │ │ -00046640: 3020 3120 7c20 207c 2031 2030 2030 207c 0 1 | | 1 0 0 | │ │ │ │ -00046650: 2020 7c20 3020 3120 3020 7c20 2020 2020 | 0 1 0 | │ │ │ │ +00046620: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046630: 2020 2020 2020 2020 2020 207c 2031 2030 | 1 0 │ │ │ │ +00046640: 2030 207c 2020 7c20 3020 3120 3020 7c20 0 | | 0 1 0 | │ │ │ │ +00046650: 207c 2030 2030 2031 207c 2020 2020 2020 | 0 0 1 | │ │ │ │ 00046660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046680: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00046690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000466a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046670: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046680: 2020 2020 2020 2020 2020 207c 2030 2030 | 0 0 │ │ │ │ +00046690: 2031 207c 2020 7c20 3120 3020 3020 7c20 1 | | 1 0 0 | │ │ │ │ +000466a0: 207c 2030 2031 2030 207c 2020 2020 2020 | 0 1 0 | │ │ │ │ 000466b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000466c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000466d0: 2020 207c 0a7c 6f32 203a 204c 6973 7420 |.|o2 : List │ │ │ │ +000466c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000466d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000466e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000466f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046720: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ -00046730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046770: 2d2d 2d7c 0a7c 6d61 7472 6978 207b 7b31 ---|.|matrix {{1 │ │ │ │ -00046780: 2c30 2c30 7d2c 7b30 2c30 2c31 7d2c 7b30 ,0,0},{0,0,1},{0 │ │ │ │ -00046790: 2c31 2c30 7d7d 207d 2020 2020 2020 2020 ,1,0}} } │ │ │ │ -000467a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000467b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000467c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -000467d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000467e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000467f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046810: 2d2d 2d2b 0a7c 6933 203a 2047 203d 2066 ---+.|i3 : G = f │ │ │ │ -00046820: 696e 6974 6541 6374 696f 6e28 4c2c 2052 initeAction(L, R │ │ │ │ -00046830: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00046710: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046720: 6f33 203a 2046 696e 6974 6547 726f 7570 o3 : FiniteGroup │ │ │ │ +00046730: 4163 7469 6f6e 2020 2020 2020 2020 2020 Action │ │ │ │ +00046740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046760: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00046770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000467a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000467b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000467c0: 6934 203a 2072 656c 6174 696f 6e73 2047 i4 : relations G │ │ │ │ +000467d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000467e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000467f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046800: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046860: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00046870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000468a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000468b0: 2020 207c 0a7c 6f33 203d 2052 203c 2d20 |.|o3 = R <- │ │ │ │ -000468c0: 3c7c 2030 2031 2030 207c 2c20 7c20 3020 <| 0 1 0 |, | 0 │ │ │ │ -000468d0: 3020 3120 7c2c 207c 2031 2030 2030 207c 0 1 |, | 1 0 0 | │ │ │ │ -000468e0: 3e20 2020 2020 2020 2020 2020 2020 2020 > │ │ │ │ -000468f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046900: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00046910: 207c 2031 2030 2030 207c 2020 7c20 3020 | 1 0 0 | | 0 │ │ │ │ -00046920: 3120 3020 7c20 207c 2030 2030 2031 207c 1 0 | | 0 0 1 | │ │ │ │ -00046930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046950: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00046960: 207c 2030 2030 2031 207c 2020 7c20 3120 | 0 0 1 | | 1 │ │ │ │ -00046970: 3020 3020 7c20 207c 2030 2031 2030 207c 0 0 | | 0 1 0 | │ │ │ │ -00046980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000469a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000469b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000469c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000469d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000469e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000469f0: 2020 207c 0a7c 6f33 203a 2046 696e 6974 |.|o3 : Finit │ │ │ │ -00046a00: 6547 726f 7570 4163 7469 6f6e 2020 2020 eGroupAction │ │ │ │ +00046850: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046860: 6f34 203d 207b 7b7b 7d2c 207b 312c 2031 o4 = {{{}, {1, 1 │ │ │ │ +00046870: 7d7d 2c20 7b7b 7d2c 207b 322c 2032 7d7d }}, {{}, {2, 2}} │ │ │ │ +00046880: 2c20 7b7b 317d 2c20 7b30 2c20 312c 2032 , {{1}, {0, 1, 2 │ │ │ │ +00046890: 7d7d 2c20 7b7b 317d 2c20 7b30 2c20 322c }}, {{1}, {0, 2, │ │ │ │ +000468a0: 2030 7d7d 2c20 7b7b 7d2c 2020 207c 0a7c 0}}, {{}, |.| │ │ │ │ +000468b0: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ +000468c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000468d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000468e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000468f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +00046900: 2020 2020 207b 302c 2030 7d7d 2c20 7b7b {0, 0}}, {{ │ │ │ │ +00046910: 302c 2032 7d2c 207b 312c 2030 7d7d 2c20 0, 2}, {1, 0}}, │ │ │ │ +00046920: 7b7b 302c 2032 7d2c 207b 322c 2031 7d7d {{0, 2}, {2, 1}} │ │ │ │ +00046930: 2c20 7b7b 302c 2031 7d2c 207b 312c 2032 , {{0, 1}, {1, 2 │ │ │ │ +00046940: 7d7d 2c20 7b7b 302c 2031 7d2c 207c 0a7c }}, {{0, 1}, |.| │ │ │ │ +00046950: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ +00046960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +000469a0: 2020 2020 207b 322c 2030 7d7d 2c20 7b7b {2, 0}}, {{ │ │ │ │ +000469b0: 327d 2c20 7b30 2c20 312c 2030 7d7d 2c20 2}, {0, 1, 0}}, │ │ │ │ +000469c0: 7b7b 327d 2c20 7b30 2c20 322c 2031 7d7d {{2}, {0, 2, 1}} │ │ │ │ +000469d0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +000469e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000469f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046a00: 2020 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 2020 2020 2020 2020 │ │ │ │ -00046a40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00046a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046a90: 2d2d 2d2b 0a7c 6934 203a 2072 656c 6174 ---+.|i4 : relat │ │ │ │ -00046aa0: 696f 6e73 2047 2020 2020 2020 2020 2020 ions G │ │ │ │ -00046ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046ae0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00046af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046b30: 2020 207c 0a7c 6f34 203d 207b 7b7b 7d2c |.|o4 = {{{}, │ │ │ │ -00046b40: 207b 312c 2031 7d7d 2c20 7b7b 7d2c 207b {1, 1}}, {{}, { │ │ │ │ -00046b50: 322c 2032 7d7d 2c20 7b7b 317d 2c20 7b30 2, 2}}, {{1}, {0 │ │ │ │ -00046b60: 2c20 312c 2032 7d7d 2c20 7b7b 317d 2c20 , 1, 2}}, {{1}, │ │ │ │ -00046b70: 7b30 2c20 322c 2030 7d7d 2c20 7b7b 7d2c {0, 2, 0}}, {{}, │ │ │ │ -00046b80: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ -00046b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046bd0: 2d2d 2d7c 0a7c 2020 2020 207b 302c 2030 ---|.| {0, 0 │ │ │ │ -00046be0: 7d7d 2c20 7b7b 302c 2032 7d2c 207b 312c }}, {{0, 2}, {1, │ │ │ │ -00046bf0: 2030 7d7d 2c20 7b7b 302c 2032 7d2c 207b 0}}, {{0, 2}, { │ │ │ │ -00046c00: 322c 2031 7d7d 2c20 7b7b 302c 2031 7d2c 2, 1}}, {{0, 1}, │ │ │ │ -00046c10: 207b 312c 2032 7d7d 2c20 7b7b 302c 2031 {1, 2}}, {{0, 1 │ │ │ │ -00046c20: 7d2c 207c 0a7c 2020 2020 202d 2d2d 2d2d }, |.| ----- │ │ │ │ -00046c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046c70: 2d2d 2d7c 0a7c 2020 2020 207b 322c 2030 ---|.| {2, 0 │ │ │ │ -00046c80: 7d7d 2c20 7b7b 327d 2c20 7b30 2c20 312c }}, {{2}, {0, 1, │ │ │ │ -00046c90: 2030 7d7d 2c20 7b7b 327d 2c20 7b30 2c20 0}}, {{2}, {0, │ │ │ │ -00046ca0: 322c 2031 7d7d 7d20 2020 2020 2020 2020 2, 1}}} │ │ │ │ -00046cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046cc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00046cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046d10: 2020 207c 0a7c 6f34 203a 204c 6973 7420 |.|o4 : List │ │ │ │ -00046d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046d60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00046d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046db0: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ -00046dc0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -00046dd0: 7465 2067 726f 7570 3a20 6772 6f75 702c te group: group, │ │ │ │ -00046de0: 202d 2d20 6c69 7374 2061 6c6c 2065 6c65 -- list all ele │ │ │ │ -00046df0: 6d65 6e74 7320 6f66 2074 6865 2067 726f ments of the gro │ │ │ │ -00046e00: 7570 206f 6620 6120 6669 6e69 7465 2067 up of a finite g │ │ │ │ -00046e10: 726f 7570 0a20 2020 2061 6374 696f 6e0a roup. action. │ │ │ │ -00046e20: 2020 2a20 2a6e 6f74 6520 7363 6872 6569 * *note schrei │ │ │ │ -00046e30: 6572 4772 6170 683a 2073 6368 7265 6965 erGraph: schreie │ │ │ │ -00046e40: 7247 7261 7068 2c20 2d2d 2053 6368 7265 rGraph, -- Schre │ │ │ │ -00046e50: 6965 7220 6772 6170 6820 6f66 2061 2066 ier graph of a f │ │ │ │ -00046e60: 696e 6974 6520 6772 6f75 700a 2020 2a20 inite group. * │ │ │ │ -00046e70: 2a6e 6f74 6520 776f 7264 733a 2077 6f72 *note words: wor │ │ │ │ -00046e80: 6473 2c20 2d2d 2061 7373 6f63 6961 7465 ds, -- associate │ │ │ │ -00046e90: 2061 2077 6f72 6420 696e 2074 6865 2067 a word in the g │ │ │ │ -00046ea0: 656e 6572 6174 6f72 7320 6f66 2061 2067 enerators of a g │ │ │ │ -00046eb0: 726f 7570 2074 6f0a 2020 2020 6561 6368 roup to. each │ │ │ │ -00046ec0: 2065 6c65 6d65 6e74 0a0a 5761 7973 2074 element..Ways t │ │ │ │ -00046ed0: 6f20 7573 6520 7468 6973 206d 6574 686f o use this metho │ │ │ │ -00046ee0: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d d:.============= │ │ │ │ -00046ef0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00046f00: 202a 6e6f 7465 2072 656c 6174 696f 6e73 *note relations │ │ │ │ -00046f10: 2846 696e 6974 6547 726f 7570 4163 7469 (FiniteGroupActi │ │ │ │ -00046f20: 6f6e 293a 2072 656c 6174 696f 6e73 5f6c on): relations_l │ │ │ │ -00046f30: 7046 696e 6974 6547 726f 7570 4163 7469 pFiniteGroupActi │ │ │ │ -00046f40: 6f6e 5f72 702c 202d 2d0a 2020 2020 7265 on_rp, --. re │ │ │ │ -00046f50: 6c61 7469 6f6e 7320 6f66 2061 2066 696e lations of a fin │ │ │ │ -00046f60: 6974 6520 6772 6f75 700a 2d2d 2d2d 2d2d ite group.------ │ │ │ │ -00046f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046fb0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00046fc0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00046fd0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -00046fe0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -00046ff0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -00047000: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -00047010: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -00047020: 2f0a 496e 7661 7269 616e 7452 696e 672f /.InvariantRing/ │ │ │ │ -00047030: 4669 6e69 7465 4772 6f75 7073 446f 632e FiniteGroupsDoc. │ │ │ │ -00047040: 6d32 3a33 3838 3a30 2e0a 1f0a 4669 6c65 m2:388:0....File │ │ │ │ -00047050: 3a20 496e 7661 7269 616e 7452 696e 672e : InvariantRing. │ │ │ │ -00047060: 696e 666f 2c20 4e6f 6465 3a20 7265 796e info, Node: reyn │ │ │ │ -00047070: 6f6c 6473 4f70 6572 6174 6f72 2c20 4e65 oldsOperator, Ne │ │ │ │ -00047080: 7874 3a20 7269 6e67 5f6c 7047 726f 7570 xt: ring_lpGroup │ │ │ │ -00047090: 4163 7469 6f6e 5f72 702c 2050 7265 763a Action_rp, Prev: │ │ │ │ -000470a0: 2072 656c 6174 696f 6e73 5f6c 7046 696e relations_lpFin │ │ │ │ -000470b0: 6974 6547 726f 7570 4163 7469 6f6e 5f72 iteGroupAction_r │ │ │ │ -000470c0: 702c 2055 703a 2054 6f70 0a0a 7265 796e p, Up: Top..reyn │ │ │ │ -000470d0: 6f6c 6473 4f70 6572 6174 6f72 202d 2d20 oldsOperator -- │ │ │ │ -000470e0: 7468 6520 696d 6167 6520 6f66 2061 2070 the image of a p │ │ │ │ -000470f0: 6f6c 796e 6f6d 6961 6c20 756e 6465 7220 olynomial under │ │ │ │ -00047100: 7468 6520 5265 796e 6f6c 6473 206f 7065 the Reynolds ope │ │ │ │ -00047110: 7261 746f 720a 2a2a 2a2a 2a2a 2a2a 2a2a rator.********** │ │ │ │ -00047120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00047130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00047140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00047150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00047160: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -00047170: 2020 2020 2072 6579 6e6f 6c64 734f 7065 reynoldsOpe │ │ │ │ -00047180: 7261 746f 7228 662c 2047 292c 2072 6579 rator(f, G), rey │ │ │ │ -00047190: 6e6f 6c64 734f 7065 7261 746f 7228 662c noldsOperator(f, │ │ │ │ -000471a0: 2044 290a 2020 2a20 496e 7075 7473 3a0a D). * Inputs:. │ │ │ │ -000471b0: 2020 2020 2020 2a20 662c 2061 202a 6e6f * f, a *no │ │ │ │ -000471c0: 7465 2072 696e 6720 656c 656d 656e 743a te ring element: │ │ │ │ -000471d0: 2028 4d61 6361 756c 6179 3244 6f63 2952 (Macaulay2Doc)R │ │ │ │ -000471e0: 696e 6745 6c65 6d65 6e74 2c2c 2061 2070 ingElement,, a p │ │ │ │ -000471f0: 6f6c 796e 6f6d 6961 6c20 696e 0a20 2020 olynomial in. │ │ │ │ -00047200: 2020 2020 2074 6865 2070 6f6c 796e 6f6d the polynom │ │ │ │ -00047210: 6961 6c20 7269 6e67 206f 6620 7468 6520 ial ring of the │ │ │ │ -00047220: 6769 7665 6e20 6772 6f75 7020 6163 7469 given group acti │ │ │ │ -00047230: 6f6e 0a20 2020 2020 202a 2047 2c20 616e on. * G, an │ │ │ │ -00047240: 2069 6e73 7461 6e63 6520 6f66 2074 6865 instance of the │ │ │ │ -00047250: 2074 7970 6520 2a6e 6f74 6520 4669 6e69 type *note Fini │ │ │ │ -00047260: 7465 4772 6f75 7041 6374 696f 6e3a 2046 teGroupAction: F │ │ │ │ -00047270: 696e 6974 6547 726f 7570 4163 7469 6f6e initeGroupAction │ │ │ │ -00047280: 2c0a 2020 2020 2020 2a20 442c 2061 6e20 ,. * D, an │ │ │ │ -00047290: 696e 7374 616e 6365 206f 6620 7468 6520 instance of the │ │ │ │ -000472a0: 7479 7065 202a 6e6f 7465 2044 6961 676f type *note Diago │ │ │ │ -000472b0: 6e61 6c41 6374 696f 6e3a 2044 6961 676f nalAction: Diago │ │ │ │ -000472c0: 6e61 6c41 6374 696f 6e2c 0a20 202a 204f nalAction,. * O │ │ │ │ -000472d0: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -000472e0: 6120 2a6e 6f74 6520 7269 6e67 2065 6c65 a *note ring ele │ │ │ │ -000472f0: 6d65 6e74 3a20 284d 6163 6175 6c61 7932 ment: (Macaulay2 │ │ │ │ -00047300: 446f 6329 5269 6e67 456c 656d 656e 742c Doc)RingElement, │ │ │ │ -00047310: 2c20 7468 6520 696e 7661 7269 616e 740a , the invariant. │ │ │ │ -00047320: 2020 2020 2020 2020 706f 6c79 6e6f 6d69 polynomi │ │ │ │ -00047330: 616c 2077 6869 6368 2069 7320 7468 6520 al which is the │ │ │ │ -00047340: 696d 6167 6520 6f66 2074 6865 2067 6976 image of the giv │ │ │ │ -00047350: 656e 2070 6f6c 796e 6f6d 6961 6c20 756e en polynomial un │ │ │ │ -00047360: 6465 7220 7468 650a 2020 2020 2020 2020 der the. │ │ │ │ -00047370: 5265 796e 6f6c 6473 206f 7065 7261 746f Reynolds operato │ │ │ │ -00047380: 7220 6f66 2074 6865 2067 6976 656e 2066 r of the given f │ │ │ │ -00047390: 696e 6974 6520 6772 6f75 7020 6163 7469 inite group acti │ │ │ │ -000473a0: 6f6e 206f 7220 7468 6520 6769 7665 6e20 on or the given │ │ │ │ -000473b0: 746f 7275 730a 2020 2020 2020 2020 6163 torus. ac │ │ │ │ -000473c0: 7469 6f6e 0a0a 4465 7363 7269 7074 696f tion..Descriptio │ │ │ │ -000473d0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 n.===========..T │ │ │ │ -000473e0: 6869 7320 6675 6e63 7469 6f6e 2069 7320 his function is │ │ │ │ -000473f0: 7072 6f76 6964 6564 2062 7920 7468 6520 provided by the │ │ │ │ -00047400: 7061 636b 6167 6520 2a6e 6f74 6520 496e package *note In │ │ │ │ -00047410: 7661 7269 616e 7452 696e 673a 2054 6f70 variantRing: Top │ │ │ │ -00047420: 2c2e 200a 0a54 6865 2066 6f6c 6c6f 7769 ,. ..The followi │ │ │ │ -00047430: 6e67 2065 7861 6d70 6c65 2063 6f6d 7075 ng example compu │ │ │ │ -00047440: 7465 7320 7468 6520 696d 6167 6520 6f66 tes the image of │ │ │ │ -00047450: 2061 2070 6f6c 796e 6f6d 6961 6c20 756e a polynomial un │ │ │ │ -00047460: 6465 7220 7468 6520 5265 796e 6f6c 6473 der the Reynolds │ │ │ │ -00047470: 0a6f 7065 7261 746f 7220 666f 7220 6120 .operator for a │ │ │ │ -00047480: 6379 636c 6963 2070 6572 6d75 7461 7469 cyclic permutati │ │ │ │ -00047490: 6f6e 206f 6620 7468 6520 7661 7269 6162 on of the variab │ │ │ │ -000474a0: 6c65 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d les...+--------- │ │ │ │ -000474b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000474c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000474d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000474e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -000474f0: 5220 3d20 5a5a 2f33 5b78 5f30 2e2e 785f R = ZZ/3[x_0..x_ │ │ │ │ -00047500: 365d 2020 2020 2020 2020 2020 2020 2020 6] │ │ │ │ -00047510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047520: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00046a30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00046a40: 6f34 203a 204c 6973 7420 2020 2020 2020 o4 : List │ │ │ │ +00046a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046a80: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00046a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00046ae0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +00046af0: 3d0a 0a20 202a 202a 6e6f 7465 2067 726f =.. * *note gro │ │ │ │ +00046b00: 7570 3a20 6772 6f75 702c 202d 2d20 6c69 up: group, -- li │ │ │ │ +00046b10: 7374 2061 6c6c 2065 6c65 6d65 6e74 7320 st all elements │ │ │ │ +00046b20: 6f66 2074 6865 2067 726f 7570 206f 6620 of the group of │ │ │ │ +00046b30: 6120 6669 6e69 7465 2067 726f 7570 0a20 a finite group. │ │ │ │ +00046b40: 2020 2061 6374 696f 6e0a 2020 2a20 2a6e action. * *n │ │ │ │ +00046b50: 6f74 6520 7363 6872 6569 6572 4772 6170 ote schreierGrap │ │ │ │ +00046b60: 683a 2073 6368 7265 6965 7247 7261 7068 h: schreierGraph │ │ │ │ +00046b70: 2c20 2d2d 2053 6368 7265 6965 7220 6772 , -- Schreier gr │ │ │ │ +00046b80: 6170 6820 6f66 2061 2066 696e 6974 6520 aph of a finite │ │ │ │ +00046b90: 6772 6f75 700a 2020 2a20 2a6e 6f74 6520 group. * *note │ │ │ │ +00046ba0: 776f 7264 733a 2077 6f72 6473 2c20 2d2d words: words, -- │ │ │ │ +00046bb0: 2061 7373 6f63 6961 7465 2061 2077 6f72 associate a wor │ │ │ │ +00046bc0: 6420 696e 2074 6865 2067 656e 6572 6174 d in the generat │ │ │ │ +00046bd0: 6f72 7320 6f66 2061 2067 726f 7570 2074 ors of a group t │ │ │ │ +00046be0: 6f0a 2020 2020 6561 6368 2065 6c65 6d65 o. each eleme │ │ │ │ +00046bf0: 6e74 0a0a 5761 7973 2074 6f20 7573 6520 nt..Ways to use │ │ │ │ +00046c00: 7468 6973 206d 6574 686f 643a 0a3d 3d3d this method:.=== │ │ │ │ +00046c10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00046c20: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ +00046c30: 2072 656c 6174 696f 6e73 2846 696e 6974 relations(Finit │ │ │ │ +00046c40: 6547 726f 7570 4163 7469 6f6e 293a 2072 eGroupAction): r │ │ │ │ +00046c50: 656c 6174 696f 6e73 5f6c 7046 696e 6974 elations_lpFinit │ │ │ │ +00046c60: 6547 726f 7570 4163 7469 6f6e 5f72 702c eGroupAction_rp, │ │ │ │ +00046c70: 202d 2d0a 2020 2020 7265 6c61 7469 6f6e --. relation │ │ │ │ +00046c80: 7320 6f66 2061 2066 696e 6974 6520 6772 s of a finite gr │ │ │ │ +00046c90: 6f75 700a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d oup.------------ │ │ │ │ +00046ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046ce0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +00046cf0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +00046d00: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +00046d10: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +00046d20: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +00046d30: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +00046d40: 322f 7061 636b 6167 6573 2f0a 496e 7661 2/packages/.Inva │ │ │ │ +00046d50: 7269 616e 7452 696e 672f 4669 6e69 7465 riantRing/Finite │ │ │ │ +00046d60: 4772 6f75 7073 446f 632e 6d32 3a33 3838 GroupsDoc.m2:388 │ │ │ │ +00046d70: 3a30 2e0a 1f0a 4669 6c65 3a20 496e 7661 :0....File: Inva │ │ │ │ +00046d80: 7269 616e 7452 696e 672e 696e 666f 2c20 riantRing.info, │ │ │ │ +00046d90: 4e6f 6465 3a20 7265 796e 6f6c 6473 4f70 Node: reynoldsOp │ │ │ │ +00046da0: 6572 6174 6f72 2c20 4e65 7874 3a20 7269 erator, Next: ri │ │ │ │ +00046db0: 6e67 5f6c 7047 726f 7570 4163 7469 6f6e ng_lpGroupAction │ │ │ │ +00046dc0: 5f72 702c 2050 7265 763a 2072 656c 6174 _rp, Prev: relat │ │ │ │ +00046dd0: 696f 6e73 5f6c 7046 696e 6974 6547 726f ions_lpFiniteGro │ │ │ │ +00046de0: 7570 4163 7469 6f6e 5f72 702c 2055 703a upAction_rp, Up: │ │ │ │ +00046df0: 2054 6f70 0a0a 7265 796e 6f6c 6473 4f70 Top..reynoldsOp │ │ │ │ +00046e00: 6572 6174 6f72 202d 2d20 7468 6520 696d erator -- the im │ │ │ │ +00046e10: 6167 6520 6f66 2061 2070 6f6c 796e 6f6d age of a polynom │ │ │ │ +00046e20: 6961 6c20 756e 6465 7220 7468 6520 5265 ial under the Re │ │ │ │ +00046e30: 796e 6f6c 6473 206f 7065 7261 746f 720a ynolds operator. │ │ │ │ +00046e40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00046e50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00046e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00046e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00046e80: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00046e90: 7361 6765 3a20 0a20 2020 2020 2020 2072 sage: . r │ │ │ │ +00046ea0: 6579 6e6f 6c64 734f 7065 7261 746f 7228 eynoldsOperator( │ │ │ │ +00046eb0: 662c 2047 292c 2072 6579 6e6f 6c64 734f f, G), reynoldsO │ │ │ │ +00046ec0: 7065 7261 746f 7228 662c 2044 290a 2020 perator(f, D). │ │ │ │ +00046ed0: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +00046ee0: 2a20 662c 2061 202a 6e6f 7465 2072 696e * f, a *note rin │ │ │ │ +00046ef0: 6720 656c 656d 656e 743a 2028 4d61 6361 g element: (Maca │ │ │ │ +00046f00: 756c 6179 3244 6f63 2952 696e 6745 6c65 ulay2Doc)RingEle │ │ │ │ +00046f10: 6d65 6e74 2c2c 2061 2070 6f6c 796e 6f6d ment,, a polynom │ │ │ │ +00046f20: 6961 6c20 696e 0a20 2020 2020 2020 2074 ial in. t │ │ │ │ +00046f30: 6865 2070 6f6c 796e 6f6d 6961 6c20 7269 he polynomial ri │ │ │ │ +00046f40: 6e67 206f 6620 7468 6520 6769 7665 6e20 ng of the given │ │ │ │ +00046f50: 6772 6f75 7020 6163 7469 6f6e 0a20 2020 group action. │ │ │ │ +00046f60: 2020 202a 2047 2c20 616e 2069 6e73 7461 * G, an insta │ │ │ │ +00046f70: 6e63 6520 6f66 2074 6865 2074 7970 6520 nce of the type │ │ │ │ +00046f80: 2a6e 6f74 6520 4669 6e69 7465 4772 6f75 *note FiniteGrou │ │ │ │ +00046f90: 7041 6374 696f 6e3a 2046 696e 6974 6547 pAction: FiniteG │ │ │ │ +00046fa0: 726f 7570 4163 7469 6f6e 2c0a 2020 2020 roupAction,. │ │ │ │ +00046fb0: 2020 2a20 442c 2061 6e20 696e 7374 616e * D, an instan │ │ │ │ +00046fc0: 6365 206f 6620 7468 6520 7479 7065 202a ce of the type * │ │ │ │ +00046fd0: 6e6f 7465 2044 6961 676f 6e61 6c41 6374 note DiagonalAct │ │ │ │ +00046fe0: 696f 6e3a 2044 6961 676f 6e61 6c41 6374 ion: DiagonalAct │ │ │ │ +00046ff0: 696f 6e2c 0a20 202a 204f 7574 7075 7473 ion,. * Outputs │ │ │ │ +00047000: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ +00047010: 6520 7269 6e67 2065 6c65 6d65 6e74 3a20 e ring element: │ │ │ │ +00047020: 284d 6163 6175 6c61 7932 446f 6329 5269 (Macaulay2Doc)Ri │ │ │ │ +00047030: 6e67 456c 656d 656e 742c 2c20 7468 6520 ngElement,, the │ │ │ │ +00047040: 696e 7661 7269 616e 740a 2020 2020 2020 invariant. │ │ │ │ +00047050: 2020 706f 6c79 6e6f 6d69 616c 2077 6869 polynomial whi │ │ │ │ +00047060: 6368 2069 7320 7468 6520 696d 6167 6520 ch is the image │ │ │ │ +00047070: 6f66 2074 6865 2067 6976 656e 2070 6f6c of the given pol │ │ │ │ +00047080: 796e 6f6d 6961 6c20 756e 6465 7220 7468 ynomial under th │ │ │ │ +00047090: 650a 2020 2020 2020 2020 5265 796e 6f6c e. Reynol │ │ │ │ +000470a0: 6473 206f 7065 7261 746f 7220 6f66 2074 ds operator of t │ │ │ │ +000470b0: 6865 2067 6976 656e 2066 696e 6974 6520 he given finite │ │ │ │ +000470c0: 6772 6f75 7020 6163 7469 6f6e 206f 7220 group action or │ │ │ │ +000470d0: 7468 6520 6769 7665 6e20 746f 7275 730a the given torus. │ │ │ │ +000470e0: 2020 2020 2020 2020 6163 7469 6f6e 0a0a action.. │ │ │ │ +000470f0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00047100: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6675 =======..This fu │ │ │ │ +00047110: 6e63 7469 6f6e 2069 7320 7072 6f76 6964 nction is provid │ │ │ │ +00047120: 6564 2062 7920 7468 6520 7061 636b 6167 ed by the packag │ │ │ │ +00047130: 6520 2a6e 6f74 6520 496e 7661 7269 616e e *note Invarian │ │ │ │ +00047140: 7452 696e 673a 2054 6f70 2c2e 200a 0a54 tRing: Top,. ..T │ │ │ │ +00047150: 6865 2066 6f6c 6c6f 7769 6e67 2065 7861 he following exa │ │ │ │ +00047160: 6d70 6c65 2063 6f6d 7075 7465 7320 7468 mple computes th │ │ │ │ +00047170: 6520 696d 6167 6520 6f66 2061 2070 6f6c e image of a pol │ │ │ │ +00047180: 796e 6f6d 6961 6c20 756e 6465 7220 7468 ynomial under th │ │ │ │ +00047190: 6520 5265 796e 6f6c 6473 0a6f 7065 7261 e Reynolds.opera │ │ │ │ +000471a0: 746f 7220 666f 7220 6120 6379 636c 6963 tor for a cyclic │ │ │ │ +000471b0: 2070 6572 6d75 7461 7469 6f6e 206f 6620 permutation of │ │ │ │ +000471c0: 7468 6520 7661 7269 6162 6c65 732e 0a0a the variables... │ │ │ │ +000471d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +000471e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000471f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047210: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 5a5a --+.|i1 : R = ZZ │ │ │ │ +00047220: 2f33 5b78 5f30 2e2e 785f 365d 2020 2020 /3[x_0..x_6] │ │ │ │ +00047230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047250: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00047260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047290: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +000472a0: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +000472b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000472c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000472d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000472e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000472f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047320: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f |.|o1 : Polyno │ │ │ │ +00047330: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +00047340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047360: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00047370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000473a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +000473b0: 3a20 5020 3d20 7065 726d 7574 6174 696f : P = permutatio │ │ │ │ +000473c0: 6e4d 6174 7269 7820 5b32 2c20 332c 2034 nMatrix [2, 3, 4 │ │ │ │ +000473d0: 2c20 352c 2036 2c20 372c 2031 5d20 2020 , 5, 6, 7, 1] │ │ │ │ +000473e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000473f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00047400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047430: 2020 7c0a 7c6f 3220 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2020 2020 2020 2020 2020 7c20 |.| | │ │ │ │ -00047ad0: 3120 3020 3020 3020 3020 3020 3020 7c20 1 0 0 0 0 0 0 | │ │ │ │ -00047ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047b00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00047b10: 2020 7c20 3020 3120 3020 3020 3020 3020 | 0 1 0 0 0 0 │ │ │ │ -00047b20: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -00047b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047b40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00047b50: 2020 2020 2020 7c20 3020 3020 3120 3020 | 0 0 1 0 │ │ │ │ -00047b60: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ -00047b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00047b90: 2020 2020 2020 2020 2020 7c20 3020 3020 | 0 0 │ │ │ │ -00047ba0: 3020 3120 3020 3020 3020 7c20 2020 2020 0 1 0 0 0 | │ │ │ │ +00047ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047ad0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00047ae0: 2020 2032 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00047af0: 2020 2020 2032 2020 2020 2020 2020 3220 2 2 │ │ │ │ +00047b00: 2020 2020 2032 2020 2020 2020 3220 2020 2 2 │ │ │ │ +00047b10: 2020 2020 2020 2020 2032 7c0a 7c6f 3420 2|.|o4 │ │ │ │ +00047b20: 3d20 7820 7820 7820 202b 2078 2078 2078 = x x x + x x x │ │ │ │ +00047b30: 2020 2b20 7820 7820 7820 202b 2078 2078 + x x x + x x │ │ │ │ +00047b40: 2078 2020 2b20 7820 7820 7820 202b 2078 x + x x x + x │ │ │ │ +00047b50: 2078 2078 2020 2b20 7820 7820 7820 7c0a x x + x x x |. │ │ │ │ +00047b60: 7c20 2020 2020 2030 2031 2032 2020 2020 | 0 1 2 │ │ │ │ +00047b70: 3120 3220 3320 2020 2032 2033 2034 2020 1 2 3 2 3 4 │ │ │ │ +00047b80: 2020 3320 3420 3520 2020 2030 2031 2036 3 4 5 0 1 6 │ │ │ │ +00047b90: 2020 2020 3020 3520 3620 2020 2034 2035 0 5 6 4 5 │ │ │ │ +00047ba0: 2036 7c0a 7c20 2020 2020 2020 2020 2020 6|.| │ │ │ │ 00047bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047bd0: 7c0a 7c20 2020 2020 2020 2020 2020 7c20 |.| | │ │ │ │ -00047be0: 3020 3020 3020 3020 3120 3020 3020 7c20 0 0 0 0 1 0 0 | │ │ │ │ +00047bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047be0: 2020 2020 2020 7c0a 7c6f 3420 3a20 5220 |.|o4 : R │ │ │ │ 00047bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00047c20: 2020 7c20 3020 3020 3020 3020 3020 3120 | 0 0 0 0 0 1 │ │ │ │ -00047c30: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -00047c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c50: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00047c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00047ca0: 3320 3a20 4669 6e69 7465 4772 6f75 7041 3 : FiniteGroupA │ │ │ │ -00047cb0: 6374 696f 6e20 2020 2020 2020 2020 2020 ction │ │ │ │ -00047cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047ce0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00047cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047c20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00047c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00047c70: 0a48 6572 6520 6973 2061 6e20 6578 616d .Here is an exam │ │ │ │ +00047c80: 706c 6520 636f 6d70 7574 696e 6720 7468 ple computing th │ │ │ │ +00047c90: 6520 696d 6167 6520 6f66 2061 2070 6f6c e image of a pol │ │ │ │ +00047ca0: 796e 6f6d 6961 6c20 756e 6465 7220 7468 ynomial under th │ │ │ │ +00047cb0: 6520 5265 796e 6f6c 6473 0a6f 7065 7261 e Reynolds.opera │ │ │ │ +00047cc0: 746f 7220 666f 7220 6120 7477 6f2d 6469 tor for a two-di │ │ │ │ +00047cd0: 6d65 6e73 696f 6e61 6c20 746f 7275 7320 mensional torus │ │ │ │ +00047ce0: 6163 7469 6e67 206f 6e20 706f 6c79 6e6f acting on polyno │ │ │ │ +00047cf0: 6d69 616c 2072 696e 6720 696e 2066 6f75 mial ring in fou │ │ │ │ +00047d00: 720a 7661 7269 6162 6c65 733a 0a0a 2b2d r.variables:..+- │ │ │ │ 00047d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047d20: 2d2d 2d2d 2b0a 7c69 3420 3a20 7265 796e ----+.|i4 : reyn │ │ │ │ -00047d30: 6f6c 6473 4f70 6572 6174 6f72 2878 5f30 oldsOperator(x_0 │ │ │ │ -00047d40: 2a78 5f31 2a78 5f32 5e32 2c20 4337 2920 *x_1*x_2^2, C7) │ │ │ │ -00047d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047d60: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00047d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047d40: 2d2d 2b0a 7c69 3520 3a20 5220 3d20 5151 --+.|i5 : R = QQ │ │ │ │ +00047d50: 5b78 5f31 2e2e 785f 345d 2020 2020 2020 [x_1..x_4] │ │ │ │ +00047d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047d70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00047d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00047db0: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00047dc0: 2020 3220 2020 2020 2020 2032 2020 2020 2 2 │ │ │ │ -00047dd0: 2020 2020 3220 2020 2020 2032 2020 2020 2 2 │ │ │ │ -00047de0: 2020 3220 2020 2020 2020 2020 2020 2032 2 2 │ │ │ │ -00047df0: 7c0a 7c6f 3420 3d20 7820 7820 7820 202b |.|o4 = x x x + │ │ │ │ -00047e00: 2078 2078 2078 2020 2b20 7820 7820 7820 x x x + x x x │ │ │ │ -00047e10: 202b 2078 2078 2078 2020 2b20 7820 7820 + x x x + x x │ │ │ │ -00047e20: 7820 202b 2078 2078 2078 2020 2b20 7820 x + x x x + x │ │ │ │ -00047e30: 7820 7820 7c0a 7c20 2020 2020 2030 2031 x x |.| 0 1 │ │ │ │ -00047e40: 2032 2020 2020 3120 3220 3320 2020 2032 2 1 2 3 2 │ │ │ │ -00047e50: 2033 2034 2020 2020 3320 3420 3520 2020 3 4 3 4 5 │ │ │ │ -00047e60: 2030 2031 2036 2020 2020 3020 3520 3620 0 1 6 0 5 6 │ │ │ │ -00047e70: 2020 2034 2035 2036 7c0a 7c20 2020 2020 4 5 6|.| │ │ │ │ -00047e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00047ec0: 3420 3a20 5220 2020 2020 2020 2020 2020 4 : R │ │ │ │ +00047da0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00047db0: 7c6f 3520 3d20 5220 2020 2020 2020 2020 |o5 = R │ │ │ │ +00047dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047de0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00047df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047e10: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +00047e20: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +00047e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047e50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00047e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047e80: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 5720 ------+.|i6 : W │ │ │ │ +00047e90: 3d20 6d61 7472 6978 7b7b 302c 312c 2d31 = matrix{{0,1,-1 │ │ │ │ +00047ea0: 2c31 7d2c 207b 312c 302c 2d31 2c2d 317d ,1}, {1,0,-1,-1} │ │ │ │ +00047eb0: 7d20 2020 2020 2020 2020 2020 7c0a 7c20 } |.| │ │ │ │ +00047ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047f00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00047f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00047f40: 2d2d 2d2d 2b0a 0a48 6572 6520 6973 2061 ----+..Here is a │ │ │ │ -00047f50: 6e20 6578 616d 706c 6520 636f 6d70 7574 n example comput │ │ │ │ -00047f60: 696e 6720 7468 6520 696d 6167 6520 6f66 ing the image of │ │ │ │ -00047f70: 2061 2070 6f6c 796e 6f6d 6961 6c20 756e a polynomial un │ │ │ │ -00047f80: 6465 7220 7468 6520 5265 796e 6f6c 6473 der the Reynolds │ │ │ │ -00047f90: 0a6f 7065 7261 746f 7220 666f 7220 6120 .operator for a │ │ │ │ -00047fa0: 7477 6f2d 6469 6d65 6e73 696f 6e61 6c20 two-dimensional │ │ │ │ -00047fb0: 746f 7275 7320 6163 7469 6e67 206f 6e20 torus acting on │ │ │ │ -00047fc0: 706f 6c79 6e6f 6d69 616c 2072 696e 6720 polynomial ring │ │ │ │ -00047fd0: 696e 2066 6f75 720a 7661 7269 6162 6c65 in four.variable │ │ │ │ -00047fe0: 733a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s:..+----------- │ │ │ │ -00047ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048010: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ -00048020: 5220 3d20 5151 5b78 5f31 2e2e 785f 345d R = QQ[x_1..x_4] │ │ │ │ -00048030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048040: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00048050: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00048060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047ef0: 2020 7c0a 7c6f 3620 3d20 7c20 3020 3120 |.|o6 = | 0 1 │ │ │ │ +00047f00: 2d31 2031 2020 7c20 2020 2020 2020 2020 -1 1 | │ │ │ │ +00047f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047f20: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00047f30: 7c20 3120 3020 2d31 202d 3120 7c20 2020 | 1 0 -1 -1 | │ │ │ │ +00047f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047f50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00047f60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00047f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047f90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00047fa0: 2020 2020 2032 2020 2020 2020 2034 2020 2 4 │ │ │ │ +00047fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047fc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +00047fd0: 3a20 4d61 7472 6978 205a 5a20 203c 2d2d : Matrix ZZ <-- │ │ │ │ +00047fe0: 205a 5a20 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00047ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048000: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00048010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048030: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 5420 ------+.|i7 : T │ │ │ │ +00048040: 3d20 6469 6167 6f6e 616c 4163 7469 6f6e = diagonalAction │ │ │ │ +00048050: 2857 2c20 5229 2020 2020 2020 2020 2020 (W, R) │ │ │ │ +00048060: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00048070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048080: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020 |.|o5 = R │ │ │ │ +00048080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00048090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000480a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000480b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000480a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000480b0: 2020 2a20 3220 2020 2020 2020 2020 2020 * 2 │ │ │ │ 000480c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000480d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000480e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000480f0: 7c0a 7c6f 3520 3a20 506f 6c79 6e6f 6d69 |.|o5 : Polynomi │ │ │ │ -00048100: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ -00048110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048120: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -00048130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00048160: 3620 3a20 5720 3d20 6d61 7472 6978 7b7b 6 : W = matrix{{ │ │ │ │ -00048170: 302c 312c 2d31 2c31 7d2c 207b 312c 302c 0,1,-1,1}, {1,0, │ │ │ │ -00048180: 2d31 2c2d 317d 7d20 2020 2020 2020 2020 -1,-1}} │ │ │ │ -00048190: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000480d0: 2020 2020 2020 2020 7c0a 7c6f 3720 3d20 |.|o7 = │ │ │ │ +000480e0: 5220 3c2d 2028 5151 2029 2020 7669 6120 R <- (QQ ) via │ │ │ │ +000480f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048100: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00048110: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00048120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048140: 2020 2020 7c0a 7c20 2020 2020 7c20 3020 |.| | 0 │ │ │ │ +00048150: 3120 2d31 2031 2020 7c20 2020 2020 2020 1 -1 1 | │ │ │ │ +00048160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048170: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00048180: 2020 7c20 3120 3020 2d31 202d 3120 7c20 | 1 0 -1 -1 | │ │ │ │ +00048190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000481a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000481b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000481c0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -000481d0: 7c20 3020 3120 2d31 2031 2020 7c20 2020 | 0 1 -1 1 | │ │ │ │ -000481e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000481f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00048200: 7c20 2020 2020 7c20 3120 3020 2d31 202d | | 1 0 -1 - │ │ │ │ -00048210: 3120 7c20 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -00048220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048230: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00048240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048260: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00048270: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -00048280: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +000481b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000481c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000481d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000481e0: 2020 2020 2020 7c0a 7c6f 3720 3a20 4469 |.|o7 : Di │ │ │ │ +000481f0: 6167 6f6e 616c 4163 7469 6f6e 2020 2020 agonalAction │ │ │ │ +00048200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048210: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00048220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048250: 2d2d 2b0a 7c69 3820 3a20 7265 796e 6f6c --+.|i8 : reynol │ │ │ │ +00048260: 6473 4f70 6572 6174 6f72 2878 5f31 2a78 dsOperator(x_1*x │ │ │ │ +00048270: 5f32 2a78 5f33 202b 2078 5f31 2a78 5f32 _2*x_3 + x_1*x_2 │ │ │ │ +00048280: 2a78 5f34 2c20 5429 7c0a 7c20 2020 2020 *x_4, T)|.| │ │ │ │ 00048290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000482a0: 7c0a 7c6f 3620 3a20 4d61 7472 6978 205a |.|o6 : Matrix Z │ │ │ │ -000482b0: 5a20 203c 2d2d 205a 5a20 2020 2020 2020 Z <-- ZZ │ │ │ │ -000482c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000482d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -000482e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000482f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00048310: 3720 3a20 5420 3d20 6469 6167 6f6e 616c 7 : T = diagonal │ │ │ │ -00048320: 4163 7469 6f6e 2857 2c20 5229 2020 2020 Action(W, R) │ │ │ │ +000482a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000482b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000482c0: 7c6f 3820 3d20 7820 7820 7820 2020 2020 |o8 = x x x │ │ │ │ +000482d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000482e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000482f0: 2020 2020 7c0a 7c20 2020 2020 2031 2032 |.| 1 2 │ │ │ │ +00048300: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00048310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048320: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00048330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048340: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00048340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00048350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048370: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00048380: 2020 2020 2020 2020 2a20 3220 2020 2020 * 2 │ │ │ │ -00048390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000483a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000483b0: 7c6f 3720 3d20 5220 3c2d 2028 5151 2029 |o7 = R <- (QQ ) │ │ │ │ -000483c0: 2020 7669 6120 2020 2020 2020 2020 2020 via │ │ │ │ -000483d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000483e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000483f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048410: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00048420: 2020 7c20 3020 3120 2d31 2031 2020 7c20 | 0 1 -1 1 | │ │ │ │ -00048430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048450: 7c0a 7c20 2020 2020 7c20 3120 3020 2d31 |.| | 1 0 -1 │ │ │ │ -00048460: 202d 3120 7c20 2020 2020 2020 2020 2020 -1 | │ │ │ │ -00048470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048480: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00048490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000484a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000484b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000484c0: 3720 3a20 4469 6167 6f6e 616c 4163 7469 7 : DiagonalActi │ │ │ │ -000484d0: 6f6e 2020 2020 2020 2020 2020 2020 2020 on │ │ │ │ -000484e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000484f0: 2020 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+00048420: 7228 5269 6e67 456c 656d 656e 742c 4469 r(RingElement,Di │ │ │ │ +00048430: 6167 6f6e 616c 4163 7469 6f6e 2922 0a20 agonalAction)". │ │ │ │ +00048440: 202a 2022 7265 796e 6f6c 6473 4f70 6572 * "reynoldsOper │ │ │ │ +00048450: 6174 6f72 2852 696e 6745 6c65 6d65 6e74 ator(RingElement │ │ │ │ +00048460: 2c46 696e 6974 6547 726f 7570 4163 7469 ,FiniteGroupActi │ │ │ │ +00048470: 6f6e 2922 0a0a 466f 7220 7468 6520 7072 on)"..For the pr │ │ │ │ +00048480: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +00048490: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +000484a0: 206f 626a 6563 7420 2a6e 6f74 6520 7265 object *note re │ │ │ │ +000484b0: 796e 6f6c 6473 4f70 6572 6174 6f72 3a20 ynoldsOperator: │ │ │ │ +000484c0: 7265 796e 6f6c 6473 4f70 6572 6174 6f72 reynoldsOperator │ │ │ │ +000484d0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +000484e0: 686f 640a 6675 6e63 7469 6f6e 3a20 284d hod.function: (M │ │ │ │ +000484f0: 6163 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reynoldsOp │ │ │ │ -000487a0: 6572 6174 6f72 2c20 6973 2061 202a 6e6f erator, is a *no │ │ │ │ -000487b0: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ -000487c0: 6f6e 3a20 284d 6163 6175 6c61 7932 446f on: (Macaulay2Do │ │ │ │ -000487d0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -000487e0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ -000487f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048830: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -00048840: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -00048850: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -00048860: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -00048870: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -00048880: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -00048890: 322f 7061 636b 6167 6573 2f0a 496e 7661 2/packages/.Inva │ │ │ │ -000488a0: 7269 616e 7452 696e 672f 496e 7661 7269 riantRing/Invari │ │ │ │ -000488b0: 616e 7473 446f 632e 6d32 3a39 3134 3a30 antsDoc.m2:914:0 │ │ │ │ -000488c0: 2e0a 1f0a 4669 6c65 3a20 496e 7661 7269 ....File: Invari │ │ │ │ -000488d0: 616e 7452 696e 672e 696e 666f 2c20 4e6f antRing.info, No │ │ │ │ -000488e0: 6465 3a20 7269 6e67 5f6c 7047 726f 7570 de: ring_lpGroup │ │ │ │ -000488f0: 4163 7469 6f6e 5f72 702c 204e 6578 743a Action_rp, Next: │ │ │ │ -00048900: 2052 696e 674f 6649 6e76 6172 6961 6e74 RingOfInvariant │ │ │ │ -00048910: 732c 2050 7265 763a 2072 6579 6e6f 6c64 s, Prev: reynold │ │ │ │ -00048920: 734f 7065 7261 746f 722c 2055 703a 2054 sOperator, Up: T │ │ │ │ -00048930: 6f70 0a0a 7269 6e67 2847 726f 7570 4163 op..ring(GroupAc │ │ │ │ -00048940: 7469 6f6e 2920 2d2d 2074 6865 2070 6f6c tion) -- the pol │ │ │ │ -00048950: 796e 6f6d 6961 6c20 7269 6e67 2062 6569 ynomial ring bei │ │ │ │ -00048960: 6e67 2061 6374 6564 2075 706f 6e0a 2a2a ng acted upon.** │ │ │ │ -00048970: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00048980: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00048990: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000489a0: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e *******.. * Fun │ │ │ │ -000489b0: 6374 696f 6e3a 202a 6e6f 7465 2072 696e ction: *note rin │ │ │ │ -000489c0: 673a 2028 4d61 6361 756c 6179 3244 6f63 g: (Macaulay2Doc │ │ │ │ -000489d0: 2972 696e 672c 0a20 202a 2055 7361 6765 )ring,. * Usage │ │ │ │ -000489e0: 3a20 0a20 2020 2020 2020 2072 696e 6720 : . ring │ │ │ │ -000489f0: 470a 2020 2a20 496e 7075 7473 3a0a 2020 G. * Inputs:. │ │ │ │ -00048a00: 2020 2020 2a20 472c 2061 6e20 696e 7374 * G, an inst │ │ │ │ -00048a10: 616e 6365 206f 6620 7468 6520 7479 7065 ance of the type │ │ │ │ -00048a20: 202a 6e6f 7465 2047 726f 7570 4163 7469 *note GroupActi │ │ │ │ -00048a30: 6f6e 3a20 4772 6f75 7041 6374 696f 6e2c on: GroupAction, │ │ │ │ -00048a40: 2c20 6120 6772 6f75 700a 2020 2020 2020 , a group. │ │ │ │ -00048a50: 2020 6163 7469 6f6e 206f 6e20 6120 706f action on a po │ │ │ │ -00048a60: 6c79 6e6f 6d69 616c 2072 696e 670a 2020 lynomial ring. │ │ │ │ -00048a70: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -00048a80: 202a 2061 202a 6e6f 7465 2072 696e 673a * a *note ring: │ │ │ │ -00048a90: 2028 4d61 6361 756c 6179 3244 6f63 2952 (Macaulay2Doc)R │ │ │ │ -00048aa0: 696e 672c 2c20 7468 6520 706f 6c79 6e6f ing,, the polyno │ │ │ │ -00048ab0: 6d69 616c 2072 696e 6720 6265 696e 6720 mial ring being │ │ │ │ -00048ac0: 6163 7465 6420 7570 6f6e 0a0a 4465 7363 acted upon..Desc │ │ │ │ -00048ad0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -00048ae0: 3d3d 3d0a 0a54 6869 7320 6675 6e63 7469 ===..This functi │ │ │ │ -00048af0: 6f6e 2069 7320 7072 6f76 6964 6564 2062 on is provided b │ │ │ │ -00048b00: 7920 7468 6520 7061 636b 6167 6520 2a6e y the package *n │ │ │ │ -00048b10: 6f74 6520 496e 7661 7269 616e 7452 696e ote InvariantRin │ │ │ │ -00048b20: 673a 2054 6f70 2c2e 0a0a 2b2d 2d2d 2d2d g: Top,...+----- │ │ │ │ -00048b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048b60: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5220 ------+.|i1 : R │ │ │ │ -00048b70: 3d20 5151 5b78 5f31 2e2e 785f 345d 2020 = QQ[x_1..x_4] │ │ │ │ -00048b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +00048560: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +00048570: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +00048580: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +00048590: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +000485a0: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ +000485b0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +000485c0: 6167 6573 2f0a 496e 7661 7269 616e 7452 ages/.InvariantR │ │ │ │ +000485d0: 696e 672f 496e 7661 7269 616e 7473 446f ing/InvariantsDo │ │ │ │ +000485e0: 632e 6d32 3a39 3134 3a30 2e0a 1f0a 4669 c.m2:914:0....Fi │ │ │ │ +000485f0: 6c65 3a20 496e 7661 7269 616e 7452 696e le: InvariantRin │ │ │ │ +00048600: 672e 696e 666f 2c20 4e6f 6465 3a20 7269 g.info, Node: ri │ │ │ │ +00048610: 6e67 5f6c 7047 726f 7570 4163 7469 6f6e ng_lpGroupAction │ │ │ │ +00048620: 5f72 702c 204e 6578 743a 2052 696e 674f _rp, Next: RingO │ │ │ │ +00048630: 6649 6e76 6172 6961 6e74 732c 2050 7265 fInvariants, Pre │ │ │ │ +00048640: 763a 2072 6579 6e6f 6c64 734f 7065 7261 v: reynoldsOpera │ │ │ │ +00048650: 746f 722c 2055 703a 2054 6f70 0a0a 7269 tor, Up: Top..ri │ │ │ │ +00048660: 6e67 2847 726f 7570 4163 7469 6f6e 2920 ng(GroupAction) │ │ │ │ +00048670: 2d2d 2074 6865 2070 6f6c 796e 6f6d 6961 -- the polynomia │ │ │ │ +00048680: 6c20 7269 6e67 2062 6569 6e67 2061 6374 l ring being act │ │ │ │ +00048690: 6564 2075 706f 6e0a 2a2a 2a2a 2a2a 2a2a ed upon.******** │ │ │ │ +000486a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000486b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000486c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000486d0: 2a0a 0a20 202a 2046 756e 6374 696f 6e3a *.. * Function: │ │ │ │ +000486e0: 202a 6e6f 7465 2072 696e 673a 2028 4d61 *note ring: (Ma │ │ │ │ +000486f0: 6361 756c 6179 3244 6f63 2972 696e 672c caulay2Doc)ring, │ │ │ │ +00048700: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +00048710: 2020 2020 2072 696e 6720 470a 2020 2a20 ring G. * │ │ │ │ +00048720: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ +00048730: 472c 2061 6e20 696e 7374 616e 6365 206f G, an instance o │ │ │ │ +00048740: 6620 7468 6520 7479 7065 202a 6e6f 7465 f the type *note │ │ │ │ +00048750: 2047 726f 7570 4163 7469 6f6e 3a20 4772 GroupAction: Gr │ │ │ │ +00048760: 6f75 7041 6374 696f 6e2c 2c20 6120 6772 oupAction,, a gr │ │ │ │ +00048770: 6f75 700a 2020 2020 2020 2020 6163 7469 oup. acti │ │ │ │ +00048780: 6f6e 206f 6e20 6120 706f 6c79 6e6f 6d69 on on a polynomi │ │ │ │ +00048790: 616c 2072 696e 670a 2020 2a20 4f75 7470 al ring. * Outp │ │ │ │ +000487a0: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ +000487b0: 6e6f 7465 2072 696e 673a 2028 4d61 6361 note ring: (Maca │ │ │ │ +000487c0: 756c 6179 3244 6f63 2952 696e 672c 2c20 ulay2Doc)Ring,, │ │ │ │ +000487d0: 7468 6520 706f 6c79 6e6f 6d69 616c 2072 the polynomial r │ │ │ │ +000487e0: 696e 6720 6265 696e 6720 6163 7465 6420 ing being acted │ │ │ │ +000487f0: 7570 6f6e 0a0a 4465 7363 7269 7074 696f upon..Descriptio │ │ │ │ +00048800: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 n.===========..T │ │ │ │ +00048810: 6869 7320 6675 6e63 7469 6f6e 2069 7320 his function is │ │ │ │ +00048820: 7072 6f76 6964 6564 2062 7920 7468 6520 provided by the │ │ │ │ +00048830: 7061 636b 6167 6520 2a6e 6f74 6520 496e package *note In │ │ │ │ +00048840: 7661 7269 616e 7452 696e 673a 2054 6f70 variantRing: Top │ │ │ │ +00048850: 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...+----------- │ │ │ │ +00048860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048890: 2b0a 7c69 3120 3a20 5220 3d20 5151 5b78 +.|i1 : R = QQ[x │ │ │ │ +000488a0: 5f31 2e2e 785f 345d 2020 2020 2020 2020 _1..x_4] │ │ │ │ +000488b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000488c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000488d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000488e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000488f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048900: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00048910: 3120 3d20 5220 2020 2020 2020 2020 2020 1 = R │ │ │ │ +00048920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048940: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00048950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048980: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +00048990: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +000489a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000489b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000489c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +000489d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000489e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000489f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048a00: 2d2d 2d2d 2b0a 7c69 3220 3a20 5420 3d20 ----+.|i2 : T = │ │ │ │ +00048a10: 6469 6167 6f6e 616c 4163 7469 6f6e 286d diagonalAction(m │ │ │ │ +00048a20: 6174 7269 7820 7b7b 302c 312c 2d31 2c31 atrix {{0,1,-1,1 │ │ │ │ +00048a30: 7d2c 7b31 2c30 2c2d 312c 2d31 7d7d 2c20 },{1,0,-1,-1}}, │ │ │ │ +00048a40: 5229 7c0a 7c20 2020 2020 2020 2020 2020 R)|.| │ │ │ │ +00048a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048a80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00048a90: 2a20 3220 2020 2020 2020 2020 2020 2020 * 2 │ │ │ │ +00048aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00048ac0: 7c6f 3220 3d20 5220 3c2d 2028 5151 2029 |o2 = R <- (QQ ) │ │ │ │ +00048ad0: 2020 7669 6120 2020 2020 2020 2020 2020 via │ │ │ │ +00048ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00048b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048b30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00048b40: 2020 7c20 3020 3120 2d31 2031 2020 7c20 | 0 1 -1 1 | │ │ │ │ +00048b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048b70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00048b80: 7c20 3120 3020 2d31 202d 3120 7c20 2020 | 1 0 -1 -1 | │ │ │ │ 00048b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048ba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00048bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048bb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00048bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00048bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048be0: 2020 7c0a 7c6f 3120 3d20 5220 2020 2020 |.|o1 = R │ │ │ │ -00048bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048bf0: 2020 2020 7c0a 7c6f 3220 3a20 4469 6167 |.|o2 : Diag │ │ │ │ +00048c00: 6f6e 616c 4163 7469 6f6e 2020 2020 2020 onalAction │ │ │ │ 00048c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048c20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00048c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048c50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00048c60: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c |o1 : Polynomial │ │ │ │ -00048c70: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ -00048c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048c90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00048ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ -00048ce0: 3a20 5420 3d20 6469 6167 6f6e 616c 4163 : T = diagonalAc │ │ │ │ -00048cf0: 7469 6f6e 286d 6174 7269 7820 7b7b 302c tion(matrix {{0, │ │ │ │ -00048d00: 312c 2d31 2c31 7d2c 7b31 2c30 2c2d 312c 1,-1,1},{1,0,-1, │ │ │ │ -00048d10: 2d31 7d7d 2c20 5229 7c0a 7c20 2020 2020 -1}}, R)|.| │ │ │ │ -00048d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048d50: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00048d60: 2020 2020 2020 2a20 3220 2020 2020 2020 * 2 │ │ │ │ -00048d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048d90: 2020 2020 7c0a 7c6f 3220 3d20 5220 3c2d |.|o2 = R <- │ │ │ │ -00048da0: 2028 5151 2029 2020 7669 6120 2020 2020 (QQ ) via │ │ │ │ -00048db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048dd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00048de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048e10: 7c0a 7c20 2020 2020 7c20 3020 3120 2d31 |.| | 0 1 -1 │ │ │ │ -00048e20: 2031 2020 7c20 2020 2020 2020 2020 2020 1 | │ │ │ │ -00048e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048e40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00048e50: 7c20 2020 2020 7c20 3120 3020 2d31 202d | | 1 0 -1 - │ │ │ │ -00048e60: 3120 7c20 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -00048e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048e80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00048e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048ec0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -00048ed0: 3a20 4469 6167 6f6e 616c 4163 7469 6f6e : DiagonalAction │ │ │ │ -00048ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048f00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00048f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048f40: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 7269 ------+.|i3 : ri │ │ │ │ -00048f50: 6e67 2054 203d 3d3d 2052 2020 2020 2020 ng T === R │ │ │ │ -00048f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048f80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00048f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048fc0: 2020 7c0a 7c6f 3320 3d20 7472 7565 2020 |.|o3 = true │ │ │ │ -00048fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049000: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00049010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00049040: 0a57 6179 7320 746f 2075 7365 2074 6869 .Ways to use thi │ │ │ │ -00049050: 7320 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d s method:.====== │ │ │ │ -00049060: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00049070: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7269 ==.. * *note ri │ │ │ │ -00049080: 6e67 2847 726f 7570 4163 7469 6f6e 293a ng(GroupAction): │ │ │ │ -00049090: 2072 696e 675f 6c70 4772 6f75 7041 6374 ring_lpGroupAct │ │ │ │ -000490a0: 696f 6e5f 7270 2c20 2d2d 2074 6865 2070 ion_rp, -- the p │ │ │ │ -000490b0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a20 olynomial ring. │ │ │ │ -000490c0: 2020 2062 6569 6e67 2061 6374 6564 2075 being acted u │ │ │ │ -000490d0: 706f 6e0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d pon.------------ │ │ │ │ -000490e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000490f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049120: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -00049130: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -00049140: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -00049150: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -00049160: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -00049170: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -00049180: 322f 7061 636b 6167 6573 2f0a 496e 7661 2/packages/.Inva │ │ │ │ -00049190: 7269 616e 7452 696e 672f 496e 7661 7269 riantRing/Invari │ │ │ │ -000491a0: 616e 7452 696e 6744 6f63 2e6d 323a 3139 antRingDoc.m2:19 │ │ │ │ -000491b0: 393a 302e 0a1f 0a46 696c 653a 2049 6e76 9:0....File: Inv │ │ │ │ -000491c0: 6172 6961 6e74 5269 6e67 2e69 6e66 6f2c ariantRing.info, │ │ │ │ -000491d0: 204e 6f64 653a 2052 696e 674f 6649 6e76 Node: RingOfInv │ │ │ │ -000491e0: 6172 6961 6e74 732c 204e 6578 743a 2073 ariants, Next: s │ │ │ │ -000491f0: 6368 7265 6965 7247 7261 7068 2c20 5072 chreierGraph, Pr │ │ │ │ -00049200: 6576 3a20 7269 6e67 5f6c 7047 726f 7570 ev: ring_lpGroup │ │ │ │ -00049210: 4163 7469 6f6e 5f72 702c 2055 703a 2054 Action_rp, Up: T │ │ │ │ -00049220: 6f70 0a0a 5269 6e67 4f66 496e 7661 7269 op..RingOfInvari │ │ │ │ -00049230: 616e 7473 202d 2d20 7468 6520 636c 6173 ants -- the clas │ │ │ │ -00049240: 7320 6f66 2074 6865 2072 696e 6773 206f s of the rings o │ │ │ │ -00049250: 6620 696e 7661 7269 616e 7473 2075 6e64 f invariants und │ │ │ │ -00049260: 6572 2074 6865 2061 6374 696f 6e20 6f66 er the action of │ │ │ │ -00049270: 2061 2066 696e 6974 6520 6772 6f75 702c a finite group, │ │ │ │ -00049280: 2061 6e20 4162 656c 6961 6e20 6772 6f75 an Abelian grou │ │ │ │ -00049290: 7020 6f72 2061 206c 696e 6561 726c 7920 p or a linearly │ │ │ │ -000492a0: 7265 6475 6374 6976 6520 6772 6f75 700a reductive group. │ │ │ │ -000492b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000492c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000492d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000492e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000492f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 ***********..Des │ │ │ │ -00049340: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00049350: 3d3d 3d3d 0a0a 5468 6973 2063 6c61 7373 ====..This class │ │ │ │ -00049360: 2069 7320 7072 6f76 6964 6564 2062 7920 is provided by │ │ │ │ -00049370: 7468 6520 7061 636b 6167 6520 2a6e 6f74 the package *not │ │ │ │ -00049380: 6520 496e 7661 7269 616e 7452 696e 673a e InvariantRing: │ │ │ │ -00049390: 2054 6f70 2c2e 0a0a 5269 6e67 4f66 496e Top,...RingOfIn │ │ │ │ -000493a0: 7661 7269 616e 7473 2069 7320 7468 6520 variants is the │ │ │ │ -000493b0: 636c 6173 7320 6f66 2072 696e 6773 206f class of rings o │ │ │ │ -000493c0: 6620 696e 7661 7269 616e 7473 2077 6865 f invariants whe │ │ │ │ -000493d0: 6e20 6120 6669 6e69 7465 2067 726f 7570 n a finite group │ │ │ │ -000493e0: 2c20 616e 0a41 6265 6c69 616e 2067 726f , an.Abelian gro │ │ │ │ -000493f0: 7570 206f 7220 6120 6c69 6e65 6172 6c79 up or a linearly │ │ │ │ -00049400: 2072 6564 7563 7469 7665 2067 726f 7570 reductive group │ │ │ │ -00049410: 2061 6374 7320 6f6e 2061 2070 6f6c 796e acts on a polyn │ │ │ │ -00049420: 6f6d 6961 6c20 7269 6e67 2e0a 0a46 756e omial ring...Fun │ │ │ │ -00049430: 6374 696f 6e73 2061 6e64 206d 6574 686f ctions and metho │ │ │ │ -00049440: 6473 2072 6574 7572 6e69 6e67 2061 6e20 ds returning an │ │ │ │ -00049450: 6f62 6a65 6374 206f 6620 636c 6173 7320 object of class │ │ │ │ -00049460: 5269 6e67 4f66 496e 7661 7269 616e 7473 RingOfInvariants │ │ │ │ -00049470: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00049480: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00049490: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000494a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000494b0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 6e76 ======.. * "inv │ │ │ │ -000494c0: 6172 6961 6e74 5269 6e67 2847 726f 7570 ariantRing(Group │ │ │ │ -000494d0: 4163 7469 6f6e 2922 202d 2d20 7365 6520 Action)" -- see │ │ │ │ -000494e0: 2a6e 6f74 6520 696e 7661 7269 616e 7452 *note invariantR │ │ │ │ -000494f0: 696e 673a 2069 6e76 6172 6961 6e74 5269 ing: invariantRi │ │ │ │ -00049500: 6e67 2c20 2d2d 0a20 2020 2074 6865 2072 ng, --. the r │ │ │ │ -00049510: 696e 6720 6f66 2069 6e76 6172 6961 6e74 ing of invariant │ │ │ │ -00049520: 7320 6f66 2061 2067 726f 7570 2061 6374 s of a group act │ │ │ │ -00049530: 696f 6e0a 2020 2a20 2250 6f6c 796e 6f6d ion. * "Polynom │ │ │ │ -00049540: 6961 6c52 696e 6720 5e20 4772 6f75 7041 ialRing ^ GroupA │ │ │ │ -00049550: 6374 696f 6e22 202d 2d20 7365 6520 2a6e ction" -- see *n │ │ │ │ -00049560: 6f74 6520 696e 7661 7269 616e 7452 696e ote invariantRin │ │ │ │ -00049570: 673a 2069 6e76 6172 6961 6e74 5269 6e67 g: invariantRing │ │ │ │ -00049580: 2c0a 2020 2020 2d2d 2074 6865 2072 696e ,. -- the rin │ │ │ │ -00049590: 6720 6f66 2069 6e76 6172 6961 6e74 7320 g of invariants │ │ │ │ -000495a0: 6f66 2061 2067 726f 7570 2061 6374 696f of a group actio │ │ │ │ -000495b0: 6e0a 2020 2a20 2251 756f 7469 656e 7452 n. * "QuotientR │ │ │ │ -000495c0: 696e 6720 5e20 4c69 6e65 6172 6c79 5265 ing ^ LinearlyRe │ │ │ │ -000495d0: 6475 6374 6976 6541 6374 696f 6e22 202d ductiveAction" - │ │ │ │ -000495e0: 2d20 7365 6520 2a6e 6f74 6520 696e 7661 - see *note inva │ │ │ │ -000495f0: 7269 616e 7452 696e 673a 0a20 2020 2069 riantRing:. i │ │ │ │ -00049600: 6e76 6172 6961 6e74 5269 6e67 2c20 2d2d nvariantRing, -- │ │ │ │ -00049610: 2074 6865 2072 696e 6720 6f66 2069 6e76 the ring of inv │ │ │ │ -00049620: 6172 6961 6e74 7320 6f66 2061 2067 726f ariants of a gro │ │ │ │ -00049630: 7570 2061 6374 696f 6e0a 0a4d 6574 686f up action..Metho │ │ │ │ -00049640: 6473 2074 6861 7420 7573 6520 616e 206f ds that use an o │ │ │ │ -00049650: 626a 6563 7420 6f66 2063 6c61 7373 2052 bject of class R │ │ │ │ -00049660: 696e 674f 6649 6e76 6172 6961 6e74 733a ingOfInvariants: │ │ │ │ -00049670: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00049680: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00049690: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000496a0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2261 6374 ======.. * "act │ │ │ │ -000496b0: 696f 6e28 5269 6e67 4f66 496e 7661 7269 ion(RingOfInvari │ │ │ │ -000496c0: 616e 7473 2922 202d 2d20 7365 6520 2a6e ants)" -- see *n │ │ │ │ -000496d0: 6f74 6520 6163 7469 6f6e 3a20 6163 7469 ote action: acti │ │ │ │ -000496e0: 6f6e 2c20 2d2d 2074 6865 2067 726f 7570 on, -- the group │ │ │ │ -000496f0: 2061 6374 696f 6e0a 2020 2020 7468 6174 action. that │ │ │ │ -00049700: 2070 726f 6475 6365 6420 6120 7269 6e67 produced a ring │ │ │ │ -00049710: 206f 6620 696e 7661 7269 616e 7473 0a20 of invariants. │ │ │ │ -00049720: 202a 202a 6e6f 7465 2061 6d62 6965 6e74 * *note ambient │ │ │ │ -00049730: 2852 696e 674f 6649 6e76 6172 6961 6e74 (RingOfInvariant │ │ │ │ -00049740: 7329 3a20 616d 6269 656e 745f 6c70 5269 s): ambient_lpRi │ │ │ │ -00049750: 6e67 4f66 496e 7661 7269 616e 7473 5f72 ngOfInvariants_r │ │ │ │ -00049760: 702c 202d 2d20 7468 650a 2020 2020 616d p, -- the. am │ │ │ │ -00049770: 6269 656e 7420 706f 6c79 6e6f 6d69 616c bient polynomial │ │ │ │ -00049780: 2072 696e 6720 7768 6572 6520 7468 6520 ring where the │ │ │ │ -00049790: 6772 6f75 7020 6163 7465 6420 7570 6f6e group acted upon │ │ │ │ -000497a0: 0a20 202a 2022 6465 6669 6e69 6e67 4964 . * "definingId │ │ │ │ -000497b0: 6561 6c28 5269 6e67 4f66 496e 7661 7269 eal(RingOfInvari │ │ │ │ -000497c0: 616e 7473 2922 202d 2d20 7365 6520 2a6e ants)" -- see *n │ │ │ │ -000497d0: 6f74 6520 6465 6669 6e69 6e67 4964 6561 ote definingIdea │ │ │ │ -000497e0: 6c3a 0a20 2020 2064 6566 696e 696e 6749 l:. definingI │ │ │ │ -000497f0: 6465 616c 2c20 2d2d 2070 7265 7365 6e74 deal, -- present │ │ │ │ -00049800: 6174 696f 6e20 6f66 2061 2072 696e 6720 ation of a ring │ │ │ │ -00049810: 6f66 2069 6e76 6172 6961 6e74 7320 6173 of invariants as │ │ │ │ -00049820: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ -00049830: 0a20 2020 206d 6f64 756c 6f20 7468 6520 . modulo the │ │ │ │ -00049840: 6465 6669 6e69 6e67 2069 6465 616c 0a20 defining ideal. │ │ │ │ -00049850: 202a 202a 6e6f 7465 2067 656e 6572 6174 * *note generat │ │ │ │ -00049860: 6f72 7328 5269 6e67 4f66 496e 7661 7269 ors(RingOfInvari │ │ │ │ -00049870: 616e 7473 293a 2067 656e 6572 6174 6f72 ants): generator │ │ │ │ -00049880: 735f 6c70 5269 6e67 4f66 496e 7661 7269 s_lpRingOfInvari │ │ │ │ -00049890: 616e 7473 5f72 702c 202d 2d0a 2020 2020 ants_rp, --. │ │ │ │ -000498a0: 7468 6520 6765 6e65 7261 746f 7273 2066 the generators f │ │ │ │ -000498b0: 6f72 2061 2072 696e 6720 6f66 2069 6e76 or a ring of inv │ │ │ │ -000498c0: 6172 6961 6e74 730a 2020 2a20 2a6e 6f74 ariants. * *not │ │ │ │ -000498d0: 6520 6869 6c62 6572 7453 6572 6965 7328 e hilbertSeries( │ │ │ │ -000498e0: 5269 6e67 4f66 496e 7661 7269 616e 7473 RingOfInvariants │ │ │ │ -000498f0: 293a 2068 696c 6265 7274 5365 7269 6573 ): hilbertSeries │ │ │ │ -00049900: 5f6c 7052 696e 674f 6649 6e76 6172 6961 _lpRingOfInvaria │ │ │ │ -00049910: 6e74 735f 7270 2c0a 2020 2020 2d2d 2048 nts_rp,. -- H │ │ │ │ -00049920: 696c 6265 7274 2073 6572 6965 7320 6f66 ilbert series of │ │ │ │ -00049930: 2074 6865 2069 6e76 6172 6961 6e74 2072 the invariant r │ │ │ │ -00049940: 696e 670a 2020 2a20 2a6e 6f74 6520 6e65 ing. * *note ne │ │ │ │ -00049950: 7428 5269 6e67 4f66 496e 7661 7269 616e t(RingOfInvarian │ │ │ │ -00049960: 7473 293a 206e 6574 5f6c 7052 696e 674f ts): net_lpRingO │ │ │ │ -00049970: 6649 6e76 6172 6961 6e74 735f 7270 2c20 fInvariants_rp, │ │ │ │ -00049980: 2d2d 2066 6f72 6d61 7420 666f 720a 2020 -- format for. │ │ │ │ -00049990: 2020 7072 696e 7469 6e67 2c20 6173 2061 printing, as a │ │ │ │ -000499a0: 206e 6574 0a20 202a 202a 6e6f 7465 2074 net. * *note t │ │ │ │ -000499b0: 6578 4d61 7468 2852 696e 674f 6649 6e76 exMath(RingOfInv │ │ │ │ -000499c0: 6172 6961 6e74 7329 3a20 7465 784d 6174 ariants): texMat │ │ │ │ -000499d0: 685f 6c70 5269 6e67 4f66 496e 7661 7269 h_lpRingOfInvari │ │ │ │ -000499e0: 616e 7473 5f72 702c 202d 2d20 666f 726d ants_rp, -- form │ │ │ │ -000499f0: 6174 0a20 2020 2066 6f72 2070 7269 6e74 at. for print │ │ │ │ -00049a00: 696e 672c 2061 7320 5465 5820 7374 7269 ing, as TeX stri │ │ │ │ -00049a10: 6e67 730a 0a46 6f72 2074 6865 2070 726f ngs..For the pro │ │ │ │ -00049a20: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -00049a30: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -00049a40: 6f62 6a65 6374 202a 6e6f 7465 2052 696e object *note Rin │ │ │ │ -00049a50: 674f 6649 6e76 6172 6961 6e74 733a 2052 gOfInvariants: R │ │ │ │ -00049a60: 696e 674f 6649 6e76 6172 6961 6e74 732c ingOfInvariants, │ │ │ │ -00049a70: 2069 7320 6120 2a6e 6f74 6520 7479 7065 is a *note type │ │ │ │ -00049a80: 3a0a 284d 6163 6175 6c61 7932 446f 6329 :.(Macaulay2Doc) │ │ │ │ -00049a90: 5479 7065 2c2c 2077 6974 6820 616e 6365 Type,, with ance │ │ │ │ -00049aa0: 7374 6f72 2063 6c61 7373 6573 202a 6e6f stor classes *no │ │ │ │ -00049ab0: 7465 2048 6173 6854 6162 6c65 3a0a 284d te HashTable:.(M │ │ │ │ -00049ac0: 6163 6175 6c61 7932 446f 6329 4861 7368 acaulay2Doc)Hash │ │ │ │ -00049ad0: 5461 626c 652c 203c 202a 6e6f 7465 2054 Table, < *note T │ │ │ │ -00049ae0: 6869 6e67 3a20 284d 6163 6175 6c61 7932 hing: (Macaulay2 │ │ │ │ -00049af0: 446f 6329 5468 696e 672c 2e0a 0a2d 2d2d Doc)Thing,...--- │ │ │ │ -00049b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00049b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00049b50: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00049b60: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00049b70: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00049b80: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00049b90: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00049ba0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00049bb0: 6765 732f 0a49 6e76 6172 6961 6e74 5269 ges/.InvariantRi │ │ │ │ -00049bc0: 6e67 2f49 6e76 6172 6961 6e74 7344 6f63 ng/InvariantsDoc │ │ │ │ -00049bd0: 2e6d 323a 3936 313a 302e 0a1f 0a46 696c .m2:961:0....Fil │ │ │ │ -00049be0: 653a 2049 6e76 6172 6961 6e74 5269 6e67 e: InvariantRing │ │ │ │ -00049bf0: 2e69 6e66 6f2c 204e 6f64 653a 2073 6368 .info, Node: sch │ │ │ │ -00049c00: 7265 6965 7247 7261 7068 2c20 4e65 7874 reierGraph, Next │ │ │ │ -00049c10: 3a20 7365 636f 6e64 6172 7949 6e76 6172 : secondaryInvar │ │ │ │ -00049c20: 6961 6e74 732c 2050 7265 763a 2052 696e iants, Prev: Rin │ │ │ │ -00049c30: 674f 6649 6e76 6172 6961 6e74 732c 2055 gOfInvariants, U │ │ │ │ -00049c40: 703a 2054 6f70 0a0a 7363 6872 6569 6572 p: Top..schreier │ │ │ │ -00049c50: 4772 6170 6820 2d2d 2053 6368 7265 6965 Graph -- Schreie │ │ │ │ -00049c60: 7220 6772 6170 6820 6f66 2061 2066 696e r graph of a fin │ │ │ │ -00049c70: 6974 6520 6772 6f75 700a 2a2a 2a2a 2a2a ite group.****** │ │ │ │ -00049c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -00049cb0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -00049cc0: 2073 6368 7265 6965 7247 7261 7068 2047 schreierGraph G │ │ │ │ -00049cd0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -00049ce0: 2020 202a 2047 2c20 616e 2069 6e73 7461 * G, an insta │ │ │ │ -00049cf0: 6e63 6520 6f66 2074 6865 2074 7970 6520 nce of the type │ │ │ │ -00049d00: 2a6e 6f74 6520 4669 6e69 7465 4772 6f75 *note FiniteGrou │ │ │ │ -00049d10: 7041 6374 696f 6e3a 2046 696e 6974 6547 pAction: FiniteG │ │ │ │ -00049d20: 726f 7570 4163 7469 6f6e 2c2c 0a20 2020 roupAction,,. │ │ │ │ -00049d30: 2020 2020 2061 2066 696e 6974 6520 6772 a finite gr │ │ │ │ -00049d40: 6f75 7020 6163 7469 6f6e 0a20 202a 204f oup action. * O │ │ │ │ -00049d50: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -00049d60: 6120 2a6e 6f74 6520 6861 7368 2074 6162 a *note hash tab │ │ │ │ -00049d70: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -00049d80: 6329 4861 7368 5461 626c 652c 2c20 7265 c)HashTable,, re │ │ │ │ -00049d90: 7072 6573 656e 7469 6e67 2074 6865 2053 presenting the S │ │ │ │ -00049da0: 6368 7265 6965 720a 2020 2020 2020 2020 chreier. │ │ │ │ -00049db0: 6772 6170 6820 6f66 2074 6865 2067 726f graph of the gro │ │ │ │ -00049dc0: 7570 0a0a 4465 7363 7269 7074 696f 6e0a up..Description. │ │ │ │ -00049dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -00049de0: 7320 6675 6e63 7469 6f6e 2069 7320 7072 s function is pr │ │ │ │ -00049df0: 6f76 6964 6564 2062 7920 7468 6520 7061 ovided by the pa │ │ │ │ -00049e00: 636b 6167 6520 2a6e 6f74 6520 496e 7661 ckage *note Inva │ │ │ │ -00049e10: 7269 616e 7452 696e 673a 2054 6f70 2c2e riantRing: Top,. │ │ │ │ -00049e20: 0a0a 466f 7220 6120 6669 6e69 7465 2067 ..For a finite g │ │ │ │ -00049e30: 726f 7570 2061 6374 696f 6e2c 2077 6520 roup action, we │ │ │ │ -00049e40: 666f 726d 2061 202a 6e6f 7465 2048 6173 form a *note Has │ │ │ │ -00049e50: 6854 6162 6c65 3a20 284d 6163 6175 6c61 hTable: (Macaula │ │ │ │ -00049e60: 7932 446f 6329 4861 7368 5461 626c 652c y2Doc)HashTable, │ │ │ │ -00049e70: 0a77 686f 7365 206b 6579 7320 6172 6520 .whose keys are │ │ │ │ -00049e80: 7468 6520 6765 6e65 7261 746f 7273 2070 the generators p │ │ │ │ -00049e90: 726f 7669 6465 6420 6279 2074 6865 2075 rovided by the u │ │ │ │ -00049ea0: 7365 722e 2054 6865 2076 616c 7565 2063 ser. The value c │ │ │ │ -00049eb0: 6f72 7265 7370 6f6e 6469 6e67 2074 6f0a orresponding to. │ │ │ │ -00049ec0: 6120 6765 6e65 7261 746f 7220 6720 6973 a generator g is │ │ │ │ -00049ed0: 2061 202a 6e6f 7465 2048 6173 6854 6162 a *note HashTab │ │ │ │ -00049ee0: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -00049ef0: 6329 4861 7368 5461 626c 652c 2063 6f6e c)HashTable, con │ │ │ │ -00049f00: 7461 696e 696e 6720 616c 6c0a 7061 6972 taining all.pair │ │ │ │ -00049f10: 7320 6120 3d3e 2062 2073 7563 6820 7468 s a => b such th │ │ │ │ -00049f20: 6174 2061 2a67 203d 3d20 622e 2054 6869 at a*g == b. Thi │ │ │ │ -00049f30: 7320 7265 7072 6573 656e 7473 2074 6865 s represents the │ │ │ │ -00049f40: 2053 6368 7265 6965 7220 6772 6170 6820 Schreier graph │ │ │ │ -00049f50: 6f66 2074 6865 0a67 726f 7570 2072 656c of the.group rel │ │ │ │ -00049f60: 6174 6976 6520 746f 2074 6865 2067 656e ative to the gen │ │ │ │ -00049f70: 6572 6174 696e 6720 7365 7420 7072 6f76 erating set prov │ │ │ │ -00049f80: 6964 6564 2062 7920 7468 6520 7573 6572 ided by the user │ │ │ │ -00049f90: 2e0a 0a54 6865 2066 6f6c 6c6f 7769 6e67 ...The following │ │ │ │ -00049fa0: 2065 7861 6d70 6c65 2064 6566 696e 6573 example defines │ │ │ │ -00049fb0: 2074 6865 2070 6572 6d75 7461 7469 6f6e the permutation │ │ │ │ -00049fc0: 2061 6374 696f 6e20 6f66 2061 2073 796d action of a sym │ │ │ │ -00049fd0: 6d65 7472 6963 2067 726f 7570 206f 6e0a metric group on. │ │ │ │ -00049fe0: 7468 7265 6520 656c 656d 656e 7473 2075 three elements u │ │ │ │ -00049ff0: 7369 6e67 206f 6e6c 7920 7477 6f20 6765 sing only two ge │ │ │ │ -0004a000: 6e65 7261 746f 7273 2c20 6120 7472 616e nerators, a tran │ │ │ │ -0004a010: 7370 6f73 6974 696f 6e20 616e 6420 6120 sposition and a │ │ │ │ -0004a020: 332d 6379 636c 652e 0a0a 2b2d 2d2d 2d2d 3-cycle...+----- │ │ │ │ -0004a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a070: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -0004a080: 5220 3d20 5151 5b78 5f31 2e2e 785f 335d R = QQ[x_1..x_3] │ │ │ │ +00048c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048c30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00048c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048c70: 2b0a 7c69 3320 3a20 7269 6e67 2054 203d +.|i3 : ring T = │ │ │ │ +00048c80: 3d3d 2052 2020 2020 2020 2020 2020 2020 == R │ │ │ │ +00048c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048ca0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00048cb0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00048cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048ce0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00048cf0: 3320 3d20 7472 7565 2020 2020 2020 2020 3 = true │ │ │ │ +00048d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00048d20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00048d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048d60: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 --------+..Ways │ │ │ │ +00048d70: 746f 2075 7365 2074 6869 7320 6d65 7468 to use this meth │ │ │ │ +00048d80: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d od:.============ │ │ │ │ +00048d90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +00048da0: 2a20 2a6e 6f74 6520 7269 6e67 2847 726f * *note ring(Gro │ │ │ │ +00048db0: 7570 4163 7469 6f6e 293a 2072 696e 675f upAction): ring_ │ │ │ │ +00048dc0: 6c70 4772 6f75 7041 6374 696f 6e5f 7270 lpGroupAction_rp │ │ │ │ +00048dd0: 2c20 2d2d 2074 6865 2070 6f6c 796e 6f6d , -- the polynom │ │ │ │ +00048de0: 6961 6c20 7269 6e67 0a20 2020 2062 6569 ial ring. bei │ │ │ │ +00048df0: 6e67 2061 6374 6564 2075 706f 6e0a 2d2d ng acted upon.-- │ │ │ │ +00048e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +00048e50: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +00048e60: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +00048e70: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +00048e80: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +00048e90: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ +00048ea0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +00048eb0: 6167 6573 2f0a 496e 7661 7269 616e 7452 ages/.InvariantR │ │ │ │ +00048ec0: 696e 672f 496e 7661 7269 616e 7452 696e ing/InvariantRin │ │ │ │ +00048ed0: 6744 6f63 2e6d 323a 3139 393a 302e 0a1f gDoc.m2:199:0... │ │ │ │ +00048ee0: 0a46 696c 653a 2049 6e76 6172 6961 6e74 .File: Invariant │ │ │ │ +00048ef0: 5269 6e67 2e69 6e66 6f2c 204e 6f64 653a Ring.info, Node: │ │ │ │ +00048f00: 2052 696e 674f 6649 6e76 6172 6961 6e74 RingOfInvariant │ │ │ │ +00048f10: 732c 204e 6578 743a 2073 6368 7265 6965 s, Next: schreie │ │ │ │ +00048f20: 7247 7261 7068 2c20 5072 6576 3a20 7269 rGraph, Prev: ri │ │ │ │ +00048f30: 6e67 5f6c 7047 726f 7570 4163 7469 6f6e ng_lpGroupAction │ │ │ │ +00048f40: 5f72 702c 2055 703a 2054 6f70 0a0a 5269 _rp, Up: Top..Ri │ │ │ │ +00048f50: 6e67 4f66 496e 7661 7269 616e 7473 202d ngOfInvariants - │ │ │ │ +00048f60: 2d20 7468 6520 636c 6173 7320 6f66 2074 - the class of t │ │ │ │ +00048f70: 6865 2072 696e 6773 206f 6620 696e 7661 he rings of inva │ │ │ │ +00048f80: 7269 616e 7473 2075 6e64 6572 2074 6865 riants under the │ │ │ │ +00048f90: 2061 6374 696f 6e20 6f66 2061 2066 696e action of a fin │ │ │ │ +00048fa0: 6974 6520 6772 6f75 702c 2061 6e20 4162 ite group, an Ab │ │ │ │ +00048fb0: 656c 6961 6e20 6772 6f75 7020 6f72 2061 elian group or a │ │ │ │ +00048fc0: 206c 696e 6561 726c 7920 7265 6475 6374 linearly reduct │ │ │ │ +00048fd0: 6976 6520 6772 6f75 700a 2a2a 2a2a 2a2a ive group.****** │ │ │ │ +00048fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00048ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00049000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00049010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00049020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00049030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00049040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00049050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00049060: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ +00049070: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +00049080: 5468 6973 2063 6c61 7373 2069 7320 7072 This class is pr │ │ │ │ +00049090: 6f76 6964 6564 2062 7920 7468 6520 7061 ovided by the pa │ │ │ │ +000490a0: 636b 6167 6520 2a6e 6f74 6520 496e 7661 ckage *note Inva │ │ │ │ +000490b0: 7269 616e 7452 696e 673a 2054 6f70 2c2e riantRing: Top,. │ │ │ │ +000490c0: 0a0a 5269 6e67 4f66 496e 7661 7269 616e ..RingOfInvarian │ │ │ │ +000490d0: 7473 2069 7320 7468 6520 636c 6173 7320 ts is the class │ │ │ │ +000490e0: 6f66 2072 696e 6773 206f 6620 696e 7661 of rings of inva │ │ │ │ +000490f0: 7269 616e 7473 2077 6865 6e20 6120 6669 riants when a fi │ │ │ │ +00049100: 6e69 7465 2067 726f 7570 2c20 616e 0a41 nite group, an.A │ │ │ │ +00049110: 6265 6c69 616e 2067 726f 7570 206f 7220 belian group or │ │ │ │ +00049120: 6120 6c69 6e65 6172 6c79 2072 6564 7563 a linearly reduc │ │ │ │ +00049130: 7469 7665 2067 726f 7570 2061 6374 7320 tive group acts │ │ │ │ +00049140: 6f6e 2061 2070 6f6c 796e 6f6d 6961 6c20 on a polynomial │ │ │ │ +00049150: 7269 6e67 2e0a 0a46 756e 6374 696f 6e73 ring...Functions │ │ │ │ +00049160: 2061 6e64 206d 6574 686f 6473 2072 6574 and methods ret │ │ │ │ +00049170: 7572 6e69 6e67 2061 6e20 6f62 6a65 6374 urning an object │ │ │ │ +00049180: 206f 6620 636c 6173 7320 5269 6e67 4f66 of class RingOf │ │ │ │ +00049190: 496e 7661 7269 616e 7473 3a0a 3d3d 3d3d Invariants:.==== │ │ │ │ +000491a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000491b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000491c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000491d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000491e0: 0a0a 2020 2a20 2269 6e76 6172 6961 6e74 .. * "invariant │ │ │ │ +000491f0: 5269 6e67 2847 726f 7570 4163 7469 6f6e Ring(GroupAction │ │ │ │ +00049200: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ +00049210: 696e 7661 7269 616e 7452 696e 673a 2069 invariantRing: i │ │ │ │ +00049220: 6e76 6172 6961 6e74 5269 6e67 2c20 2d2d nvariantRing, -- │ │ │ │ +00049230: 0a20 2020 2074 6865 2072 696e 6720 6f66 . the ring of │ │ │ │ +00049240: 2069 6e76 6172 6961 6e74 7320 6f66 2061 invariants of a │ │ │ │ +00049250: 2067 726f 7570 2061 6374 696f 6e0a 2020 group action. │ │ │ │ +00049260: 2a20 2250 6f6c 796e 6f6d 6961 6c52 696e * "PolynomialRin │ │ │ │ +00049270: 6720 5e20 4772 6f75 7041 6374 696f 6e22 g ^ GroupAction" │ │ │ │ +00049280: 202d 2d20 7365 6520 2a6e 6f74 6520 696e -- see *note in │ │ │ │ +00049290: 7661 7269 616e 7452 696e 673a 2069 6e76 variantRing: inv │ │ │ │ +000492a0: 6172 6961 6e74 5269 6e67 2c0a 2020 2020 ariantRing,. │ │ │ │ +000492b0: 2d2d 2074 6865 2072 696e 6720 6f66 2069 -- the ring of i │ │ │ │ +000492c0: 6e76 6172 6961 6e74 7320 6f66 2061 2067 nvariants of a g │ │ │ │ +000492d0: 726f 7570 2061 6374 696f 6e0a 2020 2a20 roup action. * │ │ │ │ +000492e0: 2251 756f 7469 656e 7452 696e 6720 5e20 "QuotientRing ^ │ │ │ │ +000492f0: 4c69 6e65 6172 6c79 5265 6475 6374 6976 LinearlyReductiv │ │ │ │ +00049300: 6541 6374 696f 6e22 202d 2d20 7365 6520 eAction" -- see │ │ │ │ +00049310: 2a6e 6f74 6520 696e 7661 7269 616e 7452 *note invariantR │ │ │ │ +00049320: 696e 673a 0a20 2020 2069 6e76 6172 6961 ing:. invaria │ │ │ │ +00049330: 6e74 5269 6e67 2c20 2d2d 2074 6865 2072 ntRing, -- the r │ │ │ │ +00049340: 696e 6720 6f66 2069 6e76 6172 6961 6e74 ing of invariant │ │ │ │ +00049350: 7320 6f66 2061 2067 726f 7570 2061 6374 s of a group act │ │ │ │ +00049360: 696f 6e0a 0a4d 6574 686f 6473 2074 6861 ion..Methods tha │ │ │ │ +00049370: 7420 7573 6520 616e 206f 626a 6563 7420 t use an object │ │ │ │ +00049380: 6f66 2063 6c61 7373 2052 696e 674f 6649 of class RingOfI │ │ │ │ +00049390: 6e76 6172 6961 6e74 733a 0a3d 3d3d 3d3d nvariants:.===== │ │ │ │ +000493a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000493b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000493c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000493d0: 0a0a 2020 2a20 2261 6374 696f 6e28 5269 .. * "action(Ri │ │ │ │ +000493e0: 6e67 4f66 496e 7661 7269 616e 7473 2922 ngOfInvariants)" │ │ │ │ +000493f0: 202d 2d20 7365 6520 2a6e 6f74 6520 6163 -- see *note ac │ │ │ │ +00049400: 7469 6f6e 3a20 6163 7469 6f6e 2c20 2d2d tion: action, -- │ │ │ │ +00049410: 2074 6865 2067 726f 7570 2061 6374 696f the group actio │ │ │ │ +00049420: 6e0a 2020 2020 7468 6174 2070 726f 6475 n. that produ │ │ │ │ +00049430: 6365 6420 6120 7269 6e67 206f 6620 696e ced a ring of in │ │ │ │ +00049440: 7661 7269 616e 7473 0a20 202a 202a 6e6f variants. * *no │ │ │ │ +00049450: 7465 2061 6d62 6965 6e74 2852 696e 674f te ambient(RingO │ │ │ │ +00049460: 6649 6e76 6172 6961 6e74 7329 3a20 616d fInvariants): am │ │ │ │ +00049470: 6269 656e 745f 6c70 5269 6e67 4f66 496e bient_lpRingOfIn │ │ │ │ +00049480: 7661 7269 616e 7473 5f72 702c 202d 2d20 variants_rp, -- │ │ │ │ +00049490: 7468 650a 2020 2020 616d 6269 656e 7420 the. ambient │ │ │ │ +000494a0: 706f 6c79 6e6f 6d69 616c 2072 696e 6720 polynomial ring │ │ │ │ +000494b0: 7768 6572 6520 7468 6520 6772 6f75 7020 where the group │ │ │ │ +000494c0: 6163 7465 6420 7570 6f6e 0a20 202a 2022 acted upon. * " │ │ │ │ +000494d0: 6465 6669 6e69 6e67 4964 6561 6c28 5269 definingIdeal(Ri │ │ │ │ +000494e0: 6e67 4f66 496e 7661 7269 616e 7473 2922 ngOfInvariants)" │ │ │ │ +000494f0: 202d 2d20 7365 6520 2a6e 6f74 6520 6465 -- see *note de │ │ │ │ +00049500: 6669 6e69 6e67 4964 6561 6c3a 0a20 2020 finingIdeal:. │ │ │ │ +00049510: 2064 6566 696e 696e 6749 6465 616c 2c20 definingIdeal, │ │ │ │ +00049520: 2d2d 2070 7265 7365 6e74 6174 696f 6e20 -- presentation │ │ │ │ +00049530: 6f66 2061 2072 696e 6720 6f66 2069 6e76 of a ring of inv │ │ │ │ +00049540: 6172 6961 6e74 7320 6173 2070 6f6c 796e ariants as polyn │ │ │ │ +00049550: 6f6d 6961 6c20 7269 6e67 0a20 2020 206d omial ring. m │ │ │ │ +00049560: 6f64 756c 6f20 7468 6520 6465 6669 6e69 odulo the defini │ │ │ │ +00049570: 6e67 2069 6465 616c 0a20 202a 202a 6e6f ng ideal. * *no │ │ │ │ +00049580: 7465 2067 656e 6572 6174 6f72 7328 5269 te generators(Ri │ │ │ │ +00049590: 6e67 4f66 496e 7661 7269 616e 7473 293a ngOfInvariants): │ │ │ │ +000495a0: 2067 656e 6572 6174 6f72 735f 6c70 5269 generators_lpRi │ │ │ │ +000495b0: 6e67 4f66 496e 7661 7269 616e 7473 5f72 ngOfInvariants_r │ │ │ │ +000495c0: 702c 202d 2d0a 2020 2020 7468 6520 6765 p, --. the ge │ │ │ │ +000495d0: 6e65 7261 746f 7273 2066 6f72 2061 2072 nerators for a r │ │ │ │ +000495e0: 696e 6720 6f66 2069 6e76 6172 6961 6e74 ing of invariant │ │ │ │ +000495f0: 730a 2020 2a20 2a6e 6f74 6520 6869 6c62 s. * *note hilb │ │ │ │ +00049600: 6572 7453 6572 6965 7328 5269 6e67 4f66 ertSeries(RingOf │ │ │ │ +00049610: 496e 7661 7269 616e 7473 293a 2068 696c Invariants): hil │ │ │ │ +00049620: 6265 7274 5365 7269 6573 5f6c 7052 696e bertSeries_lpRin │ │ │ │ +00049630: 674f 6649 6e76 6172 6961 6e74 735f 7270 gOfInvariants_rp │ │ │ │ +00049640: 2c0a 2020 2020 2d2d 2048 696c 6265 7274 ,. -- Hilbert │ │ │ │ +00049650: 2073 6572 6965 7320 6f66 2074 6865 2069 series of the i │ │ │ │ +00049660: 6e76 6172 6961 6e74 2072 696e 670a 2020 nvariant ring. │ │ │ │ +00049670: 2a20 2a6e 6f74 6520 6e65 7428 5269 6e67 * *note net(Ring │ │ │ │ +00049680: 4f66 496e 7661 7269 616e 7473 293a 206e OfInvariants): n │ │ │ │ +00049690: 6574 5f6c 7052 696e 674f 6649 6e76 6172 et_lpRingOfInvar │ │ │ │ +000496a0: 6961 6e74 735f 7270 2c20 2d2d 2066 6f72 iants_rp, -- for │ │ │ │ +000496b0: 6d61 7420 666f 720a 2020 2020 7072 696e mat for. prin │ │ │ │ +000496c0: 7469 6e67 2c20 6173 2061 206e 6574 0a20 ting, as a net. │ │ │ │ +000496d0: 202a 202a 6e6f 7465 2074 6578 4d61 7468 * *note texMath │ │ │ │ +000496e0: 2852 696e 674f 6649 6e76 6172 6961 6e74 (RingOfInvariant │ │ │ │ +000496f0: 7329 3a20 7465 784d 6174 685f 6c70 5269 s): texMath_lpRi │ │ │ │ +00049700: 6e67 4f66 496e 7661 7269 616e 7473 5f72 ngOfInvariants_r │ │ │ │ +00049710: 702c 202d 2d20 666f 726d 6174 0a20 2020 p, -- format. │ │ │ │ +00049720: 2066 6f72 2070 7269 6e74 696e 672c 2061 for printing, a │ │ │ │ +00049730: 7320 5465 5820 7374 7269 6e67 730a 0a46 s TeX strings..F │ │ │ │ +00049740: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +00049750: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00049760: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00049770: 202a 6e6f 7465 2052 696e 674f 6649 6e76 *note RingOfInv │ │ │ │ +00049780: 6172 6961 6e74 733a 2052 696e 674f 6649 ariants: RingOfI │ │ │ │ +00049790: 6e76 6172 6961 6e74 732c 2069 7320 6120 nvariants, is a │ │ │ │ +000497a0: 2a6e 6f74 6520 7479 7065 3a0a 284d 6163 *note type:.(Mac │ │ │ │ +000497b0: 6175 6c61 7932 446f 6329 5479 7065 2c2c aulay2Doc)Type,, │ │ │ │ +000497c0: 2077 6974 6820 616e 6365 7374 6f72 2063 with ancestor c │ │ │ │ +000497d0: 6c61 7373 6573 202a 6e6f 7465 2048 6173 lasses *note Has │ │ │ │ +000497e0: 6854 6162 6c65 3a0a 284d 6163 6175 6c61 hTable:.(Macaula │ │ │ │ +000497f0: 7932 446f 6329 4861 7368 5461 626c 652c y2Doc)HashTable, │ │ │ │ +00049800: 203c 202a 6e6f 7465 2054 6869 6e67 3a20 < *note Thing: │ │ │ │ +00049810: 284d 6163 6175 6c61 7932 446f 6329 5468 (Macaulay2Doc)Th │ │ │ │ +00049820: 696e 672c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ing,...--------- │ │ │ │ +00049830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049870: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +00049880: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +00049890: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +000498a0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +000498b0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +000498c0: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +000498d0: 6c61 7932 2f70 6163 6b61 6765 732f 0a49 lay2/packages/.I │ │ │ │ +000498e0: 6e76 6172 6961 6e74 5269 6e67 2f49 6e76 nvariantRing/Inv │ │ │ │ +000498f0: 6172 6961 6e74 7344 6f63 2e6d 323a 3936 ariantsDoc.m2:96 │ │ │ │ +00049900: 313a 302e 0a1f 0a46 696c 653a 2049 6e76 1:0....File: Inv │ │ │ │ +00049910: 6172 6961 6e74 5269 6e67 2e69 6e66 6f2c ariantRing.info, │ │ │ │ +00049920: 204e 6f64 653a 2073 6368 7265 6965 7247 Node: schreierG │ │ │ │ +00049930: 7261 7068 2c20 4e65 7874 3a20 7365 636f raph, Next: seco │ │ │ │ +00049940: 6e64 6172 7949 6e76 6172 6961 6e74 732c ndaryInvariants, │ │ │ │ +00049950: 2050 7265 763a 2052 696e 674f 6649 6e76 Prev: RingOfInv │ │ │ │ +00049960: 6172 6961 6e74 732c 2055 703a 2054 6f70 ariants, Up: Top │ │ │ │ +00049970: 0a0a 7363 6872 6569 6572 4772 6170 6820 ..schreierGraph │ │ │ │ +00049980: 2d2d 2053 6368 7265 6965 7220 6772 6170 -- Schreier grap │ │ │ │ +00049990: 6820 6f66 2061 2066 696e 6974 6520 6772 h of a finite gr │ │ │ │ +000499a0: 6f75 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a oup.************ │ │ │ │ +000499b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000499c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +000499d0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +000499e0: 3a20 0a20 2020 2020 2020 2073 6368 7265 : . schre │ │ │ │ +000499f0: 6965 7247 7261 7068 2047 0a20 202a 2049 ierGraph G. * I │ │ │ │ +00049a00: 6e70 7574 733a 0a20 2020 2020 202a 2047 nputs:. * G │ │ │ │ +00049a10: 2c20 616e 2069 6e73 7461 6e63 6520 6f66 , an instance of │ │ │ │ +00049a20: 2074 6865 2074 7970 6520 2a6e 6f74 6520 the type *note │ │ │ │ +00049a30: 4669 6e69 7465 4772 6f75 7041 6374 696f FiniteGroupActio │ │ │ │ +00049a40: 6e3a 2046 696e 6974 6547 726f 7570 4163 n: FiniteGroupAc │ │ │ │ +00049a50: 7469 6f6e 2c2c 0a20 2020 2020 2020 2061 tion,,. a │ │ │ │ +00049a60: 2066 696e 6974 6520 6772 6f75 7020 6163 finite group ac │ │ │ │ +00049a70: 7469 6f6e 0a20 202a 204f 7574 7075 7473 tion. * Outputs │ │ │ │ +00049a80: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ +00049a90: 6520 6861 7368 2074 6162 6c65 3a20 284d e hash table: (M │ │ │ │ +00049aa0: 6163 6175 6c61 7932 446f 6329 4861 7368 acaulay2Doc)Hash │ │ │ │ +00049ab0: 5461 626c 652c 2c20 7265 7072 6573 656e Table,, represen │ │ │ │ +00049ac0: 7469 6e67 2074 6865 2053 6368 7265 6965 ting the Schreie │ │ │ │ +00049ad0: 720a 2020 2020 2020 2020 6772 6170 6820 r. graph │ │ │ │ +00049ae0: 6f66 2074 6865 2067 726f 7570 0a0a 4465 of the group..De │ │ │ │ +00049af0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00049b00: 3d3d 3d3d 3d0a 0a54 6869 7320 6675 6e63 =====..This func │ │ │ │ +00049b10: 7469 6f6e 2069 7320 7072 6f76 6964 6564 tion is provided │ │ │ │ +00049b20: 2062 7920 7468 6520 7061 636b 6167 6520 by the package │ │ │ │ +00049b30: 2a6e 6f74 6520 496e 7661 7269 616e 7452 *note InvariantR │ │ │ │ +00049b40: 696e 673a 2054 6f70 2c2e 0a0a 466f 7220 ing: Top,...For │ │ │ │ +00049b50: 6120 6669 6e69 7465 2067 726f 7570 2061 a finite group a │ │ │ │ +00049b60: 6374 696f 6e2c 2077 6520 666f 726d 2061 ction, we form a │ │ │ │ +00049b70: 202a 6e6f 7465 2048 6173 6854 6162 6c65 *note HashTable │ │ │ │ +00049b80: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00049b90: 4861 7368 5461 626c 652c 0a77 686f 7365 HashTable,.whose │ │ │ │ +00049ba0: 206b 6579 7320 6172 6520 7468 6520 6765 keys are the ge │ │ │ │ +00049bb0: 6e65 7261 746f 7273 2070 726f 7669 6465 nerators provide │ │ │ │ +00049bc0: 6420 6279 2074 6865 2075 7365 722e 2054 d by the user. T │ │ │ │ +00049bd0: 6865 2076 616c 7565 2063 6f72 7265 7370 he value corresp │ │ │ │ +00049be0: 6f6e 6469 6e67 2074 6f0a 6120 6765 6e65 onding to.a gene │ │ │ │ +00049bf0: 7261 746f 7220 6720 6973 2061 202a 6e6f rator g is a *no │ │ │ │ +00049c00: 7465 2048 6173 6854 6162 6c65 3a20 284d te HashTable: (M │ │ │ │ +00049c10: 6163 6175 6c61 7932 446f 6329 4861 7368 acaulay2Doc)Hash │ │ │ │ +00049c20: 5461 626c 652c 2063 6f6e 7461 696e 696e Table, containin │ │ │ │ +00049c30: 6720 616c 6c0a 7061 6972 7320 6120 3d3e g all.pairs a => │ │ │ │ +00049c40: 2062 2073 7563 6820 7468 6174 2061 2a67 b such that a*g │ │ │ │ +00049c50: 203d 3d20 622e 2054 6869 7320 7265 7072 == b. 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│ +00049d30: 7273 2c20 6120 7472 616e 7370 6f73 6974 rs, a transposit │ │ │ │ +00049d40: 696f 6e20 616e 6420 6120 332d 6379 636c ion and a 3-cycl │ │ │ │ +00049d50: 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d e...+----------- │ │ │ │ +00049d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049da0: 2d2d 2b0a 7c69 3120 3a20 5220 3d20 5151 --+.|i1 : R = QQ │ │ │ │ +00049db0: 5b78 5f31 2e2e 785f 335d 2020 2020 2020 [x_1..x_3] │ │ │ │ +00049dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049df0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00049e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049e40: 2020 7c0a 7c6f 3120 3d20 5220 2020 2020 |.|o1 = R │ │ │ │ +00049e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049e90: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00049ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049ee0: 2020 7c0a 7c6f 3120 3a20 506f 6c79 6e6f |.|o1 : Polyno │ │ │ │ +00049ef0: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +00049f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049f30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00049f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049f80: 2d2d 2b0a 7c69 3220 3a20 4c20 3d20 7b6d --+.|i2 : L = {m │ │ │ │ +00049f90: 6174 7269 7820 7b7b 302c 312c 307d 2c7b atrix {{0,1,0},{ │ │ │ │ +00049fa0: 312c 302c 307d 2c7b 302c 302c 317d 7d2c 1,0,0},{0,0,1}}, │ │ │ │ +00049fb0: 206d 6174 7269 7820 7b7b 302c 302c 317d matrix {{0,0,1} │ │ │ │ +00049fc0: 2c7b 302c 312c 307d 2c7b 312c 302c 307d ,{0,1,0},{1,0,0} │ │ │ │ +00049fd0: 7d20 7c0a 7c20 2020 2020 2020 2020 2020 } |.| │ │ │ │ +00049fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00049ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a020: 2020 7c0a 7c6f 3220 3d20 7b7c 2030 2031 |.|o2 = {| 0 1 │ │ │ │ +0004a030: 2030 207c 2c20 7c20 3020 3020 3120 7c7d 0 |, | 0 0 1 |} │ │ │ │ +0004a040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a070: 2020 7c0a 7c20 2020 2020 207c 2031 2030 |.| | 1 0 │ │ │ │ +0004a080: 2030 207c 2020 7c20 3020 3120 3020 7c20 0 | | 0 1 0 | │ │ │ │ 0004a090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a0c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a0c0: 2020 7c0a 7c20 2020 2020 207c 2030 2030 |.| | 0 0 │ │ │ │ +0004a0d0: 2031 207c 2020 7c20 3120 3020 3020 7c20 1 | | 1 0 0 | │ │ │ │ 0004a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a110: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -0004a120: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0004a110: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a160: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0004a160: 2020 7c0a 7c6f 3220 3a20 4c69 7374 2020 |.|o2 : List │ │ │ │ 0004a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a1b0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -0004a1c0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ -0004a1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a200: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0004a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a250: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -0004a260: 4c20 3d20 7b6d 6174 7269 7820 7b7b 302c L = {matrix {{0, │ │ │ │ -0004a270: 312c 307d 2c7b 312c 302c 307d 2c7b 302c 1,0},{1,0,0},{0, │ │ │ │ -0004a280: 302c 317d 7d2c 206d 6174 7269 7820 7b7b 0,1}}, matrix {{ │ │ │ │ -0004a290: 302c 302c 317d 2c7b 302c 312c 307d 2c7b 0,0,1},{0,1,0},{ │ │ │ │ -0004a2a0: 312c 302c 307d 7d20 7c0a 7c20 2020 2020 1,0,0}} |.| │ │ │ │ -0004a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a1b0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ +0004a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a200: 2d2d 7c0a 7c7d 2020 2020 2020 2020 2020 --|.|} │ │ │ │ +0004a210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a250: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0004a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004a2a0: 2d2d 2b0a 7c69 3320 3a20 4720 3d20 6669 --+.|i3 : G = fi │ │ │ │ +0004a2b0: 6e69 7465 4163 7469 6f6e 284c 2c20 5229 niteAction(L, R) │ │ │ │ 0004a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a2f0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -0004a300: 7b7c 2030 2031 2030 207c 2c20 7c20 3020 {| 0 1 0 |, | 0 │ │ │ │ -0004a310: 3020 3120 7c7d 2020 2020 2020 2020 2020 0 1 |} │ │ │ │ +0004a2f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004a300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a340: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004a350: 207c 2031 2030 2030 207c 2020 7c20 3020 | 1 0 0 | | 0 │ │ │ │ -0004a360: 3120 3020 7c20 2020 2020 2020 2020 2020 1 0 | │ │ │ │ +0004a340: 2020 7c0a 7c6f 3320 3d20 5220 3c2d 203c |.|o3 = R <- < │ │ │ │ +0004a350: 7c20 3020 3120 3020 7c2c 207c 2030 2030 | 0 1 0 |, | 0 0 │ │ │ │ +0004a360: 2031 207c 3e20 2020 2020 2020 2020 2020 1 |> │ │ │ │ 0004a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a390: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004a3a0: 207c 2030 2030 2031 207c 2020 7c20 3120 | 0 0 1 | | 1 │ │ │ │ -0004a3b0: 3020 3020 7c20 2020 2020 2020 2020 2020 0 0 | │ │ │ │ +0004a390: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004a3a0: 7c20 3120 3020 3020 7c20 207c 2030 2031 | 1 0 0 | | 0 1 │ │ │ │ +0004a3b0: 2030 207c 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2020 2020 2020 2020 │ │ │ │ 0004a470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a480: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d |.|----- │ │ │ │ -0004a490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a4d0: 2d2d 2d2d 2d2d 2d2d 7c0a 7c7d 2020 2020 --------|.|} │ │ │ │ -0004a4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a520: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0004a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a550: 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+0004a5d0: 626c 657b 3020 3d3e 2048 6173 6854 6162 ble{0 => HashTab │ │ │ │ +0004a5e0: 6c65 7b7c 2030 2030 2031 207c 203d 3e20 le{| 0 0 1 | => │ │ │ │ +0004a5f0: 7c20 3020 3020 3120 7c7d 7d20 2020 2020 | 0 0 1 |}} │ │ │ │ 0004a600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a610: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -0004a620: 5220 3c2d 203c 7c20 3020 3120 3020 7c2c R <- <| 0 1 0 |, │ │ │ │ -0004a630: 207c 2030 2030 2031 207c 3e20 2020 2020 | 0 0 1 |> │ │ │ │ -0004a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a610: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a630: 2020 207c 2030 2031 2030 207c 2020 2020 | 0 1 0 | │ │ │ │ +0004a640: 7c20 3120 3020 3020 7c20 2020 2020 2020 | 1 0 0 | │ │ │ │ 0004a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a660: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004a670: 2020 2020 2020 7c20 3120 3020 3020 7c20 | 1 0 0 | │ │ │ │ -0004a680: 207c 2030 2031 2030 207c 2020 2020 2020 | 0 1 0 | │ │ │ │ -0004a690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a660: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004a670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a680: 2020 207c 2031 2030 2030 207c 2020 2020 | 1 0 0 | │ │ │ │ +0004a690: 7c20 3020 3120 3020 7c20 2020 2020 2020 | 0 1 0 | │ │ │ │ 0004a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a6b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004a6c0: 2020 2020 2020 7c20 3020 3020 3120 7c20 | 0 0 1 | │ │ │ │ -0004a6d0: 207c 2031 2030 2030 207c 2020 2020 2020 | 1 0 0 | │ │ │ │ -0004a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a6b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a6d0: 2020 207c 2030 2030 2031 207c 203d 3e20 | 0 0 1 | => │ │ │ │ +0004a6e0: 7c20 3020 3020 3120 7c20 2020 2020 2020 | 0 0 1 | │ │ │ │ 0004a6f0: 2020 2020 2020 2020 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-0004ba30: 6c79 6e6f 6d69 616c 3a0a 2020 2020 2020 lynomial:. │ │ │ │ -0004ba40: 2020 7365 636f 6e64 6172 7949 6e76 6172 secondaryInvar │ │ │ │ -0004ba50: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ -0004ba60: 645f 636d 5072 696e 7444 6567 7265 6550 d_cmPrintDegreeP │ │ │ │ -0004ba70: 6f6c 796e 6f6d 6961 6c3d 3e5f 7064 5f70 olynomial=>_pd_p │ │ │ │ -0004ba80: 645f 7064 5f72 702c 0a20 2020 2020 2020 d_pd_rp,. │ │ │ │ -0004ba90: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -0004baa0: 2076 616c 7565 2066 616c 7365 2c20 616e value false, an │ │ │ │ -0004bab0: 206f 7074 696f 6e61 6c20 6172 6775 6d65 optional argume │ │ │ │ -0004bac0: 6e74 2066 6f72 0a20 2020 2020 2020 2073 nt for. s │ │ │ │ -0004bad0: 6563 6f6e 6461 7279 496e 7661 7269 616e econdaryInvarian │ │ │ │ -0004bae0: 7473 2074 6861 7420 6465 7465 726d 696e ts that determin │ │ │ │ -0004baf0: 6573 2074 6865 2070 7269 6e74 696e 6720 es the printing │ │ │ │ -0004bb00: 6f66 2061 6e20 696e 666f 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computes │ │ │ │ +0004b4a0: 7365 636f 6e64 6172 7920 696e 7661 7269 secondary invari │ │ │ │ +0004b4b0: 616e 7473 2066 6f72 2074 6865 2069 6e76 ants for the inv │ │ │ │ +0004b4c0: 6172 6961 6e74 2072 696e 6720 6f66 2061 ariant ring of a │ │ │ │ +0004b4d0: 2066 696e 6974 6520 6772 6f75 700a 2a2a finite group.** │ │ │ │ +0004b4e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004b4f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004b500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004b510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004b520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004b530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0004b540: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0004b550: 2073 6563 6f6e 6461 7279 496e 7661 7269 secondaryInvari │ │ │ │ +0004b560: 616e 7473 2850 2c47 290a 2020 2a20 496e ants(P,G). * In │ │ │ │ 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*note seconda │ │ │ │ -0004c3a0: 7279 496e 7661 7269 616e 7473 3a20 7365 ryInvariants: se │ │ │ │ -0004c3b0: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ -0004c3c0: 732c 0a6f 6e6c 7920 776f 726b 7320 7768 s,.only works wh │ │ │ │ -0004c3d0: 656e 2052 2069 7320 6465 6669 6e65 6420 en R is defined │ │ │ │ -0004c3e0: 6f76 6572 2061 2066 6965 6c64 206f 6620 over a field of │ │ │ │ -0004c3f0: 6368 6172 6163 7465 7269 7374 6963 207a characteristic z │ │ │ │ -0004c400: 6572 6f2e 0a0a 5761 7973 2074 6f20 7573 ero...Ways to us │ │ │ │ -0004c410: 6520 7365 636f 6e64 6172 7949 6e76 6172 e secondaryInvar │ │ │ │ -0004c420: 6961 6e74 733a 0a3d 3d3d 3d3d 3d3d 3d3d iants:.========= │ │ │ │ -0004c430: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004c440: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7365 =======.. * "se │ │ │ │ -0004c450: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ -0004c460: 7328 4c69 7374 2c46 696e 6974 6547 726f s(List,FiniteGro │ 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-0004c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004c580: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0004c590: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0004c5a0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0004c5b0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0004c5c0: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ -0004c5d0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0004c5e0: 7061 636b 6167 6573 2f0a 496e 7661 7269 packages/.Invari │ │ │ │ -0004c5f0: 616e 7452 696e 672f 4861 7765 7344 6f63 antRing/HawesDoc │ │ │ │ -0004c600: 2e6d 323a 3438 373a 302e 0a1f 0a46 696c .m2:487:0....Fil │ │ │ │ -0004c610: 653a 2049 6e76 6172 6961 6e74 5269 6e67 e: InvariantRing │ │ │ │ -0004c620: 2e69 6e66 6f2c 204e 6f64 653a 2073 6563 .info, Node: sec │ │ │ │ -0004c630: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ -0004c640: 5f6c 705f 7064 5f70 645f 7064 5f63 6d50 _lp_pd_pd_pd_cmP │ │ │ │ -0004c650: 7269 6e74 4465 6772 6565 506f 6c79 6e6f rintDegreePolyno │ │ │ │ -0004c660: 6d69 616c 3d3e 5f70 645f 7064 5f70 645f mial=>_pd_pd_pd_ │ │ │ │ -0004c670: 7270 2c20 4e65 7874 3a20 7465 784d 6174 rp, Next: texMat │ │ │ │ -0004c680: 685f 6c70 5269 6e67 4f66 496e 7661 7269 h_lpRingOfInvari │ │ │ │ -0004c690: 616e 7473 5f72 702c 2050 7265 763a 2073 ants_rp, Prev: s │ │ │ │ -0004c6a0: 6563 6f6e 6461 7279 496e 7661 7269 616e econdaryInvarian │ │ │ │ -0004c6b0: 7473 2c20 5570 3a20 546f 700a 0a73 6563 ts, Up: Top..sec │ │ │ │ -0004c6c0: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ -0004c6d0: 282e 2e2e 2c50 7269 6e74 4465 6772 6565 (...,PrintDegree │ │ │ │ -0004c6e0: 506f 6c79 6e6f 6d69 616c 3d3e 2e2e 2e29 Polynomial=>...) │ │ │ │ -0004c6f0: 202d 2d20 616e 206f 7074 696f 6e61 6c20 -- an optional │ │ │ │ -0004c700: 6172 6775 6d65 6e74 2066 6f72 2073 6563 argument for sec │ │ │ │ -0004c710: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ -0004c720: 2074 6861 7420 6465 7465 726d 696e 6573 that determines │ │ │ │ -0004c730: 2074 6865 2070 7269 6e74 696e 6720 6f66 the printing of │ │ │ │ -0004c740: 2061 6e20 696e 666f 726d 6174 6976 6520 an informative │ │ │ │ -0004c750: 706f 6c79 6e6f 6d69 616c 0a2a 2a2a 2a2a polynomial.***** │ │ │ │ -0004c760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004c770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004c780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004c790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004c7a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004c7b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004c7c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004c7d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004c7e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004c7f0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ -0004c800: 6167 653a 200a 2020 2020 2020 2020 7365 age: . se │ │ │ │ -0004c810: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ -0004c820: 7328 502c 4729 0a20 202a 2049 6e70 7574 s(P,G). * Input │ │ │ │ -0004c830: 733a 0a20 2020 2020 202a 2050 2c20 6120 s:. * P, a │ │ │ │ -0004c840: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ -0004c850: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ -0004c860: 2061 206c 6973 7420 6f66 2070 7269 6d61 a list of prima │ │ │ │ -0004c870: 7279 2069 6e76 6172 6961 6e74 7320 696e ry invariants in │ │ │ │ -0004c880: 206e 0a20 2020 2020 2020 2020 2020 2020 n. │ │ │ │ -0004c890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c8d0: 2020 0a20 2020 2020 2020 2076 6172 6961 . varia │ │ │ │ -0004c8e0: 626c 6573 2066 202c 2e2e 2e2c 6620 2066 bles f ,...,f f │ │ │ │ -0004c8f0: 6f72 2074 6865 2069 6e76 6172 6961 6e74 or the invariant │ │ │ │ -0004c900: 2072 696e 6720 6f66 2047 2064 6566 696e ring of G defin │ │ │ │ -0004c910: 6564 206f 7665 7220 6120 6669 656c 6420 ed over a field │ │ │ │ -0004c920: 4b0a 2020 2020 2020 2020 2020 2020 2020 K. │ │ │ │ -0004c930: 2020 2020 2031 2020 2020 2020 6e0a 2020 1 n. │ │ │ │ -0004c940: 2020 2020 2020 6f66 2063 6861 7261 6374 of charact │ │ │ │ -0004c950: 6572 6973 7469 6320 7a65 726f 0a20 2020 eristic zero. │ │ │ │ -0004c960: 2020 202a 2047 2c20 616e 2069 6e73 7461 * G, an insta │ │ │ │ -0004c970: 6e63 6520 6f66 2074 6865 2074 7970 6520 nce of the type │ │ │ │ -0004c980: 2a6e 6f74 6520 4669 6e69 7465 4772 6f75 *note FiniteGrou │ │ │ │ -0004c990: 7041 6374 696f 6e3a 2046 696e 6974 6547 pAction: FiniteG │ │ │ │ -0004c9a0: 726f 7570 4163 7469 6f6e 2c0a 2020 2a20 roupAction,. * │ │ │ │ -0004c9b0: 4f75 7470 7574 733a 0a20 2020 2020 2020 Outputs:. │ │ │ │ -0004c9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004bb70: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0004bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004bb90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +0004bba0: 413d 6d61 7472 6978 7b7b 612c 307d 2c7b A=matrix{{a,0},{ │ │ │ │ +0004bbb0: 302c 615e 327d 7d3b 2020 2020 207c 0a7c 0,a^2}}; |.| │ │ │ │ +0004bbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004bbe0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004bbf0: 2020 3220 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ +0004bc00: 2020 2020 2020 207c 0a7c 6f33 203a 204d |.|o3 : M │ │ │ │ +0004bc10: 6174 7269 7820 4b20 203c 2d2d 204b 2020 atrix K <-- K │ │ │ │ +0004bc20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0004bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004bc50: 2d2b 0a7c 6934 203a 2042 3d73 7562 286d -+.|i4 : B=sub(m │ │ │ │ +0004bc60: 6174 7269 787b 7b30 2c31 7d2c 7b31 2c30 atrix{{0,1},{1,0 │ │ │ │ +0004bc70: 7d7d 2c4b 293b 7c0a 7c20 2020 2020 2020 }},K);|.| │ │ │ │ +0004bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004bc90: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0004bca0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +0004bcb0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0004bcc0: 7c0a 7c6f 3420 3a20 4d61 7472 6978 204b |.|o4 : Matrix K │ │ │ │ +0004bcd0: 2020 3c2d 2d20 4b20 2020 2020 2020 2020 <-- K │ │ │ │ +0004bce0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0004bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +0004bd10: 3a20 4436 3d66 696e 6974 6541 6374 696f : D6=finiteActio │ │ │ │ +0004bd20: 6e28 7b41 2c42 7d2c 5229 2020 2020 207c n({A,B},R) | │ │ │ │ +0004bd30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0004bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004bd50: 2020 2020 7c0a 7c6f 3520 3d20 5220 3c2d |.|o5 = R <- │ │ │ │ +0004bd60: 203c 7c20 6120 3020 2020 207c 2c20 7c20 <| a 0 |, | │ │ │ │ +0004bd70: 3020 3120 7c3e 2020 207c 0a7c 2020 2020 0 1 |> |.| │ │ │ │ +0004bd80: 2020 2020 2020 207c 2030 202d 612d 3120 | 0 -a-1 │ │ │ │ +0004bd90: 7c20 207c 2031 2030 207c 2020 2020 7c0a | | 1 0 | |. │ │ │ │ +0004bda0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0004bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004bdc0: 2020 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+0004bea0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004beb0: 2020 2032 2032 2020 2020 2020 2020 2020 2 2 │ │ │ │ +0004bec0: 2020 2020 2020 7c0a 7c6f 3720 3d20 7b31 |.|o7 = {1 │ │ │ │ +0004bed0: 2c20 782a 792c 2078 2079 207d 2020 2020 , x*y, x y } │ │ │ │ +0004bee0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0004bef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004bf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004bf10: 7c0a 7c6f 3720 3a20 4c69 7374 2020 2020 |.|o7 : List │ │ │ │ +0004bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004bf30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0004bf40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004bf50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a54 6869 ----------+..Thi │ │ │ │ +0004bf60: 7320 6675 6e63 7469 6f6e 2069 7320 7072 s function is pr │ │ │ │ +0004bf70: 6f76 6964 6564 2062 7920 7468 6520 7061 ovided by the pa │ │ │ │ +0004bf80: 636b 6167 6520 2a6e 6f74 6520 496e 7661 ckage *note Inva │ │ │ │ +0004bf90: 7269 616e 7452 696e 673a 2054 6f70 2c2e riantRing: Top,. │ │ │ │ +0004bfa0: 0a0a 4361 7665 6174 0a3d 3d3d 3d3d 3d0a ..Caveat.======. │ │ │ │ +0004bfb0: 0a43 7572 7265 6e74 6c79 2c20 6120 7573 .Currently, a us │ │ │ │ +0004bfc0: 6572 206e 6565 6473 2074 6f20 656e 7375 er needs to ensu │ │ │ │ +0004bfd0: 7265 2074 6861 7420 7468 6520 616c 6c20 re that the all │ │ │ │ +0004bfe0: 7072 696d 6172 7920 696e 7661 7269 616e primary invarian │ │ │ │ +0004bff0: 7473 2061 7265 2064 6566 696e 6564 0a77 ts are defined.w │ │ │ │ +0004c000: 6974 6820 636f 6566 6669 6369 656e 7473 ith coefficients │ │ │ │ +0004c010: 2069 6e20 6120 7269 6e67 2074 6861 7420 in a ring that │ │ │ │ +0004c020: 4d61 6361 756c 6179 3220 7265 636f 676e Macaulay2 recogn │ │ │ │ +0004c030: 6973 6573 2061 7320 6120 6368 6172 6163 ises as a charac │ │ │ │ +0004c040: 7465 7269 7374 6963 207a 6572 6f0a 6669 teristic zero.fi │ │ │ │ +0004c050: 656c 6420 2873 6565 202a 6e6f 7465 2074 eld (see *note t │ │ │ │ +0004c060: 6f46 6965 6c64 3a20 284d 6163 6175 6c61 oField: (Macaula │ │ │ │ +0004c070: 7932 446f 6329 746f 4669 656c 645f 6c70 y2Doc)toField_lp │ │ │ │ +0004c080: 5269 6e67 5f72 702c 2066 6f72 2061 2077 Ring_rp, for a w │ │ │ │ +0004c090: 6179 2074 6f20 646f 0a74 6869 7329 2e0a ay to do.this).. │ │ │ │ +0004c0a0: 0a4e 6f74 6520 616c 736f 2074 6861 7420 .Note also that │ │ │ │ +0004c0b0: 7468 6520 6675 6e63 7469 6f6e 202a 6e6f the function *no │ │ │ │ +0004c0c0: 7465 2073 6563 6f6e 6461 7279 496e 7661 te secondaryInva │ │ │ │ +0004c0d0: 7269 616e 7473 3a20 7365 636f 6e64 6172 riants: secondar │ │ │ │ +0004c0e0: 7949 6e76 6172 6961 6e74 732c 0a6f 6e6c yInvariants,.onl │ │ │ │ +0004c0f0: 7920 776f 726b 7320 7768 656e 2052 2069 y works when R i │ │ │ │ +0004c100: 7320 6465 6669 6e65 6420 6f76 6572 2061 s defined over a │ │ │ │ +0004c110: 2066 6965 6c64 206f 6620 6368 6172 6163 field of charac │ │ │ │ +0004c120: 7465 7269 7374 6963 207a 6572 6f2e 0a0a teristic zero... │ │ │ │ +0004c130: 5761 7973 2074 6f20 7573 6520 7365 636f Ways to use seco │ │ │ │ +0004c140: 6e64 6172 7949 6e76 6172 6961 6e74 733a ndaryInvariants: │ │ │ │ +0004c150: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +0004c160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004c170: 3d0a 0a20 202a 2022 7365 636f 6e64 6172 =.. * "secondar │ │ │ │ +0004c180: 7949 6e76 6172 6961 6e74 7328 4c69 7374 yInvariants(List │ │ │ │ +0004c190: 2c46 696e 6974 6547 726f 7570 4163 7469 ,FiniteGroupActi │ │ │ │ +0004c1a0: 6f6e 2922 0a0a 466f 7220 7468 6520 7072 on)"..For the pr │ │ │ │ +0004c1b0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +0004c1c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +0004c1d0: 206f 626a 6563 7420 2a6e 6f74 6520 7365 object *note se │ │ │ │ +0004c1e0: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ +0004c1f0: 733a 2073 6563 6f6e 6461 7279 496e 7661 s: secondaryInva │ │ │ │ +0004c200: 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7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +0004c2e0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +0004c2f0: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +0004c300: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +0004c310: 6573 2f0a 496e 7661 7269 616e 7452 696e es/.InvariantRin │ │ │ │ +0004c320: 672f 4861 7765 7344 6f63 2e6d 323a 3438 g/HawesDoc.m2:48 │ │ │ │ +0004c330: 373a 302e 0a1f 0a46 696c 653a 2049 6e76 7:0....File: Inv │ │ │ │ +0004c340: 6172 6961 6e74 5269 6e67 2e69 6e66 6f2c ariantRing.info, │ │ │ │ +0004c350: 204e 6f64 653a 2073 6563 6f6e 6461 7279 Node: secondary │ │ │ │ +0004c360: 496e 7661 7269 616e 7473 5f6c 705f 7064 Invariants_lp_pd │ │ │ │ +0004c370: 5f70 645f 7064 5f63 6d50 7269 6e74 4465 _pd_pd_cmPrintDe │ │ │ │ +0004c380: 6772 6565 506f 6c79 6e6f 6d69 616c 3d3e greePolynomial=> │ │ │ │ +0004c390: 5f70 645f 7064 5f70 645f 7270 2c20 4e65 _pd_pd_pd_rp, Ne │ │ │ │ +0004c3a0: 7874 3a20 7465 784d 6174 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G │ │ │ │ +0004d2c0: 2064 2020 2020 2020 2020 2020 2064 0a48 d d.H │ │ │ │ +0004d2d0: 2853 202c 5429 2a28 312d 5420 3129 2a2e (S ,T)*(1-T 1)*. │ │ │ │ +0004d2e0: 2e2e 2a28 312d 5420 6e29 2e0a 0a54 6865 ..*(1-T n)...The │ │ │ │ +0004d2f0: 2065 7861 6d70 6c65 2062 656c 6f77 2063 example below c │ │ │ │ +0004d300: 6f6d 7075 7465 7320 7468 6520 7365 636f omputes the seco │ │ │ │ +0004d310: 6e64 6172 7920 696e 7661 7269 616e 7473 ndary invariants │ │ │ │ +0004d320: 2066 6f72 2074 6865 2064 6968 6564 7261 for the dihedra │ │ │ │ +0004d330: 6c20 6772 6f75 7020 7769 7468 0a36 2065 l group with.6 e │ │ │ │ +0004d340: 6c65 6d65 6e74 732c 2067 6976 656e 2061 lements, given a │ │ │ │ +0004d350: 2073 6574 206f 6620 7072 696d 6172 7920 set of primary │ │ │ │ +0004d360: 696e 7661 7269 616e 7473 2050 2e20 5468 invariants P. Th │ │ │ │ +0004d370: 6520 6f70 7469 6f6e 616c 2061 7267 756d e optional argum │ │ │ │ +0004d380: 656e 7420 2a6e 6f74 650a 5072 696e 7444 ent *note.PrintD │ │ │ │ +0004d390: 6567 7265 6550 6f6c 796e 6f6d 6961 6c3a egreePolynomial: │ │ │ │ +0004d3a0: 2073 6563 6f6e 6461 7279 496e 7661 7269 secondaryInvari │ │ │ │ +0004d3b0: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ +0004d3c0: 5f63 6d50 7269 6e74 4465 6772 6565 506f _cmPrintDegreePo │ │ │ │ +0004d3d0: 6c79 6e6f 6d69 616c 0a3d 3e5f 7064 5f70 lynomial.=>_pd_p │ │ │ │ +0004d3e0: 645f 7064 5f72 702c 2069 7320 7365 7420 d_pd_rp, is set │ │ │ │ +0004d3f0: 746f 202a 6e6f 7465 2074 7275 653a 2028 to *note true: ( │ │ │ │ +0004d400: 4d61 6361 756c 6179 3244 6f63 2974 7275 Macaulay2Doc)tru │ │ │ │ +0004d410: 652c 2069 6e20 6f72 6465 7220 746f 2073 e, in order to s │ │ │ │ +0004d420: 6565 2077 6869 6368 0a64 6567 7265 6573 ee which.degrees │ │ │ │ +0004d430: 2074 6865 2073 6563 6f6e 6461 7279 2069 the secondary i │ │ │ │ +0004d440: 6e76 6172 6961 6e74 7320 7368 6f75 6c64 nvariants should │ │ │ │ +0004d450: 2068 6176 652e 0a0a 2b2d 2d2d 2d2d 2d2d have...+------- │ │ │ │ +0004d460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d490: 2d2d 2d2b 0a7c 6931 203a 204b 3d74 6f46 ---+.|i1 : K=toF │ │ │ │ +0004d4a0: 6965 6c64 2851 515b 615d 2f28 615e 322b ield(QQ[a]/(a^2+ │ │ │ │ +0004d4b0: 612b 3129 293b 2020 2020 2020 2020 2020 a+1)); │ │ │ │ +0004d4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d4d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0004d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0004d510: 6932 203a 2052 3d4b 5b78 2c79 5d3b 2020 i2 : R=K[x,y]; │ │ │ │ 0004d520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004d530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d540: 2020 2020 0a73 6563 6f6e 6461 7279 496e .secondaryIn │ │ │ │ -0004d550: 7661 7269 616e 7473 3a20 7365 636f 6e64 variants: second │ │ │ │ -0004d560: 6172 7949 6e76 6172 6961 6e74 732c 2077 aryInvariants, w │ │ │ │ -0004d570: 696c 6c20 636f 6d70 7574 6520 7468 6520 ill compute the │ │ │ │ -0004d580: 706f 6c79 6e6f 6d69 616c 0a20 2020 4720 polynomial. 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│ │ │ -0004d720: 7368 6f75 6c64 2068 6176 652e 0a0a 2b2d should have...+- │ │ │ │ -0004d730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d760: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ -0004d770: 204b 3d74 6f46 6965 6c64 2851 515b 615d K=toField(QQ[a] │ │ │ │ -0004d780: 2f28 615e 322b 612b 3129 293b 2020 2020 /(a^2+a+1)); │ │ │ │ +0004d540: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0004d550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d580: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2041 -------+.|i3 : A │ │ │ │ +0004d590: 3d6d 6174 7269 787b 7b61 2c30 7d2c 7b30 =matrix{{a,0},{0 │ │ │ │ +0004d5a0: 2c61 5e32 7d7d 3b20 2020 2020 2020 2020 ,a^2}}; │ │ │ │ +0004d5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d5c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0004d5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d600: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004d610: 2032 2020 2020 2020 3220 2020 2020 2020 2 2 │ │ │ │ +0004d620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d630: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0004d640: 7c6f 3320 3a20 4d61 7472 6978 204b 2020 |o3 : Matrix K │ │ │ │ +0004d650: 3c2d 2d20 4b20 2020 2020 2020 2020 2020 <-- K │ │ │ │ +0004d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d670: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0004d680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d6b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ +0004d6c0: 423d 7375 6228 6d61 7472 6978 7b7b 302c B=sub(matrix{{0, │ │ │ │ +0004d6d0: 317d 2c7b 312c 307d 7d2c 4b29 3b20 2020 1},{1,0}},K); │ │ │ │ +0004d6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d6f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0004d700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d730: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004d740: 2020 3220 2020 2020 2032 2020 2020 2020 2 2 │ │ │ │ +0004d750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004d770: 0a7c 6f34 203a 204d 6174 7269 7820 4b20 .|o4 : Matrix K │ │ │ │ +0004d780: 203c 2d2d 204b 2020 2020 2020 2020 2020 <-- K │ │ │ │ 0004d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d7a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0004d7a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0004d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d7e0: 2d2d 2d2b 0a7c 6932 203a 2052 3d4b 5b78 ---+.|i2 : R=K[x │ │ │ │ -0004d7f0: 2c79 5d3b 2020 2020 2020 2020 2020 2020 ,y]; │ │ │ │ -0004d800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d7e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ +0004d7f0: 2044 363d 6669 6e69 7465 4163 7469 6f6e D6=finiteAction │ │ │ │ +0004d800: 287b 412c 427d 2c52 2920 2020 2020 2020 ({A,B},R) │ │ │ │ 0004d810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d820: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0004d830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0004d860: 6933 203a 2041 3d6d 6174 7269 787b 7b61 i3 : A=matrix{{a │ │ │ │ -0004d870: 2c30 7d2c 7b30 2c61 5e32 7d7d 3b20 2020 ,0},{0,a^2}}; │ │ │ │ -0004d880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d890: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0004d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d8d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0004d8e0: 2020 2020 2020 2032 2020 2020 2020 3220 2 2 │ │ │ │ +0004d820: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0004d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d860: 2020 207c 0a7c 6f35 203d 2052 203c 2d20 |.|o5 = R <- │ │ │ │ +0004d870: 3c7c 2061 2030 2020 2020 7c2c 207c 2030 <| a 0 |, | 0 │ │ │ │ +0004d880: 2031 207c 3e20 2020 2020 2020 2020 2020 1 |> │ │ │ │ +0004d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d8a0: 7c0a 7c20 2020 2020 2020 2020 2020 7c20 |.| | │ │ │ │ +0004d8b0: 3020 2d61 2d31 207c 2020 7c20 3120 3020 0 -a-1 | | 1 0 │ │ │ │ +0004d8c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0004d8d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0004d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004d8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d910: 2020 2020 7c0a 7c6f 3320 3a20 4d61 7472 |.|o3 : Matr │ │ │ │ -0004d920: 6978 204b 2020 3c2d 2d20 4b20 2020 2020 ix K <-- K │ │ │ │ -0004d930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d910: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +0004d920: 3a20 4669 6e69 7465 4772 6f75 7041 6374 : FiniteGroupAct │ │ │ │ +0004d930: 696f 6e20 2020 2020 2020 2020 2020 2020 ion │ │ │ │ 0004d940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d950: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0004d950: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0004d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0004d990: 7c69 3420 3a20 423d 7375 6228 6d61 7472 |i4 : B=sub(matr │ │ │ │ -0004d9a0: 6978 7b7b 302c 317d 2c7b 312c 307d 7d2c ix{{0,1},{1,0}}, │ │ │ │ -0004d9b0: 4b29 3b20 2020 2020 2020 2020 2020 2020 K); │ │ │ │ -0004d9c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0004d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004d9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004da00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004da10: 2020 2020 2020 2020 3220 2020 2020 2032 2 2 │ │ │ │ -0004da20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004da40: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ -0004da50: 7269 7820 4b20 203c 2d2d 204b 2020 2020 rix K <-- K │ │ │ │ +0004d980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d990: 2d2d 2d2d 2b0a 7c69 3620 3a20 503d 7b78 ----+.|i6 : P={x │ │ │ │ +0004d9a0: 5e33 2b79 5e33 2c2d 2878 5e33 2d79 5e33 ^3+y^3,-(x^3-y^3 │ │ │ │ +0004d9b0: 295e 327d 3b20 2020 2020 2020 2020 2020 )^2}; │ │ │ │ +0004d9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004d9d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0004d9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004da00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0004da10: 7c69 3720 3a20 7365 636f 6e64 6172 7949 |i7 : secondaryI │ │ │ │ +0004da20: 6e76 6172 6961 6e74 7328 502c 4436 2c50 nvariants(P,D6,P │ │ │ │ +0004da30: 7269 6e74 4465 6772 6565 506f 6c79 6e6f rintDegreePolyno │ │ │ │ +0004da40: 6d69 616c 3d3e 7472 7565 297c 0a7c 2034 mial=>true)|.| 4 │ │ │ │ +0004da50: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ 0004da60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004da80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -0004da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0004dac0: 0a7c 6935 203a 2044 363d 6669 6e69 7465 .|i5 : D6=finite │ │ │ │ -0004dad0: 4163 7469 6f6e 287b 412c 427d 2c52 2920 Action({A,B},R) │ │ │ │ +0004da80: 2020 2020 2020 2020 7c0a 7c74 2020 2b20 |.|t + │ │ │ │ +0004da90: 7420 202b 2031 2020 2020 2020 2020 2020 t + 1 │ │ │ │ +0004daa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dac0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0004dad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004daf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004db00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004db10: 2020 2020 3220 3220 2020 2020 2020 2020 2 2 │ │ │ │ 0004db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004db30: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ -0004db40: 2052 203c 2d20 3c7c 2061 2030 2020 2020 R <- <| a 0 │ │ │ │ -0004db50: 7c2c 207c 2030 2031 207c 3e20 2020 2020 |, | 0 1 |> │ │ │ │ +0004db30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004db40: 0a7c 6f37 203d 207b 312c 2078 2a79 2c20 .|o7 = {1, x*y, │ │ │ │ +0004db50: 7820 7920 7d20 2020 2020 2020 2020 2020 x y } │ │ │ │ 0004db60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004db70: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0004db80: 2020 2020 7c20 3020 2d61 2d31 207c 2020 | 0 -a-1 | │ │ │ │ -0004db90: 7c20 3120 3020 7c20 2020 2020 2020 2020 | 1 0 | │ │ │ │ +0004db70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0004db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004db90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004dbb0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004dbb0: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ +0004dbc0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 0004dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004dbf0: 7c0a 7c6f 3520 3a20 4669 6e69 7465 4772 |.|o5 : FiniteGr │ │ │ │ -0004dc00: 6f75 7041 6374 696f 6e20 2020 2020 2020 oupAction │ │ │ │ -0004dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004dc20: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0004dc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -0004dc70: 3a20 503d 7b78 5e33 2b79 5e33 2c2d 2878 : P={x^3+y^3,-(x │ │ │ │ -0004dc80: 5e33 2d79 5e33 295e 327d 3b20 2020 2020 ^3-y^3)^2}; │ │ │ │ -0004dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004dca0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0004dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dce0: 2d2d 2d2d 2b0a 7c69 3720 3a20 7365 636f ----+.|i7 : seco │ │ │ │ -0004dcf0: 6e64 6172 7949 6e76 6172 6961 6e74 7328 ndaryInvariants( │ │ │ │ -0004dd00: 502c 4436 2c50 7269 6e74 4465 6772 6565 P,D6,PrintDegree │ │ │ │ -0004dd10: 506f 6c79 6e6f 6d69 616c 3d3e 7472 7565 Polynomial=>true │ │ │ │ -0004dd20: 297c 0a7c 2034 2020 2020 3220 2020 2020 )|.| 4 2 │ │ │ │ -0004dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004dd50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004dd60: 7c74 2020 2b20 7420 202b 2031 2020 2020 |t + t + 1 │ │ │ │ -0004dd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004dd90: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0004dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ddd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004dde0: 2020 2020 2020 2020 2020 3220 3220 2020 2 2 │ │ │ │ -0004ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004de00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004de10: 2020 2020 207c 0a7c 6f37 203d 207b 312c |.|o7 = {1, │ │ │ │ -0004de20: 2078 2a79 2c20 7820 7920 7d20 2020 2020 x*y, x y } │ │ │ │ -0004de30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004de40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004de50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004de70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004de80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004de90: 0a7c 6f37 203a 204c 6973 7420 2020 2020 .|o7 : List │ │ │ │ -0004dea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004deb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004dec0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0004ded0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004dee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004df00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 ---------+..Cave │ │ │ │ -0004df10: 6174 0a3d 3d3d 3d3d 3d0a 0a43 7572 7265 at.======..Curre │ │ │ │ -0004df20: 6e74 6c79 2c20 6120 7573 6572 206e 6565 ntly, a user nee │ │ │ │ -0004df30: 6473 2074 6f20 656e 7375 7265 2074 6861 ds to ensure tha │ │ │ │ -0004df40: 7420 7468 6520 616c 6c20 7072 696d 6172 t the all primar │ │ │ │ -0004df50: 7920 696e 7661 7269 616e 7473 2061 7265 y invariants are │ │ │ │ -0004df60: 2064 6566 696e 6564 0a77 6974 6820 636f defined.with co │ │ │ │ -0004df70: 6566 6669 6369 656e 7473 2069 6e20 6120 efficients in a │ │ │ │ -0004df80: 7269 6e67 2074 6861 7420 4d61 6361 756c ring that Macaul │ │ │ │ -0004df90: 6179 3220 7265 636f 676e 6973 6573 2061 ay2 recognises a │ │ │ │ -0004dfa0: 7320 6120 6368 6172 6163 7465 7269 7374 s a characterist │ │ │ │ -0004dfb0: 6963 207a 6572 6f0a 6669 656c 6420 2873 ic zero.field (s │ │ │ │ -0004dfc0: 6565 202a 6e6f 7465 2074 6f46 6965 6c64 ee *note toField │ │ │ │ -0004dfd0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0004dfe0: 746f 4669 656c 645f 6c70 5269 6e67 5f72 toField_lpRing_r │ │ │ │ -0004dff0: 702c 2066 6f72 2061 2077 6179 2074 6f20 p, for a way to │ │ │ │ -0004e000: 646f 0a74 6869 7329 2e0a 0a46 756e 6374 do.this)...Funct │ │ │ │ -0004e010: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ -0004e020: 616c 2061 7267 756d 656e 7420 6e61 6d65 al argument name │ │ │ │ -0004e030: 6420 5072 696e 7444 6567 7265 6550 6f6c d PrintDegreePol │ │ │ │ -0004e040: 796e 6f6d 6961 6c3a 0a3d 3d3d 3d3d 3d3d ynomial:.======= │ │ │ │ -0004e050: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004e060: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004e070: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004e080: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2268 6972 ======.. * "hir │ │ │ │ -0004e090: 6f6e 616b 6144 6563 6f6d 706f 7369 7469 onakaDecompositi │ │ │ │ -0004e0a0: 6f6e 282e 2e2e 2c50 7269 6e74 4465 6772 on(...,PrintDegr │ │ │ │ -0004e0b0: 6565 506f 6c79 6e6f 6d69 616c 3d3e 2e2e eePolynomial=>.. │ │ │ │ -0004e0c0: 2e29 220a 2020 2a20 2a6e 6f74 6520 7365 .)". * *note se │ │ │ │ -0004e0d0: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ -0004e0e0: 7328 2e2e 2e2c 5072 696e 7444 6567 7265 s(...,PrintDegre │ │ │ │ -0004e0f0: 6550 6f6c 796e 6f6d 6961 6c3d 3e2e 2e2e ePolynomial=>... │ │ │ │ -0004e100: 293a 0a20 2020 2073 6563 6f6e 6461 7279 ):. secondary │ │ │ │ -0004e110: 496e 7661 7269 616e 7473 5f6c 705f 7064 Invariants_lp_pd │ │ │ │ -0004e120: 5f70 645f 7064 5f63 6d50 7269 6e74 4465 _pd_pd_cmPrintDe │ │ │ │ -0004e130: 6772 6565 506f 6c79 6e6f 6d69 616c 3d3e greePolynomial=> │ │ │ │ -0004e140: 5f70 645f 7064 5f70 645f 7270 2c20 2d2d _pd_pd_pd_rp, -- │ │ │ │ -0004e150: 0a20 2020 2061 6e20 6f70 7469 6f6e 616c . an optional │ │ │ │ -0004e160: 2061 7267 756d 656e 7420 666f 7220 7365 argument for se │ │ │ │ -0004e170: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ -0004e180: 7320 7468 6174 2064 6574 6572 6d69 6e65 s that determine │ │ │ │ -0004e190: 7320 7468 6520 7072 696e 7469 6e67 0a20 s the printing. │ │ │ │ -0004e1a0: 2020 206f 6620 616e 2069 6e66 6f72 6d61 of an informa │ │ │ │ -0004e1b0: 7469 7665 2070 6f6c 796e 6f6d 6961 6c0a tive polynomial. │ │ │ │ -0004e1c0: 0a46 7572 7468 6572 2069 6e66 6f72 6d61 .Further informa │ │ │ │ -0004e1d0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -0004e1e0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 4465 ========.. * De │ │ │ │ -0004e1f0: 6661 756c 7420 7661 6c75 653a 202a 6e6f fault value: *no │ │ │ │ -0004e200: 7465 2066 616c 7365 3a20 284d 6163 6175 te false: (Macau │ │ │ │ -0004e210: 6c61 7932 446f 6329 6661 6c73 652c 0a20 lay2Doc)false,. │ │ │ │ -0004e220: 202a 2046 756e 6374 696f 6e3a 202a 6e6f * Function: *no │ │ │ │ -0004e230: 7465 2073 6563 6f6e 6461 7279 496e 7661 te secondaryInva │ │ │ │ -0004e240: 7269 616e 7473 3a20 7365 636f 6e64 6172 riants: secondar │ │ │ │ -0004e250: 7949 6e76 6172 6961 6e74 732c 202d 2d20 yInvariants, -- │ │ │ │ -0004e260: 636f 6d70 7574 6573 0a20 2020 2073 6563 computes. sec │ │ │ │ -0004e270: 6f6e 6461 7279 2069 6e76 6172 6961 6e74 ondary invariant │ │ │ │ -0004e280: 7320 666f 7220 7468 6520 696e 7661 7269 s for the invari │ │ │ │ -0004e290: 616e 7420 7269 6e67 206f 6620 6120 6669 ant ring of a fi │ │ │ │ -0004e2a0: 6e69 7465 2067 726f 7570 0a20 202a 204f nite group. * O │ │ │ │ -0004e2b0: 7074 696f 6e20 6b65 793a 202a 6e6f 7465 ption key: *note │ │ │ │ -0004e2c0: 2050 7269 6e74 4465 6772 6565 506f 6c79 PrintDegreePoly │ │ │ │ -0004e2d0: 6e6f 6d69 616c 3a0a 2020 2020 7365 636f nomial:. seco │ │ │ │ -0004e2e0: 6e64 6172 7949 6e76 6172 6961 6e74 735f ndaryInvariants_ │ │ │ │ -0004e2f0: 6c70 5f70 645f 7064 5f70 645f 636d 5072 lp_pd_pd_pd_cmPr │ │ │ │ -0004e300: 696e 7444 6567 7265 6550 6f6c 796e 6f6d intDegreePolynom │ │ │ │ -0004e310: 6961 6c3d 3e5f 7064 5f70 645f 7064 5f72 ial=>_pd_pd_pd_r │ │ │ │ -0004e320: 702c 202d 2d0a 2020 2020 616e 206f 7074 p, --. an opt │ │ │ │ -0004e330: 696f 6e61 6c20 6172 6775 6d65 6e74 2066 ional argument f │ │ │ │ -0004e340: 6f72 2073 6563 6f6e 6461 7279 496e 7661 or secondaryInva │ │ │ │ -0004e350: 7269 616e 7473 2074 6861 7420 6465 7465 riants that dete │ │ │ │ -0004e360: 726d 696e 6573 2074 6865 2070 7269 6e74 rmines the print │ │ │ │ -0004e370: 696e 670a 2020 2020 6f66 2061 6e20 696e ing. of an in │ │ │ │ -0004e380: 666f 726d 6174 6976 6520 706f 6c79 6e6f formative polyno │ │ │ │ -0004e390: 6d69 616c 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d mial.----------- │ │ │ │ -0004e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e3e0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -0004e3f0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -0004e400: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -0004e410: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -0004e420: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -0004e430: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -0004e440: 7932 2f70 6163 6b61 6765 732f 0a49 6e76 y2/packages/.Inv │ │ │ │ -0004e450: 6172 6961 6e74 5269 6e67 2f48 6177 6573 ariantRing/Hawes │ │ │ │ -0004e460: 446f 632e 6d32 3a35 3535 3a30 2e0a 1f0a Doc.m2:555:0.... │ │ │ │ -0004e470: 4669 6c65 3a20 496e 7661 7269 616e 7452 File: InvariantR │ │ │ │ -0004e480: 696e 672e 696e 666f 2c20 4e6f 6465 3a20 ing.info, Node: │ │ │ │ -0004e490: 7465 784d 6174 685f 6c70 5269 6e67 4f66 texMath_lpRingOf │ │ │ │ -0004e4a0: 496e 7661 7269 616e 7473 5f72 702c 204e Invariants_rp, N │ │ │ │ -0004e4b0: 6578 743a 2077 6569 6768 7473 2c20 5072 ext: weights, Pr │ │ │ │ -0004e4c0: 6576 3a20 7365 636f 6e64 6172 7949 6e76 ev: secondaryInv │ │ │ │ -0004e4d0: 6172 6961 6e74 735f 6c70 5f70 645f 7064 ariants_lp_pd_pd │ │ │ │ -0004e4e0: 5f70 645f 636d 5072 696e 7444 6567 7265 _pd_cmPrintDegre │ │ │ │ -0004e4f0: 6550 6f6c 796e 6f6d 6961 6c3d 3e5f 7064 ePolynomial=>_pd │ │ │ │ -0004e500: 5f70 645f 7064 5f72 702c 2055 703a 2054 _pd_pd_rp, Up: T │ │ │ │ -0004e510: 6f70 0a0a 7465 784d 6174 6828 5269 6e67 op..texMath(Ring │ │ │ │ -0004e520: 4f66 496e 7661 7269 616e 7473 2920 2d2d OfInvariants) -- │ │ │ │ -0004e530: 2066 6f72 6d61 7420 666f 7220 7072 696e format for prin │ │ │ │ -0004e540: 7469 6e67 2c20 6173 2054 6558 2073 7472 ting, as TeX str │ │ │ │ -0004e550: 696e 6773 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a ings.*********** │ │ │ │ -0004e560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004e570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004e580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004e590: 2a2a 2a2a 2a0a 0a20 202a 2046 756e 6374 *****.. * Funct │ │ │ │ -0004e5a0: 696f 6e3a 202a 6e6f 7465 2074 6578 4d61 ion: *note texMa │ │ │ │ -0004e5b0: 7468 3a20 284d 6163 6175 6c61 7932 446f th: (Macaulay2Do │ │ │ │ -0004e5c0: 6329 7465 784d 6174 682c 0a0a 4465 7363 c)texMath,..Desc │ │ │ │ -0004e5d0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -0004e5e0: 3d3d 3d0a 0a46 6f72 6d61 7420 6f62 6a65 ===..Format obje │ │ │ │ -0004e5f0: 6374 7320 6f66 2074 6865 2070 6163 6b61 cts of the packa │ │ │ │ -0004e600: 6765 202a 6e6f 7465 2049 6e76 6172 6961 ge *note Invaria │ │ │ │ -0004e610: 6e74 5269 6e67 3a20 546f 702c 2066 6f72 ntRing: Top, for │ │ │ │ -0004e620: 2070 7269 6e74 696e 6720 696e 2054 6558 printing in TeX │ │ │ │ -0004e630: 2e0a 5365 6520 2a6e 6f74 6520 7465 784d ..See *note texM │ │ │ │ -0004e640: 6174 683a 2028 4d61 6361 756c 6179 3244 ath: (Macaulay2D │ │ │ │ -0004e650: 6f63 2974 6578 4d61 7468 2c20 666f 7220 oc)texMath, for │ │ │ │ -0004e660: 6d6f 7265 2069 6e66 6f72 6d61 7469 6f6e more information │ │ │ │ -0004e670: 2e0a 0a57 6179 7320 746f 2075 7365 2074 ...Ways to use t │ │ │ │ -0004e680: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ -0004e690: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004e6a0: 3d3d 3d3d 0a0a 2020 2a20 2274 6578 4d61 ====.. * "texMa │ │ │ │ -0004e6b0: 7468 2844 6961 676f 6e61 6c41 6374 696f th(DiagonalActio │ │ │ │ -0004e6c0: 6e29 220a 2020 2a20 2274 6578 4d61 7468 n)". * "texMath │ │ │ │ -0004e6d0: 2846 696e 6974 6547 726f 7570 4163 7469 (FiniteGroupActi │ │ │ │ -0004e6e0: 6f6e 2922 0a20 202a 2022 7465 784d 6174 on)". * "texMat │ │ │ │ -0004e6f0: 6828 4c69 6e65 6172 6c79 5265 6475 6374 h(LinearlyReduct │ │ │ │ -0004e700: 6976 6541 6374 696f 6e29 220a 2020 2a20 iveAction)". * │ │ │ │ -0004e710: 2a6e 6f74 6520 7465 784d 6174 6828 5269 *note texMath(Ri │ │ │ │ -0004e720: 6e67 4f66 496e 7661 7269 616e 7473 293a ngOfInvariants): │ │ │ │ -0004e730: 2074 6578 4d61 7468 5f6c 7052 696e 674f texMath_lpRingO │ │ │ │ -0004e740: 6649 6e76 6172 6961 6e74 735f 7270 2c20 fInvariants_rp, │ │ │ │ -0004e750: 2d2d 2066 6f72 6d61 740a 2020 2020 666f -- format. fo │ │ │ │ -0004e760: 7220 7072 696e 7469 6e67 2c20 6173 2054 r printing, as T │ │ │ │ -0004e770: 6558 2073 7472 696e 6773 0a2d 2d2d 2d2d eX strings.----- │ │ │ │ -0004e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004e7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -0004e7d0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -0004e7e0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -0004e7f0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -0004e800: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -0004e810: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -0004e820: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -0004e830: 732f 0a49 6e76 6172 6961 6e74 5269 6e67 s/.InvariantRing │ │ │ │ -0004e840: 2f49 6e76 6172 6961 6e74 5269 6e67 446f /InvariantRingDo │ │ │ │ -0004e850: 632e 6d32 3a32 3235 3a30 2e0a 1f0a 4669 c.m2:225:0....Fi │ │ │ │ -0004e860: 6c65 3a20 496e 7661 7269 616e 7452 696e le: InvariantRin │ │ │ │ -0004e870: 672e 696e 666f 2c20 4e6f 6465 3a20 7765 g.info, Node: we │ │ │ │ -0004e880: 6967 6874 732c 204e 6578 743a 2077 6f72 ights, Next: wor │ │ │ │ -0004e890: 6473 2c20 5072 6576 3a20 7465 784d 6174 ds, Prev: texMat │ │ │ │ -0004e8a0: 685f 6c70 5269 6e67 4f66 496e 7661 7269 h_lpRingOfInvari │ │ │ │ -0004e8b0: 616e 7473 5f72 702c 2055 703a 2054 6f70 ants_rp, Up: Top │ │ │ │ -0004e8c0: 0a0a 7765 6967 6874 7320 2d2d 206f 6620 ..weights -- of │ │ │ │ -0004e8d0: 6120 6469 6167 6f6e 616c 2061 6374 696f a diagonal actio │ │ │ │ -0004e8e0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a n.************** │ │ │ │ -0004e8f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004e900: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ -0004e910: 2020 2020 2020 2077 6569 6768 7473 2044 weights D │ │ │ │ -0004e920: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -0004e930: 2020 202a 2044 2c20 616e 2069 6e73 7461 * D, an insta │ │ │ │ -0004e940: 6e63 6520 6f66 2074 6865 2074 7970 6520 nce of the type │ │ │ │ -0004e950: 2a6e 6f74 6520 4469 6167 6f6e 616c 4163 *note DiagonalAc │ │ │ │ -0004e960: 7469 6f6e 3a20 4469 6167 6f6e 616c 4163 tion: DiagonalAc │ │ │ │ -0004e970: 7469 6f6e 2c0a 2020 2a20 4f75 7470 7574 tion,. * Output │ │ │ │ -0004e980: 733a 0a20 2020 2020 202a 2061 202a 6e6f s:. * a *no │ │ │ │ -0004e990: 7465 206d 6174 7269 783a 2028 4d61 6361 te matrix: (Maca │ │ │ │ -0004e9a0: 756c 6179 3244 6f63 294d 6174 7269 782c ulay2Doc)Matrix, │ │ │ │ -0004e9b0: 2c20 7468 6520 7765 6967 6874 206d 6174 , the weight mat │ │ │ │ -0004e9c0: 7269 7820 6f66 2074 6865 2067 726f 7570 rix of the group │ │ │ │ -0004e9d0: 0a20 2020 2020 2020 2061 6374 696f 6e0a . action. │ │ │ │ -0004e9e0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ -0004e9f0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2066 ========..This f │ │ │ │ -0004ea00: 756e 6374 696f 6e20 6973 2070 726f 7669 unction is provi │ │ │ │ -0004ea10: 6465 6420 6279 2074 6865 2070 6163 6b61 ded by the packa │ │ │ │ -0004ea20: 6765 202a 6e6f 7465 2049 6e76 6172 6961 ge *note Invaria │ │ │ │ -0004ea30: 6e74 5269 6e67 3a20 546f 702c 2e20 0a0a ntRing: Top,. .. │ │ │ │ -0004ea40: 5573 6520 7468 6973 2066 756e 6374 696f Use this functio │ │ │ │ -0004ea50: 6e20 746f 2072 6563 6f76 6572 2074 6865 n to recover the │ │ │ │ -0004ea60: 2077 6569 6768 7420 6d61 7472 6978 206f weight matrix o │ │ │ │ -0004ea70: 6620 6120 6469 6167 6f6e 616c 2061 6374 f a diagonal act │ │ │ │ -0004ea80: 696f 6e20 6f6e 2061 0a70 6f6c 796e 6f6d ion on a.polynom │ │ │ │ -0004ea90: 6961 6c20 7269 6e67 2e20 466f 7220 6120 ial ring. For a │ │ │ │ -0004eaa0: 6469 6167 6f6e 616c 2061 6374 696f 6e20 diagonal action │ │ │ │ -0004eab0: 6f6e 2061 2070 6f6c 796e 6f6d 6961 6c20 on a polynomial │ │ │ │ -0004eac0: 7269 6e67 2020 246b 5b78 5f31 2c20 5c64 ring $k[x_1, \d │ │ │ │ -0004ead0: 6f74 732c 0a78 5f6e 5d24 202c 2074 6865 ots,.x_n]$ , the │ │ │ │ -0004eae0: 2020 246a 2420 2d74 6820 636f 6c75 6d6e $j$ -th column │ │ │ │ -0004eaf0: 206f 6620 7468 6520 7765 6967 6874 206d of the weight m │ │ │ │ -0004eb00: 6174 7269 7820 6973 2074 6865 2077 6569 atrix is the wei │ │ │ │ -0004eb10: 6768 7420 6f66 2074 6865 2076 6172 6961 ght of the varia │ │ │ │ -0004eb20: 626c 650a 2478 5f6a 2420 2e0a 0a54 6865 ble.$x_j$ ...The │ │ │ │ -0004eb30: 2066 6f6c 6c6f 7769 6e67 2065 7861 6d70 following examp │ │ │ │ -0004eb40: 6c65 2064 6566 696e 6573 2061 6e20 6163 le defines an ac │ │ │ │ -0004eb50: 7469 6f6e 206f 6620 6120 7477 6f2d 6469 tion of a two-di │ │ │ │ -0004eb60: 6d65 6e73 696f 6e61 6c20 746f 7275 7320 mensional torus │ │ │ │ -0004eb70: 6f6e 2061 0a70 6f6c 796e 6f6d 6961 6c20 on a.polynomial │ │ │ │ -0004eb80: 7269 6e67 2069 6e20 666f 7572 2076 6172 ring in four var │ │ │ │ -0004eb90: 6961 626c 6573 2e0a 0a2b 2d2d 2d2d 2d2d iables...+------ │ │ │ │ -0004eba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ebb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ebc0: 2d2b 0a7c 6931 203a 2052 203d 2051 515b -+.|i1 : R = QQ[ │ │ │ │ -0004ebd0: 785f 312e 2e78 5f34 5d20 2020 2020 2020 x_1..x_4] │ │ │ │ -0004ebe0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0004dbf0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0004dc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004dc10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004dc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004dc30: 2d2d 2d2b 0a0a 4361 7665 6174 0a3d 3d3d ---+..Caveat.=== │ │ │ │ +0004dc40: 3d3d 3d0a 0a43 7572 7265 6e74 6c79 2c20 ===..Currently, │ │ │ │ +0004dc50: 6120 7573 6572 206e 6565 6473 2074 6f20 a user needs to │ │ │ │ +0004dc60: 656e 7375 7265 2074 6861 7420 7468 6520 ensure that the │ │ │ │ +0004dc70: 616c 6c20 7072 696d 6172 7920 696e 7661 all primary inva │ │ │ │ +0004dc80: 7269 616e 7473 2061 7265 2064 6566 696e riants are defin │ │ │ │ +0004dc90: 6564 0a77 6974 6820 636f 6566 6669 6369 ed.with coeffici │ │ │ │ +0004dca0: 656e 7473 2069 6e20 6120 7269 6e67 2074 ents in a ring t │ │ │ │ +0004dcb0: 6861 7420 4d61 6361 756c 6179 3220 7265 hat Macaulay2 re │ │ │ │ +0004dcc0: 636f 676e 6973 6573 2061 7320 6120 6368 cognises as a ch │ │ │ │ +0004dcd0: 6172 6163 7465 7269 7374 6963 207a 6572 aracteristic zer │ │ │ │ +0004dce0: 6f0a 6669 656c 6420 2873 6565 202a 6e6f o.field (see *no │ │ │ │ +0004dcf0: 7465 2074 6f46 6965 6c64 3a20 284d 6163 te toField: (Mac │ │ │ │ +0004dd00: 6175 6c61 7932 446f 6329 746f 4669 656c aulay2Doc)toFiel │ │ │ │ +0004dd10: 645f 6c70 5269 6e67 5f72 702c 2066 6f72 d_lpRing_rp, for │ │ │ │ +0004dd20: 2061 2077 6179 2074 6f20 646f 0a74 6869 a way to do.thi │ │ │ │ +0004dd30: 7329 2e0a 0a46 756e 6374 696f 6e73 2077 s)...Functions w │ │ │ │ +0004dd40: 6974 6820 6f70 7469 6f6e 616c 2061 7267 ith optional arg │ │ │ │ +0004dd50: 756d 656e 7420 6e61 6d65 6420 5072 696e ument named Prin │ │ │ │ +0004dd60: 7444 6567 7265 6550 6f6c 796e 6f6d 6961 tDegreePolynomia │ │ │ │ +0004dd70: 6c3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d l:.============= │ │ │ │ +0004dd80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004dd90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004dda0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004ddb0: 0a0a 2020 2a20 2268 6972 6f6e 616b 6144 .. * "hironakaD │ │ │ │ +0004ddc0: 6563 6f6d 706f 7369 7469 6f6e 282e 2e2e ecomposition(... │ │ │ │ +0004ddd0: 2c50 7269 6e74 4465 6772 6565 506f 6c79 ,PrintDegreePoly │ │ │ │ +0004dde0: 6e6f 6d69 616c 3d3e 2e2e 2e29 220a 2020 nomial=>...)". │ │ │ │ +0004ddf0: 2a20 2a6e 6f74 6520 7365 636f 6e64 6172 * *note secondar │ │ │ │ +0004de00: 7949 6e76 6172 6961 6e74 7328 2e2e 2e2c yInvariants(..., │ │ │ │ +0004de10: 5072 696e 7444 6567 7265 6550 6f6c 796e PrintDegreePolyn │ │ │ │ +0004de20: 6f6d 6961 6c3d 3e2e 2e2e 293a 0a20 2020 omial=>...):. │ │ │ │ +0004de30: 2073 6563 6f6e 6461 7279 496e 7661 7269 secondaryInvari │ │ │ │ +0004de40: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ +0004de50: 5f63 6d50 7269 6e74 4465 6772 6565 506f _cmPrintDegreePo │ │ │ │ +0004de60: 6c79 6e6f 6d69 616c 3d3e 5f70 645f 7064 lynomial=>_pd_pd │ │ │ │ +0004de70: 5f70 645f 7270 2c20 2d2d 0a20 2020 2061 _pd_rp, --. a │ │ │ │ +0004de80: 6e20 6f70 7469 6f6e 616c 2061 7267 756d n optional argum │ │ │ │ +0004de90: 656e 7420 666f 7220 7365 636f 6e64 6172 ent for secondar │ │ │ │ +0004dea0: 7949 6e76 6172 6961 6e74 7320 7468 6174 yInvariants that │ │ │ │ +0004deb0: 2064 6574 6572 6d69 6e65 7320 7468 6520 determines the │ │ │ │ +0004dec0: 7072 696e 7469 6e67 0a20 2020 206f 6620 printing. of │ │ │ │ +0004ded0: 616e 2069 6e66 6f72 6d61 7469 7665 2070 an informative p │ │ │ │ +0004dee0: 6f6c 796e 6f6d 6961 6c0a 0a46 7572 7468 olynomial..Furth │ │ │ │ +0004def0: 6572 2069 6e66 6f72 6d61 7469 6f6e 0a3d er information.= │ │ │ │ +0004df00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004df10: 3d3d 0a0a 2020 2a20 4465 6661 756c 7420 ==.. * Default │ │ │ │ +0004df20: 7661 6c75 653a 202a 6e6f 7465 2066 616c value: *note fal │ │ │ │ +0004df30: 7365 3a20 284d 6163 6175 6c61 7932 446f se: (Macaulay2Do │ │ │ │ +0004df40: 6329 6661 6c73 652c 0a20 202a 2046 756e c)false,. * Fun │ │ │ │ +0004df50: 6374 696f 6e3a 202a 6e6f 7465 2073 6563 ction: *note sec │ │ │ │ +0004df60: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ +0004df70: 3a20 7365 636f 6e64 6172 7949 6e76 6172 : secondaryInvar │ │ │ │ +0004df80: 6961 6e74 732c 202d 2d20 636f 6d70 7574 iants, -- comput │ │ │ │ +0004df90: 6573 0a20 2020 2073 6563 6f6e 6461 7279 es. secondary │ │ │ │ +0004dfa0: 2069 6e76 6172 6961 6e74 7320 666f 7220 invariants for │ │ │ │ +0004dfb0: 7468 6520 696e 7661 7269 616e 7420 7269 the invariant ri │ │ │ │ +0004dfc0: 6e67 206f 6620 6120 6669 6e69 7465 2067 ng of a finite g │ │ │ │ +0004dfd0: 726f 7570 0a20 202a 204f 7074 696f 6e20 roup. * Option │ │ │ │ +0004dfe0: 6b65 793a 202a 6e6f 7465 2050 7269 6e74 key: *note Print │ │ │ │ +0004dff0: 4465 6772 6565 506f 6c79 6e6f 6d69 616c DegreePolynomial │ │ │ │ +0004e000: 3a0a 2020 2020 7365 636f 6e64 6172 7949 :. secondaryI │ │ │ │ +0004e010: 6e76 6172 6961 6e74 735f 6c70 5f70 645f nvariants_lp_pd_ │ │ │ │ +0004e020: 7064 5f70 645f 636d 5072 696e 7444 6567 pd_pd_cmPrintDeg │ │ │ │ +0004e030: 7265 6550 6f6c 796e 6f6d 6961 6c3d 3e5f reePolynomial=>_ │ │ │ │ +0004e040: 7064 5f70 645f 7064 5f72 702c 202d 2d0a pd_pd_pd_rp, --. │ │ │ │ +0004e050: 2020 2020 616e 206f 7074 696f 6e61 6c20 an optional │ │ │ │ +0004e060: 6172 6775 6d65 6e74 2066 6f72 2073 6563 argument for sec │ │ │ │ +0004e070: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ +0004e080: 2074 6861 7420 6465 7465 726d 696e 6573 that determines │ │ │ │ +0004e090: 2074 6865 2070 7269 6e74 696e 670a 2020 the printing. │ │ │ │ +0004e0a0: 2020 6f66 2061 6e20 696e 666f 726d 6174 of an informat │ │ │ │ +0004e0b0: 6976 6520 706f 6c79 6e6f 6d69 616c 0a2d ive polynomial.- │ │ │ │ +0004e0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0004e110: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0004e120: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0004e130: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0004e140: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0004e150: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +0004e160: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0004e170: 6b61 6765 732f 0a49 6e76 6172 6961 6e74 kages/.Invariant │ │ │ │ +0004e180: 5269 6e67 2f48 6177 6573 446f 632e 6d32 Ring/HawesDoc.m2 │ │ │ │ +0004e190: 3a35 3535 3a30 2e0a 1f0a 4669 6c65 3a20 :555:0....File: │ │ │ │ +0004e1a0: 496e 7661 7269 616e 7452 696e 672e 696e InvariantRing.in │ │ │ │ +0004e1b0: 666f 2c20 4e6f 6465 3a20 7465 784d 6174 fo, Node: texMat │ │ │ │ +0004e1c0: 685f 6c70 5269 6e67 4f66 496e 7661 7269 h_lpRingOfInvari │ │ │ │ +0004e1d0: 616e 7473 5f72 702c 204e 6578 743a 2077 ants_rp, Next: w │ │ │ │ +0004e1e0: 6569 6768 7473 2c20 5072 6576 3a20 7365 eights, Prev: se │ │ │ │ +0004e1f0: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ +0004e200: 735f 6c70 5f70 645f 7064 5f70 645f 636d s_lp_pd_pd_pd_cm │ │ │ │ +0004e210: 5072 696e 7444 6567 7265 6550 6f6c 796e PrintDegreePolyn │ │ │ │ +0004e220: 6f6d 6961 6c3d 3e5f 7064 5f70 645f 7064 omial=>_pd_pd_pd │ │ │ │ +0004e230: 5f72 702c 2055 703a 2054 6f70 0a0a 7465 _rp, Up: Top..te │ │ │ │ +0004e240: 784d 6174 6828 5269 6e67 4f66 496e 7661 xMath(RingOfInva │ │ │ │ +0004e250: 7269 616e 7473 2920 2d2d 2066 6f72 6d61 riants) -- forma │ │ │ │ +0004e260: 7420 666f 7220 7072 696e 7469 6e67 2c20 t for printing, │ │ │ │ +0004e270: 6173 2054 6558 2073 7472 696e 6773 0a2a as TeX strings.* │ │ │ │ +0004e280: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004e290: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004e2a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004e2b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +0004e2c0: 0a20 202a 2046 756e 6374 696f 6e3a 202a . * Function: * │ │ │ │ +0004e2d0: 6e6f 7465 2074 6578 4d61 7468 3a20 284d note texMath: (M │ │ │ │ +0004e2e0: 6163 6175 6c61 7932 446f 6329 7465 784d acaulay2Doc)texM │ │ │ │ +0004e2f0: 6174 682c 0a0a 4465 7363 7269 7074 696f ath,..Descriptio │ │ │ │ +0004e300: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a46 n.===========..F │ │ │ │ +0004e310: 6f72 6d61 7420 6f62 6a65 6374 7320 6f66 ormat objects of │ │ │ │ +0004e320: 2074 6865 2070 6163 6b61 6765 202a 6e6f the package *no │ │ │ │ +0004e330: 7465 2049 6e76 6172 6961 6e74 5269 6e67 te InvariantRing │ │ │ │ +0004e340: 3a20 546f 702c 2066 6f72 2070 7269 6e74 : Top, for print │ │ │ │ +0004e350: 696e 6720 696e 2054 6558 2e0a 5365 6520 ing in TeX..See │ │ │ │ +0004e360: 2a6e 6f74 6520 7465 784d 6174 683a 2028 *note texMath: ( │ │ │ │ +0004e370: 4d61 6361 756c 6179 3244 6f63 2974 6578 Macaulay2Doc)tex │ │ │ │ +0004e380: 4d61 7468 2c20 666f 7220 6d6f 7265 2069 Math, for more i │ │ │ │ +0004e390: 6e66 6f72 6d61 7469 6f6e 2e0a 0a57 6179 nformation...Way │ │ │ │ +0004e3a0: 7320 746f 2075 7365 2074 6869 7320 6d65 s to use this me │ │ │ │ +0004e3b0: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d thod:.========== │ │ │ │ +0004e3c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0004e3d0: 2020 2a20 2274 6578 4d61 7468 2844 6961 * "texMath(Dia │ │ │ │ +0004e3e0: 676f 6e61 6c41 6374 696f 6e29 220a 2020 gonalAction)". │ │ │ │ +0004e3f0: 2a20 2274 6578 4d61 7468 2846 696e 6974 * "texMath(Finit │ │ │ │ +0004e400: 6547 726f 7570 4163 7469 6f6e 2922 0a20 eGroupAction)". │ │ │ │ +0004e410: 202a 2022 7465 784d 6174 6828 4c69 6e65 * "texMath(Line │ │ │ │ +0004e420: 6172 6c79 5265 6475 6374 6976 6541 6374 arlyReductiveAct │ │ │ │ +0004e430: 696f 6e29 220a 2020 2a20 2a6e 6f74 6520 ion)". * *note │ │ │ │ +0004e440: 7465 784d 6174 6828 5269 6e67 4f66 496e texMath(RingOfIn │ │ │ │ +0004e450: 7661 7269 616e 7473 293a 2074 6578 4d61 variants): texMa │ │ │ │ +0004e460: 7468 5f6c 7052 696e 674f 6649 6e76 6172 th_lpRingOfInvar │ │ │ │ +0004e470: 6961 6e74 735f 7270 2c20 2d2d 2066 6f72 iants_rp, -- for │ │ │ │ +0004e480: 6d61 740a 2020 2020 666f 7220 7072 696e mat. for prin │ │ │ │ +0004e490: 7469 6e67 2c20 6173 2054 6558 2073 7472 ting, as TeX str │ │ │ │ +0004e4a0: 696e 6773 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d ings.----------- │ │ │ │ +0004e4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e4f0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +0004e500: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +0004e510: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +0004e520: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +0004e530: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +0004e540: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +0004e550: 7932 2f70 6163 6b61 6765 732f 0a49 6e76 y2/packages/.Inv │ │ │ │ +0004e560: 6172 6961 6e74 5269 6e67 2f49 6e76 6172 ariantRing/Invar │ │ │ │ +0004e570: 6961 6e74 5269 6e67 446f 632e 6d32 3a32 iantRingDoc.m2:2 │ │ │ │ +0004e580: 3235 3a30 2e0a 1f0a 4669 6c65 3a20 496e 25:0....File: In │ │ │ │ +0004e590: 7661 7269 616e 7452 696e 672e 696e 666f variantRing.info │ │ │ │ +0004e5a0: 2c20 4e6f 6465 3a20 7765 6967 6874 732c , Node: weights, │ │ │ │ +0004e5b0: 204e 6578 743a 2077 6f72 6473 2c20 5072 Next: words, Pr │ │ │ │ +0004e5c0: 6576 3a20 7465 784d 6174 685f 6c70 5269 ev: texMath_lpRi │ │ │ │ +0004e5d0: 6e67 4f66 496e 7661 7269 616e 7473 5f72 ngOfInvariants_r │ │ │ │ +0004e5e0: 702c 2055 703a 2054 6f70 0a0a 7765 6967 p, Up: Top..weig │ │ │ │ +0004e5f0: 6874 7320 2d2d 206f 6620 6120 6469 6167 hts -- of a diag │ │ │ │ +0004e600: 6f6e 616c 2061 6374 696f 6e0a 2a2a 2a2a onal action.**** │ │ │ │ +0004e610: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004e620: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0004e630: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0004e640: 2077 6569 6768 7473 2044 0a20 202a 2049 weights D. * I │ │ │ │ +0004e650: 6e70 7574 733a 0a20 2020 2020 202a 2044 nputs:. * D │ │ │ │ +0004e660: 2c20 616e 2069 6e73 7461 6e63 6520 6f66 , an instance of │ │ │ │ +0004e670: 2074 6865 2074 7970 6520 2a6e 6f74 6520 the type *note │ │ │ │ +0004e680: 4469 6167 6f6e 616c 4163 7469 6f6e 3a20 DiagonalAction: │ │ │ │ +0004e690: 4469 6167 6f6e 616c 4163 7469 6f6e 2c0a DiagonalAction,. │ │ │ │ +0004e6a0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +0004e6b0: 2020 202a 2061 202a 6e6f 7465 206d 6174 * a *note mat │ │ │ │ +0004e6c0: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ +0004e6d0: 6f63 294d 6174 7269 782c 2c20 7468 6520 oc)Matrix,, the │ │ │ │ +0004e6e0: 7765 6967 6874 206d 6174 7269 7820 6f66 weight matrix of │ │ │ │ +0004e6f0: 2074 6865 2067 726f 7570 0a20 2020 2020 the group. │ │ │ │ +0004e700: 2020 2061 6374 696f 6e0a 0a44 6573 6372 action..Descr │ │ │ │ +0004e710: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +0004e720: 3d3d 0a0a 5468 6973 2066 756e 6374 696f ==..This functio │ │ │ │ +0004e730: 6e20 6973 2070 726f 7669 6465 6420 6279 n is provided by │ │ │ │ +0004e740: 2074 6865 2070 6163 6b61 6765 202a 6e6f the package *no │ │ │ │ +0004e750: 7465 2049 6e76 6172 6961 6e74 5269 6e67 te InvariantRing │ │ │ │ +0004e760: 3a20 546f 702c 2e20 0a0a 5573 6520 7468 : Top,. ..Use th │ │ │ │ +0004e770: 6973 2066 756e 6374 696f 6e20 746f 2072 is function to r │ │ │ │ +0004e780: 6563 6f76 6572 2074 6865 2077 6569 6768 ecover the weigh │ │ │ │ +0004e790: 7420 6d61 7472 6978 206f 6620 6120 6469 t matrix of a di │ │ │ │ +0004e7a0: 6167 6f6e 616c 2061 6374 696f 6e20 6f6e agonal action on │ │ │ │ +0004e7b0: 2061 0a70 6f6c 796e 6f6d 6961 6c20 7269 a.polynomial ri │ │ │ │ +0004e7c0: 6e67 2e20 466f 7220 6120 6469 6167 6f6e ng. For a diagon │ │ │ │ +0004e7d0: 616c 2061 6374 696f 6e20 6f6e 2061 2070 al action on a p │ │ │ │ +0004e7e0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2020 olynomial ring │ │ │ │ +0004e7f0: 246b 5b78 5f31 2c20 5c64 6f74 732c 0a78 $k[x_1, \dots,.x │ │ │ │ +0004e800: 5f6e 5d24 202c 2074 6865 2020 246a 2420 _n]$ , the $j$ │ │ │ │ +0004e810: 2d74 6820 636f 6c75 6d6e 206f 6620 7468 -th column of th │ │ │ │ +0004e820: 6520 7765 6967 6874 206d 6174 7269 7820 e weight matrix │ │ │ │ +0004e830: 6973 2074 6865 2077 6569 6768 7420 6f66 is the weight of │ │ │ │ +0004e840: 2074 6865 2076 6172 6961 626c 650a 2478 the variable.$x │ │ │ │ +0004e850: 5f6a 2420 2e0a 0a54 6865 2066 6f6c 6c6f _j$ ...The follo │ │ │ │ +0004e860: 7769 6e67 2065 7861 6d70 6c65 2064 6566 wing example def │ │ │ │ +0004e870: 696e 6573 2061 6e20 6163 7469 6f6e 206f ines an action o │ │ │ │ +0004e880: 6620 6120 7477 6f2d 6469 6d65 6e73 696f f a two-dimensio │ │ │ │ +0004e890: 6e61 6c20 746f 7275 7320 6f6e 2061 0a70 nal torus on a.p │ │ │ │ +0004e8a0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 2069 olynomial ring i │ │ │ │ +0004e8b0: 6e20 666f 7572 2076 6172 6961 626c 6573 n four variables │ │ │ │ +0004e8c0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +0004e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0004e8f0: 203a 2052 203d 2051 515b 785f 312e 2e78 : R = QQ[x_1..x │ │ │ │ +0004e900: 5f34 5d20 2020 2020 2020 2020 2020 2020 _4] │ │ │ │ +0004e910: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0004e920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e930: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004e940: 0a7c 6f31 203d 2052 2020 2020 2020 2020 .|o1 = R │ │ │ │ +0004e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e960: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0004e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004e990: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ +0004e9a0: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ +0004e9b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0004e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004e9e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2057 -------+.|i2 : W │ │ │ │ +0004e9f0: 203d 206d 6174 7269 787b 7b30 2c31 2c2d = matrix{{0,1,- │ │ │ │ +0004ea00: 312c 317d 2c7b 312c 302c 2d31 2c2d 317d 1,1},{1,0,-1,-1} │ │ │ │ +0004ea10: 7d7c 0a7c 2020 2020 2020 2020 2020 2020 }|.| │ │ │ │ +0004ea20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ea30: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0004ea40: 203d 207c 2030 2031 202d 3120 3120 207c = | 0 1 -1 1 | │ │ │ │ +0004ea50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ea60: 2020 2020 207c 0a7c 2020 2020 207c 2031 |.| | 1 │ │ │ │ +0004ea70: 2030 202d 3120 2d31 207c 2020 2020 2020 0 -1 -1 | │ │ │ │ +0004ea80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004ea90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0004eaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004eab0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0004eac0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0004ead0: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +0004eae0: 2020 207c 0a7c 6f32 203a 204d 6174 7269 |.|o2 : Matri │ │ │ │ +0004eaf0: 7820 5a5a 2020 3c2d 2d20 5a5a 2020 2020 x ZZ <-- ZZ │ │ │ │ +0004eb00: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0004eb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004eb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004eb30: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2054 -------+.|i3 : T │ │ │ │ +0004eb40: 203d 2064 6961 676f 6e61 6c41 6374 696f = diagonalActio │ │ │ │ +0004eb50: 6e28 572c 2052 2920 2020 2020 2020 2020 n(W, R) │ │ │ │ +0004eb60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004eb80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0004eb90: 2020 2020 2020 2020 2020 202a 2032 2020 * 2 │ │ │ │ +0004eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ebb0: 2020 2020 207c 0a7c 6f33 203d 2052 203c |.|o3 = R < │ │ │ │ +0004ebc0: 2d20 2851 5120 2920 2076 6961 2020 2020 - (QQ ) via │ │ │ │ +0004ebd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004ebe0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0004ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ec10: 2020 2020 207c 0a7c 6f31 203d 2052 2020 |.|o1 = R │ │ │ │ +0004ec00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0004ec10: 207c 2030 2031 202d 3120 3120 207c 2020 | 0 1 -1 1 | │ │ │ │ 0004ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ec30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004ec40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0004ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ec60: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ -0004ec70: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ -0004ec80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ec90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0004eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0004ecc0: 6932 203a 2057 203d 206d 6174 7269 787b i2 : W = matrix{ │ │ │ │ -0004ecd0: 7b30 2c31 2c2d 312c 317d 2c7b 312c 302c {0,1,-1,1},{1,0, │ │ │ │ -0004ece0: 2d31 2c2d 317d 7d7c 0a7c 2020 2020 2020 -1,-1}}|.| │ │ │ │ +0004ec30: 2020 207c 0a7c 2020 2020 207c 2031 2030 |.| | 1 0 │ │ │ │ +0004ec40: 202d 3120 2d31 207c 2020 2020 2020 2020 -1 -1 | │ │ │ │ +0004ec50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0004ec60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ec80: 2020 2020 2020 207c 0a7c 6f33 203a 2044 |.|o3 : D │ │ │ │ +0004ec90: 6961 676f 6e61 6c41 6374 696f 6e20 2020 iagonalAction │ │ │ │ +0004eca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ecb0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0004ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ +0004ece0: 203a 2077 6569 6768 7473 2054 2020 2020 : weights T │ │ │ │ 0004ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ed00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ed10: 207c 0a7c 6f32 203d 207c 2030 2031 202d |.|o2 = | 0 1 - │ │ │ │ -0004ed20: 3120 3120 207c 2020 2020 2020 2020 2020 1 1 | │ │ │ │ -0004ed30: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0004ed40: 2020 207c 2031 2030 202d 3120 2d31 207c | 1 0 -1 -1 | │ │ │ │ -0004ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ed60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0004ed00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0004ed10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ed20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004ed30: 0a7c 6f34 203d 2028 7c20 3020 3120 2d31 .|o4 = (| 0 1 -1 │ │ │ │ +0004ed40: 2031 2020 7c2c 2030 2920 2020 2020 2020 1 |, 0) │ │ │ │ +0004ed50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0004ed60: 2020 7c20 3120 3020 2d31 202d 3120 7c20 | 1 0 -1 -1 | │ │ │ │ 0004ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ed80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004ed90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0004eda0: 3220 2020 2020 2020 3420 2020 2020 2020 2 4 │ │ │ │ -0004edb0: 2020 2020 2020 2020 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ -0004edc0: 204d 6174 7269 7820 5a5a 2020 3c2d 2d20 Matrix ZZ <-- │ │ │ │ -0004edd0: 5a5a 2020 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ -0004ede0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0004ed80: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004ed90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004eda0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0004edb0: 6f34 203a 2053 6571 7565 6e63 6520 2020 o4 : Sequence │ │ │ │ +0004edc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004edd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0004ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004ee00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0004ee10: 6933 203a 2054 203d 2064 6961 676f 6e61 i3 : T = diagona │ │ │ │ -0004ee20: 6c41 6374 696f 6e28 572c 2052 2920 2020 lAction(W, R) │ │ │ │ -0004ee30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0004ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ee60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004ee70: 202a 2032 2020 2020 2020 2020 2020 2020 * 2 │ │ │ │ -0004ee80: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -0004ee90: 203d 2052 203c 2d20 2851 5120 2920 2076 = R <- (QQ ) v │ │ │ │ -0004eea0: 6961 2020 2020 2020 2020 2020 2020 2020 ia │ │ │ │ +0004ee00: 2d2b 0a0a 4865 7265 2069 7320 616e 2065 -+..Here is an e │ │ │ │ +0004ee10: 7861 6d70 6c65 206f 6620 6120 7072 6f64 xample of a prod │ │ │ │ +0004ee20: 7563 7420 6f66 2074 776f 2063 7963 6c69 uct of two cycli │ │ │ │ +0004ee30: 6320 6772 6f75 7073 206f 6620 6f72 6465 c groups of orde │ │ │ │ +0004ee40: 7220 3320 6163 7469 6e67 206f 6e20 610a r 3 acting on a. │ │ │ │ +0004ee50: 706f 6c79 6e6f 6d69 616c 2072 696e 6720 polynomial ring │ │ │ │ +0004ee60: 696e 2033 2076 6172 6961 626c 6573 2e0a in 3 variables.. │ │ │ │ +0004ee70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0004ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004ee90: 2d2d 2b0a 7c69 3520 3a20 5220 3d20 5151 --+.|i5 : R = QQ │ │ │ │ +0004eea0: 5b78 5f31 2e2e 785f 335d 2020 2020 2020 [x_1..x_3] │ │ │ │ 0004eeb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0004eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004eed0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004eee0: 0a7c 2020 2020 207c 2030 2031 202d 3120 .| | 0 1 -1 │ │ │ │ -0004eef0: 3120 207c 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -0004ef00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0004ef10: 207c 2031 2030 202d 3120 2d31 207c 2020 | 1 0 -1 -1 | │ │ │ │ -0004ef20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ef30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004ef50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0004ef60: 6f33 203a 2044 6961 676f 6e61 6c41 6374 o3 : DiagonalAct │ │ │ │ -0004ef70: 696f 6e20 2020 2020 2020 2020 2020 2020 ion │ │ │ │ -0004ef80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0004ef90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004efa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004efb0: 2d2b 0a7c 6934 203a 2077 6569 6768 7473 -+.|i4 : weights │ │ │ │ -0004efc0: 2054 2020 2020 2020 2020 2020 2020 2020 T │ │ │ │ -0004efd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0004eed0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ +0004eee0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0004eef0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0004ef00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004ef10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0004ef20: 7c6f 3520 3a20 506f 6c79 6e6f 6d69 616c |o5 : Polynomial │ │ │ │ +0004ef30: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +0004ef40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0004ef50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004ef60: 2d2d 2d2d 2b0a 7c69 3620 3a20 6420 3d20 ----+.|i6 : d = │ │ │ │ +0004ef70: 7b33 2c33 7d20 2020 2020 2020 2020 2020 {3,3} │ │ │ │ +0004ef80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0004ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004efa0: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +0004efb0: 3d20 7b33 2c20 337d 2020 2020 2020 2020 = {3, 3} │ │ │ │ +0004efc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0004efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f000: 2020 2020 207c 0a7c 6f34 203d 2028 7c20 |.|o4 = (| │ │ │ │ -0004f010: 3020 3120 2d31 2031 2020 7c2c 2030 2920 0 1 -1 1 |, 0) │ │ │ │ -0004f020: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004f030: 0a7c 2020 2020 2020 7c20 3120 3020 2d31 .| | 1 0 -1 │ │ │ │ -0004f040: 202d 3120 7c20 2020 2020 2020 2020 2020 -1 | │ │ │ │ -0004f050: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0004eff0: 7c0a 7c6f 3620 3a20 4c69 7374 2020 2020 |.|o6 : List │ │ │ │ +0004f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f010: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0004f020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f030: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 5720 ------+.|i7 : W │ │ │ │ +0004f040: 3d20 6d61 7472 6978 7b7b 312c 302c 317d = matrix{{1,0,1} │ │ │ │ +0004f050: 2c7b 302c 312c 317d 7d7c 0a7c 2020 2020 ,{0,1,1}}|.| │ │ │ │ 0004f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f080: 2020 207c 0a7c 6f34 203a 2053 6571 7565 |.|o4 : Seque │ │ │ │ -0004f090: 6e63 6520 2020 2020 2020 2020 2020 2020 nce │ │ │ │ -0004f0a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -0004f0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f0d0: 2d2d 2d2d 2d2d 2d2b 0a0a 4865 7265 2069 -------+..Here i │ │ │ │ -0004f0e0: 7320 616e 2065 7861 6d70 6c65 206f 6620 s an example of │ │ │ │ -0004f0f0: 6120 7072 6f64 7563 7420 6f66 2074 776f a product of two │ │ │ │ -0004f100: 2063 7963 6c69 6320 6772 6f75 7073 206f cyclic groups o │ │ │ │ -0004f110: 6620 6f72 6465 7220 3320 6163 7469 6e67 f order 3 acting │ │ │ │ -0004f120: 206f 6e20 610a 706f 6c79 6e6f 6d69 616c on a.polynomial │ │ │ │ -0004f130: 2072 696e 6720 696e 2033 2076 6172 6961 ring in 3 varia │ │ │ │ -0004f140: 626c 6573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d bles...+-------- │ │ │ │ -0004f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f160: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ -0004f170: 5220 3d20 5151 5b78 5f31 2e2e 785f 335d R = QQ[x_1..x_3] │ │ │ │ -0004f180: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0004f190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f1a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004f1b0: 7c6f 3520 3d20 5220 2020 2020 2020 2020 |o5 = R │ │ │ │ +0004f070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0004f080: 3720 3d20 7c20 3120 3020 3120 7c20 2020 7 = | 1 0 1 | │ │ │ │ +0004f090: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004f0a0: 0a7c 2020 2020 207c 2030 2031 2031 207c .| | 0 1 1 | │ │ │ │ +0004f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f0c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f0e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0004f0f0: 2020 2020 2020 3220 2020 2020 2020 3320 2 3 │ │ │ │ +0004f100: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20 |.|o7 : │ │ │ │ +0004f110: 4d61 7472 6978 205a 5a20 203c 2d2d 205a Matrix ZZ <-- Z │ │ │ │ +0004f120: 5a20 2020 2020 2020 2020 207c 0a2b 2d2d Z |.+-- │ │ │ │ +0004f130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0004f150: 7c69 3820 3a20 4120 3d20 6469 6167 6f6e |i8 : A = diagon │ │ │ │ +0004f160: 616c 4163 7469 6f6e 2857 2c20 642c 2052 alAction(W, d, R │ │ │ │ +0004f170: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ +0004f180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f190: 2020 2020 7c0a 7c6f 3820 3d20 5220 3c2d |.|o8 = R <- │ │ │ │ +0004f1a0: 205a 5a2f 3320 7820 5a5a 2f33 2076 6961 ZZ/3 x ZZ/3 via │ │ │ │ +0004f1b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0004f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f1d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f1f0: 2020 2020 7c0a 7c6f 3520 3a20 506f 6c79 |.|o5 : Poly │ │ │ │ -0004f200: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ -0004f210: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0004f220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f230: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -0004f240: 3a20 6420 3d20 7b33 2c33 7d20 2020 2020 : d = {3,3} │ │ │ │ -0004f250: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0004f260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f280: 7c0a 7c6f 3620 3d20 7b33 2c20 337d 2020 |.|o6 = {3, 3} │ │ │ │ -0004f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f2a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004f1d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0004f1e0: 2020 7c20 3120 3020 3120 7c20 2020 2020 | 1 0 1 | │ │ │ │ +0004f1f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0004f200: 2020 2020 207c 2030 2031 2031 207c 2020 | 0 1 1 | │ │ │ │ +0004f210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f220: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004f230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f240: 2020 207c 0a7c 6f38 203a 2044 6961 676f |.|o8 : Diago │ │ │ │ +0004f250: 6e61 6c41 6374 696f 6e20 2020 2020 2020 nalAction │ │ │ │ +0004f260: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0004f270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f280: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a ---------+.|i9 : │ │ │ │ +0004f290: 2077 6569 6768 7473 2041 2020 2020 2020 weights A │ │ │ │ +0004f2a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0004f2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f2c0: 2020 2020 2020 7c0a 7c6f 3620 3a20 4c69 |.|o6 : Li │ │ │ │ -0004f2d0: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ -0004f2e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0004f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0004f310: 3720 3a20 5720 3d20 6d61 7472 6978 7b7b 7 : W = matrix{{ │ │ │ │ -0004f320: 312c 302c 317d 2c7b 302c 312c 317d 7d7c 1,0,1},{0,1,1}}| │ │ │ │ -0004f330: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0004f340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f350: 2020 7c0a 7c6f 3720 3d20 7c20 3120 3020 |.|o7 = | 1 0 │ │ │ │ -0004f360: 3120 7c20 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -0004f370: 2020 2020 207c 0a7c 2020 2020 207c 2030 |.| | 0 │ │ │ │ -0004f380: 2031 2031 207c 2020 2020 2020 2020 2020 1 1 | │ │ │ │ -0004f390: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004f3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f3b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0004f3c0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ -0004f3d0: 2020 2020 3320 2020 2020 2020 2020 7c0a 3 |. │ │ │ │ -0004f3e0: 7c6f 3720 3a20 4d61 7472 6978 205a 5a20 |o7 : Matrix ZZ │ │ │ │ -0004f3f0: 203c 2d2d 205a 5a20 2020 2020 2020 2020 <-- ZZ │ │ │ │ -0004f400: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0004f410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f420: 2d2d 2d2d 2b0a 7c69 3820 3a20 4120 3d20 ----+.|i8 : A = │ │ │ │ -0004f430: 6469 6167 6f6e 616c 4163 7469 6f6e 2857 diagonalAction(W │ │ │ │ -0004f440: 2c20 642c 2052 297c 0a7c 2020 2020 2020 , d, R)|.| │ │ │ │ -0004f450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f460: 2020 2020 2020 2020 2020 7c0a 7c6f 3820 |.|o8 │ │ │ │ -0004f470: 3d20 5220 3c2d 205a 5a2f 3320 7820 5a5a = R <- ZZ/3 x ZZ │ │ │ │ -0004f480: 2f33 2076 6961 2020 2020 2020 207c 0a7c /3 via |.| │ │ │ │ -0004f490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f4b0: 7c0a 7c20 2020 2020 7c20 3120 3020 3120 |.| | 1 0 1 │ │ │ │ -0004f4c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0004f4d0: 2020 207c 0a7c 2020 2020 207c 2030 2031 |.| | 0 1 │ │ │ │ -0004f4e0: 2031 207c 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -0004f4f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0004f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f510: 2020 2020 2020 2020 207c 0a7c 6f38 203a |.|o8 : │ │ │ │ -0004f520: 2044 6961 676f 6e61 6c41 6374 696f 6e20 DiagonalAction │ │ │ │ -0004f530: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0004f2c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0004f2d0: 0a7c 6f39 203d 2028 302c 207c 2031 2030 .|o9 = (0, | 1 0 │ │ │ │ +0004f2e0: 2031 207c 2920 2020 2020 2020 2020 2020 1 |) │ │ │ │ +0004f2f0: 2020 7c0a 7c20 2020 2020 2020 2020 7c20 |.| | │ │ │ │ +0004f300: 3020 3120 3120 7c20 2020 2020 2020 2020 0 1 1 | │ │ │ │ +0004f310: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0004f320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f330: 2020 2020 2020 2020 7c0a 7c6f 3920 3a20 |.|o9 : │ │ │ │ +0004f340: 5365 7175 656e 6365 2020 2020 2020 2020 Sequence │ │ │ │ +0004f350: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0004f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0004f380: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +0004f390: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 4469 ==.. * *note Di │ │ │ │ +0004f3a0: 6167 6f6e 616c 4163 7469 6f6e 3a20 4469 agonalAction: Di │ │ │ │ +0004f3b0: 6167 6f6e 616c 4163 7469 6f6e 2c20 2d2d agonalAction, -- │ │ │ │ +0004f3c0: 2074 6865 2063 6c61 7373 206f 6620 616c the class of al │ │ │ │ +0004f3d0: 6c20 6469 6167 6f6e 616c 2061 6374 696f l diagonal actio │ │ │ │ +0004f3e0: 6e73 0a20 202a 202a 6e6f 7465 2064 6961 ns. * *note dia │ │ │ │ +0004f3f0: 676f 6e61 6c41 6374 696f 6e3a 2064 6961 gonalAction: dia │ │ │ │ +0004f400: 676f 6e61 6c41 6374 696f 6e2c 202d 2d20 gonalAction, -- │ │ │ │ +0004f410: 6469 6167 6f6e 616c 2067 726f 7570 2061 diagonal group a │ │ │ │ +0004f420: 6374 696f 6e20 7669 6120 7765 6967 6874 ction via weight │ │ │ │ +0004f430: 730a 0a57 6179 7320 746f 2075 7365 2077 s..Ways to use w │ │ │ │ +0004f440: 6569 6768 7473 3a0a 3d3d 3d3d 3d3d 3d3d eights:.======== │ │ │ │ +0004f450: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +0004f460: 2a20 2277 6569 6768 7473 2844 6961 676f * "weights(Diago │ │ │ │ +0004f470: 6e61 6c41 6374 696f 6e29 220a 0a46 6f72 nalAction)"..For │ │ │ │ +0004f480: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0004f490: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0004f4a0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0004f4b0: 6e6f 7465 2077 6569 6768 7473 3a20 7765 note weights: we │ │ │ │ +0004f4c0: 6967 6874 732c 2069 7320 6120 2a6e 6f74 ights, is a *not │ │ │ │ +0004f4d0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +0004f4e0: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +0004f4f0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +0004f500: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0004f510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004f540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0004f560: 0a7c 6939 203a 2077 6569 6768 7473 2041 .|i9 : weights A │ │ │ │ -0004f570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f580: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0004f590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f5a0: 2020 2020 207c 0a7c 6f39 203d 2028 302c |.|o9 = (0, │ │ │ │ -0004f5b0: 207c 2031 2030 2031 207c 2920 2020 2020 | 1 0 1 |) │ │ │ │ -0004f5c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0004f5d0: 2020 2020 7c20 3020 3120 3120 7c20 2020 | 0 1 1 | │ │ │ │ -0004f5e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0004f5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f600: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0004f610: 7c6f 3920 3a20 5365 7175 656e 6365 2020 |o9 : Sequence │ │ │ │ -0004f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004f630: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0004f640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f650: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a ----+..See also. │ │ │ │ -0004f660: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -0004f670: 6f74 6520 4469 6167 6f6e 616c 4163 7469 ote DiagonalActi │ │ │ │ -0004f680: 6f6e 3a20 4469 6167 6f6e 616c 4163 7469 on: DiagonalActi │ │ │ │ -0004f690: 6f6e 2c20 2d2d 2074 6865 2063 6c61 7373 on, -- the class │ │ │ │ -0004f6a0: 206f 6620 616c 6c20 6469 6167 6f6e 616c of all diagonal │ │ │ │ -0004f6b0: 2061 6374 696f 6e73 0a20 202a 202a 6e6f actions. * *no │ │ │ │ -0004f6c0: 7465 2064 6961 676f 6e61 6c41 6374 696f te diagonalActio │ │ │ │ -0004f6d0: 6e3a 2064 6961 676f 6e61 6c41 6374 696f n: diagonalActio │ │ │ │ -0004f6e0: 6e2c 202d 2d20 6469 6167 6f6e 616c 2067 n, -- diagonal g │ │ │ │ -0004f6f0: 726f 7570 2061 6374 696f 6e20 7669 6120 roup action via │ │ │ │ -0004f700: 7765 6967 6874 730a 0a57 6179 7320 746f weights..Ways to │ │ │ │ -0004f710: 2075 7365 2077 6569 6768 7473 3a0a 3d3d use weights:.== │ │ │ │ -0004f720: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004f730: 3d3d 0a0a 2020 2a20 2277 6569 6768 7473 ==.. * "weights │ │ │ │ -0004f740: 2844 6961 676f 6e61 6c41 6374 696f 6e29 (DiagonalAction) │ │ │ │ -0004f750: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ -0004f760: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -0004f770: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -0004f780: 6a65 6374 202a 6e6f 7465 2077 6569 6768 ject *note weigh │ │ │ │ -0004f790: 7473 3a20 7765 6967 6874 732c 2069 7320 ts: weights, is │ │ │ │ -0004f7a0: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ -0004f7b0: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ -0004f7c0: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ -0004f7d0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ -0004f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004f820: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -0004f830: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -0004f840: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -0004f850: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -0004f860: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -0004f870: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -0004f880: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -0004f890: 0a49 6e76 6172 6961 6e74 5269 6e67 2f41 .InvariantRing/A │ │ │ │ -0004f8a0: 6265 6c69 616e 4772 6f75 7073 446f 632e belianGroupsDoc. │ │ │ │ -0004f8b0: 6d32 3a33 3138 3a30 2e0a 1f0a 4669 6c65 m2:318:0....File │ │ │ │ -0004f8c0: 3a20 496e 7661 7269 616e 7452 696e 672e : InvariantRing. │ │ │ │ -0004f8d0: 696e 666f 2c20 4e6f 6465 3a20 776f 7264 info, Node: word │ │ │ │ -0004f8e0: 732c 2050 7265 763a 2077 6569 6768 7473 s, Prev: weights │ │ │ │ -0004f8f0: 2c20 5570 3a20 546f 700a 0a77 6f72 6473 , Up: Top..words │ │ │ │ -0004f900: 202d 2d20 6173 736f 6369 6174 6520 6120 -- associate a │ │ │ │ -0004f910: 776f 7264 2069 6e20 7468 6520 6765 6e65 word in the gene │ │ │ │ -0004f920: 7261 746f 7273 206f 6620 6120 6772 6f75 rators of a grou │ │ │ │ -0004f930: 7020 746f 2065 6163 6820 656c 656d 656e p to each elemen │ │ │ │ -0004f940: 740a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a t.************** │ │ │ │ -0004f950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004f960: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004f970: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0004f980: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ -0004f990: 6167 653a 200a 2020 2020 2020 2020 776f age: . wo │ │ │ │ -0004f9a0: 7264 7320 470a 2020 2a20 496e 7075 7473 rds G. * Inputs │ │ │ │ -0004f9b0: 3a0a 2020 2020 2020 2a20 472c 2061 6e20 :. * G, an │ │ │ │ -0004f9c0: 696e 7374 616e 6365 206f 6620 7468 6520 instance of the │ │ │ │ -0004f9d0: 7479 7065 202a 6e6f 7465 2046 696e 6974 type *note Finit │ │ │ │ -0004f9e0: 6547 726f 7570 4163 7469 6f6e 3a20 4669 eGroupAction: Fi │ │ │ │ -0004f9f0: 6e69 7465 4772 6f75 7041 6374 696f 6e2c niteGroupAction, │ │ │ │ -0004fa00: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -0004fa10: 2020 2020 2a20 6120 2a6e 6f74 6520 6861 * a *note ha │ │ │ │ -0004fa20: 7368 2074 6162 6c65 3a20 284d 6163 6175 sh table: (Macau │ │ │ │ -0004fa30: 6c61 7932 446f 6329 4861 7368 5461 626c lay2Doc)HashTabl │ │ │ │ -0004fa40: 652c 2c20 6173 736f 6369 6174 696e 6720 e,, associating │ │ │ │ -0004fa50: 746f 2065 6163 680a 2020 2020 2020 2020 to each. │ │ │ │ -0004fa60: 656c 656d 656e 7420 696e 2074 6865 2067 element in the g │ │ │ │ -0004fa70: 726f 7570 206f 6620 7468 6520 6163 7469 roup of the acti │ │ │ │ -0004fa80: 6f6e 2061 2077 6f72 6420 6f66 206d 696e on a word of min │ │ │ │ -0004fa90: 696d 616c 206c 656e 6774 6820 696e 2028 imal length in ( │ │ │ │ -0004faa0: 7468 650a 2020 2020 2020 2020 696e 6469 the. indi │ │ │ │ -0004fab0: 6365 7320 6f66 2920 7468 6520 6765 6e65 ces of) the gene │ │ │ │ -0004fac0: 7261 746f 7273 206f 6620 7468 6520 6772 rators of the gr │ │ │ │ -0004fad0: 6f75 700a 0a44 6573 6372 6970 7469 6f6e oup..Description │ │ │ │ -0004fae0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -0004faf0: 6973 2066 756e 6374 696f 6e20 6973 2070 is function is p │ │ │ │ -0004fb00: 726f 7669 6465 6420 6279 2074 6865 2070 rovided by the p │ │ │ │ -0004fb10: 6163 6b61 6765 202a 6e6f 7465 2049 6e76 ackage *note Inv │ │ │ │ -0004fb20: 6172 6961 6e74 5269 6e67 3a20 546f 702c ariantRing: Top, │ │ │ │ -0004fb30: 2e0a 0a54 6865 2066 6f6c 6c6f 7769 6e67 ...The following │ │ │ │ -0004fb40: 2065 7861 6d70 6c65 2063 6f6d 7075 7465 example compute │ │ │ │ -0004fb50: 732c 2066 6f72 2065 6163 6820 7065 726d s, for each perm │ │ │ │ -0004fb60: 7574 6174 696f 6e20 696e 2074 6865 2073 utation in the s │ │ │ │ -0004fb70: 796d 6d65 7472 6963 2067 726f 7570 206f ymmetric group o │ │ │ │ -0004fb80: 6e0a 7468 7265 6520 656c 656d 656e 7473 n.three elements │ │ │ │ -0004fb90: 2c20 6120 776f 7264 206f 6620 6d69 6e69 , a word of mini │ │ │ │ -0004fba0: 6d61 6c20 6c65 6e67 7468 2069 6e20 7468 mal length in th │ │ │ │ -0004fbb0: 6520 436f 7865 7465 7220 6765 6e65 7261 e Coxeter genera │ │ │ │ -0004fbc0: 746f 7273 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d tors...+-------- │ │ │ │ -0004fbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fc00: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5220 ------+.|i1 : R │ │ │ │ -0004fc10: 3d20 5151 5b78 5f31 2e2e 785f 335d 2020 = QQ[x_1..x_3] │ │ │ │ +0004f550: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0004f560: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0004f570: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0004f580: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0004f590: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +0004f5a0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0004f5b0: 2f70 6163 6b61 6765 732f 0a49 6e76 6172 /packages/.Invar │ │ │ │ +0004f5c0: 6961 6e74 5269 6e67 2f41 6265 6c69 616e iantRing/Abelian │ │ │ │ +0004f5d0: 4772 6f75 7073 446f 632e 6d32 3a33 3138 GroupsDoc.m2:318 │ │ │ │ +0004f5e0: 3a30 2e0a 1f0a 4669 6c65 3a20 496e 7661 :0....File: Inva │ │ │ │ +0004f5f0: 7269 616e 7452 696e 672e 696e 666f 2c20 riantRing.info, │ │ │ │ +0004f600: 4e6f 6465 3a20 776f 7264 732c 2050 7265 Node: words, Pre │ │ │ │ +0004f610: 763a 2077 6569 6768 7473 2c20 5570 3a20 v: weights, Up: │ │ │ │ +0004f620: 546f 700a 0a77 6f72 6473 202d 2d20 6173 Top..words -- as │ │ │ │ +0004f630: 736f 6369 6174 6520 6120 776f 7264 2069 sociate a word i │ │ │ │ +0004f640: 6e20 7468 6520 6765 6e65 7261 746f 7273 n the generators │ │ │ │ +0004f650: 206f 6620 6120 6772 6f75 7020 746f 2065 of a group to e │ │ │ │ +0004f660: 6163 6820 656c 656d 656e 740a 2a2a 2a2a ach element.**** │ │ │ │ +0004f670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004f680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004f690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004f6a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0004f6b0: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +0004f6c0: 2020 2020 2020 2020 776f 7264 7320 470a words G. │ │ │ │ +0004f6d0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0004f6e0: 2020 2a20 472c 2061 6e20 696e 7374 616e * G, an instan │ │ │ │ +0004f6f0: 6365 206f 6620 7468 6520 7479 7065 202a ce of the type * │ │ │ │ +0004f700: 6e6f 7465 2046 696e 6974 6547 726f 7570 note FiniteGroup │ │ │ │ +0004f710: 4163 7469 6f6e 3a20 4669 6e69 7465 4772 Action: FiniteGr │ │ │ │ +0004f720: 6f75 7041 6374 696f 6e2c 0a20 202a 204f oupAction,. * O │ │ │ │ +0004f730: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +0004f740: 6120 2a6e 6f74 6520 6861 7368 2074 6162 a *note hash tab │ │ │ │ +0004f750: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ +0004f760: 6329 4861 7368 5461 626c 652c 2c20 6173 c)HashTable,, as │ │ │ │ +0004f770: 736f 6369 6174 696e 6720 746f 2065 6163 sociating to eac │ │ │ │ +0004f780: 680a 2020 2020 2020 2020 656c 656d 656e h. elemen │ │ │ │ +0004f790: 7420 696e 2074 6865 2067 726f 7570 206f t in the group o │ │ │ │ +0004f7a0: 6620 7468 6520 6163 7469 6f6e 2061 2077 f the action a w │ │ │ │ +0004f7b0: 6f72 6420 6f66 206d 696e 696d 616c 206c ord of minimal l │ │ │ │ +0004f7c0: 656e 6774 6820 696e 2028 7468 650a 2020 ength in (the. │ │ │ │ +0004f7d0: 2020 2020 2020 696e 6469 6365 7320 6f66 indices of │ │ │ │ +0004f7e0: 2920 7468 6520 6765 6e65 7261 746f 7273 ) the generators │ │ │ │ +0004f7f0: 206f 6620 7468 6520 6772 6f75 700a 0a44 of the group..D │ │ │ │ +0004f800: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0004f810: 3d3d 3d3d 3d3d 0a0a 5468 6973 2066 756e ======..This fun │ │ │ │ +0004f820: 6374 696f 6e20 6973 2070 726f 7669 6465 ction is provide │ │ │ │ +0004f830: 6420 6279 2074 6865 2070 6163 6b61 6765 d by the package │ │ │ │ +0004f840: 202a 6e6f 7465 2049 6e76 6172 6961 6e74 *note Invariant │ │ │ │ +0004f850: 5269 6e67 3a20 546f 702c 2e0a 0a54 6865 Ring: Top,...The │ │ │ │ +0004f860: 2066 6f6c 6c6f 7769 6e67 2065 7861 6d70 following examp │ │ │ │ +0004f870: 6c65 2063 6f6d 7075 7465 732c 2066 6f72 le computes, for │ │ │ │ +0004f880: 2065 6163 6820 7065 726d 7574 6174 696f each permutatio │ │ │ │ +0004f890: 6e20 696e 2074 6865 2073 796d 6d65 7472 n in the symmetr │ │ │ │ +0004f8a0: 6963 2067 726f 7570 206f 6e0a 7468 7265 ic group on.thre │ │ │ │ +0004f8b0: 6520 656c 656d 656e 7473 2c20 6120 776f e elements, a wo │ │ │ │ +0004f8c0: 7264 206f 6620 6d69 6e69 6d61 6c20 6c65 rd of minimal le │ │ │ │ +0004f8d0: 6e67 7468 2069 6e20 7468 6520 436f 7865 ngth in the Coxe │ │ │ │ +0004f8e0: 7465 7220 6765 6e65 7261 746f 7273 2e0a ter generators.. │ │ │ │ +0004f8f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0004f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004f930: 2b0a 7c69 3120 3a20 5220 3d20 5151 5b78 +.|i1 : R = QQ[x │ │ │ │ +0004f940: 5f31 2e2e 785f 335d 2020 2020 2020 2020 _1..x_3] │ │ │ │ +0004f950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f970: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f9b0: 2020 7c0a 7c6f 3120 3d20 5220 2020 2020 |.|o1 = R │ │ │ │ +0004f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004f9f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004fa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004fa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004fa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004fa30: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79 |.|o1 : Poly │ │ │ │ +0004fa40: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +0004fa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004fa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004fa70: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0004fa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004fa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004faa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004fab0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 4c20 ------+.|i2 : L │ │ │ │ +0004fac0: 3d20 6170 706c 7928 322c 2069 202d 3e20 = apply(2, i -> │ │ │ │ +0004fad0: 7065 726d 7574 6174 696f 6e4d 6174 7269 permutationMatri │ │ │ │ +0004fae0: 7828 332c 205b 6920 2b20 312c 2069 202b x(3, [i + 1, i + │ │ │ │ +0004faf0: 2032 5d20 2920 297c 0a7c 2020 2020 2020 2] ) )|.| │ │ │ │ +0004fb00: 2020 2020 2020 2020 2020 2020 2020 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2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004fc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fc40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0004fc30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0004fc40: 3220 3a20 4c69 7374 2020 2020 2020 2020 2 : List │ │ │ │ 0004fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fc80: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -0004fc90: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -0004fca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fcc0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0004fcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004fc70: 2020 2020 2020 2020 2020 2020 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2020 2020 2020 2020 2020 │ │ │ │ -0004fd40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0004fd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004fd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0004fd90: 3220 3a20 4c20 3d20 6170 706c 7928 322c 2 : L = apply(2, │ │ │ │ -0004fda0: 2069 202d 3e20 7065 726d 7574 6174 696f i -> permutatio │ │ │ │ -0004fdb0: 6e4d 6174 7269 7828 332c 205b 6920 2b20 nMatrix(3, [i + │ │ │ │ -0004fdc0: 312c 2069 202b 2032 5d20 2920 297c 0a7c 1, i + 2] ) )|.| │ │ │ │ -0004fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004fd40: 7c0a 7c6f 3320 3d20 5220 3c2d 203c 7c20 |.|o3 = R <- <| │ │ │ │ +0004fd50: 3020 3120 3020 7c2c 207c 2031 2030 2030 0 1 0 |, | 1 0 0 │ │ │ │ +0004fd60: 207c 3e20 2020 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│ │ 0004fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fe40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0004fe50: 0a7c 2020 2020 2020 7c20 3120 3020 3020 .| | 1 0 0 │ │ │ │ -0004fe60: 7c20 207c 2030 2030 2031 207c 2020 2020 | | 0 0 1 | │ │ │ │ +0004fe40: 2020 2020 7c0a 7c6f 3320 3a20 4669 6e69 |.|o3 : Fini │ │ │ │ +0004fe50: 7465 4772 6f75 7041 6374 696f 6e20 2020 teGroupAction │ │ │ │ +0004fe60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004fe70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fe80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fe90: 7c0a 7c20 2020 2020 207c 2030 2030 2031 |.| | 0 0 1 │ │ │ │ -0004fea0: 207c 2020 7c20 3020 3120 3020 7c20 2020 | | 0 1 0 | │ │ │ │ -0004feb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004fed0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004fe80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0004fe90: 2d2d 2d2d 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│ │ │ -00050150: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00050160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00050170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00050180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00050190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -000501a0: 3420 3a20 776f 7264 7320 5333 2020 2020 4 : words S3 │ │ │ │ -000501b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00050150: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00050160: 2020 7c20 3120 3020 3020 7c20 2020 2020 | 1 0 0 | │ │ │ │ +00050170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00050180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00050190: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000501a0: 2020 207c 2030 2031 2030 207c 203d 3e20 | 0 1 0 | => │ │ │ │ +000501b0: 7b30 7d20 2020 2020 2020 2020 2020 2020 {0} │ │ │ │ 000501c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000501d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000501e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000501d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000501e0: 2020 2020 7c20 3120 3020 3020 7c20 2020 | 1 0 0 | │ │ │ │ 000501f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00050200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050210: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00050220: 7c6f 3420 3d20 4861 7368 5461 626c 657b |o4 = HashTable{ │ │ │ │ -00050230: 7c20 3020 3020 3120 7c20 3d3e 207b 302c | 0 0 1 | => {0, │ │ │ │ -00050240: 2031 2c20 307d 7d20 2020 2020 2020 2020 1, 0}} │ │ │ │ -00050250: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00050260: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00050270: 207c 2030 2031 2030 207c 2020 2020 2020 | 0 1 0 | │ │ │ │ +00050210: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00050220: 2020 2020 207c 2030 2030 2031 207c 2020 | 0 0 1 | │ │ │ │ +00050230: 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2020 |.| │ │ │ │ +00050360: 2020 2020 2020 2020 2020 7c20 3020 3120 | 0 1 │ │ │ │ +00050370: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ 00050380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000503a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000503b0: 2020 2020 2020 7c20 3020 3120 3020 7c20 | 0 1 0 | │ │ │ │ -000503c0: 3d3e 207b 312c 2030 7d20 2020 2020 2020 => {1, 0} │ │ │ │ -000503d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000503e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000503f0: 2020 2020 2020 207c 2030 2030 2031 207c | 0 0 1 | │ │ │ │ +00050390: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000503a0: 2020 2020 2020 2020 2020 207c 2030 2030 | 0 0 │ │ │ │ +000503b0: 2031 207c 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ +000503c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000503d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000503e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000503f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00050400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050420: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00050430: 2020 2020 2020 2020 7c20 3120 3020 3020 | 1 0 0 │ │ │ │ -00050440: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00050450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00050470: 2020 2020 2020 2020 207c 2030 2031 2030 | 0 1 0 │ │ │ │ -00050480: 207c 203d 3e20 7b30 7d20 2020 2020 2020 | => {0} │ │ │ │ -00050490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000504a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000504b0: 2020 2020 2020 2020 2020 7c20 3120 3020 | 1 0 │ │ │ │ -000504c0: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -000504d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000504e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000504f0: 2020 2020 2020 2020 2020 207c 2030 2030 | 0 0 │ │ │ │ -00050500: 2031 207c 2020 2020 2020 2020 2020 2020 1 | │ │ │ │ -00050510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050520: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00050530: 2020 2020 2020 2020 2020 2020 7c20 3120 | 1 │ │ │ │ -00050540: 3020 3020 7c20 3d3e 207b 317d 2020 2020 0 0 | => {1} │ │ │ │ -00050550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050560: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00050570: 2020 2020 2020 2020 2020 2020 207c 2030 | 0 │ │ │ │ -00050580: 2030 2031 207c 2020 2020 2020 2020 2020 0 1 | │ │ │ │ -00050590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000505a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000505b0: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ -000505c0: 3020 3120 3020 7c20 2020 2020 2020 2020 0 1 0 | │ │ │ │ -000505d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000505e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000505f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00050600: 2031 2030 2030 207c 203d 3e20 7b7d 2020 1 0 0 | => {} │ │ │ │ -00050610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050620: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00050630: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00050640: 7c20 3020 3120 3020 7c20 2020 2020 2020 | 0 1 0 | │ │ │ │ -00050650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050660: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00050670: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00050680: 207c 2030 2030 2031 207c 2020 2020 2020 | 0 0 1 | │ │ │ │ -00050690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000506a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000506b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000506c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000506d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000506e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000506f0: 207c 0a7c 6f34 203a 2048 6173 6854 6162 |.|o4 : HashTab │ │ │ │ -00050700: 6c65 2020 2020 2020 2020 2020 2020 2020 le │ │ │ │ -00050710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00050730: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00050410: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +00050420: 203a 2048 6173 6854 6162 6c65 2020 2020 : HashTable │ │ │ │ +00050430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00050440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00050450: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00050460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00050470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00050480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00050490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +000504a0: 5468 6520 636f 6d70 7574 6174 696f 6e20 The computation │ │ │ │ +000504b0: 6f66 2074 6865 2077 6f72 6473 2061 6464 of the words add │ │ │ │ +000504c0: 7265 7373 696e 6720 6561 6368 2065 6c65 ressing each ele │ │ │ │ +000504d0: 6d65 6e74 2069 6e20 7468 6520 6772 6f75 ment in the grou │ │ │ │ +000504e0: 7020 6973 2061 6374 7561 6c6c 790a 7065 p is actually.pe │ │ │ │ +000504f0: 7266 6f72 6d65 6420 6279 2074 6865 206d rformed by the m │ │ │ │ +00050500: 6574 686f 6420 2a6e 6f74 6520 7363 6872 ethod *note schr │ │ │ │ +00050510: 6569 6572 4772 6170 683a 2073 6368 7265 eierGraph: schre │ │ │ │ +00050520: 6965 7247 7261 7068 2c20 7369 6e63 6520 ierGraph, since │ │ │ │ +00050530: 7468 6520 7072 6f63 6573 730a 6f66 2063 the process.of c │ │ │ │ +00050540: 6f6d 7075 7469 6e67 2074 6865 2053 6368 omputing the Sch │ │ │ │ +00050550: 7265 6965 7220 6772 6170 6820 6f66 2074 reier graph of t │ │ │ │ +00050560: 6865 2067 726f 7570 2079 6965 6c64 7320 he group yields │ │ │ │ +00050570: 6f74 6865 7220 7573 6566 756c 2069 6e66 other useful inf │ │ │ │ +00050580: 6f72 6d61 7469 6f6e 0a61 626f 7574 2074 ormation.about t │ │ │ │ +00050590: 6865 2067 726f 7570 2e0a 0a53 6565 2061 he group...See a │ │ │ │ +000505a0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +000505b0: 2a20 2a6e 6f74 6520 6772 6f75 703a 2067 * *note group: g │ │ │ │ +000505c0: 726f 7570 2c20 2d2d 206c 6973 7420 616c roup, -- list al │ │ │ │ +000505d0: 6c20 656c 656d 656e 7473 206f 6620 7468 l elements of th │ │ │ │ +000505e0: 6520 6772 6f75 7020 6f66 2061 2066 696e e group of a fin │ │ │ │ +000505f0: 6974 6520 6772 6f75 700a 2020 2020 6163 ite group. ac │ │ │ │ +00050600: 7469 6f6e 0a20 202a 202a 6e6f 7465 2073 tion. * *note s │ │ │ │ +00050610: 6368 7265 6965 7247 7261 7068 3a20 7363 chreierGraph: sc │ │ │ │ +00050620: 6872 6569 6572 4772 6170 682c 202d 2d20 hreierGraph, -- │ │ │ │ +00050630: 5363 6872 6569 6572 2067 7261 7068 206f Schreier graph o │ │ │ │ +00050640: 6620 6120 6669 6e69 7465 2067 726f 7570 f a finite group │ │ │ │ +00050650: 0a0a 5761 7973 2074 6f20 7573 6520 776f ..Ways to use wo │ │ │ │ +00050660: 7264 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d rds:.=========== │ │ │ │ +00050670: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 776f =======.. * "wo │ │ │ │ +00050680: 7264 7328 4669 6e69 7465 4772 6f75 7041 rds(FiniteGroupA │ │ │ │ +00050690: 6374 696f 6e29 220a 0a46 6f72 2074 6865 ction)"..For the │ │ │ │ +000506a0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +000506b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +000506c0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +000506d0: 2077 6f72 6473 3a20 776f 7264 732c 2069 words: words, i │ │ │ │ +000506e0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +000506f0: 2066 756e 6374 696f 6e20 7769 7468 206f function with o │ │ │ │ +00050700: 7074 696f 6e73 3a0a 284d 6163 6175 6c61 ptions:.(Macaula │ │ │ │ +00050710: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +00050720: 7469 6f6e 5769 7468 4f70 7469 6f6e 732c tionWithOptions, │ │ │ │ +00050730: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 00050740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00050750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00050760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00050770: 2d2d 2d2b 0a0a 5468 6520 636f 6d70 7574 ---+..The comput │ │ │ │ -00050780: 6174 696f 6e20 6f66 2074 6865 2077 6f72 ation of the wor │ │ │ │ -00050790: 6473 2061 6464 7265 7373 696e 6720 6561 ds addressing ea │ │ │ │ -000507a0: 6368 2065 6c65 6d65 6e74 2069 6e20 7468 ch element in th │ │ │ │ -000507b0: 6520 6772 6f75 7020 6973 2061 6374 7561 e group is actua │ │ │ │ -000507c0: 6c6c 790a 7065 7266 6f72 6d65 6420 6279 lly.performed by │ │ │ │ -000507d0: 2074 6865 206d 6574 686f 6420 2a6e 6f74 the method *not │ │ │ │ -000507e0: 6520 7363 6872 6569 6572 4772 6170 683a e schreierGraph: │ │ │ │ -000507f0: 2073 6368 7265 6965 7247 7261 7068 2c20 schreierGraph, │ │ │ │ -00050800: 7369 6e63 6520 7468 6520 7072 6f63 6573 since the proces │ │ │ │ -00050810: 730a 6f66 2063 6f6d 7075 7469 6e67 2074 s.of computing t │ │ │ │ -00050820: 6865 2053 6368 7265 6965 7220 6772 6170 he Schreier grap │ │ │ │ -00050830: 6820 6f66 2074 6865 2067 726f 7570 2079 h of the group y │ │ │ │ -00050840: 6965 6c64 7320 6f74 6865 7220 7573 6566 ields other usef │ │ │ │ -00050850: 756c 2069 6e66 6f72 6d61 7469 6f6e 0a61 ul information.a │ │ │ │ -00050860: 626f 7574 2074 6865 2067 726f 7570 2e0a bout the group.. │ │ │ │ -00050870: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -00050880: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6772 ==.. * *note gr │ │ │ │ -00050890: 6f75 703a 2067 726f 7570 2c20 2d2d 206c oup: group, -- l │ │ │ │ -000508a0: 6973 7420 616c 6c20 656c 656d 656e 7473 ist all elements │ │ │ │ -000508b0: 206f 6620 7468 6520 6772 6f75 7020 6f66 of the group of │ │ │ │ -000508c0: 2061 2066 696e 6974 6520 6772 6f75 700a a finite group. │ │ │ │ -000508d0: 2020 2020 6163 7469 6f6e 0a20 202a 202a action. * * │ │ │ │ -000508e0: 6e6f 7465 2073 6368 7265 6965 7247 7261 note schreierGra │ │ │ │ -000508f0: 7068 3a20 7363 6872 6569 6572 4772 6170 ph: schreierGrap │ │ │ │ -00050900: 682c 202d 2d20 5363 6872 6569 6572 2067 h, -- Schreier g │ │ │ │ -00050910: 7261 7068 206f 6620 6120 6669 6e69 7465 raph of a finite │ │ │ │ -00050920: 2067 726f 7570 0a0a 5761 7973 2074 6f20 group..Ways to │ │ │ │ -00050930: 7573 6520 776f 7264 733a 0a3d 3d3d 3d3d use words:.===== │ │ │ │ -00050940: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -00050950: 202a 2022 776f 7264 7328 4669 6e69 7465 * "words(Finite │ │ │ │ -00050960: 4772 6f75 7041 6374 696f 6e29 220a 0a46 GroupAction)"..F │ │ │ │ -00050970: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00050980: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00050990: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -000509a0: 202a 6e6f 7465 2077 6f72 6473 3a20 776f *note words: wo │ │ │ │ -000509b0: 7264 732c 2069 7320 6120 2a6e 6f74 6520 rds, is a *note │ │ │ │ -000509c0: 6d65 7468 6f64 2066 756e 6374 696f 6e20 method function │ │ │ │ -000509d0: 7769 7468 206f 7074 696f 6e73 3a0a 284d with options:.(M │ │ │ │ -000509e0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -000509f0: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70 odFunctionWithOp │ │ │ │ -00050a00: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d tions,...------- │ │ │ │ -00050a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00050a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00050a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00050a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00050a50: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00050a60: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00050a70: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00050a80: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00050a90: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00050aa0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -00050ab0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00050ac0: 0a49 6e76 6172 6961 6e74 5269 6e67 2f46 .InvariantRing/F │ │ │ │ -00050ad0: 696e 6974 6547 726f 7570 7344 6f63 2e6d initeGroupsDoc.m │ │ │ │ -00050ae0: 323a 3239 383a 302e 0a1f 0a54 6167 2054 2:298:0....Tag T │ │ │ │ -00050af0: 6162 6c65 3a0a 4e6f 6465 3a20 546f 707f able:.Node: Top. │ │ │ │ -00050b00: 3233 390a 4e6f 6465 3a20 6163 7469 6f6e 239.Node: action │ │ │ │ -00050b10: 7f32 3139 3231 0a4e 6f64 653a 2061 6374 .21921.Node: act │ │ │ │ -00050b20: 696f 6e4d 6174 7269 787f 3235 3331 310a ionMatrix.25311. │ │ │ │ -00050b30: 4e6f 6465 3a20 616d 6269 656e 745f 6c70 Node: ambient_lp │ │ │ │ -00050b40: 5269 6e67 4f66 496e 7661 7269 616e 7473 RingOfInvariants │ │ │ │ -00050b50: 5f72 707f 3239 3138 390a 4e6f 6465 3a20 _rp.29189.Node: │ │ │ │ -00050b60: 6379 636c 6963 4661 6374 6f72 737f 3332 cyclicFactors.32 │ │ │ │ -00050b70: 3330 310a 4e6f 6465 3a20 6465 6669 6e69 301.Node: defini │ │ │ │ -00050b80: 6e67 4964 6561 6c7f 3334 3938 360a 4e6f ngIdeal.34986.No │ │ │ │ -00050b90: 6465 3a20 6465 6772 6565 7352 696e 675f de: degreesRing_ │ │ │ │ -00050ba0: 6c70 4469 6167 6f6e 616c 4163 7469 6f6e lpDiagonalAction │ │ │ │ -00050bb0: 5f72 707f 3338 3232 310a 4e6f 6465 3a20 _rp.38221.Node: │ │ │ │ -00050bc0: 4469 6167 6f6e 616c 4163 7469 6f6e 7f34 DiagonalAction.4 │ │ │ │ -00050bd0: 3133 3632 0a4e 6f64 653a 2064 6961 676f 1362.Node: diago │ │ │ │ -00050be0: 6e61 6c41 6374 696f 6e7f 3434 3539 390a nalAction.44599. │ │ │ │ -00050bf0: 4e6f 6465 3a20 6469 6d5f 6c70 4772 6f75 Node: dim_lpGrou │ │ │ │ -00050c00: 7041 6374 696f 6e5f 7270 7f35 3139 3837 pAction_rp.51987 │ │ │ │ -00050c10: 0a4e 6f64 653a 2065 7175 6976 6172 6961 .Node: equivaria │ │ │ │ -00050c20: 6e74 4869 6c62 6572 747f 3534 3333 360a ntHilbert.54336. │ │ │ │ -00050c30: 4e6f 6465 3a20 6571 7569 7661 7269 616e Node: equivarian │ │ │ │ -00050c40: 7448 696c 6265 7274 5365 7269 6573 7f35 tHilbertSeries.5 │ │ │ │ -00050c50: 3939 3737 0a4e 6f64 653a 2066 696e 6974 9977.Node: finit │ │ │ │ -00050c60: 6541 6374 696f 6e7f 3639 3036 380a 4e6f eAction.69068.No │ │ │ │ -00050c70: 6465 3a20 4669 6e69 7465 4772 6f75 7041 de: FiniteGroupA │ │ │ │ -00050c80: 6374 696f 6e7f 3732 3932 380a 4e6f 6465 ction.72928.Node │ │ │ │ -00050c90: 3a20 6765 6e65 7261 746f 7273 5f6c 7046 : generators_lpF │ │ │ │ -00050ca0: 696e 6974 6547 726f 7570 4163 7469 6f6e initeGroupAction │ │ │ │ -00050cb0: 5f72 707f 3737 3237 380a 4e6f 6465 3a20 _rp.77278.Node: │ │ │ │ -00050cc0: 6765 6e65 7261 746f 7273 5f6c 7052 696e generators_lpRin │ │ │ │ -00050cd0: 674f 6649 6e76 6172 6961 6e74 735f 7270 gOfInvariants_rp │ │ │ │ -00050ce0: 7f38 3133 3035 0a4e 6f64 653a 2067 726f .81305.Node: gro │ │ │ │ -00050cf0: 7570 7f38 3433 3732 0a4e 6f64 653a 2047 up.84372.Node: G │ │ │ │ -00050d00: 726f 7570 4163 7469 6f6e 7f38 3832 3533 roupAction.88253 │ │ │ │ -00050d10: 0a4e 6f64 653a 2067 726f 7570 4964 6561 .Node: groupIdea │ │ │ │ -00050d20: 6c7f 3931 3436 320a 4e6f 6465 3a20 6869 l.91462.Node: hi │ │ │ │ -00050d30: 6c62 6572 7449 6465 616c 7f39 3532 3633 lbertIdeal.95263 │ │ │ │ -00050d40: 0a4e 6f64 653a 2068 696c 6265 7274 5365 .Node: hilbertSe │ │ │ │ -00050d50: 7269 6573 5f6c 7052 696e 674f 6649 6e76 ries_lpRingOfInv │ │ │ │ -00050d60: 6172 6961 6e74 735f 7270 7f31 3033 3532 ariants_rp.10352 │ │ │ │ -00050d70: 360a 4e6f 6465 3a20 6869 726f 6e61 6b61 6.Node: hironaka │ │ │ │ -00050d80: 4465 636f 6d70 6f73 6974 696f 6e7f 3130 Decomposition.10 │ │ │ │ -00050d90: 3636 3530 0a4e 6f64 653a 2068 736f 7020 6650.Node: hsop │ │ │ │ -00050da0: 616c 676f 7269 7468 6d73 7f31 3132 3038 algorithms.11208 │ │ │ │ -00050db0: 340a 4e6f 6465 3a20 696e 7661 7269 616e 4.Node: invarian │ │ │ │ -00050dc0: 7452 696e 677f 3133 3439 3236 0a4e 6f64 tRing.134926.Nod │ │ │ │ -00050dd0: 653a 2069 6e76 6172 6961 6e74 737f 3133 e: invariants.13 │ │ │ │ -00050de0: 3933 3236 0a4e 6f64 653a 2069 6e76 6172 9326.Node: invar │ │ │ │ -00050df0: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ -00050e00: 645f 636d 4465 6772 6565 426f 756e 643d d_cmDegreeBound= │ │ │ │ -00050e10: 3e5f 7064 5f70 645f 7064 5f72 707f 3134 >_pd_pd_pd_rp.14 │ │ │ │ -00050e20: 3433 3039 0a4e 6f64 653a 2069 6e76 6172 4309.Node: invar │ │ │ │ -00050e30: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ -00050e40: 645f 636d 4465 6772 6565 4c69 6d69 743d d_cmDegreeLimit= │ │ │ │ -00050e50: 3e5f 7064 5f70 645f 7064 5f72 707f 3135 >_pd_pd_pd_rp.15 │ │ │ │ -00050e60: 3037 3638 0a4e 6f64 653a 2069 6e76 6172 0768.Node: invar │ │ │ │ -00050e70: 6961 6e74 735f 6c70 5f70 645f 7064 5f70 iants_lp_pd_pd_p │ │ │ │ -00050e80: 645f 636d 5374 7261 7465 6779 3d3e 5f70 d_cmStrategy=>_p │ │ │ │ -00050e90: 645f 7064 5f70 645f 7270 7f31 3534 3532 d_pd_pd_rp.15452 │ │ │ │ -00050ea0: 310a 4e6f 6465 3a20 696e 7661 7269 616e 1.Node: invarian │ │ │ │ -00050eb0: 7473 5f6c 705f 7064 5f70 645f 7064 5f63 ts_lp_pd_pd_pd_c │ │ │ │ -00050ec0: 6d53 7562 7269 6e67 4c69 6d69 743d 3e5f mSubringLimit=>_ │ │ │ │ -00050ed0: 7064 5f70 645f 7064 5f72 707f 3136 3532 pd_pd_pd_rp.1652 │ │ │ │ -00050ee0: 3037 0a4e 6f64 653a 2069 6e76 6172 6961 07.Node: invaria │ │ │ │ -00050ef0: 6e74 735f 6c70 5f70 645f 7064 5f70 645f nts_lp_pd_pd_pd_ │ │ │ │ -00050f00: 636d 5573 6543 6f65 6666 6963 6965 6e74 cmUseCoefficient │ │ │ │ -00050f10: 5269 6e67 3d3e 5f70 645f 7064 5f70 645f Ring=>_pd_pd_pd_ │ │ │ │ -00050f20: 7270 7f31 3637 3337 300a 4e6f 6465 3a20 rp.167370.Node: │ │ │ │ -00050f30: 696e 7661 7269 616e 7473 5f6c 705f 7064 invariants_lp_pd │ │ │ │ -00050f40: 5f70 645f 7064 5f63 6d55 7365 506f 6c79 _pd_pd_cmUsePoly │ │ │ │ -00050f50: 6865 6472 613d 3e5f 7064 5f70 645f 7064 hedra=>_pd_pd_pd │ │ │ │ -00050f60: 5f72 707f 3137 3136 3832 0a4e 6f64 653a _rp.171682.Node: │ │ │ │ -00050f70: 2069 6e76 6172 6961 6e74 735f 6c70 4469 invariants_lpDi │ │ │ │ -00050f80: 6167 6f6e 616c 4163 7469 6f6e 5f72 707f agonalAction_rp. │ │ │ │ -00050f90: 3137 3339 3132 0a4e 6f64 653a 2069 6e76 173912.Node: inv │ │ │ │ -00050fa0: 6172 6961 6e74 735f 6c70 4669 6e69 7465 ariants_lpFinite │ │ │ │ -00050fb0: 4772 6f75 7041 6374 696f 6e5f 7270 7f31 GroupAction_rp.1 │ │ │ │ -00050fc0: 3833 3733 390a 4e6f 6465 3a20 696e 7661 83739.Node: inva │ │ │ │ -00050fd0: 7269 616e 7473 5f6c 7046 696e 6974 6547 riants_lpFiniteG │ │ │ │ -00050fe0: 726f 7570 4163 7469 6f6e 5f63 6d5a 5a5f roupAction_cmZZ_ │ │ │ │ -00050ff0: 7270 7f31 3931 3537 310a 4e6f 6465 3a20 rp.191571.Node: │ │ │ │ -00051000: 696e 7661 7269 616e 7473 5f6c 704c 696e invariants_lpLin │ │ │ │ -00051010: 6561 726c 7952 6564 7563 7469 7665 4163 earlyReductiveAc │ │ │ │ -00051020: 7469 6f6e 5f72 707f 3139 3639 3031 0a4e tion_rp.196901.N │ │ │ │ -00051030: 6f64 653a 2069 6e76 6172 6961 6e74 735f ode: invariants_ │ │ │ │ -00051040: 6c70 4c69 6e65 6172 6c79 5265 6475 6374 lpLinearlyReduct │ │ │ │ -00051050: 6976 6541 6374 696f 6e5f 636d 5a5a 5f72 iveAction_cmZZ_r │ │ │ │ -00051060: 707f 3230 3339 3538 0a4e 6f64 653a 2069 p.203958.Node: i │ │ │ │ -00051070: 7341 6265 6c69 616e 7f32 3131 3934 310a sAbelian.211941. │ │ │ │ -00051080: 4e6f 6465 3a20 6973 496e 7661 7269 616e Node: isInvarian │ │ │ │ -00051090: 747f 3231 3735 3734 0a4e 6f64 653a 204c t.217574.Node: L │ │ │ │ -000510a0: 696e 6561 726c 7952 6564 7563 7469 7665 inearlyReductive │ │ │ │ -000510b0: 4163 7469 6f6e 7f32 3238 3938 310a 4e6f Action.228981.No │ │ │ │ -000510c0: 6465 3a20 6c69 6e65 6172 6c79 5265 6475 de: linearlyRedu │ │ │ │ -000510d0: 6374 6976 6541 6374 696f 6e7f 3233 3234 ctiveAction.2324 │ │ │ │ -000510e0: 3935 0a4e 6f64 653a 206d 6f6c 6965 6e53 95.Node: molienS │ │ │ │ -000510f0: 6572 6965 737f 3233 3931 3132 0a4e 6f64 eries.239112.Nod │ │ │ │ -00051100: 653a 206e 6574 5f6c 7052 696e 674f 6649 e: net_lpRingOfI │ │ │ │ -00051110: 6e76 6172 6961 6e74 735f 7270 7f32 3431 nvariants_rp.241 │ │ │ │ -00051120: 3538 390a 4e6f 6465 3a20 6e75 6d67 656e 589.Node: numgen │ │ │ │ -00051130: 735f 6c70 4469 6167 6f6e 616c 4163 7469 s_lpDiagonalActi │ │ │ │ -00051140: 6f6e 5f72 707f 3234 3234 3835 0a4e 6f64 on_rp.242485.Nod │ │ │ │ -00051150: 653a 206e 756d 6765 6e73 5f6c 7046 696e e: numgens_lpFin │ │ │ │ -00051160: 6974 6547 726f 7570 4163 7469 6f6e 5f72 iteGroupAction_r │ │ │ │ -00051170: 707f 3234 3530 3931 0a4e 6f64 653a 2070 p.245091.Node: p │ │ │ │ -00051180: 6572 6d75 7461 7469 6f6e 4d61 7472 6978 ermutationMatrix │ │ │ │ -00051190: 7f32 3438 3838 340a 4e6f 6465 3a20 7072 .248884.Node: pr │ │ │ │ -000511a0: 696d 6172 7949 6e76 6172 6961 6e74 737f imaryInvariants. │ │ │ │ -000511b0: 3235 3239 3530 0a4e 6f64 653a 2070 7269 252950.Node: pri │ │ │ │ -000511c0: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ -000511d0: 705f 7064 5f70 645f 7064 5f63 6d44 6164 p_pd_pd_pd_cmDad │ │ │ │ -000511e0: 653d 3e5f 7064 5f70 645f 7064 5f72 707f e=>_pd_pd_pd_rp. │ │ │ │ -000511f0: 3236 3232 3139 0a4e 6f64 653a 2070 7269 262219.Node: pri │ │ │ │ -00051200: 6d61 7279 496e 7661 7269 616e 7473 5f6c maryInvariants_l │ │ │ │ -00051210: 705f 7064 5f70 645f 7064 5f63 6d44 6567 p_pd_pd_pd_cmDeg │ │ │ │ -00051220: 7265 6556 6563 746f 723d 3e5f 7064 5f70 reeVector=>_pd_p │ │ │ │ -00051230: 645f 7064 5f72 707f 3237 3733 3135 0a4e d_pd_rp.277315.N │ │ │ │ -00051240: 6f64 653a 2072 616e 6b5f 6c70 4469 6167 ode: rank_lpDiag │ │ │ │ -00051250: 6f6e 616c 4163 7469 6f6e 5f72 707f 3238 onalAction_rp.28 │ │ │ │ -00051260: 3338 3030 0a4e 6f64 653a 2072 656c 6174 3800.Node: relat │ │ │ │ -00051270: 696f 6e73 5f6c 7046 696e 6974 6547 726f ions_lpFiniteGro │ │ │ │ -00051280: 7570 4163 7469 6f6e 5f72 707f 3238 3633 upAction_rp.2863 │ │ │ │ -00051290: 3834 0a4e 6f64 653a 2072 6579 6e6f 6c64 84.Node: reynold │ │ │ │ -000512a0: 734f 7065 7261 746f 727f 3239 3038 3930 sOperator.290890 │ │ │ │ -000512b0: 0a4e 6f64 653a 2072 696e 675f 6c70 4772 .Node: ring_lpGr │ │ │ │ -000512c0: 6f75 7041 6374 696f 6e5f 7270 7f32 3937 oupAction_rp.297 │ │ │ │ -000512d0: 3135 340a 4e6f 6465 3a20 5269 6e67 4f66 154.Node: RingOf │ │ │ │ -000512e0: 496e 7661 7269 616e 7473 7f32 3939 3434 Invariants.29944 │ │ │ │ -000512f0: 350a 4e6f 6465 3a20 7363 6872 6569 6572 5.Node: schreier │ │ │ │ -00051300: 4772 6170 687f 3330 3230 3433 0a4e 6f64 Graph.302043.Nod │ │ │ │ -00051310: 653a 2073 6563 6f6e 6461 7279 496e 7661 e: secondaryInva │ │ │ │ -00051320: 7269 616e 7473 7f33 3038 3931 360a 4e6f riants.308916.No │ │ │ │ -00051330: 6465 3a20 7365 636f 6e64 6172 7949 6e76 de: secondaryInv │ │ │ │ -00051340: 6172 6961 6e74 735f 6c70 5f70 645f 7064 ariants_lp_pd_pd │ │ │ │ -00051350: 5f70 645f 636d 5072 696e 7444 6567 7265 _pd_cmPrintDegre │ │ │ │ -00051360: 6550 6f6c 796e 6f6d 6961 6c3d 3e5f 7064 ePolynomial=>_pd │ │ │ │ -00051370: 5f70 645f 7064 5f72 707f 3331 3238 3433 _pd_pd_rp.312843 │ │ │ │ -00051380: 0a4e 6f64 653a 2074 6578 4d61 7468 5f6c .Node: texMath_l │ │ │ │ -00051390: 7052 696e 674f 6649 6e76 6172 6961 6e74 pRingOfInvariant │ │ │ │ -000513a0: 735f 7270 7f33 3230 3632 320a 4e6f 6465 s_rp.320622.Node │ │ │ │ -000513b0: 3a20 7765 6967 6874 737f 3332 3136 3238 : weights.321628 │ │ │ │ -000513c0: 0a4e 6f64 653a 2077 6f72 6473 7f33 3235 .Node: words.325 │ │ │ │ -000513d0: 3831 380a 1f0a 456e 6420 5461 6720 5461 818...End Tag Ta │ │ │ │ -000513e0: 626c 650a ble. │ │ │ │ +00050770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00050780: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00050790: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +000507a0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +000507b0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +000507c0: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +000507d0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +000507e0: 2f70 6163 6b61 6765 732f 0a49 6e76 6172 /packages/.Invar │ │ │ │ +000507f0: 6961 6e74 5269 6e67 2f46 696e 6974 6547 iantRing/FiniteG │ │ │ │ +00050800: 726f 7570 7344 6f63 2e6d 323a 3239 383a roupsDoc.m2:298: │ │ │ │ +00050810: 302e 0a1f 0a54 6167 2054 6162 6c65 3a0a 0....Tag Table:. │ │ │ │ +00050820: 4e6f 6465 3a20 546f 707f 3233 390a 4e6f Node: Top.239.No │ │ │ │ +00050830: 6465 3a20 6163 7469 6f6e 7f32 3139 3231 de: action.21921 │ │ │ │ +00050840: 0a4e 6f64 653a 2061 6374 696f 6e4d 6174 .Node: actionMat │ │ │ │ +00050850: 7269 787f 3235 3331 310a 4e6f 6465 3a20 rix.25311.Node: │ │ │ │ +00050860: 616d 6269 656e 745f 6c70 5269 6e67 4f66 ambient_lpRingOf │ │ │ │ +00050870: 496e 7661 7269 616e 7473 5f72 707f 3239 Invariants_rp.29 │ │ │ │ +00050880: 3138 390a 4e6f 6465 3a20 6379 636c 6963 189.Node: cyclic │ │ │ │ +00050890: 4661 6374 6f72 737f 3332 3330 310a 4e6f Factors.32301.No │ │ │ │ +000508a0: 6465 3a20 6465 6669 6e69 6e67 4964 6561 de: definingIdea │ │ │ │ +000508b0: 6c7f 3334 3938 360a 4e6f 6465 3a20 6465 l.34986.Node: de │ │ │ │ +000508c0: 6772 6565 7352 696e 675f 6c70 4469 6167 greesRing_lpDiag │ │ │ │ +000508d0: 6f6e 616c 4163 7469 6f6e 5f72 707f 3338 onalAction_rp.38 │ │ │ │ +000508e0: 3232 310a 4e6f 6465 3a20 4469 6167 6f6e 221.Node: Diagon │ │ │ │ +000508f0: 616c 4163 7469 6f6e 7f34 3133 3632 0a4e alAction.41362.N │ │ │ │ +00050900: 6f64 653a 2064 6961 676f 6e61 6c41 6374 ode: diagonalAct │ │ │ │ +00050910: 696f 6e7f 3434 3539 390a 4e6f 6465 3a20 ion.44599.Node: │ │ │ │ +00050920: 6469 6d5f 6c70 4772 6f75 7041 6374 696f dim_lpGroupActio │ │ │ │ +00050930: 6e5f 7270 7f35 3139 3837 0a4e 6f64 653a n_rp.51987.Node: │ │ │ │ +00050940: 2065 7175 6976 6172 6961 6e74 4869 6c62 equivariantHilb │ │ │ │ +00050950: 6572 747f 3534 3333 360a 4e6f 6465 3a20 ert.54336.Node: │ │ │ │ +00050960: 6571 7569 7661 7269 616e 7448 696c 6265 equivariantHilbe │ │ │ │ +00050970: 7274 5365 7269 6573 7f35 3939 3737 0a4e rtSeries.59977.N │ │ │ │ +00050980: 6f64 653a 2066 696e 6974 6541 6374 696f ode: finiteActio │ │ │ │ +00050990: 6e7f 3639 3036 380a 4e6f 6465 3a20 4669 n.69068.Node: Fi │ │ │ │ +000509a0: 6e69 7465 4772 6f75 7041 6374 696f 6e7f niteGroupAction. │ │ │ │ +000509b0: 3732 3932 380a 4e6f 6465 3a20 6765 6e65 72928.Node: gene │ │ │ │ +000509c0: 7261 746f 7273 5f6c 7046 696e 6974 6547 rators_lpFiniteG │ │ │ │ +000509d0: 726f 7570 4163 7469 6f6e 5f72 707f 3737 roupAction_rp.77 │ │ │ │ +000509e0: 3237 380a 4e6f 6465 3a20 6765 6e65 7261 278.Node: genera │ │ │ │ +000509f0: 746f 7273 5f6c 7052 696e 674f 6649 6e76 tors_lpRingOfInv │ │ │ │ +00050a00: 6172 6961 6e74 735f 7270 7f38 3133 3035 ariants_rp.81305 │ │ │ │ +00050a10: 0a4e 6f64 653a 2067 726f 7570 7f38 3433 .Node: group.843 │ │ │ │ +00050a20: 3732 0a4e 6f64 653a 2047 726f 7570 4163 72.Node: GroupAc │ │ │ │ +00050a30: 7469 6f6e 7f38 3832 3533 0a4e 6f64 653a tion.88253.Node: │ │ │ │ +00050a40: 2067 726f 7570 4964 6561 6c7f 3931 3436 groupIdeal.9146 │ │ │ │ +00050a50: 320a 4e6f 6465 3a20 6869 6c62 6572 7449 2.Node: hilbertI │ │ │ │ +00050a60: 6465 616c 7f39 3532 3633 0a4e 6f64 653a deal.95263.Node: │ │ │ │ +00050a70: 2068 696c 6265 7274 5365 7269 6573 5f6c hilbertSeries_l │ │ │ │ +00050a80: 7052 696e 674f 6649 6e76 6172 6961 6e74 pRingOfInvariant │ │ │ │ +00050a90: 735f 7270 7f31 3033 3532 360a 4e6f 6465 s_rp.103526.Node │ │ │ │ +00050aa0: 3a20 6869 726f 6e61 6b61 4465 636f 6d70 : hironakaDecomp │ │ │ │ +00050ab0: 6f73 6974 696f 6e7f 3130 3636 3530 0a4e osition.106650.N │ │ │ │ +00050ac0: 6f64 653a 2068 736f 7020 616c 676f 7269 ode: hsop algori │ │ │ │ +00050ad0: 7468 6d73 7f31 3132 3038 340a 4e6f 6465 thms.112084.Node │ │ │ │ +00050ae0: 3a20 696e 7661 7269 616e 7452 696e 677f : invariantRing. │ │ │ │ +00050af0: 3133 3439 3236 0a4e 6f64 653a 2069 6e76 134926.Node: inv │ │ │ │ +00050b00: 6172 6961 6e74 737f 3133 3933 3236 0a4e ariants.139326.N │ │ │ │ +00050b10: 6f64 653a 2069 6e76 6172 6961 6e74 735f ode: invariants_ │ │ │ │ +00050b20: 6c70 5f70 645f 7064 5f70 645f 636d 4465 lp_pd_pd_pd_cmDe │ │ │ │ +00050b30: 6772 6565 426f 756e 643d 3e5f 7064 5f70 greeBound=>_pd_p │ │ │ │ +00050b40: 645f 7064 5f72 707f 3134 3433 3039 0a4e d_pd_rp.144309.N │ │ │ │ +00050b50: 6f64 653a 2069 6e76 6172 6961 6e74 735f ode: invariants_ │ │ │ │ +00050b60: 6c70 5f70 645f 7064 5f70 645f 636d 4465 lp_pd_pd_pd_cmDe │ │ │ │ +00050b70: 6772 6565 4c69 6d69 743d 3e5f 7064 5f70 greeLimit=>_pd_p │ │ │ │ +00050b80: 645f 7064 5f72 707f 3135 3037 3638 0a4e d_pd_rp.150768.N │ │ │ │ +00050b90: 6f64 653a 2069 6e76 6172 6961 6e74 735f ode: invariants_ │ │ │ │ +00050ba0: 6c70 5f70 645f 7064 5f70 645f 636d 5374 lp_pd_pd_pd_cmSt │ │ │ │ +00050bb0: 7261 7465 6779 3d3e 5f70 645f 7064 5f70 rategy=>_pd_pd_p │ │ │ │ +00050bc0: 645f 7270 7f31 3534 3532 310a 4e6f 6465 d_rp.154521.Node │ │ │ │ +00050bd0: 3a20 696e 7661 7269 616e 7473 5f6c 705f : invariants_lp_ │ │ │ │ +00050be0: 7064 5f70 645f 7064 5f63 6d53 7562 7269 pd_pd_pd_cmSubri │ │ │ │ +00050bf0: 6e67 4c69 6d69 743d 3e5f 7064 5f70 645f ngLimit=>_pd_pd_ │ │ │ │ +00050c00: 7064 5f72 707f 3136 3532 3037 0a4e 6f64 pd_rp.165207.Nod │ │ │ │ +00050c10: 653a 2069 6e76 6172 6961 6e74 735f 6c70 e: invariants_lp │ │ │ │ +00050c20: 5f70 645f 7064 5f70 645f 636d 5573 6543 _pd_pd_pd_cmUseC │ │ │ │ +00050c30: 6f65 6666 6963 6965 6e74 5269 6e67 3d3e oefficientRing=> │ │ │ │ +00050c40: 5f70 645f 7064 5f70 645f 7270 7f31 3637 _pd_pd_pd_rp.167 │ │ │ │ +00050c50: 3337 300a 4e6f 6465 3a20 696e 7661 7269 370.Node: invari │ │ │ │ +00050c60: 616e 7473 5f6c 705f 7064 5f70 645f 7064 ants_lp_pd_pd_pd │ │ │ │ +00050c70: 5f63 6d55 7365 506f 6c79 6865 6472 613d _cmUsePolyhedra= │ │ │ │ +00050c80: 3e5f 7064 5f70 645f 7064 5f72 707f 3137 >_pd_pd_pd_rp.17 │ │ │ │ +00050c90: 3136 3832 0a4e 6f64 653a 2069 6e76 6172 1682.Node: invar │ │ │ │ +00050ca0: 6961 6e74 735f 6c70 4469 6167 6f6e 616c iants_lpDiagonal │ │ │ │ +00050cb0: 4163 7469 6f6e 5f72 707f 3137 3339 3132 Action_rp.173912 │ │ │ │ +00050cc0: 0a4e 6f64 653a 2069 6e76 6172 6961 6e74 .Node: invariant │ │ │ │ +00050cd0: 735f 6c70 4669 6e69 7465 4772 6f75 7041 s_lpFiniteGroupA │ │ │ │ +00050ce0: 6374 696f 6e5f 7270 7f31 3833 3733 390a ction_rp.183739. │ │ │ │ +00050cf0: 4e6f 6465 3a20 696e 7661 7269 616e 7473 Node: invariants │ │ │ │ +00050d00: 5f6c 7046 696e 6974 6547 726f 7570 4163 _lpFiniteGroupAc │ │ │ │ +00050d10: 7469 6f6e 5f63 6d5a 5a5f 7270 7f31 3931 tion_cmZZ_rp.191 │ │ │ │ +00050d20: 3537 310a 4e6f 6465 3a20 696e 7661 7269 571.Node: invari │ │ │ │ +00050d30: 616e 7473 5f6c 704c 696e 6561 726c 7952 ants_lpLinearlyR │ │ │ │ +00050d40: 6564 7563 7469 7665 4163 7469 6f6e 5f72 eductiveAction_r │ │ │ │ +00050d50: 707f 3139 3639 3031 0a4e 6f64 653a 2069 p.196901.Node: i │ │ │ │ +00050d60: 6e76 6172 6961 6e74 735f 6c70 4c69 6e65 nvariants_lpLine │ │ │ │ +00050d70: 6172 6c79 5265 6475 6374 6976 6541 6374 arlyReductiveAct │ │ │ │ +00050d80: 696f 6e5f 636d 5a5a 5f72 707f 3230 3339 ion_cmZZ_rp.2039 │ │ │ │ +00050d90: 3538 0a4e 6f64 653a 2069 7341 6265 6c69 58.Node: isAbeli │ │ │ │ +00050da0: 616e 7f32 3131 3934 310a 4e6f 6465 3a20 an.211941.Node: │ │ │ │ +00050db0: 6973 496e 7661 7269 616e 747f 3231 3735 isInvariant.2175 │ │ │ │ +00050dc0: 3734 0a4e 6f64 653a 204c 696e 6561 726c 74.Node: Linearl │ │ │ │ +00050dd0: 7952 6564 7563 7469 7665 4163 7469 6f6e yReductiveAction │ │ │ │ +00050de0: 7f32 3238 3938 310a 4e6f 6465 3a20 6c69 .228981.Node: li │ │ │ │ +00050df0: 6e65 6172 6c79 5265 6475 6374 6976 6541 nearlyReductiveA │ │ │ │ +00050e00: 6374 696f 6e7f 3233 3234 3935 0a4e 6f64 ction.232495.Nod │ │ │ │ +00050e10: 653a 206d 6f6c 6965 6e53 6572 6965 737f e: molienSeries. │ │ │ │ +00050e20: 3233 3931 3132 0a4e 6f64 653a 206e 6574 239112.Node: net │ │ │ │ +00050e30: 5f6c 7052 696e 674f 6649 6e76 6172 6961 _lpRingOfInvaria │ │ │ │ +00050e40: 6e74 735f 7270 7f32 3431 3538 390a 4e6f nts_rp.241589.No │ │ │ │ +00050e50: 6465 3a20 6e75 6d67 656e 735f 6c70 4469 de: numgens_lpDi │ │ │ │ +00050e60: 6167 6f6e 616c 4163 7469 6f6e 5f72 707f agonalAction_rp. │ │ │ │ +00050e70: 3234 3234 3835 0a4e 6f64 653a 206e 756d 242485.Node: num │ │ │ │ +00050e80: 6765 6e73 5f6c 7046 696e 6974 6547 726f gens_lpFiniteGro │ │ │ │ +00050e90: 7570 4163 7469 6f6e 5f72 707f 3234 3530 upAction_rp.2450 │ │ │ │ +00050ea0: 3931 0a4e 6f64 653a 2070 6572 6d75 7461 91.Node: permuta │ │ │ │ +00050eb0: 7469 6f6e 4d61 7472 6978 7f32 3438 3838 tionMatrix.24888 │ │ │ │ +00050ec0: 340a 4e6f 6465 3a20 7072 696d 6172 7949 4.Node: primaryI │ │ │ │ +00050ed0: 6e76 6172 6961 6e74 737f 3235 3239 3530 nvariants.252950 │ │ │ │ +00050ee0: 0a4e 6f64 653a 2070 7269 6d61 7279 496e .Node: primaryIn │ │ │ │ +00050ef0: 7661 7269 616e 7473 5f6c 705f 7064 5f70 variants_lp_pd_p │ │ │ │ +00050f00: 645f 7064 5f63 6d44 6164 653d 3e5f 7064 d_pd_cmDade=>_pd │ │ │ │ +00050f10: 5f70 645f 7064 5f72 707f 3236 3230 3131 _pd_pd_rp.262011 │ │ │ │ +00050f20: 0a4e 6f64 653a 2070 7269 6d61 7279 496e .Node: primaryIn │ │ │ │ +00050f30: 7661 7269 616e 7473 5f6c 705f 7064 5f70 variants_lp_pd_p │ │ │ │ +00050f40: 645f 7064 5f63 6d44 6567 7265 6556 6563 d_pd_cmDegreeVec │ │ │ │ +00050f50: 746f 723d 3e5f 7064 5f70 645f 7064 5f72 tor=>_pd_pd_pd_r │ │ │ │ +00050f60: 707f 3237 3731 3837 0a4e 6f64 653a 2072 p.277187.Node: r │ │ │ │ +00050f70: 616e 6b5f 6c70 4469 6167 6f6e 616c 4163 ank_lpDiagonalAc │ │ │ │ +00050f80: 7469 6f6e 5f72 707f 3238 3330 3734 0a4e tion_rp.283074.N │ │ │ │ +00050f90: 6f64 653a 2072 656c 6174 696f 6e73 5f6c ode: relations_l │ │ │ │ +00050fa0: 7046 696e 6974 6547 726f 7570 4163 7469 pFiniteGroupActi │ │ │ │ +00050fb0: 6f6e 5f72 707f 3238 3536 3538 0a4e 6f64 on_rp.285658.Nod │ │ │ │ +00050fc0: 653a 2072 6579 6e6f 6c64 734f 7065 7261 e: reynoldsOpera │ │ │ │ +00050fd0: 746f 727f 3239 3031 3634 0a4e 6f64 653a tor.290164.Node: │ │ │ │ +00050fe0: 2072 696e 675f 6c70 4772 6f75 7041 6374 ring_lpGroupAct │ │ │ │ +00050ff0: 696f 6e5f 7270 7f32 3936 3432 380a 4e6f ion_rp.296428.No │ │ │ │ +00051000: 6465 3a20 5269 6e67 4f66 496e 7661 7269 de: RingOfInvari │ │ │ │ +00051010: 616e 7473 7f32 3938 3731 390a 4e6f 6465 ants.298719.Node │ │ │ │ +00051020: 3a20 7363 6872 6569 6572 4772 6170 687f : schreierGraph. │ │ │ │ +00051030: 3330 3133 3137 0a4e 6f64 653a 2073 6563 301317.Node: sec │ │ │ │ +00051040: 6f6e 6461 7279 496e 7661 7269 616e 7473 ondaryInvariants │ │ │ │ +00051050: 7f33 3038 3139 300a 4e6f 6465 3a20 7365 .308190.Node: se │ │ │ │ +00051060: 636f 6e64 6172 7949 6e76 6172 6961 6e74 condaryInvariant │ │ │ │ +00051070: 735f 6c70 5f70 645f 7064 5f70 645f 636d s_lp_pd_pd_pd_cm │ │ │ │ +00051080: 5072 696e 7444 6567 7265 6550 6f6c 796e PrintDegreePolyn │ │ │ │ +00051090: 6f6d 6961 6c3d 3e5f 7064 5f70 645f 7064 omial=>_pd_pd_pd │ │ │ │ +000510a0: 5f72 707f 3331 3231 3137 0a4e 6f64 653a _rp.312117.Node: │ │ │ │ +000510b0: 2074 6578 4d61 7468 5f6c 7052 696e 674f texMath_lpRingO │ │ │ │ +000510c0: 6649 6e76 6172 6961 6e74 735f 7270 7f33 fInvariants_rp.3 │ │ │ │ +000510d0: 3139 3839 360a 4e6f 6465 3a20 7765 6967 19896.Node: weig │ │ │ │ +000510e0: 6874 737f 3332 3039 3032 0a4e 6f64 653a hts.320902.Node: │ │ │ │ +000510f0: 2077 6f72 6473 7f33 3235 3039 320a 1f0a words.325092... │ │ │ │ +00051100: 456e 6420 5461 6720 5461 626c 650a End Tag Table. │ │ ├── ./usr/share/info/Isomorphism.info.gz │ │ │ ├── Isomorphism.info │ │ │ │ @@ -4544,15 +4544,15 @@ │ │ │ │ 00011bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011c00: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3320 3a20 -------+.|i23 : │ │ │ │ 00011c10: 656c 6170 7365 6454 696d 6520 6973 4973 elapsedTime isIs │ │ │ │ 00011c20: 6f6d 6f72 7068 6963 2854 312c 2054 3229 omorphic(T1, T2) │ │ │ │ 00011c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c50: 2020 2020 2020 207c 0a7c 202d 2d20 312e |.| -- 1. │ │ │ │ -00011c60: 3639 3430 3773 2065 6c61 7073 6564 2020 69407s elapsed │ │ │ │ +00011c60: 3438 3937 3773 2065 6c61 7073 6564 2020 48977s elapsed │ │ │ │ 00011c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011ca0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00011cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4569,15 +4569,15 @@ │ │ │ │ 00011d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011d90: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420 3a20 -------+.|i24 : │ │ │ │ 00011da0: 656c 6170 7365 6454 696d 6520 6973 6f6d elapsedTime isom │ │ │ │ 00011db0: 6f72 7068 6973 6d28 5431 2c20 5432 2920 orphism(T1, T2) │ │ │ │ 00011dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011de0: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00011df0: 3030 3032 3930 3234 7320 656c 6170 7365 00029024s elapse │ │ │ │ +00011df0: 3030 3032 3131 3938 7320 656c 6170 7365 00021198s elapse │ │ │ │ 00011e00: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 00011e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00011e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/JSON.info.gz │ │ │ ├── JSON.info │ │ │ │ @@ -458,27 +458,27 @@ │ │ │ │ 00001c90: 2d2d 2d2d 2b0a 7c69 3920 3a20 6a73 6f6e ----+.|i9 : json │ │ │ │ 00001ca0: 4669 6c65 203d 2074 656d 706f 7261 7279 File = temporary │ │ │ │ 00001cb0: 4669 6c65 4e61 6d65 2829 207c 2022 2e6a FileName() | ".j │ │ │ │ 00001cc0: 736f 6e22 207c 0a7c 2020 2020 2020 2020 son" |.| │ │ │ │ 00001cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001cf0: 2020 2020 2020 7c0a 7c6f 3920 3d20 2f74 |.|o9 = /t │ │ │ │ -00001d00: 6d70 2f4d 322d 3335 3435 312d 302f 302e mp/M2-35451-0/0. │ │ │ │ +00001d00: 6d70 2f4d 322d 3438 3831 302d 302f 302e mp/M2-48810-0/0. │ │ │ │ 00001d10: 6a73 6f6e 2020 2020 2020 2020 2020 2020 json │ │ │ │ 00001d20: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00001d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001d50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a --------+.|i10 : │ │ │ │ 00001d60: 206a 736f 6e46 696c 6520 3c3c 2022 5b31 jsonFile << "[1 │ │ │ │ 00001d70: 2c20 322c 2033 5d22 203c 3c20 656e 646c , 2, 3]" << endl │ │ │ │ 00001d80: 203c 3c20 636c 6f73 657c 0a7c 2020 2020 << close|.| │ │ │ │ 00001d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001db0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130 |.|o10 │ │ │ │ -00001dc0: 203d 202f 746d 702f 4d32 2d33 3534 3531 = /tmp/M2-35451 │ │ │ │ +00001dc0: 203d 202f 746d 702f 4d32 2d34 3838 3130 = /tmp/M2-48810 │ │ │ │ 00001dd0: 2d30 2f30 2e6a 736f 6e20 2020 2020 2020 -0/0.json │ │ │ │ 00001de0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00001df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001e10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ 00001e20: 3130 203a 2046 696c 6520 2020 2020 2020 10 : File │ │ │ │ 00001e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/Jets.info.gz │ │ │ ├── Jets.info │ │ │ │ @@ -5260,16 +5260,16 @@ │ │ │ │ 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 3233 3131 3173 2065 6c61 7073 6564 0023111s elapsed │ │ │ │ -00014930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014920: 3030 3234 3835 3539 7320 656c 6170 7365 00248559s 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 │ │ │ │ 000149a0: 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: 3432 3433 3636 7320 656c 6170 7365 6420 424366s elapsed │ │ │ │ +00014bf0: 3231 3839 3273 2065 6c61 7073 6564 2020 21892s elapsed │ │ │ │ 00014c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00014c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8104,16 +8104,16 @@ │ │ │ │ 0001fa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa90: 2d2d 2b0a 7c69 3820 3a20 656c 6170 7365 --+.|i8 : elapse │ │ │ │ 0001faa0: 6454 696d 6520 6a65 7473 2833 2c49 2920 dTime jets(3,I) │ │ │ │ 0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fae0: 207c 0a7c 202d 2d20 2e30 3037 3636 3535 |.| -- .0076655 │ │ │ │ -0001faf0: 3273 2065 6c61 7073 6564 2020 2020 2020 2s elapsed │ │ │ │ +0001fae0: 207c 0a7c 202d 2d20 2e30 3036 3737 3736 |.| -- .0067776 │ │ │ │ +0001faf0: 3173 2065 6c61 7073 6564 2020 2020 2020 1s elapsed │ │ │ │ 0001fb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001fb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -8203,16 +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 3032 3530 3732 7320 656c -- .0025072s el │ │ │ │ -00020120: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +00020110: 202d 2d20 2e30 3032 3432 3234 3473 2065 -- .00242244s e │ │ │ │ +00020120: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 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2d20 2e30 3131 3337 3137 7320 .| -- .0113717s │ │ │ │ +00022380: 0a7c 202d 2d20 2e30 3130 3733 3938 7320 .| -- .0107398s │ │ │ │ 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 │ │ │ │ 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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: 3234 3932 3639 7320 656c 6170 7365 6420 249269s elapsed │ │ │ │ +000074a0: 3234 3831 3531 7320 656c 6170 7365 6420 248151s 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 3239 3436 3373 2065 6c61 7073 6564 0229463s elapsed │ │ │ │ +00007d10: 3032 3537 3337 3373 2065 6c61 7073 6564 0257373s 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: 3039 3336 3231 3273 2065 6c61 7073 6564 0936212s elapsed │ │ │ │ +00007d60: 3035 3936 3830 3673 2065 6c61 7073 6564 0596806s 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 │ │ │ │ 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│ 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 │ │ │ │ @@ -3677,34 +3677,34 @@ │ │ │ │ 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 3233 |.| -- .0023 │ │ │ │ -0000e630: 3130 3036 7320 656c 6170 7365 6420 2020 1006s elapsed │ │ │ │ +0000e630: 3437 3434 7320 656c 6170 7365 6420 2020 4744s 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 3231 3738 3336 |.| -- .0217836 │ │ │ │ -0000e670: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ +0000e660: 207c 0a7c 202d 2d20 2e30 3036 3939 3035 |.| -- .0069905 │ │ │ │ +0000e670: 3273 2065 6c61 7073 6564 2020 2020 2020 2s elapsed │ │ │ │ 0000e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e690: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0000e6a0: 7c20 2d2d 202e 3032 3330 3839 3573 2065 | -- .0230895s e │ │ │ │ +0000e6a0: 7c20 2d2d 202e 3032 3337 3433 3673 2065 | -- .0237436s 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 3130 3138 3338 7320 656c 6170 - .0101838s elap │ │ │ │ +0000e6e0: 2d20 2e30 3130 3435 3239 7320 656c 6170 - .0104529s 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 3335 3333 3337 7320 656c 6170 7365 00353337s elapse │ │ │ │ +0000e720: 3030 3338 3639 3339 7320 656c 6170 7365 00386939s 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 2e32 3934 |.| -- .294 │ │ │ │ -0000e760: 3931 3773 2065 6c61 7073 6564 2020 2020 917s elapsed │ │ │ │ +0000e750: 2020 2020 207c 0a7c 202d 2d20 2e33 3038 |.| -- .308 │ │ │ │ +0000e760: 3336 3773 2065 6c61 7073 6564 2020 2020 367s elapsed │ │ │ │ 0000e770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e790: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000e7c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000e7d0: 0a7c 2020 2020 2020 2020 2020 2020 3020 .| 0 │ │ │ │ @@ -3764,15 +3764,15 @@ │ │ │ │ 0000eb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000eb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000eb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000eb60: 2d2d 2b0a 7c69 3420 3a20 656c 6170 7365 --+.|i4 : elapse │ │ │ │ 0000eb70: 6454 696d 6520 5427 3d6d 696e 696d 616c dTime T'=minimal │ │ │ │ 0000eb80: 4265 7474 6920 4a20 2020 2020 2020 2020 Betti J │ │ │ │ 0000eb90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000eba0: 0a7c 202d 2d20 2e31 3235 3139 3673 2065 .| -- .125196s e │ │ │ │ +0000eba0: 0a7c 202d 2d20 2e31 3632 3135 3573 2065 .| -- .162155s 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 3735 3633 3373 2065 -- .00475633s e │ │ │ │ +0000f0e0: 202d 2d20 2e30 3038 3632 3633 3173 2065 -- .00862631s 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 3039 3638 3330 3673 2065 6c61 7073 .0968306s elaps │ │ │ │ +0000f120: 202e 3033 3139 3939 3873 2065 6c61 7073 .0319998s 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: 3131 3433 3373 2065 6c61 7073 6564 2020 11433s elapsed │ │ │ │ +0000f150: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ +0000f160: 3934 3633 3831 7320 656c 6170 7365 6420 946381s 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 2e34 3735 |.| -- 1.475 │ │ │ │ -0000f1a0: 3739 7320 656c 6170 7365 6420 2020 2020 79s elapsed │ │ │ │ +0000f190: 2020 2020 7c0a 7c20 2d2d 202e 3936 3330 |.| -- .9630 │ │ │ │ +0000f1a0: 3339 7320 656c 6170 7365 6420 2020 2020 39s 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 3531 3732 3273 |.| -- .551722s │ │ │ │ +0000f1d0: 207c 0a7c 202d 2d20 2e34 3433 3733 3473 |.| -- .443734s │ │ │ │ 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 3033 3938 3830 3473 2065 | -- .0398804s e │ │ │ │ +0000f210: 7c20 2d2d 202e 3034 3034 3832 3473 2065 | -- .0404824s 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 3037 3237 3631 3373 2065 6c61 - .00727613s ela │ │ │ │ +0000f250: 2d20 2e30 3036 3838 3039 3173 2065 6c61 - .00688091s 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: 2e30 3530 3634 7320 656c 6170 7365 6420 .05064s elapsed │ │ │ │ +0000f280: 2020 2020 2020 2020 7c0a 7c20 2d2d 2034 |.| -- 4 │ │ │ │ +0000f290: 2e37 3933 3731 7320 656c 6170 7365 6420 .79371s elapsed │ │ │ │ 0000f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000f2c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0000f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000f300: 2d2d 2b0a 7c69 3720 3a20 6361 7270 6574 --+.|i7 : carpet │ │ │ │ @@ -4108,31 +4108,31 @@ │ │ │ │ 000100b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000100c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000100d0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2068 3d63 -----+.|i2 : h=c │ │ │ │ 000100e0: 6172 7065 7442 6574 7469 5461 626c 6573 arpetBettiTables │ │ │ │ 000100f0: 2861 2c62 2920 2020 2020 2020 2020 2020 (a,b) │ │ │ │ 00010100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010110: 2020 2020 207c 0a7c 202d 2d20 2e30 3032 |.| -- .002 │ │ │ │ -00010120: 3234 3634 3273 2065 6c61 7073 6564 2020 24642s elapsed │ │ │ │ +00010120: 3231 3332 7320 656c 6170 7365 6420 2020 2132s 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 3130 |.| -- .010 │ │ │ │ -00010160: 3035 3373 2065 6c61 7073 6564 2020 2020 053s elapsed │ │ │ │ +00010150: 2020 2020 207c 0a7c 202d 2d20 2e30 3035 |.| -- .005 │ │ │ │ +00010160: 3839 3331 3373 2065 6c61 7073 6564 2020 89313s 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 3336 |.| -- .036 │ │ │ │ -000101a0: 3535 3633 7320 656c 6170 7365 6420 2020 5563s elapsed │ │ │ │ +00010190: 2020 2020 207c 0a7c 202d 2d20 2e30 3232 |.| -- .022 │ │ │ │ +000101a0: 3932 3134 7320 656c 6170 7365 6420 2020 9214s 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 3130 |.| -- .010 │ │ │ │ -000101e0: 3034 3938 7320 656c 6170 7365 6420 2020 0498s elapsed │ │ │ │ +000101d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3039 |.| -- .009 │ │ │ │ +000101e0: 3635 3139 3673 2065 6c61 7073 6564 2020 65196s 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: 3533 3934 3173 2065 6c61 7073 6564 2020 53941s elapsed │ │ │ │ +00010220: 3432 3237 3973 2065 6c61 7073 6564 2020 42279s elapsed │ │ │ │ 00010230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010250: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00010260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010290: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ @@ -4287,16 +4287,16 @@ │ │ │ │ 00010be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010c10: 2d2d 2d2d 2d2b 0a7c 6935 203a 2065 6c61 -----+.|i5 : ela │ │ │ │ 00010c20: 7073 6564 5469 6d65 2054 273d 6d69 6e69 psedTime T'=mini │ │ │ │ 00010c30: 6d61 6c42 6574 7469 204a 2020 2020 2020 malBetti J │ │ │ │ 00010c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00010c50: 2020 2020 207c 0a7c 202d 2d20 2e31 3936 |.| -- .196 │ │ │ │ -00010c60: 3231 3573 2065 6c61 7073 6564 2020 2020 215s elapsed │ │ │ │ +00010c50: 2020 2020 207c 0a7c 202d 2d20 2e31 3633 |.| -- .163 │ │ │ │ +00010c60: 3638 3673 2065 6c61 7073 6564 2020 2020 686s 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 |.| │ │ │ │ @@ -4376,43 +4376,43 @@ │ │ │ │ 00011170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011190: 2d2d 2d2d 2d2b 0a7c 6937 203a 2065 6c61 -----+.|i7 : ela │ │ │ │ 000111a0: 7073 6564 5469 6d65 2068 3d63 6172 7065 psedTime h=carpe │ │ │ │ 000111b0: 7442 6574 7469 5461 626c 6573 2836 2c36 tBettiTables(6,6 │ │ │ │ 000111c0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ 000111d0: 2020 2020 207c 0a7c 202d 2d20 2e30 3034 |.| -- .004 │ │ │ │ -000111e0: 3930 3038 3773 2065 6c61 7073 6564 2020 90087s elapsed │ │ │ │ +000111e0: 3835 3738 3473 2065 6c61 7073 6564 2020 85784s 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 3138 |.| -- .018 │ │ │ │ -00011220: 3934 3931 7320 656c 6170 7365 6420 2020 9491s elapsed │ │ │ │ +00011210: 2020 2020 207c 0a7c 202d 2d20 2e30 3136 |.| -- .016 │ │ │ │ +00011220: 3138 3173 2065 6c61 7073 6564 2020 2020 181s 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 2e30 3936 |.| -- .096 │ │ │ │ -00011260: 3634 3839 7320 656c 6170 7365 6420 2020 6489s elapsed │ │ │ │ +00011250: 2020 2020 207c 0a7c 202d 2d20 2e31 3032 |.| -- .102 │ │ │ │ +00011260: 3139 3973 2065 6c61 7073 6564 2020 2020 199s elapsed │ │ │ │ 00011270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011290: 2020 2020 207c 0a7c 202d 2d20 312e 3139 |.| -- 1.19 │ │ │ │ -000112a0: 3135 3973 2065 6c61 7073 6564 2020 2020 159s elapsed │ │ │ │ +00011290: 2020 2020 207c 0a7c 202d 2d20 2e38 3334 |.| -- .834 │ │ │ │ +000112a0: 3532 3773 2065 6c61 7073 6564 2020 2020 527s elapsed │ │ │ │ 000112b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000112c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000112d0: 2020 2020 207c 0a7c 202d 2d20 2e35 3234 |.| -- .524 │ │ │ │ -000112e0: 3534 3973 2065 6c61 7073 6564 2020 2020 549s elapsed │ │ │ │ +000112d0: 2020 2020 207c 0a7c 202d 2d20 2e33 3737 |.| -- .377 │ │ │ │ +000112e0: 3232 3373 2065 6c61 7073 6564 2020 2020 223s 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 3430 |.| -- .040 │ │ │ │ -00011320: 3536 3332 7320 656c 6170 7365 6420 2020 5632s elapsed │ │ │ │ +00011310: 2020 2020 207c 0a7c 202d 2d20 2e30 3336 |.| -- .036 │ │ │ │ +00011320: 3436 3531 7320 656c 6170 7365 6420 2020 4651s 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: 3830 3935 3173 2065 6c61 7073 6564 2020 80951s elapsed │ │ │ │ +00011360: 3535 3839 3873 2065 6c61 7073 6564 2020 55898s elapsed │ │ │ │ 00011370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011390: 2020 2020 207c 0a7c 202d 2d20 352e 3337 |.| -- 5.37 │ │ │ │ -000113a0: 3830 3573 2065 6c61 7073 6564 2020 2020 805s elapsed │ │ │ │ +00011390: 2020 2020 207c 0a7c 202d 2d20 342e 3338 |.| -- 4.38 │ │ │ │ +000113a0: 3630 3173 2065 6c61 7073 6564 2020 2020 601s 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 3633 |.| -- .0063 │ │ │ │ -00012210: 3134 3734 7320 656c 6170 7365 6420 2020 1474s elapsed │ │ │ │ +00012200: 2020 2020 7c0a 7c20 2d2d 202e 3030 3639 |.| -- .0069 │ │ │ │ +00012210: 3339 3034 7320 656c 6170 7365 6420 2020 3904s elapsed │ │ │ │ 00012220: 2020 2020 2020 207c 0a7c 202d 2d20 2e30 |.| -- .0 │ │ │ │ -00012230: 3132 3533 3132 7320 656c 6170 7365 6420 125312s elapsed │ │ │ │ +00012230: 3132 3930 3135 7320 656c 6170 7365 6420 129015s 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 3934 3731 7320 -- .000299471s │ │ │ │ +00012270: 202d 2d20 2e30 3030 3237 3139 3232 7320 -- .000271922s │ │ │ │ 00012280: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 00012290: 7c0a 7c31 2020 2020 2020 2020 2020 2020 |.|1 │ │ │ │ 000122a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000122b0: 2020 207c 0a7c 202d 2d20 2e30 3030 3134 |.| -- .00014 │ │ │ │ -000122c0: 3639 3834 7320 656c 6170 7365 6420 2020 6984s elapsed │ │ │ │ +000122b0: 2020 207c 0a7c 202d 2d20 2e30 3030 3137 |.| -- .00017 │ │ │ │ +000122c0: 3435 3873 2065 6c61 7073 6564 2020 2020 458s 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 3132 3832 3173 2065 6c61 7073 .00012821s elaps │ │ │ │ -00012310: 6564 2020 2020 2020 2020 2020 7c0a 7c31 ed |.|1 │ │ │ │ +00012300: 2e30 3030 3136 3434 3631 7320 656c 6170 .000164461s 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 3437 3933 .| -- .000124793 │ │ │ │ +00012340: 0a7c 202d 2d20 2e30 3030 3137 3239 3833 .| -- .000172983 │ │ │ │ 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: 3132 3934 3233 7320 656c 6170 7365 6420 129423s elapsed │ │ │ │ +00012390: 3136 3931 3631 7320 656c 6170 7365 6420 169161s 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 3133 3637 3936 7320 656c - .000136796s el │ │ │ │ +000123d0: 2d20 2e30 3030 3136 3730 3636 7320 656c - .000167066s 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: 3337 3633 3773 2065 6c61 7073 6564 2020 37637s elapsed │ │ │ │ +00012440: 3638 3930 3973 2065 6c61 7073 6564 2020 68909s 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 3134 3833 3639 7320 -- .000148369s │ │ │ │ +000124a0: 202d 2d20 2e30 3030 3138 3136 3332 7320 -- .000181632s │ │ │ │ 000124b0: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ 000124c0: 7c0a 7c20 3220 2020 2020 2020 2020 2020 |.| 2 │ │ │ │ 000124d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000124e0: 2020 207c 0a7c 3220 3320 2020 2020 2020 |.|2 3 │ │ │ │ 000124f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012500: 2020 2020 2020 7c0a 7c20 2d2d 202e 3030 |.| -- .00 │ │ │ │ -00012510: 3031 3434 3639 3173 2065 6c61 7073 6564 0144691s elapsed │ │ │ │ +00012510: 3032 3137 3735 3573 2065 6c61 7073 6564 0217755s 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 3237 3139 .| -- .000132719 │ │ │ │ +00012570: 0a7c 202d 2d20 2e30 3030 3136 3836 3332 .| -- .000168632 │ │ │ │ 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 3235 3836 3673 2065 6c61 7073 000125866s elaps │ │ │ │ +000125e0: 3030 3031 3633 3934 3673 2065 6c61 7073 000163946s 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 3232 |.| -- .0001322 │ │ │ │ -00012650: 3937 7320 656c 6170 7365 6420 2020 2020 97s elapsed │ │ │ │ +00012640: 207c 0a7c 202d 2d20 2e30 3030 3135 3531 |.| -- .0001551 │ │ │ │ +00012650: 3137 7320 656c 6170 7365 6420 2020 2020 17s 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 3430 3234 3273 2065 6c61 .000140242s ela │ │ │ │ +000126b0: 202e 3030 3031 3739 3035 3373 2065 6c61 .000179053s 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 3432 |.| -- .00012242 │ │ │ │ -00012700: 3973 2065 6c61 7073 6564 2020 2020 2020 9s elapsed │ │ │ │ +000126f0: 7c0a 7c20 2d2d 202e 3030 3031 3433 3637 |.| -- .00014367 │ │ │ │ +00012700: 3173 2065 6c61 7073 6564 2020 2020 2020 1s 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 3333 3038 3873 2065 6c61 7073 6564 0133088s elapsed │ │ │ │ +00012740: 3031 3830 3133 3673 2065 6c61 7073 6564 0180136s 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 3034 3636 .| -- .000120466 │ │ │ │ +000127a0: 0a7c 202d 2d20 2e30 3030 3134 3934 3136 .| -- .000149416 │ │ │ │ 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 3336 3030 3473 2065 6c61 7073 000136004s elaps │ │ │ │ +00012810: 3030 3031 3635 3435 3673 2065 6c61 7073 000165456s 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 3133 |.| -- .0001313 │ │ │ │ -00012880: 3736 7320 656c 6170 7365 6420 2020 2020 76s elapsed │ │ │ │ +00012870: 207c 0a7c 202d 2d20 2e30 3030 3231 3135 |.| -- .0002115 │ │ │ │ +00012880: 3832 7320 656c 6170 7365 6420 2020 2020 82s 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 3032 3038 3036 3973 2065 6c61 .000208069s ela │ │ │ │ +000128e0: 202e 3030 3032 3237 3536 3473 2065 6c61 .000227564s 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 3134 |.| -- .00014 │ │ │ │ -00012950: 3038 3334 7320 656c 6170 7365 6420 2020 0834s elapsed │ │ │ │ +00012940: 2020 207c 0a7c 202d 2d20 2e30 3030 3139 |.| -- .00019 │ │ │ │ +00012950: 3733 3673 2065 6c61 7073 6564 2020 2020 736s 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 3334 3130 3173 2065 -- .000134101s e │ │ │ │ +000129b0: 2d2d 202e 3030 3031 3831 3035 3273 2065 -- .000181052s 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: 3132 3638 3536 7320 656c 6170 7365 6420 126856s elapsed │ │ │ │ +00012a20: 3135 3334 3333 7320 656c 6170 7365 6420 153433s 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 3132 3931 3732 7320 656c - .000129172s el │ │ │ │ +00012a60: 2d20 2e30 3030 3137 3032 3237 7320 656c - .000170227s el │ │ │ │ 00012a70: 6170 7365 6420 2020 2020 2020 2020 7c0a apsed |. │ │ │ │ 00012a80: 7c31 2020 2020 2020 2020 2020 2020 2020 |1 │ │ │ │ 00012a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012aa0: 207c 0a7c 202d 2d20 2e30 3030 3133 3037 |.| -- .0001307 │ │ │ │ -00012ab0: 3134 7320 656c 6170 7365 6420 2020 2020 14s elapsed │ │ │ │ +00012aa0: 207c 0a7c 202d 2d20 2e30 3030 3136 3039 |.| -- .0001609 │ │ │ │ +00012ab0: 3832 7320 656c 6170 7365 6420 2020 2020 82s 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 3135 3638 3233 7320 656c 6170 7365 00156823s elapse │ │ │ │ +00012af0: 3030 3137 3330 3638 7320 656c 6170 7365 00173068s 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 3132 3339 3931 7320 -- .000123991s │ │ │ │ +00012b30: 202d 2d20 2e30 3030 3136 3938 3839 7320 -- .000169889s │ │ │ │ 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 │ │ │ │ @@ -4929,26 +4929,26 @@ │ │ │ │ 00013400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013410: 2020 2020 7c0a 7c6f 3120 3a20 5365 7175 |.|o1 : Sequ │ │ │ │ 00013420: 656e 6365 2020 2020 2020 2020 2020 2020 ence │ │ │ │ 00013430: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00013440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013450: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 636f ------+.|i2 : co │ │ │ │ 00013460: 6d70 7574 6542 6f75 6e64 2033 2020 2020 mputeBound 3 │ │ │ │ -00013470: 2020 2020 2020 207c 0a7c 202d 2d20 2e32 |.| -- .2 │ │ │ │ -00013480: 3638 3932 3773 2065 6c61 7073 6564 2020 68927s elapsed │ │ │ │ +00013470: 2020 2020 2020 207c 0a7c 202d 2d20 2e31 |.| -- .1 │ │ │ │ +00013480: 3234 3634 3973 2065 6c61 7073 6564 2020 24649s elapsed │ │ │ │ 00013490: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e |.| -- . │ │ │ │ -000134a0: 3333 3239 3873 2065 6c61 7073 6564 2020 33298s elapsed │ │ │ │ +000134a0: 3135 3134 3939 7320 656c 6170 7365 6420 151499s elapsed │ │ │ │ 000134b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000134c0: 2e32 3338 3535 7320 656c 6170 7365 6420 .23855s elapsed │ │ │ │ +000134c0: 2e31 3533 3732 3173 2065 6c61 7073 6564 .153721s elapsed │ │ │ │ 000134d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -000134e0: 202e 3238 3632 3535 7320 656c 6170 7365 .286255s elapse │ │ │ │ +000134e0: 202e 3135 3033 3032 7320 656c 6170 7365 .150302s elapse │ │ │ │ 000134f0: 6420 2020 2020 2020 2020 207c 0a7c 202d d |.| - │ │ │ │ -00013500: 2d20 2e33 3832 3733 3273 2065 6c61 7073 - .382732s elaps │ │ │ │ +00013500: 2d20 2e31 3634 3232 3973 2065 6c61 7073 - .164229s elaps │ │ │ │ 00013510: 6564 2020 2020 2020 2020 2020 7c0a 7c20 ed |.| │ │ │ │ -00013520: 2d2d 202e 3239 3838 3639 7320 656c 6170 -- .298869s elap │ │ │ │ +00013520: 2d2d 202e 3138 3436 3038 7320 656c 6170 -- .184608s elap │ │ │ │ 00013530: 7365 6420 2020 2020 2020 2020 207c 0a7c sed |.| │ │ │ │ 00013540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013550: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00013560: 7c6f 3220 3d20 3620 2020 2020 2020 2020 |o2 = 6 │ │ │ │ 00013570: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00013580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00013590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -6949,33 +6949,33 @@ │ │ │ │ 0001b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b260: 2b0a 7c69 3320 3a20 683d 6465 6765 6e65 +.|i3 : h=degene │ │ │ │ 0001b270: 7261 7465 4b33 4265 7474 6954 6162 6c65 rateK3BettiTable │ │ │ │ 0001b280: 7328 612c 622c 6529 2020 2020 2020 2020 s(a,b,e) │ │ │ │ 0001b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b2a0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -0001b2b0: 2e30 3033 3032 3237 7320 656c 6170 7365 .0030227s elapse │ │ │ │ -0001b2c0: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ +0001b2b0: 2e30 3032 3335 3236 3773 2065 6c61 7073 .00235267s elaps │ │ │ │ +0001b2c0: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ 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 3739 3439 |.| -- .007949 │ │ │ │ -0001b300: 3133 7320 656c 6170 7365 6420 2020 2020 13s elapsed │ │ │ │ +0001b2f0: 2020 7c0a 7c20 2d2d 202e 3030 3536 3330 |.| -- .005630 │ │ │ │ +0001b300: 3636 7320 656c 6170 7365 6420 2020 2020 66s 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 3233 3838 3673 2065 6c61 7073 - .023886s elaps │ │ │ │ -0001b350: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ +0001b340: 2d20 2e30 3331 3938 3532 7320 656c 6170 - .0319852s 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 3031 3134 |.| -- .0114 │ │ │ │ -0001b390: 3639 3273 2065 6c61 7073 6564 2020 2020 692s elapsed │ │ │ │ +0001b380: 2020 2020 7c0a 7c20 2d2d 202e 3031 3434 |.| -- .0144 │ │ │ │ +0001b390: 3934 3273 2065 6c61 7073 6564 2020 2020 942s 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 3638 3435 3273 2065 -- .00368452s e │ │ │ │ +0001b3d0: 202d 2d20 2e30 3033 3632 3236 3373 2065 -- .00362263s e │ │ │ │ 0001b3e0: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 0001b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b410: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0001b420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7131,15 +7131,15 @@ │ │ │ │ 0001bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bdc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ 0001bdd0: 656c 6170 7365 6454 696d 6520 543d 206d elapsedTime T= m │ │ │ │ 0001bde0: 696e 696d 616c 4265 7474 6920 6465 6765 inimalBetti dege │ │ │ │ 0001bdf0: 6e65 7261 7465 4b33 2861 2c62 2c65 2c43 nerateK3(a,b,e,C │ │ │ │ 0001be00: 6861 7261 6374 6572 6973 7469 633d 3e35 haracteristic=>5 │ │ │ │ -0001be10: 297c 0a7c 202d 2d20 2e31 3437 3939 3773 )|.| -- .147997s │ │ │ │ +0001be10: 297c 0a7c 202d 2d20 2e31 3834 3636 3573 )|.| -- .184665s │ │ │ │ 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 3434 3739 |.| -- .004479 │ │ │ │ -0001c740: 3339 7320 656c 6170 7365 6420 2020 2020 39s elapsed │ │ │ │ +0001c730: 2020 7c0a 7c20 2d2d 202e 3030 3234 3739 |.| -- .002479 │ │ │ │ +0001c740: 3931 7320 656c 6170 7365 6420 2020 2020 91s 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: 3131 3832 3832 7320 656c 6170 7365 6420 118282s elapsed │ │ │ │ +0001c780: 3036 3233 3030 3373 2065 6c61 7073 6564 0623003s 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 3034 3335 3838 7320 656c 6170 -- .043588s elap │ │ │ │ -0001c7d0: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ +0001c7c0: 2d2d 202e 3032 3435 3936 3373 2065 6c61 -- .0245963s 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 3136 3731 3639 |.| -- .0167169 │ │ │ │ -0001c810: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ +0001c800: 207c 0a7c 202d 2d20 2e30 3039 3537 3735 |.| -- .0095775 │ │ │ │ +0001c810: 3973 2065 6c61 7073 6564 2020 2020 2020 9s 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: 3630 3636 3136 7320 656c 6170 7365 6420 606616s elapsed │ │ │ │ +0001c850: 3336 3931 3939 7320 656c 6170 7365 6420 369199s elapsed │ │ │ │ 0001c860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c880: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0001c890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -9875,16 +9875,16 @@ │ │ │ │ 00026920: 3d20 3420 2020 2020 2020 2020 2020 2020 = 4 │ │ │ │ 00026930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026940: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00026950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026960: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ 00026970: 3a20 2864 312c 6432 293d 7265 736f 6e61 : (d1,d2)=resona │ │ │ │ 00026980: 6e63 6544 6574 2861 2920 2020 2020 2020 nceDet(a) │ │ │ │ -00026990: 2020 7c0a 7c20 2d2d 202e 3032 3031 3030 |.| -- .020100 │ │ │ │ -000269a0: 3473 2065 6c61 7073 6564 2020 2020 2020 4s elapsed │ │ │ │ +00026990: 2020 7c0a 7c20 2d2d 202e 3032 3033 3335 |.| -- .020335 │ │ │ │ +000269a0: 3873 2065 6c61 7073 6564 2020 2020 2020 8s 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 3534 3136 3273 2065 6c61 .000054162s ela │ │ │ │ +00026b00: 202e 3030 3030 3531 3737 3273 2065 6c61 .000051772s 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 3934 |.| -- .000094 │ │ │ │ -00026c20: 3536 3673 2065 6c61 7073 6564 2020 2020 566s elapsed │ │ │ │ +00026c10: 2020 7c0a 7c20 2d2d 202e 3030 3033 3230 |.| -- .000320 │ │ │ │ +00026c20: 3233 3273 2065 6c61 7073 6564 2020 2020 232s 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 3731 3438 3473 2065 6c61 .000071484s ela │ │ │ │ +00026d80: 202e 3030 3030 3739 3238 3473 2065 6c61 .000079284s 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 3831 3038 3173 2065 6c61 .000081081s ela │ │ │ │ +00026f10: 202e 3030 3030 3830 3734 3173 2065 6c61 .000080741s 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 3932 |.| -- .000092 │ │ │ │ -000270d0: 3434 3273 2065 6c61 7073 6564 2020 2020 442s elapsed │ │ │ │ +000270c0: 2020 7c0a 7c20 2d2d 202e 3030 3030 3839 |.| -- .000089 │ │ │ │ +000270d0: 3834 3773 2065 6c61 7073 6564 2020 2020 847s 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 3932 |.| -- .000092 │ │ │ │ -00027260: 3530 3373 2065 6c61 7073 6564 2020 2020 503s elapsed │ │ │ │ +00027250: 2020 7c0a 7c20 2d2d 202e 3030 3030 3839 |.| -- .000089 │ │ │ │ +00027260: 3030 3573 2065 6c61 7073 6564 2020 2020 005s 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 3933 3773 2065 6c61 .000024937s ela │ │ │ │ +00027370: 202e 3030 3030 3237 3737 3773 2065 6c61 .000027777s 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 3937 |.| -- .000097 │ │ │ │ -00027490: 3339 3373 2065 6c61 7073 6564 2020 2020 393s elapsed │ │ │ │ +00027480: 2020 7c0a 7c20 2d2d 202e 3030 3030 3735 |.| -- 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6976 656e 732c 5265 616c 4650 7d29 3b20 ivens,RealFP}); │ │ │ │ 0000a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a350: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0000a360: 6564 2030 2e30 3130 3735 3538 7320 2863 ed 0.0107558s (c │ │ │ │ -0000a370: 7075 293b 2030 2e30 3130 3735 3673 2028 pu); 0.010756s ( │ │ │ │ +0000a360: 6564 2030 2e30 3132 3136 3837 7320 2863 ed 0.0121687s (c │ │ │ │ +0000a370: 7075 293b 2030 2e30 3132 3031 3573 2028 pu); 0.012015s ( │ │ │ │ 0000a380: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ 0000a390: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0000a3a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0000a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a3e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ @@ -2635,16 +2635,16 @@ │ │ │ │ 0000a4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ 0000a4c0: 3a20 7469 6d65 204c 4c4c 286d 2c20 5374 : time LLL(m, St │ │ │ │ 0000a4d0: 7261 7465 6779 3d3e 7b47 6976 656e 732c rategy=>{Givens, │ │ │ │ 0000a4e0: 5265 616c 5150 7d29 3b20 2020 2020 2020 RealQP}); │ │ │ │ 0000a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a500: 207c 0a7c 202d 2d20 7573 6564 2030 2e30 |.| -- used 0.0 │ │ │ │ -0000a510: 3437 3330 3239 7320 2863 7075 293b 2030 473029s (cpu); 0 │ │ │ │ -0000a520: 2e30 3437 3330 3431 7320 2874 6872 6561 .0473041s (threa │ │ │ │ +0000a510: 3534 3738 3233 7320 2863 7075 293b 2030 547823s (cpu); 0 │ │ │ │ +0000a520: 2e30 3534 3738 3637 7320 2874 6872 6561 .0547867s (threa │ │ │ │ 0000a530: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0000a540: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0000a550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a580: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000a590: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ @@ -2661,17 +2661,17 @@ │ │ │ │ 0000a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a660: 2d2d 2d2d 2b0a 7c69 3820 3a20 7469 6d65 ----+.|i8 : time │ │ │ │ 0000a670: 204c 4c4c 286d 2c20 5374 7261 7465 6779 LLL(m, Strategy │ │ │ │ 0000a680: 3d3e 7b47 6976 656e 732c 5265 616c 5844 =>{Givens,RealXD │ │ │ │ 0000a690: 7d29 3b20 2020 2020 2020 2020 2020 2020 }); │ │ │ │ 0000a6a0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -0000a6b0: 2d20 7573 6564 2030 2e30 3535 3732 3531 - used 0.0557251 │ │ │ │ -0000a6c0: 7320 2863 7075 293b 2030 2e30 3535 3732 s (cpu); 0.05572 │ │ │ │ -0000a6d0: 3731 7320 2874 6872 6561 6429 3b20 3073 71s (thread); 0s │ │ │ │ +0000a6b0: 2d20 7573 6564 2030 2e30 3533 3137 3131 - used 0.0531711 │ │ │ │ +0000a6c0: 7320 2863 7075 293b 2030 2e30 3533 3137 s (cpu); 0.05317 │ │ │ │ +0000a6d0: 3539 7320 2874 6872 6561 6429 3b20 3073 59s (thread); 0s │ │ │ │ 0000a6e0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 0000a6f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a730: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0000a740: 2020 2020 2020 2020 2020 3530 2020 2020 50 │ │ │ │ @@ -2688,16 +2688,16 @@ │ │ │ │ 0000a7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0000a810: 7c69 3920 3a20 7469 6d65 204c 4c4c 286d |i9 : time LLL(m │ │ │ │ 0000a820: 2c20 5374 7261 7465 6779 3d3e 7b47 6976 , Strategy=>{Giv │ │ │ │ 0000a830: 656e 732c 5265 616c 5252 7d29 3b20 2020 ens,RealRR}); │ │ │ │ 0000a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a850: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0000a860: 2030 2e33 3137 3135 3273 2028 6370 7529 0.317152s (cpu) │ │ │ │ -0000a870: 3b20 302e 3331 3731 3536 7320 2874 6872 ; 0.317156s (thr │ │ │ │ +0000a860: 2030 2e33 3036 3435 3773 2028 6370 7529 0.306457s (cpu) │ │ │ │ +0000a870: 3b20 302e 3330 3634 3632 7320 2874 6872 ; 0.306462s (thr │ │ │ │ 0000a880: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 0000a890: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0000a8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a8e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ @@ -2714,17 +2714,17 @@ │ │ │ │ 0000a990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a9b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a --------+.|i10 : │ │ │ │ 0000a9c0: 2074 696d 6520 4c4c 4c28 6d2c 2053 7472 time LLL(m, Str │ │ │ │ 0000a9d0: 6174 6567 793d 3e7b 424b 5a2c 4769 7665 ategy=>{BKZ,Give │ │ │ │ 0000a9e0: 6e73 2c52 6561 6c51 507d 293b 2020 2020 ns,RealQP}); │ │ │ │ 0000a9f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000aa00: 0a7c 202d 2d20 7573 6564 2030 2e31 3133 .| -- used 0.113 │ │ │ │ -0000aa10: 3039 7320 2863 7075 293b 2030 2e31 3133 09s (cpu); 0.113 │ │ │ │ -0000aa20: 3039 3273 2028 7468 7265 6164 293b 2030 092s (thread); 0 │ │ │ │ +0000aa00: 0a7c 202d 2d20 7573 6564 2030 2e31 3238 .| -- used 0.128 │ │ │ │ +0000aa10: 3531 3673 2028 6370 7529 3b20 302e 3132 516s (cpu); 0.12 │ │ │ │ +0000aa20: 3835 3273 2028 7468 7265 6164 293b 2030 852s (thread); 0 │ │ │ │ 0000aa30: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000aa40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0000aa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aa80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0000aa90: 2020 2020 2020 2020 2020 2020 2020 2035 5 │ │ ├── ./usr/share/info/LatticePolytopes.info.gz │ │ │ ├── LatticePolytopes.info │ │ │ │ @@ -1163,32 +1163,32 @@ │ │ │ │ 000048a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000048b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000048c0: 2d2d 2b0a 7c69 3620 3a20 7469 6d65 2061 --+.|i6 : time a │ │ │ │ 000048d0: 7265 4973 6f6d 6f72 7068 6963 2850 2c50 reIsomorphic(P,P │ │ │ │ 000048e0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ 000048f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00004910: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00004920: 3734 3538 3273 2028 6370 7529 3b20 302e 74582s (cpu); 0. │ │ │ │ -00004930: 3530 3639 3638 7320 2874 6872 6561 6429 506968s (thread) │ │ │ │ +00004910: 2020 7c0a 7c20 2d2d 2075 7365 6420 312e |.| -- used 1. │ │ │ │ +00004920: 3038 3732 3273 2028 6370 7529 3b20 302e 08722s (cpu); 0. │ │ │ │ +00004930: 3439 3530 3331 7320 2874 6872 6561 6429 495031s (thread) │ │ │ │ 00004940: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00004950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004960: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00004970: 2d2d 2d2d 2d2d 2d2d 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 2b0a 7c69 3720 3a20 7469 6d65 2061 --+.|i7 : time a │ │ │ │ 000049c0: 7265 4973 6f6d 6f72 7068 6963 2850 2c50 reIsomorphic(P,P │ │ │ │ 000049d0: 2c73 6d6f 6f74 6854 6573 743d 3e66 616c ,smoothTest=>fal │ │ │ │ 000049e0: 7365 293b 2020 2020 2020 2020 2020 2020 se); │ │ │ │ 000049f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004a00: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00004a10: 3336 3130 3532 7320 2863 7075 293b 2030 361052s (cpu); 0 │ │ │ │ -00004a20: 2e32 3737 3737 3173 2028 7468 7265 6164 .277771s (thread │ │ │ │ +00004a10: 3338 3339 3839 7320 2863 7075 293b 2030 383989s (cpu); 0 │ │ │ │ +00004a20: 2e32 3438 3532 3973 2028 7468 7265 6164 .248529s (thread │ │ │ │ 00004a30: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 00004a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004a50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00004a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/LinearTruncations.info.gz │ │ │ ├── LinearTruncations.info │ │ │ │ @@ -2194,16 +2194,16 @@ │ │ │ │ 00008910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008920: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ 00008930: 2065 6c61 7073 6564 5469 6d65 2066 696e elapsedTime fin │ │ │ │ 00008940: 6452 6567 696f 6e28 7b7b 302c 307d 2c7b dRegion({{0,0},{ │ │ │ │ 00008950: 342c 347d 7d2c 4d2c 6629 2020 2020 2020 4,4}},M,f) │ │ │ │ 00008960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008970: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00008980: 2e31 3334 3930 3673 2065 6c61 7073 6564 .134906s elapsed │ │ │ │ -00008990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00008980: 2e30 3730 3338 3039 7320 656c 6170 7365 .0703809s elapse │ │ │ │ +00008990: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 000089a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000089d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000089f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2229,15 +2229,15 @@ │ │ │ │ 00008b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008b50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a ---------+.|i7 : │ │ │ │ 00008b60: 2065 6c61 7073 6564 5469 6d65 2066 696e elapsedTime fin │ │ │ │ 00008b70: 6452 6567 696f 6e28 7b7b 302c 307d 2c7b dRegion({{0,0},{ │ │ │ │ 00008b80: 342c 347d 7d2c 4d2c 662c 496e 6e65 723d 4,4}},M,f,Inner= │ │ │ │ 00008b90: 3e7b 7b31 2c32 7d2c 7b33 2c31 7d7d 2c4f >{{1,2},{3,1}},O │ │ │ │ 00008ba0: 7574 6572 3d3e 7b7b 317c 0a7c 202d 2d20 uter=>{{1|.| -- │ │ │ │ -00008bb0: 2e30 3132 3733 3135 7320 656c 6170 7365 .0127315s elapse │ │ │ │ +00008bb0: 2e30 3134 3037 3839 7320 656c 6170 7365 .0140789s elapse │ │ │ │ 00008bc0: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 00008bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008bf0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00008c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00008c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4112,15 +4112,15 @@ │ │ │ │ 000100f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00010120: 2b0a 7c69 3620 3a20 656c 6170 7365 6454 +.|i6 : elapsedT │ │ │ │ 00010130: 696d 6520 6c69 6e65 6172 5472 756e 6361 ime linearTrunca │ │ │ │ 00010140: 7469 6f6e 7328 7b7b 322c 322c 327d 2c7b tions({{2,2,2},{ │ │ │ │ 00010150: 342c 342c 347d 7d2c 204d 297c 0a7c 202d 4,4,4}}, M)|.| - │ │ │ │ -00010160: 2d20 332e 3732 3431 3373 2065 6c61 7073 - 3.72413s elaps │ │ │ │ +00010160: 2d20 322e 3834 3831 3273 2065 6c61 7073 - 2.84812s elaps │ │ │ │ 00010170: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ 00010180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010190: 2020 2020 2020 7c0a 7c20 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: 207c 0a7c 6f36 203d 207b 7b34 2c20 332c |.|o6 = {{4, 3, │ │ │ │ @@ -4138,15 +4138,15 @@ │ │ │ │ 00010290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000102a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000102b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000102c0: 6937 203a 2065 6c61 7073 6564 5469 6d65 i7 : elapsedTime │ │ │ │ 000102d0: 206c 696e 6561 7254 7275 6e63 6174 696f linearTruncatio │ │ │ │ 000102e0: 6e73 426f 756e 6420 4d20 2020 2020 2020 nsBound M │ │ │ │ 000102f0: 2020 2020 2020 2020 7c0a 7c20 2d2d 202e |.| -- . │ │ │ │ -00010300: 3131 3533 3235 7320 656c 6170 7365 6420 115325s elapsed │ │ │ │ +00010300: 3035 3237 3734 3373 2065 6c61 7073 6564 0527743s elapsed │ │ │ │ 00010310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010330: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00010340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00010360: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00010370: 7c6f 3720 3d20 7b7b 342c 2033 2c20 337d |o7 = {{4, 3, 3} │ │ ├── ./usr/share/info/LocalRings.info.gz │ │ │ ├── LocalRings.info │ │ │ │ @@ -2634,15 +2634,15 @@ │ │ │ │ 0000a490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a4a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000a4c0: 2d2d 2b0a 7c69 3520 3a20 656c 6170 7365 --+.|i5 : elapse │ │ │ │ 0000a4d0: 6454 696d 6520 6869 6c62 6572 7453 616d dTime hilbertSam │ │ │ │ 0000a4e0: 7565 6c46 756e 6374 696f 6e28 4d2c 2030 uelFunction(M, 0 │ │ │ │ 0000a4f0: 2c20 3629 2020 2020 2020 7c0a 7c20 2d2d , 6) |.| -- │ │ │ │ -0000a500: 202e 3139 3234 3632 7320 656c 6170 7365 .192462s elapse │ │ │ │ +0000a500: 202e 3136 3938 3931 7320 656c 6170 7365 .169891s elapse │ │ │ │ 0000a510: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ 0000a520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a530: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000a540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000a560: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ 0000a570: 3d20 7b31 2c20 332c 2036 2c20 372c 2036 = {1, 3, 6, 7, 6 │ │ │ │ @@ -2771,16 +2771,16 @@ │ │ │ │ 0000ad20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000ad40: 2d2d 2d2b 0a7c 6931 3120 3a20 656c 6170 ---+.|i11 : elap │ │ │ │ 0000ad50: 7365 6454 696d 6520 6869 6c62 6572 7453 sedTime hilbertS │ │ │ │ 0000ad60: 616d 7565 6c46 756e 6374 696f 6e28 4e2c amuelFunction(N, │ │ │ │ 0000ad70: 2030 2c20 3529 202d 2d20 6e2b 3120 2d2d 0, 5) -- n+1 -- │ │ │ │ 0000ad80: 2030 2e30 3220 7365 636f 6e64 7320 2020 0.02 seconds │ │ │ │ -0000ad90: 2020 207c 0a7c 202d 2d20 2e30 3132 3837 |.| -- .01287 │ │ │ │ -0000ada0: 3138 7320 656c 6170 7365 6420 2020 2020 18s elapsed │ │ │ │ +0000ad90: 2020 207c 0a7c 202d 2d20 2e30 3134 3239 |.| -- .01429 │ │ │ │ +0000ada0: 3931 7320 656c 6170 7365 6420 2020 2020 91s elapsed │ │ │ │ 0000adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ade0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000adf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000ae10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -2806,16 +2806,16 @@ │ │ │ │ 0000af50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000af70: 2d2d 2d2b 0a7c 6931 3220 3a20 656c 6170 ---+.|i12 : elap │ │ │ │ 0000af80: 7365 6454 696d 6520 6869 6c62 6572 7453 sedTime hilbertS │ │ │ │ 0000af90: 616d 7565 6c46 756e 6374 696f 6e28 712c amuelFunction(q, │ │ │ │ 0000afa0: 204e 2c20 302c 2035 2920 2d2d 2036 286e N, 0, 5) -- 6(n │ │ │ │ 0000afb0: 2b31 2920 2d2d 2030 2e33 3220 7365 636f +1) -- 0.32 seco │ │ │ │ -0000afc0: 6e64 737c 0a7c 202d 2d20 2e33 3736 3439 nds|.| -- .37649 │ │ │ │ -0000afd0: 3173 2065 6c61 7073 6564 2020 2020 2020 1s elapsed │ │ │ │ +0000afc0: 6e64 737c 0a7c 202d 2d20 2e32 3534 3131 nds|.| -- .25411 │ │ │ │ +0000afd0: 3573 2065 6c61 7073 6564 2020 2020 2020 5s elapsed │ │ │ │ 0000afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b010: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0000b020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ ├── ./usr/share/info/MRDI.info.gz │ │ │ ├── MRDI.info │ │ │ │ @@ -917,16 +917,16 @@ │ │ │ │ 00003940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003950: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 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 2020 2020 2020 │ │ │ │ 000039a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000039b0: 3720 3d20 2f74 6d70 2f4d 322d 3430 3032 7 = /tmp/M2-4002 │ │ │ │ -000039c0: 372d 302f 302e 6d72 6469 2020 2020 2020 7-0/0.mrdi │ │ │ │ +000039b0: 3720 3d20 2f74 6d70 2f4d 322d 3536 3130 7 = /tmp/M2-5610 │ │ │ │ +000039c0: 352d 302f 302e 6d72 6469 2020 2020 2020 5-0/0.mrdi │ │ │ │ 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 7c0a 2b2d |.+- │ │ │ │ 00003a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -1150,29 +1150,29 @@ │ │ │ │ 000047d0: 654d 5244 497d 2020 2020 2020 2020 2020 eMRDI} │ │ │ │ 000047e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 000047f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004820: 7c0a 7c6f 3133 203d 207b 3020 3d3e 2028 |.|o13 = {0 => ( │ │ │ │ 00004830: 7b4f 7363 6172 2c20 7361 7665 4d52 4449 {Oscar, saveMRDI │ │ │ │ -00004840: 7d2c 2052 696e 6745 6c65 6d65 6e74 2920 }, RingElement) │ │ │ │ +00004840: 7d2c 2051 5129 2020 2020 2020 2020 2020 }, QQ) │ │ │ │ 00004850: 2020 7d20 2020 207c 0a7c 2020 2020 2020 } |.| │ │ │ │ 00004860: 7b31 203d 3e20 287b 4f73 6361 722c 2073 {1 => ({Oscar, s │ │ │ │ -00004870: 6176 654d 5244 497d 2c20 5269 6e67 2920 aveMRDI}, Ring) │ │ │ │ +00004870: 6176 654d 5244 497d 2c20 5a5a 2920 2020 aveMRDI}, ZZ) │ │ │ │ 00004880: 2020 2020 2020 2020 207d 2020 2020 7c0a } |. │ │ │ │ 00004890: 7c20 2020 2020 207b 3220 3d3e 2028 7b4f | {2 => ({O │ │ │ │ 000048a0: 7363 6172 2c20 7361 7665 4d52 4449 7d2c scar, saveMRDI}, │ │ │ │ -000048b0: 2050 6f6c 796e 6f6d 6961 6c52 696e 6729 PolynomialRing) │ │ │ │ +000048b0: 2052 696e 6745 6c65 6d65 6e74 2920 2020 RingElement) │ │ │ │ 000048c0: 7d20 2020 207c 0a7c 2020 2020 2020 7b33 } |.| {3 │ │ │ │ 000048d0: 203d 3e20 287b 4f73 6361 722c 2073 6176 => ({Oscar, sav │ │ │ │ -000048e0: 654d 5244 497d 2c20 5151 2920 2020 2020 eMRDI}, QQ) │ │ │ │ +000048e0: 654d 5244 497d 2c20 5269 6e67 2920 2020 eMRDI}, Ring) │ │ │ │ 000048f0: 2020 2020 2020 207d 2020 2020 7c0a 7c20 } |.| │ │ │ │ 00004900: 2020 2020 207b 3420 3d3e 2028 7b4f 7363 {4 => ({Osc │ │ │ │ -00004910: 6172 2c20 7361 7665 4d52 4449 7d2c 205a ar, saveMRDI}, Z │ │ │ │ -00004920: 5a29 2020 2020 2020 2020 2020 2020 7d20 Z) } │ │ │ │ +00004910: 6172 2c20 7361 7665 4d52 4449 7d2c 2050 ar, saveMRDI}, P │ │ │ │ +00004920: 6f6c 796e 6f6d 6961 6c52 696e 6729 7d20 olynomialRing)} │ │ │ │ 00004930: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00004940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004960: 2020 2020 2020 2020 2020 7c0a 7c6f 3133 |.|o13 │ │ │ │ 00004970: 203a 204e 756d 6265 7265 6456 6572 7469 : NumberedVerti │ │ │ │ 00004980: 6361 6c4c 6973 7420 2020 2020 2020 2020 calList │ │ │ │ 00004990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1903,15 +1903,15 @@ │ │ │ │ 000076e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000076f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00007700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007740: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ -00007750: 202f 746d 702f 4d32 2d34 3030 3038 2d30 /tmp/M2-40008-0 │ │ │ │ +00007750: 202f 746d 702f 4d32 2d35 3630 3636 2d30 /tmp/M2-56066-0 │ │ │ │ 00007760: 2f30 2e6d 7264 6920 2020 2020 2020 2020 /0.mrdi │ │ │ │ 00007770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007790: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 000077a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000077b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000077c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ ├── ./usr/share/info/Macaulay2Doc.info.gz │ │ │ ├── Macaulay2Doc.info │ │ │ │ @@ -4907,17 +4907,17 @@ │ │ │ │ 000132a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000132b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000132c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000132d0: 2d2d 2b0a 7c69 3539 203a 2074 696d 6520 --+.|i59 : time │ │ │ │ 000132e0: 4320 3d20 7265 736f 6c75 7469 6f6e 204d C = resolution M │ │ │ │ 000132f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013300: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013310: 7c20 2d2d 2075 7365 6420 302e 3030 3435 | -- used 0.0045 │ │ │ │ -00013320: 3830 3137 7320 2863 7075 293b 2030 2e30 8017s (cpu); 0.0 │ │ │ │ -00013330: 3034 3537 3436 3873 2028 7468 7265 6164 0457468s (thread │ │ │ │ +00013310: 7c20 2d2d 2075 7365 6420 302e 3030 3330 | -- used 0.0030 │ │ │ │ +00013320: 3833 3939 7320 2863 7075 293b 2030 2e30 8399s (cpu); 0.0 │ │ │ │ +00013330: 3033 3036 3239 3773 2028 7468 7265 6164 0306297s (thread │ │ │ │ 00013340: 293b 2030 7320 2867 6329 7c0a 7c20 2020 ); 0s (gc)|.| │ │ │ │ 00013350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013380: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00013390: 3320 2020 2020 2036 2020 2020 2020 3135 3 6 15 │ │ │ │ 000133a0: 2020 2020 2020 3138 2020 2020 2020 3620 18 6 │ │ │ │ @@ -26224,16 +26224,16 @@ │ │ │ │ 000666f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ 00066700: 3a20 2863 203d 2043 6f6d 6d61 6e64 2022 : (c = Command " │ │ │ │ 00066710: 6461 7465 223b 2920 2020 2020 2020 207c date";) | │ │ │ │ 00066720: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00066730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066740: 2d2d 2d2d 2b0a 7c69 3420 3a20 6320 2020 ----+.|i4 : c │ │ │ │ 00066750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00066760: 2020 2020 2020 2020 207c 0a7c 5361 7420 |.|Sat │ │ │ │ -00066770: 4f63 7420 3234 2030 333a 3131 3a35 3720 Oct 24 03:11:57 │ │ │ │ +00066760: 2020 2020 2020 2020 207c 0a7c 5468 7520 |.|Thu │ │ │ │ +00066770: 4f63 7420 3239 2031 323a 3035 3a35 3420 Oct 29 12:05:54 │ │ │ │ 00066780: 5554 4320 3230 3236 2020 2020 2020 7c0a UTC 2026 |. │ │ │ │ 00066790: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000667a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000667b0: 2020 207c 0a7c 6f34 203d 2030 2020 2020 |.|o4 = 0 │ │ │ │ 000667c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000667d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 000667e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -39875,16 +39875,16 @@ │ │ │ │ 0009bc20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009bc40: 2d2d 2b0a 7c69 3131 203a 2065 6c61 7073 --+.|i11 : elaps │ │ │ │ 0009bc50: 6564 5469 6d65 2073 6361 6e28 3130 3030 edTime scan(1000 │ │ │ │ 0009bc60: 2c20 6920 2d3e 2073 2369 203d 2069 5e32 , i -> s#i = i^2 │ │ │ │ 0009bc70: 2920 2d2d 2071 7561 6472 6174 6963 2c20 ) -- quadratic, │ │ │ │ 0009bc80: 7369 6e63 6520 7765 2067 726f 7720 7320 since we grow s │ │ │ │ -0009bc90: 6174 7c0a 7c20 2d2d 202e 3030 3630 3530 at|.| -- .006050 │ │ │ │ -0009bca0: 3438 7320 656c 6170 7365 6420 2020 2020 48s elapsed │ │ │ │ +0009bc90: 6174 7c0a 7c20 2d2d 202e 3030 3432 3134 at|.| -- .004214 │ │ │ │ +0009bca0: 3631 7320 656c 6170 7365 6420 2020 2020 61s elapsed │ │ │ │ 0009bcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009bcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009bce0: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ 0009bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009bd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -39930,16 +39930,16 @@ │ │ │ │ 0009bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009bfb0: 2d2d 2b0a 7c69 3133 203a 2065 6c61 7073 --+.|i13 : elaps │ │ │ │ 0009bfc0: 6564 5469 6d65 2073 6361 6e28 3130 3030 edTime scan(1000 │ │ │ │ 0009bfd0: 2c20 6920 2d3e 2074 2369 203d 2069 5e32 , i -> t#i = i^2 │ │ │ │ 0009bfe0: 2920 2d2d 206c 696e 6561 7220 2020 2020 ) -- linear │ │ │ │ 0009bff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0009c000: 2020 7c0a 7c20 2d2d 202e 3030 3037 3132 |.| -- .000712 │ │ │ │ -0009c010: 3137 3173 2065 6c61 7073 6564 2020 2020 171s elapsed │ │ │ │ +0009c000: 2020 7c0a 7c20 2d2d 202e 3030 3032 3732 |.| -- .000272 │ │ │ │ +0009c010: 3939 3473 2065 6c61 7073 6564 2020 2020 994s elapsed │ │ │ │ 0009c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0009c050: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0009c060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009c070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0009c080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -76569,30 +76569,30 @@ │ │ │ │ 0012b180: 2c20 4d61 7472 6978 2c20 4d61 7472 6978 , Matrix, Matrix │ │ │ │ 0012b190: 2c20 4d61 7472 6978 2c20 4d61 7472 6978 , Matrix, Matrix │ │ │ │ 0012b1a0: 297d 7c0a 7c20 2020 2020 7b33 203d 3e20 )}|.| {3 => │ │ │ │ 0012b1b0: 2869 6e74 6572 7365 6374 2c20 5252 692c (intersect, RRi, │ │ │ │ 0012b1c0: 2052 5269 2920 2020 2020 2020 2020 2020 RRi) │ │ │ │ 0012b1d0: 2020 2020 2020 2020 2020 207d 7c0a 7c20 }|.| │ │ │ │ 0012b1e0: 2020 2020 7b34 203d 3e20 2869 6e74 6572 {4 => (inter │ │ │ │ -0012b1f0: 7365 6374 2c20 4d61 7472 6978 2c20 4d61 sect, Matrix, Ma │ │ │ │ -0012b200: 7472 6978 2920 2020 2020 2020 2020 2020 trix) │ │ │ │ +0012b1f0: 7365 6374 2c20 5365 742c 2053 6574 2920 sect, Set, Set) │ │ │ │ +0012b200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012b210: 2020 2020 207d 7c0a 7c20 2020 2020 7b35 }|.| {5 │ │ │ │ 0012b220: 203d 3e20 2869 6e74 6572 7365 6374 2c20 => (intersect, │ │ │ │ -0012b230: 5365 742c 2053 6574 2920 2020 2020 2020 Set, Set) │ │ │ │ +0012b230: 4d61 7472 6978 2c20 4d61 7472 6978 2920 Matrix, Matrix) │ │ │ │ 0012b240: 2020 2020 2020 2020 2020 2020 2020 207d } │ │ │ │ 0012b250: 7c0a 7c20 2020 2020 7b36 203d 3e20 2869 |.| {6 => (i │ │ │ │ -0012b260: 6e74 6572 7365 6374 2c20 5365 7429 2020 ntersect, Set) │ │ │ │ -0012b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0012b260: 6e74 6572 7365 6374 2c20 506f 6c79 6865 ntersect, Polyhe │ │ │ │ +0012b270: 6472 6f6e 2c20 436f 6e65 2920 2020 2020 dron, Cone) │ │ │ │ 0012b280: 2020 2020 2020 2020 207d 7c0a 7c20 2020 }|.| │ │ │ │ 0012b290: 2020 7b37 203d 3e20 2869 6e74 6572 7365 {7 => (interse │ │ │ │ -0012b2a0: 6374 2c20 506f 6c79 6865 6472 6f6e 2c20 ct, Polyhedron, │ │ │ │ -0012b2b0: 436f 6e65 2920 2020 2020 2020 2020 2020 Cone) │ │ │ │ +0012b2a0: 6374 2c20 436f 6e65 2c20 506f 6c79 6865 ct, Cone, Polyhe │ │ │ │ +0012b2b0: 6472 6f6e 2920 2020 2020 2020 2020 2020 dron) │ │ │ │ 0012b2c0: 2020 207d 7c0a 7c20 2020 2020 7b38 203d }|.| {8 = │ │ │ │ -0012b2d0: 3e20 2869 6e74 6572 7365 6374 2c20 436f > (intersect, Co │ │ │ │ -0012b2e0: 6e65 2c20 506f 6c79 6865 6472 6f6e 2920 ne, Polyhedron) │ │ │ │ +0012b2d0: 3e20 2869 6e74 6572 7365 6374 2c20 5365 > (intersect, Se │ │ │ │ +0012b2e0: 7429 2020 2020 2020 2020 2020 2020 2020 t) │ │ │ │ 0012b2f0: 2020 2020 2020 2020 2020 2020 207d 7c0a }|. │ │ │ │ 0012b300: 7c20 2020 2020 7b39 203d 3e20 2869 6e74 | {9 => (int │ │ │ │ 0012b310: 6572 7365 6374 2c20 4964 6561 6c29 2020 ersect, Ideal) │ │ │ │ 0012b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012b330: 2020 2020 2020 207d 7c0a 7c20 2020 2020 }|.| │ │ │ │ 0012b340: 7b31 3020 3d3e 2028 696e 7465 7273 6563 {10 => (intersec │ │ │ │ 0012b350: 742c 2049 6465 616c 2c20 4964 6561 6c29 t, Ideal, Ideal) │ │ │ │ @@ -76602,22 +76602,22 @@ │ │ │ │ 0012b390: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0012b3a0: 2020 2020 2020 2020 2020 207d 7c0a 7c20 }|.| │ │ │ │ 0012b3b0: 2020 2020 7b31 3220 3d3e 2028 696e 7465 {12 => (inte │ │ │ │ 0012b3c0: 7273 6563 742c 2052 5269 2920 2020 2020 rsect, RRi) │ │ │ │ 0012b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012b3e0: 2020 2020 207d 7c0a 7c20 2020 2020 7b31 }|.| {1 │ │ │ │ 0012b3f0: 3320 3d3e 2028 696e 7465 7273 6563 742c 3 => (intersect, │ │ │ │ -0012b400: 2050 6f6c 7968 6564 726f 6e2c 2050 6f6c Polyhedron, Pol │ │ │ │ -0012b410: 7968 6564 726f 6e29 2020 2020 2020 207d yhedron) } │ │ │ │ +0012b400: 204d 6174 7269 7829 2020 2020 2020 2020 Matrix) │ │ │ │ +0012b410: 2020 2020 2020 2020 2020 2020 2020 207d } │ │ │ │ 0012b420: 7c0a 7c20 2020 2020 7b31 3420 3d3e 2028 |.| {14 => ( │ │ │ │ -0012b430: 696e 7465 7273 6563 742c 2043 6f6e 652c intersect, Cone, │ │ │ │ -0012b440: 2043 6f6e 6529 2020 2020 2020 2020 2020 Cone) │ │ │ │ -0012b450: 2020 2020 2020 2020 207d 7c0a 7c20 2020 }|.| │ │ │ │ +0012b430: 696e 7465 7273 6563 742c 2050 6f6c 7968 intersect, Polyh │ │ │ │ +0012b440: 6564 726f 6e2c 2050 6f6c 7968 6564 726f edron, Polyhedro │ │ │ │ +0012b450: 6e29 2020 2020 2020 207d 7c0a 7c20 2020 n) }|.| │ │ │ │ 0012b460: 2020 7b31 3520 3d3e 2028 696e 7465 7273 {15 => (inters │ │ │ │ -0012b470: 6563 742c 204d 6174 7269 7829 2020 2020 ect, Matrix) │ │ │ │ +0012b470: 6563 742c 2043 6f6e 652c 2043 6f6e 6529 ect, Cone, Cone) │ │ │ │ 0012b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0012b490: 2020 207d 7c0a 7c20 2020 2020 7b31 3620 }|.| {16 │ │ │ │ 0012b4a0: 3d3e 2028 696e 7465 7273 6563 742c 204d => (intersect, M │ │ │ │ 0012b4b0: 6f64 756c 652c 204d 6f64 756c 6529 2020 odule, Module) │ │ │ │ 0012b4c0: 2020 2020 2020 2020 2020 2020 207d 7c0a }|. │ │ │ │ 0012b4d0: 7c20 2020 2020 7b31 3720 3d3e 2028 696e | {17 => (in │ │ │ │ 0012b4e0: 7465 7273 6563 742c 2043 4369 2c20 4343 tersect, CCi, CC │ │ │ │ @@ -104123,16 +104123,16 @@ │ │ │ │ 00196ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00196bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00196bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ 00196bd0: 203a 2074 696d 6520 4a20 3d20 7472 756e : time J = trun │ │ │ │ 00196be0: 6361 7465 2838 2c20 492c 204d 696e 696d cate(8, I, Minim │ │ │ │ 00196bf0: 616c 4765 6e65 7261 746f 7273 203d 3e20 alGenerators => │ │ │ │ 00196c00: 6661 6c73 6529 3b7c 0a7c 202d 2d20 7573 false);|.| -- us │ │ │ │ -00196c10: 6564 2030 2e30 3038 3630 3232 3973 2028 ed 0.00860229s ( │ │ │ │ -00196c20: 6370 7529 3b20 302e 3030 3835 3935 3635 cpu); 0.00859565 │ │ │ │ +00196c10: 6564 2030 2e30 3034 3336 3037 3173 2028 ed 0.00436071s ( │ │ │ │ +00196c20: 6370 7529 3b20 302e 3030 3433 3537 3238 cpu); 0.00435728 │ │ │ │ 00196c30: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ 00196c40: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ 00196c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00196c80: 0a7c 6f38 203a 2049 6465 616c 206f 6620 .|o8 : Ideal of │ │ │ │ 00196c90: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ @@ -104142,17 +104142,17 @@ │ │ │ │ 00196cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00196ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00196cf0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 -------+.|i9 : t │ │ │ │ 00196d00: 696d 6520 4b20 3d20 7472 756e 6361 7465 ime K = truncate │ │ │ │ 00196d10: 2838 2c20 492c 204d 696e 696d 616c 4765 (8, I, MinimalGe │ │ │ │ 00196d20: 6e65 7261 746f 7273 203d 3e20 7472 7565 nerators => true │ │ │ │ 00196d30: 293b 207c 0a7c 202d 2d20 7573 6564 2030 ); |.| -- used 0 │ │ │ │ -00196d40: 2e30 3635 3834 3031 7320 2863 7075 293b .0658401s (cpu); │ │ │ │ -00196d50: 2030 2e30 3635 3835 3034 7320 2874 6872 0.0658504s (thr │ │ │ │ -00196d60: 6561 6429 3b20 3073 2028 6763 2920 207c ead); 0s (gc) | │ │ │ │ +00196d40: 2e30 3338 3231 3173 2028 6370 7529 3b20 .038211s (cpu); │ │ │ │ +00196d50: 302e 3033 3832 3136 3373 2028 7468 7265 0.0382163s (thre │ │ │ │ +00196d60: 6164 293b 2030 7320 2867 6329 2020 207c ad); 0s (gc) | │ │ │ │ 00196d70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00196d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196da0: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ 00196db0: 203a 2049 6465 616c 206f 6620 5220 2020 : Ideal of R │ │ │ │ 00196dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00196dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -104750,16 +104750,16 @@ │ │ │ │ 001992d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001992e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001992f0: 207c 0a7c 202d 2d20 636f 6d70 7574 696e |.| -- computin │ │ │ │ 00199300: 6720 7064 696d 2720 2020 2020 2020 2020 g pdim' │ │ │ │ 00199310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00199340: 207c 0a7c 202d 2d20 2e30 3033 3531 3732 |.| -- .0035172 │ │ │ │ -00199350: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ +00199340: 207c 0a7c 202d 2d20 2e30 3033 3338 3837 |.| -- .0033887 │ │ │ │ +00199350: 3973 2065 6c61 7073 6564 2020 2020 2020 9s elapsed │ │ │ │ 00199360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199390: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 001993a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001993b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001993c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -104775,16 +104775,16 @@ │ │ │ │ 00199460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00199470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00199480: 2d2b 0a7c 6935 203a 2065 6c61 7073 6564 -+.|i5 : elapsed │ │ │ │ 00199490: 5469 6d65 2070 6469 6d27 204d 2020 2020 Time pdim' M │ │ │ │ 001994a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001994b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001994c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -001994d0: 207c 0a7c 202d 2d20 392e 3232 652d 3773 |.| -- 9.22e-7s │ │ │ │ -001994e0: 2065 6c61 7073 6564 2020 2020 2020 2020 elapsed │ │ │ │ +001994d0: 207c 0a7c 202d 2d20 2e30 3030 3030 3234 |.| -- .0000024 │ │ │ │ +001994e0: 3734 7320 656c 6170 7365 6420 2020 2020 74s elapsed │ │ │ │ 001994f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199520: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00199530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00199550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -105279,17 +105279,17 @@ │ │ │ │ 0019b3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0019b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0019b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 0019b410: 203a 2074 696d 6520 6669 6220 3238 2020 : time fib 28 │ │ │ │ 0019b420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b440: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0019b450: 6564 2031 2e34 3733 3033 7320 2863 7075 ed 1.47303s (cpu │ │ │ │ -0019b460: 293b 2030 2e37 3930 3538 3773 2028 7468 ); 0.790587s (th │ │ │ │ -0019b470: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ +0019b450: 6564 2030 2e38 3638 3038 3473 2028 6370 ed 0.868084s (cp │ │ │ │ +0019b460: 7529 3b20 302e 3439 3336 3432 7320 2874 u); 0.493642s (t │ │ │ │ +0019b470: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0019b480: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0019b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b4b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0019b4c0: 0a7c 6f32 203d 2035 3134 3232 3920 2020 .|o2 = 514229 │ │ │ │ 0019b4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -105320,18 +105320,18 @@ │ │ │ │ 0019b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0019b680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0019b690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0019b6a0: 0a7c 6934 203a 2074 696d 6520 6669 6220 .|i4 : time fib │ │ │ │ 0019b6b0: 3238 2020 2020 2020 2020 2020 2020 2020 28 │ │ │ │ 0019b6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b6d0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -0019b6e0: 2d20 7573 6564 2036 2e36 3337 3465 2d30 - used 6.6374e-0 │ │ │ │ -0019b6f0: 3573 2028 6370 7529 3b20 362e 3538 3433 5s (cpu); 6.5843 │ │ │ │ -0019b700: 652d 3035 7320 2874 6872 6561 6429 3b20 e-05s (thread); │ │ │ │ -0019b710: 3073 2028 6763 297c 0a7c 2020 2020 2020 0s (gc)|.| │ │ │ │ +0019b6e0: 2d20 7573 6564 2036 2e30 3830 3265 2d30 - used 6.0802e-0 │ │ │ │ +0019b6f0: 3573 2028 6370 7529 3b20 352e 3838 3665 5s (cpu); 5.886e │ │ │ │ +0019b700: 2d30 3573 2028 7468 7265 6164 293b 2030 -05s (thread); 0 │ │ │ │ +0019b710: 7320 2867 6329 207c 0a7c 2020 2020 2020 s (gc) |.| │ │ │ │ 0019b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b750: 2020 207c 0a7c 6f34 203d 2035 3134 3232 |.|o4 = 51422 │ │ │ │ 0019b760: 3920 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ 0019b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b780: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -105339,16 +105339,16 @@ │ │ │ │ 0019b7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0019b7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0019b7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 0019b7d0: 203a 2074 696d 6520 6669 6220 3238 2020 : time fib 28 │ │ │ │ 0019b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b800: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0019b810: 6564 2034 2e34 3939 652d 3036 7320 2863 ed 4.499e-06s (c │ │ │ │ -0019b820: 7075 293b 2034 2e30 3738 652d 3036 7320 pu); 4.078e-06s │ │ │ │ +0019b810: 6564 2033 2e32 3639 652d 3036 7320 2863 ed 3.269e-06s (c │ │ │ │ +0019b820: 7075 293b 2032 2e37 3435 652d 3036 7320 pu); 2.745e-06s │ │ │ │ 0019b830: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 0019b840: 2920 207c 0a7c 2020 2020 2020 2020 2020 ) |.| │ │ │ │ 0019b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0019b870: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0019b880: 0a7c 6f35 203d 2035 3134 3232 3920 2020 .|o5 = 514229 │ │ │ │ 0019b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -107889,15 +107889,15 @@ │ │ │ │ 001a5700: 6c69 6e65 4e75 6d62 6572 203d 3e20 3220 lineNumber => 2 │ │ │ │ 001a5710: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 001a5720: 2020 2020 2020 2020 2020 2020 206c 6f61 loa │ │ │ │ 001a5730: 6444 6570 7468 203d 3e20 3320 2020 2020 dDepth => 3 │ │ │ │ 001a5740: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 001a5750: 2020 2020 2020 2020 2020 6d61 7841 6c6c maxAll │ │ │ │ 001a5760: 6f77 6162 6c65 5468 7265 6164 7320 3d3e owableThreads => │ │ │ │ -001a5770: 2037 2020 207c 0a7c 2020 2020 2020 2020 7 |.| │ │ │ │ +001a5770: 2031 3720 207c 0a7c 2020 2020 2020 2020 17 |.| │ │ │ │ 001a5780: 2020 2020 2020 206d 6178 4578 706f 6e65 maxExpone │ │ │ │ 001a5790: 6e74 203d 3e20 3130 3733 3734 3138 3233 nt => 1073741823 │ │ │ │ 001a57a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 001a57b0: 2020 2020 6d69 6e45 7870 6f6e 656e 7420 minExponent │ │ │ │ 001a57c0: 3d3e 202d 3130 3733 3734 3138 3234 207c => -1073741824 | │ │ │ │ 001a57d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 001a57e0: 206e 756d 5442 4254 6872 6561 6473 203d numTBBThreads = │ │ │ │ @@ -118357,16 +118357,16 @@ │ │ │ │ 001ce540: 6572 206f 6620 7365 636f 6e64 732e 0a2b er of seconds..+ │ │ │ │ 001ce550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001ce560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 001ce570: 7c69 3120 3a20 6265 6e63 686d 6172 6b20 |i1 : benchmark │ │ │ │ 001ce580: 2273 7172 7420 3270 3130 3030 3030 227c "sqrt 2p100000"| │ │ │ │ 001ce590: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 001ce5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -001ce5b0: 7c0a 7c6f 3120 3d20 2e30 3030 3337 3530 |.|o1 = .0003750 │ │ │ │ -001ce5c0: 3935 3233 3630 3430 3937 3433 2020 2020 952360409743 │ │ │ │ +001ce5b0: 7c0a 7c6f 3120 3d20 2e30 3030 3330 3634 |.|o1 = .0003064 │ │ │ │ +001ce5c0: 3134 3430 3031 3933 3339 3734 2020 2020 144001933974 │ │ │ │ 001ce5d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 001ce5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001ce5f0: 2020 7c0a 7c6f 3120 3a20 5252 2028 6f66 |.|o1 : RR (of │ │ │ │ 001ce600: 2070 7265 6369 7369 6f6e 2035 3329 2020 precision 53) │ │ │ │ 001ce610: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 001ce620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001ce630: 2d2d 2d2d 2b0a 5468 6520 736e 6970 7065 ----+.The snippe │ │ │ │ @@ -124664,38 +124664,38 @@ │ │ │ │ 001e6f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e6f80: 2020 2020 2020 2020 2020 207d 7c0a 7c20 }|.| │ │ │ │ 001e6f90: 2020 2020 7b31 3820 3d3e 2028 6d61 7468 {18 => (math │ │ │ │ 001e6fa0: 4d4c 2c20 4265 7474 6954 616c 6c79 2920 ML, BettiTally) │ │ │ │ 001e6fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e6fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e6fd0: 207d 7c0a 7c20 2020 2020 7b31 3920 3d3e }|.| {19 => │ │ │ │ -001e6fe0: 2028 7472 756e 6361 7465 2c20 4265 7474 (truncate, Bett │ │ │ │ -001e6ff0: 6954 616c 6c79 2c20 496e 6669 6e69 7465 iTally, Infinite │ │ │ │ -001e7000: 4e75 6d62 6572 2c20 5a5a 2920 2020 2020 Number, ZZ) │ │ │ │ +001e6fe0: 2028 636f 6469 6d2c 2042 6574 7469 5461 (codim, BettiTa │ │ │ │ +001e6ff0: 6c6c 7929 2020 2020 2020 2020 2020 2020 lly) │ │ │ │ +001e7000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7010: 2020 2020 2020 207d 7c0a 7c20 2020 2020 }|.| │ │ │ │ 001e7020: 7b32 3020 3d3e 2028 7472 756e 6361 7465 {20 => (truncate │ │ │ │ -001e7030: 2c20 4265 7474 6954 616c 6c79 2c20 5a5a , BettiTally, ZZ │ │ │ │ -001e7040: 2c20 496e 6669 6e69 7465 4e75 6d62 6572 , InfiniteNumber │ │ │ │ -001e7050: 2920 2020 2020 2020 2020 2020 207d 7c0a ) }|. │ │ │ │ -001e7060: 7c20 2020 2020 7b32 3120 3d3e 2028 6475 | {21 => (du │ │ │ │ -001e7070: 616c 2c20 4265 7474 6954 616c 6c79 2920 al, BettiTally) │ │ │ │ -001e7080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -001e7090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +001e7030: 2c20 4265 7474 6954 616c 6c79 2c20 496e , BettiTally, In │ │ │ │ +001e7040: 6669 6e69 7465 4e75 6d62 6572 2c20 496e finiteNumber, In │ │ │ │ +001e7050: 6669 6e69 7465 4e75 6d62 6572 297d 7c0a finiteNumber)}|. │ │ │ │ +001e7060: 7c20 2020 2020 7b32 3120 3d3e 2028 7472 | {21 => (tr │ │ │ │ +001e7070: 756e 6361 7465 2c20 4265 7474 6954 616c uncate, BettiTal │ │ │ │ +001e7080: 6c79 2c20 5a5a 2c20 496e 6669 6e69 7465 ly, ZZ, Infinite │ │ │ │ +001e7090: 4e75 6d62 6572 2920 2020 2020 2020 2020 Number) │ │ │ │ 001e70a0: 2020 207d 7c0a 7c20 2020 2020 7b32 3220 }|.| {22 │ │ │ │ 001e70b0: 3d3e 2028 7472 756e 6361 7465 2c20 4265 => (truncate, Be │ │ │ │ 001e70c0: 7474 6954 616c 6c79 2c20 496e 6669 6e69 ttiTally, Infini │ │ │ │ -001e70d0: 7465 4e75 6d62 6572 2c20 496e 6669 6e69 teNumber, Infini │ │ │ │ -001e70e0: 7465 4e75 6d62 6572 297d 7c0a 7c20 2020 teNumber)}|.| │ │ │ │ +001e70d0: 7465 4e75 6d62 6572 2c20 5a5a 2920 2020 teNumber, ZZ) │ │ │ │ +001e70e0: 2020 2020 2020 2020 207d 7c0a 7c20 2020 }|.| │ │ │ │ 001e70f0: 2020 7b32 3320 3d3e 2028 7472 756e 6361 {23 => (trunca │ │ │ │ 001e7100: 7465 2c20 4265 7474 6954 616c 6c79 2c20 te, BettiTally, │ │ │ │ 001e7110: 5a5a 2c20 5a5a 2920 2020 2020 2020 2020 ZZ, ZZ) │ │ │ │ 001e7120: 2020 2020 2020 2020 2020 2020 2020 207d } │ │ │ │ 001e7130: 7c0a 7c20 2020 2020 7b32 3420 3d3e 2028 |.| {24 => ( │ │ │ │ -001e7140: 636f 6469 6d2c 2042 6574 7469 5461 6c6c codim, BettiTall │ │ │ │ -001e7150: 7929 2020 2020 2020 2020 2020 2020 2020 y) │ │ │ │ +001e7140: 6475 616c 2c20 4265 7474 6954 616c 6c79 dual, BettiTally │ │ │ │ +001e7150: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 001e7160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7170: 2020 2020 207d 7c0a 7c20 2020 2020 7b32 }|.| {2 │ │ │ │ 001e7180: 3520 3d3e 2028 5e2c 2052 696e 672c 2042 5 => (^, Ring, B │ │ │ │ 001e7190: 6574 7469 5461 6c6c 7929 2020 2020 2020 ettiTally) │ │ │ │ 001e71a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e71b0: 2020 2020 2020 2020 2020 207d 7c0a 7c20 }|.| │ │ │ │ 001e71c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -124875,38 +124875,38 @@ │ │ │ │ 001e7ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7cc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 001e7cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7d00: 2020 2020 2020 207c 0a7c 6f35 203d 207b |.|o5 = { │ │ │ │ -001e7d10: 3020 3d3e 2028 2d2c 204d 6174 7269 782c 0 => (-, Matrix, │ │ │ │ -001e7d20: 204d 6174 7269 7829 2020 2020 2020 2020 Matrix) │ │ │ │ +001e7d10: 3020 3d3e 2028 6469 6666 272c 204d 6174 0 => (diff', Mat │ │ │ │ +001e7d20: 7269 782c 204d 6174 7269 7829 2020 2020 rix, Matrix) │ │ │ │ 001e7d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7d40: 2020 2020 2020 2020 2020 207d 7c0a 7c20 }|.| │ │ │ │ -001e7d50: 2020 2020 7b31 203d 3e20 2864 6966 6627 {1 => (diff' │ │ │ │ -001e7d60: 2c20 4d61 7472 6978 2c20 4d61 7472 6978 , Matrix, Matrix │ │ │ │ -001e7d70: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +001e7d50: 2020 2020 7b31 203d 3e20 282d 2c20 4d61 {1 => (-, Ma │ │ │ │ +001e7d60: 7472 6978 2c20 4d61 7472 6978 2920 2020 trix, Matrix) │ │ │ │ +001e7d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7d90: 7d7c 0a7c 2020 2020 207b 3220 3d3e 2028 }|.| {2 => ( │ │ │ │ -001e7da0: 2b2c 204d 6174 7269 782c 204d 6174 7269 +, Matrix, Matri │ │ │ │ -001e7db0: 7829 2020 2020 2020 2020 2020 2020 2020 x) │ │ │ │ +001e7da0: 6469 6666 2c20 4d61 7472 6978 2c20 4d61 diff, Matrix, Ma │ │ │ │ +001e7db0: 7472 6978 2920 2020 2020 2020 2020 2020 trix) │ │ │ │ 001e7dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7dd0: 2020 2020 207d 7c0a 7c20 2020 2020 7b33 }|.| {3 │ │ │ │ 001e7de0: 203d 3e20 2863 6f6e 7472 6163 742c 204d => (contract, M │ │ │ │ 001e7df0: 6174 7269 782c 204d 6174 7269 7829 2020 atrix, Matrix) │ │ │ │ 001e7e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7e10: 2020 2020 2020 2020 2020 7d7c 0a7c 2020 }|.| │ │ │ │ -001e7e20: 2020 207b 3420 3d3e 2028 6469 6666 2c20 {4 => (diff, │ │ │ │ -001e7e30: 4d61 7472 6978 2c20 4d61 7472 6978 2920 Matrix, Matrix) │ │ │ │ -001e7e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +001e7e20: 2020 207b 3420 3d3e 2028 636f 6e74 7261 {4 => (contra │ │ │ │ +001e7e30: 6374 272c 204d 6174 7269 782c 204d 6174 ct', Matrix, Mat │ │ │ │ +001e7e40: 7269 7829 2020 2020 2020 2020 2020 2020 rix) │ │ │ │ 001e7e50: 2020 2020 2020 2020 2020 2020 2020 207d } │ │ │ │ -001e7e60: 7c0a 7c20 2020 2020 7b35 203d 3e20 2863 |.| {5 => (c │ │ │ │ -001e7e70: 6f6e 7472 6163 7427 2c20 4d61 7472 6978 ontract', Matrix │ │ │ │ -001e7e80: 2c20 4d61 7472 6978 2920 2020 2020 2020 , Matrix) │ │ │ │ +001e7e60: 7c0a 7c20 2020 2020 7b35 203d 3e20 282b |.| {5 => (+ │ │ │ │ +001e7e70: 2c20 4d61 7472 6978 2c20 4d61 7472 6978 , Matrix, Matrix │ │ │ │ +001e7e80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 001e7e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7ea0: 2020 2020 7d7c 0a7c 2020 2020 207b 3620 }|.| {6 │ │ │ │ 001e7eb0: 3d3e 2028 6d61 726b 6564 4742 2c20 4d61 => (markedGB, Ma │ │ │ │ 001e7ec0: 7472 6978 2c20 4d61 7472 6978 2920 2020 trix, Matrix) │ │ │ │ 001e7ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e7ee0: 2020 2020 2020 2020 207d 7c0a 7c20 2020 }|.| │ │ │ │ 001e7ef0: 2020 7b37 203d 3e20 2848 6f6d 2c20 4d61 {7 => (Hom, Ma │ │ │ │ @@ -124978,48 +124978,48 @@ │ │ │ │ 001e8310: 4d61 7472 6978 2c20 4d61 7472 6978 2920 Matrix, Matrix) │ │ │ │ 001e8320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e8330: 2020 2020 2020 2020 207d 7c0a 7c20 2020 }|.| │ │ │ │ 001e8340: 2020 7b32 3320 3d3e 2028 7265 6d61 696e {23 => (remain │ │ │ │ 001e8350: 6465 7227 2c20 4d61 7472 6978 2c20 4d61 der', Matrix, Ma │ │ │ │ 001e8360: 7472 6978 2920 2020 2020 2020 2020 2020 trix) │ │ │ │ 001e8370: 2020 2020 2020 2020 2020 2020 2020 7d7c }| │ │ │ │ -001e8380: 0a7c 2020 2020 207b 3234 203d 3e20 2825 .| {24 => (% │ │ │ │ -001e8390: 2c20 4d61 7472 6978 2c20 4d61 7472 6978 , Matrix, Matrix │ │ │ │ -001e83a0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +001e8380: 0a7c 2020 2020 207b 3234 203d 3e20 2872 .| {24 => (r │ │ │ │ +001e8390: 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6978 2920 2020 2020 ix, Matrix) │ │ │ │ +001e8590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001e85a0: 2020 2020 2020 7d7c 0a7c 2020 2020 207b }|.| { │ │ │ │ 001e85b0: 3332 203d 3e20 2873 7562 7374 6974 7574 32 => (substitut │ │ │ │ 001e85c0: 652c 204d 6174 7269 782c 204d 6174 7269 e, Matrix, Matri │ │ │ │ 001e85d0: 7829 2020 2020 2020 2020 2020 2020 2020 x) │ │ │ │ 001e85e0: 2020 2020 2020 2020 2020 207d 7c0a 7c20 }|.| │ │ │ │ 001e85f0: 2020 2020 7b33 3320 3d3e 2028 796f 6e65 {33 => (yone │ │ │ │ 001e8600: 6461 5072 6f64 7563 742c 204d 6174 7269 daProduct, Matri │ │ │ │ @@ -125928,84 +125928,84 @@ │ │ │ │ 001ebe70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001ebe80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 001ebe90: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ 001ebea0: 203d 2023 7275 6e20 2025 7469 6d65 2020 = #run %time │ │ │ │ 001ebeb0: 2070 6f73 6974 696f 6e20 2020 2020 2020 position │ │ │ │ 001ebec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2032 282:52 |.| 2 │ │ │ │ -001ec2f0: 3020 2020 202e 3336 2020 2020 202e 2e2f 0 .36 ../ │ │ │ │ +001ec2b0: 7c0a 7c20 2020 2020 3230 2020 2020 2e38 |.| 20 .8 │ │ │ │ +001ec2c0: 3220 2020 2020 2e2e 2f2e 2e2f 6d32 2f6d 2 ../../m2/m │ │ │ │ +001ec2d0: 6174 7269 7831 2e6d 323a 3337 3a34 332d atrix1.m2:37:43- │ │ │ │ +001ec2e0: 3337 3a37 3120 207c 0a7c 2020 2020 2032 37:71 |.| 2 │ │ │ │ +001ec2f0: 3020 2020 202e 3639 2020 2020 202e 2e2f 0 .69 ../ │ │ │ │ 001ec300: 2e2e 2f6d 322f 6d61 7472 6978 312e 6d32 ../m2/matrix1.m2 │ │ │ │ -001ec310: 3a33 373a 3433 2d33 373a 3731 2020 7c0a :37:43-37:71 |. │ │ │ │ -001ec320: 7c20 2020 2020 3120 2020 2020 2e30 3034 | 1 .004 │ │ │ │ -001ec330: 3173 2020 656c 6170 7365 6420 746f 7461 1s elapsed tota │ │ │ │ +001ec310: 3a31 3830 3a31 372d 3138 303a 3239 7c0a :180:17-180:29|. │ │ │ │ +001ec320: 7c20 2020 2020 3120 2020 2020 2e30 3033 | 1 .003 │ │ │ │ +001ec330: 3073 2020 656c 6170 7365 6420 746f 7461 0s elapsed tota │ │ │ │ 001ec340: 6c20 2020 2020 2020 2020 2020 2020 2020 l │ │ │ │ 001ec350: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 001ec360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001ec370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 001ec380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 001ec390: 3320 3a20 636f 7665 7261 6765 5375 6d6d 3 : coverageSumm │ │ │ │ 001ec3a0: 6172 7920 2020 2020 2020 2020 2020 2020 ary │ │ │ │ @@ -132679,21 +132679,21 @@ │ │ │ │ 00206460: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ 00206470: 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d ========..+----- │ │ │ │ 00206480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00206490: 2d2b 0a7c 6931 203a 2073 6572 6961 6c4e -+.|i1 : serialN │ │ │ │ 002064a0: 756d 6265 7220 6173 6466 7c0a 7c20 2020 umber asdf|.| │ │ │ │ 002064b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002064c0: 2020 207c 0a7c 6f31 203d 2031 3336 3733 |.|o1 = 13673 │ │ │ │ -002064d0: 3235 2020 2020 2020 2020 2020 7c0a 2b2d 25 |.+- │ │ │ │ +002064d0: 3433 2020 2020 2020 2020 2020 7c0a 2b2d 43 |.+- │ │ │ │ 002064e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002064f0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2073 6572 -----+.|i2 : ser │ │ │ │ 00206500: 6961 6c4e 756d 6265 7220 666f 6f20 7c0a ialNumber foo |. │ │ │ │ 00206510: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00206520: 2020 2020 2020 207c 0a7c 6f32 203d 2031 |.|o2 = 1 │ │ │ │ -00206530: 3336 3733 3237 2020 2020 2020 2020 2020 367327 │ │ │ │ +00206530: 3336 3733 3435 2020 2020 2020 2020 2020 367345 │ │ │ │ 00206540: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00206550: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ 00206560: 2073 6572 6961 6c4e 756d 6265 7220 5a5a serialNumber ZZ │ │ │ │ 00206570: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00206580: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ 00206590: 203d 2031 3030 3030 3530 2020 2020 2020 = 1000050 │ │ │ │ 002065a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ @@ -136252,15 +136252,15 @@ │ │ │ │ 002143b0: 6c6f 6164 696e 6720 4669 7273 7450 6163 loading FirstPac │ │ │ │ 002143c0: 6b61 6765 3b20 7265 6372 6561 7465 2069 kage; recreate i │ │ │ │ 002143d0: 6e73 7461 6e63 6573 206f 6620 7479 7065 nstances of type │ │ │ │ 002143e0: 7320 6672 6f6d 2074 6869 7320 2020 7c0a s from this |. │ │ │ │ 002143f0: 7c20 2d2d 2063 6170 7475 7269 6e67 2063 | -- capturing c │ │ │ │ 00214400: 6865 636b 2831 2c20 2246 6972 7374 5061 heck(1, "FirstPa │ │ │ │ 00214410: 636b 6167 6522 2920 2020 2020 2020 202d ckage") - │ │ │ │ -00214420: 2d20 2e33 3133 3332 3173 2065 6c61 7073 - .313321s elaps │ │ │ │ +00214420: 2d20 2e31 3632 3135 3473 2065 6c61 7073 - .162154s elaps │ │ │ │ 00214430: 6564 2020 2020 2020 2020 2020 2020 7c0a ed |. │ │ │ │ 00214440: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ 00214450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a --------------|. │ │ │ │ 00214490: 7c70 6163 6b61 6765 2020 2020 2020 2020 |package │ │ │ │ @@ -136277,20 +136277,20 @@ │ │ │ │ 00214540: 7450 6163 6b61 6765 2020 2020 2020 2020 tPackage │ │ │ │ 00214550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214570: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00214580: 7c20 2d2d 2063 6170 7475 7269 6e67 2063 | -- capturing c │ │ │ │ 00214590: 6865 636b 2830 2c20 2246 6972 7374 5061 heck(0, "FirstPa │ │ │ │ 002145a0: 636b 6167 6522 2920 2020 2020 2020 202d ckage") - │ │ │ │ -002145b0: 2d20 2e32 3832 3736 3473 2065 6c61 7073 - .282764s elaps │ │ │ │ +002145b0: 2d20 2e31 3430 3031 3473 2065 6c61 7073 - .140014s elaps │ │ │ │ 002145c0: 6564 2020 2020 2020 2020 2020 2020 7c0a ed |. │ │ │ │ 002145d0: 7c20 2d2d 2063 6170 7475 7269 6e67 2063 | -- capturing c │ │ │ │ 002145e0: 6865 636b 2831 2c20 2246 6972 7374 5061 heck(1, "FirstPa │ │ │ │ 002145f0: 636b 6167 6522 2920 2020 2020 2020 202d ckage") - │ │ │ │ -00214600: 2d20 2e32 3833 3238 3573 2065 6c61 7073 - .283285s elaps │ │ │ │ +00214600: 2d20 2e31 3433 3236 3873 2065 6c61 7073 - .143268s elaps │ │ │ │ 00214610: 6564 2020 2020 2020 2020 2020 2020 7c0a ed |. │ │ │ │ 00214620: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00214630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00214670: 0a41 6c74 6572 6e61 7469 7665 6c79 2c20 .Alternatively, │ │ │ │ @@ -136306,33 +136306,33 @@ │ │ │ │ 00214710: 0a7c 6934 203a 2063 6865 636b 5f31 2022 .|i4 : check_1 " │ │ │ │ 00214720: 4669 7273 7450 6163 6b61 6765 2220 2020 FirstPackage" │ │ │ │ 00214730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214750: 2020 207c 0a7c 202d 2d20 6361 7074 7572 |.| -- captur │ │ │ │ 00214760: 696e 6720 6368 6563 6b28 312c 2022 4669 ing check(1, "Fi │ │ │ │ 00214770: 7273 7450 6163 6b61 6765 2229 2020 2020 rstPackage") │ │ │ │ -00214780: 2020 2020 2d2d 202e 3237 3537 3034 7320 -- .275704s │ │ │ │ +00214780: 2020 2020 2d2d 202e 3134 3234 3432 7320 -- .142442s │ │ │ │ 00214790: 656c 6170 7365 647c 0a2b 2d2d 2d2d 2d2d elapsed|.+------ │ │ │ │ 002147a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002147b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002147c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002147d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ 002147e0: 203a 2063 6865 636b 2022 4669 7273 7450 : check "FirstP │ │ │ │ 002147f0: 6163 6b61 6765 2220 2020 2020 2020 2020 ackage" │ │ │ │ 00214800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00214820: 0a7c 202d 2d20 6361 7074 7572 696e 6720 .| -- capturing │ │ │ │ 00214830: 6368 6563 6b28 302c 2022 4669 7273 7450 check(0, "FirstP │ │ │ │ 00214840: 6163 6b61 6765 2229 2020 2020 2020 2020 ackage") │ │ │ │ -00214850: 2d2d 202e 3234 3632 3935 7320 656c 6170 -- .246295s elap │ │ │ │ +00214850: 2d2d 202e 3133 3739 3338 7320 656c 6170 -- .137938s elap │ │ │ │ 00214860: 7365 647c 0a7c 202d 2d20 6361 7074 7572 sed|.| -- captur │ │ │ │ 00214870: 696e 6720 6368 6563 6b28 312c 2022 4669 ing check(1, "Fi │ │ │ │ 00214880: 7273 7450 6163 6b61 6765 2229 2020 2020 rstPackage") │ │ │ │ -00214890: 2020 2020 2d2d 202e 3231 3334 3273 2065 -- .21342s e │ │ │ │ -002148a0: 6c61 7073 6564 207c 0a2b 2d2d 2d2d 2d2d lapsed |.+------ │ │ │ │ +00214890: 2020 2020 2d2d 202e 3133 3934 3031 7320 -- .139401s │ │ │ │ +002148a0: 656c 6170 7365 647c 0a2b 2d2d 2d2d 2d2d elapsed|.+------ │ │ │ │ 002148b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002148c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002148d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002148e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4120 -----------+..A │ │ │ │ 002148f0: 2a6e 6f74 6520 5465 7374 496e 7075 743a *note TestInput: │ │ │ │ 00214900: 2074 6573 7473 2c20 6f62 6a65 6374 2028 tests, object ( │ │ │ │ 00214910: 6f72 2061 206c 6973 7420 6f66 2073 7563 or a list of suc │ │ │ │ @@ -136373,15 +136373,15 @@ │ │ │ │ 00214b40: 3720 3a20 6368 6563 6b20 6f6f 2020 2020 7 : check oo │ │ │ │ 00214b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214b80: 2020 2020 7c0a 7c20 2d2d 2063 6170 7475 |.| -- captu │ │ │ │ 00214b90: 7269 6e67 2063 6865 636b 2831 2c20 2246 ring check(1, "F │ │ │ │ 00214ba0: 6972 7374 5061 636b 6167 6522 2920 2020 irstPackage") │ │ │ │ -00214bb0: 2020 2020 202d 2d20 2e33 3032 3732 3673 -- .302726s │ │ │ │ +00214bb0: 2020 2020 202d 2d20 2e31 3437 3738 3973 -- .147789s │ │ │ │ 00214bc0: 2065 6c61 7073 6564 2020 2020 7c0a 2b2d elapsed |.+- │ │ │ │ 00214bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214c10: 2d2d 2d2d 2b0a 7c69 3820 3a20 7465 7374 ----+.|i8 : test │ │ │ │ 00214c20: 7320 2246 6972 7374 5061 636b 6167 6522 s "FirstPackage" │ │ │ │ @@ -136418,20 +136418,20 @@ │ │ │ │ 00214e10: 3920 3a20 6368 6563 6b20 6f6f 2020 2020 9 : check oo │ │ │ │ 00214e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00214e50: 2020 2020 7c0a 7c20 2d2d 2063 6170 7475 |.| -- captu │ │ │ │ 00214e60: 7269 6e67 2063 6865 636b 2830 2c20 2246 ring check(0, "F │ │ │ │ 00214e70: 6972 7374 5061 636b 6167 6522 2920 2020 irstPackage") │ │ │ │ -00214e80: 2020 2020 202d 2d20 2e32 3834 3037 7320 -- .28407s │ │ │ │ -00214e90: 656c 6170 7365 6420 2020 2020 7c0a 7c20 elapsed |.| │ │ │ │ +00214e80: 2020 2020 202d 2d20 2e31 3339 3438 3373 -- .139483s │ │ │ │ +00214e90: 2065 6c61 7073 6564 2020 2020 7c0a 7c20 elapsed |.| │ │ │ │ 00214ea0: 2d2d 2063 6170 7475 7269 6e67 2063 6865 -- capturing che │ │ │ │ 00214eb0: 636b 2831 2c20 2246 6972 7374 5061 636b ck(1, "FirstPack │ │ │ │ 00214ec0: 6167 6522 2920 2020 2020 2020 202d 2d20 age") -- │ │ │ │ -00214ed0: 2e32 3832 3338 3573 2065 6c61 7073 6564 .282385s elapsed │ │ │ │ +00214ed0: 2e31 3336 3432 3873 2065 6c61 7073 6564 .136428s elapsed │ │ │ │ 00214ee0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00214ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00214f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 ------------+..I │ │ │ │ 00214f30: 6620 6f6e 6c79 2061 6e20 696e 7465 6765 f only an intege │ │ │ │ 00214f40: 7220 6973 2070 6173 7365 6420 6173 2061 r is passed as a │ │ │ │ @@ -136481,15 +136481,15 @@ │ │ │ │ 00215200: 6b20 3120 2020 2020 2020 2020 2020 2020 k 1 │ │ │ │ 00215210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00215220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00215230: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00215240: 2d2d 2063 6170 7475 7269 6e67 2063 6865 -- capturing che │ │ │ │ 00215250: 636b 2831 2c20 2246 6972 7374 5061 636b ck(1, "FirstPack │ │ │ │ 00215260: 6167 6522 2920 2020 2020 2020 202d 2d20 age") -- │ │ │ │ -00215270: 2e32 3833 3431 3473 2065 6c61 7073 6564 .283414s elapsed │ │ │ │ +00215270: 2e31 3337 3436 3673 2065 6c61 7073 6564 .137466s elapsed │ │ │ │ 00215280: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00215290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002152a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002152b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002152c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 002152d0: 0a43 6176 6561 740a 3d3d 3d3d 3d3d 0a0a .Caveat.======.. │ │ │ │ 002152e0: 4375 7272 656e 746c 792c 2069 6620 7468 Currently, if th │ │ │ │ @@ -140441,15 +140441,15 @@ │ │ │ │ 00224980: 2020 2020 2020 2072 6177 2064 6f63 756d raw docum │ │ │ │ 00224990: 656e 7461 7469 6f6e 203d 3e20 4d75 7461 entation => Muta │ │ │ │ 002249a0: 626c 6548 6173 6854 6162 6c65 7b7d 2020 bleHashTable{} │ │ │ │ 002249b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002249c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 002249d0: 2020 2020 2020 2073 6f75 7263 6520 6469 source di │ │ │ │ 002249e0: 7265 6374 6f72 7920 3d3e 202f 746d 702f rectory => /tmp/ │ │ │ │ -002249f0: 4d32 2d31 3130 3232 2d30 2f39 362d 7275 M2-11022-0/96-ru │ │ │ │ +002249f0: 4d32 2d31 3132 3432 2d30 2f39 362d 7275 M2-11242-0/96-ru │ │ │ │ 00224a00: 6e64 6972 2f20 2020 2020 2020 2020 2020 ndir/ │ │ │ │ 00224a10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00224a20: 2020 2020 2020 2073 6f75 7263 6520 6669 source fi │ │ │ │ 00224a30: 6c65 203d 3e20 7374 6469 6f20 2020 2020 le => stdio │ │ │ │ 00224a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00224a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00224a60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ @@ -146536,26 +146536,26 @@ │ │ │ │ 0023c670: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0023c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ 0023c6b0: 203a 2065 6c61 7073 6564 5469 6d65 2020 : elapsedTime │ │ │ │ 0023c6c0: 2020 2020 2020 2061 7070 6c79 2831 2e2e apply(1.. │ │ │ │ 0023c6d0: 3130 302c 206e 202d 3e20 736f 7274 204c 100, n -> sort L │ │ │ │ -0023c6e0: 293b 7c0a 7c20 2d2d 202e 3635 3937 3335 );|.| -- .659735 │ │ │ │ +0023c6e0: 293b 7c0a 7c20 2d2d 202e 3632 3930 3035 );|.| -- .629005 │ │ │ │ 0023c6f0: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ 0023c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023c710: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0023c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023c730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023c740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023c750: 2b0a 7c69 3420 3a20 656c 6170 7365 6454 +.|i4 : elapsedT │ │ │ │ 0023c760: 696d 6520 7061 7261 6c6c 656c 4170 706c ime parallelAppl │ │ │ │ 0023c770: 7928 312e 2e31 3030 2c20 6e20 2d3e 2073 y(1..100, n -> s │ │ │ │ -0023c780: 6f72 7420 4c29 3b7c 0a7c 202d 2d20 2e32 ort L);|.| -- .2 │ │ │ │ -0023c790: 3131 3133 3473 2065 6c61 7073 6564 2020 11134s elapsed │ │ │ │ +0023c780: 6f72 7420 4c29 3b7c 0a7c 202d 2d20 2e31 ort L);|.| -- .1 │ │ │ │ +0023c790: 3636 3737 3673 2065 6c61 7073 6564 2020 66776s elapsed │ │ │ │ 0023c7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023c7b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0023c7c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0023c7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023c7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023c7f0: 2d2d 2d2d 2d2b 0a0a 596f 7520 7769 6c6c -----+..You will │ │ │ │ 0023c800: 2068 6176 6520 746f 2074 7279 2069 7420 have to try it │ │ │ │ @@ -146636,15 +146636,15 @@ │ │ │ │ 0023ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0023ccc0: 7c69 3620 3a20 616c 6c6f 7761 626c 6554 |i6 : allowableT │ │ │ │ 0023ccd0: 6872 6561 6473 203d 206d 6178 416c 6c6f hreads = maxAllo │ │ │ │ 0023cce0: 7761 626c 6554 6872 6561 6473 7c0a 7c20 wableThreads|.| │ │ │ │ 0023ccf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023cd10: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -0023cd20: 3d20 3720 2020 2020 2020 2020 2020 2020 = 7 │ │ │ │ +0023cd20: 3d20 3137 2020 2020 2020 2020 2020 2020 = 17 │ │ │ │ 0023cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0023cd40: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 0023cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023cd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023cd70: 2d2d 2d2d 2d2d 2b0a 0a54 6f20 7275 6e20 ------+..To run │ │ │ │ 0023cd80: 6120 6675 6e63 7469 6f6e 2069 6e20 616e a function in an │ │ │ │ 0023cd90: 6f74 6865 7220 7468 7265 6164 2075 7365 other thread use │ │ │ │ @@ -146769,16 +146769,16 @@ │ │ │ │ 0023d500: 2063 6f6d 706c 6574 6564 2074 6865 0a63 completed the.c │ │ │ │ 0023d510: 6f6d 7075 7461 7469 6f6e 2e0a 0a2b 2d2d omputation...+-- │ │ │ │ 0023d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023d530: 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 7420 -----+.|i11 : t │ │ │ │ 0023d540: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0023d550: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0023d560: 2020 2020 2020 2020 207c 0a7c 6f31 3120 |.|o11 │ │ │ │ -0023d570: 3d20 3c3c 7461 736b 2c20 7275 6e6e 696e = <>|.| │ │ │ │ +0023d570: 3d20 3c3c 7461 736b 2c20 6372 6561 7465 = <>|.| │ │ │ │ 0023d590: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0023d5a0: 6f31 3120 3a20 5461 736b 2020 2020 2020 o11 : Task │ │ │ │ 0023d5b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0023d5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0023d5d0: 2d2b 0a0a 5573 6520 2a6e 6f74 6520 6973 -+..Use *note is │ │ │ │ 0023d5e0: 5265 6164 793a 2069 7352 6561 6479 5f6c Ready: isReady_l │ │ │ │ 0023d5f0: 7046 696c 655f 7270 2c20 746f 2063 6865 pFile_rp, to che │ │ │ │ @@ -148156,15 +148156,15 @@ │ │ │ │ 00242bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00242bc0: 2b0a 7c69 3520 3a20 6e20 2020 2020 2020 +.|i5 : n │ │ │ │ 00242bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00242be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00242bf0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00242c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00242c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00242c20: 2020 7c0a 7c6f 3520 3d20 3333 3633 3934 |.|o5 = 336394 │ │ │ │ +00242c20: 2020 7c0a 7c6f 3520 3d20 3732 3036 3139 |.|o5 = 720619 │ │ │ │ 00242c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00242c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00242c50: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00242c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00242c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00242c80: 2d2d 2d2d 2b0a 7c69 3620 3a20 736c 6565 ----+.|i6 : slee │ │ │ │ 00242c90: 7020 3120 2020 2020 2020 2020 2020 2020 p 1 │ │ │ │ @@ -148199,15 +148199,15 @@ │ │ │ │ 00242e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00242e70: 7c69 3820 3a20 6e20 2020 2020 2020 2020 |i8 : n │ │ │ │ 00242e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00242e90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00242ea0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00242eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00242ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00242ed0: 7c0a 7c6f 3820 3d20 3932 3939 3837 2020 |.|o8 = 929987 │ │ │ │ +00242ed0: 7c0a 7c6f 3820 3d20 3136 3938 3238 3220 |.|o8 = 1698282 │ │ │ │ 00242ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00242ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00242f00: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00242f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00242f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00242f30: 2d2d 2b0a 7c69 3920 3a20 6973 5265 6164 --+.|i9 : isRead │ │ │ │ 00242f40: 7920 7420 2020 2020 2020 2020 2020 2020 y t │ │ │ │ @@ -148227,15 +148227,15 @@ │ │ │ │ 00243020: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00243030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00243040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00243050: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a --------+.|i11 : │ │ │ │ 00243060: 2073 6c65 6570 2032 2020 2020 2020 2020 sleep 2 │ │ │ │ 00243070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243080: 2020 2020 2020 2020 207c 0a7c 7374 6469 |.|stdi │ │ │ │ -00243090: 6f3a 323a 3235 3a28 3329 3a5b 315d 3a20 o:2:25:(3):[1]: │ │ │ │ +00243090: 6f3a 323a 3339 3a28 3329 3a5b 315d 3a20 o:2:39:(3):[1]: │ │ │ │ 002430a0: 6572 726f 723a 2069 6e74 6572 7275 7074 error: interrupt │ │ │ │ 002430b0: 6564 2020 2020 2020 2020 7c0a 7c20 2020 ed |.| │ │ │ │ 002430c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002430d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002430e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ 002430f0: 3120 3d20 3020 2020 2020 2020 2020 2020 1 = 0 │ │ │ │ 00243100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -148263,16 +148263,16 @@ │ │ │ │ 00243260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00243270: 2d2d 2d2b 0a7c 6931 3320 3a20 6e20 2020 ---+.|i13 : n │ │ │ │ 00243280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002432a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 002432b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002432c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002432d0: 2020 2020 207c 0a7c 6f31 3320 3d20 3933 |.|o13 = 93 │ │ │ │ -002432e0: 3032 3237 2020 2020 2020 2020 2020 2020 0227 │ │ │ │ +002432d0: 2020 2020 207c 0a7c 6f31 3320 3d20 3136 |.|o13 = 16 │ │ │ │ +002432e0: 3938 3439 3120 2020 2020 2020 2020 2020 98491 │ │ │ │ 002432f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243300: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00243310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00243320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00243330: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 -------+.|i14 : │ │ │ │ 00243340: 736c 6565 7020 3120 2020 2020 2020 2020 sleep 1 │ │ │ │ 00243350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -148288,15 +148288,15 @@ │ │ │ │ 002433f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 00243400: 3520 3a20 6e20 2020 2020 2020 2020 2020 5 : n │ │ │ │ 00243410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243420: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00243430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00243460: 6f31 3520 3d20 3933 3032 3237 2020 2020 o15 = 930227 │ │ │ │ +00243460: 6f31 3520 3d20 3136 3938 3439 3120 2020 o15 = 1698491 │ │ │ │ 00243470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243480: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00243490: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 002434a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002434b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 002434c0: 0a7c 6931 3620 3a20 6973 5265 6164 7920 .|i16 : isReady │ │ │ │ 002434d0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ @@ -148442,15 +148442,15 @@ │ │ │ │ 00243d90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00243da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00243db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00243dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ 00243dd0: 203a 2073 6c65 6570 2032 2020 2020 2020 : sleep 2 │ │ │ │ 00243de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243df0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00243e00: 7c73 7464 696f 3a32 3a32 353a 2833 293a |stdio:2:25:(3): │ │ │ │ +00243e00: 7c73 7464 696f 3a32 3a33 393a 2833 293a |stdio:2:39:(3): │ │ │ │ 00243e10: 5b31 5d3a 2065 7272 6f72 3a20 696e 7465 [1]: error: inte │ │ │ │ 00243e20: 7272 7570 7465 6420 2020 2020 2020 2020 rrupted │ │ │ │ 00243e30: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00243e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00243e60: 2020 2020 7c0a 7c6f 3720 3d20 3020 2020 |.|o7 = 0 │ │ │ │ 00243e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -149055,15 +149055,15 @@ │ │ │ │ 002463e0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ 002463f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d ========..+----- │ │ │ │ 00246400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00246410: 2d2d 2d2b 0a7c 6931 203a 206d 6178 416c ---+.|i1 : maxAl │ │ │ │ 00246420: 6c6f 7761 626c 6554 6872 6561 6473 7c0a lowableThreads|. │ │ │ │ 00246430: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00246440: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ -00246450: 2037 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ +00246450: 2031 3720 2020 2020 2020 2020 2020 2020 17 │ │ │ │ 00246460: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00246470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00246480: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ 00246490: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2070 ===.. * *note p │ │ │ │ 002464a0: 6172 616c 6c65 6c20 7072 6f67 7261 6d6d arallel programm │ │ │ │ 002464b0: 696e 6720 7769 7468 2074 6872 6561 6473 ing with threads │ │ │ │ 002464c0: 2061 6e64 2074 6173 6b73 3a20 7061 7261 and tasks: para │ │ │ │ @@ -150286,16 +150286,16 @@ │ │ │ │ 0024b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b0e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0024b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b130: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0024b140: 6f33 203d 202f 746d 702f 4d32 2d31 3138 o3 = /tmp/M2-118 │ │ │ │ -0024b150: 3033 2d30 2f30 2020 2020 2020 2020 2020 03-0/0 │ │ │ │ +0024b140: 6f33 203d 202f 746d 702f 4d32 2d31 3237 o3 = /tmp/M2-127 │ │ │ │ +0024b150: 3933 2d30 2f30 2020 2020 2020 2020 2020 93-0/0 │ │ │ │ 0024b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b180: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 0024b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0024b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0024b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0024b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -150306,16 +150306,16 @@ │ │ │ │ 0024b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b220: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0024b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b270: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0024b280: 6f34 203d 202f 746d 702f 4d32 2d31 3138 o4 = /tmp/M2-118 │ │ │ │ -0024b290: 3033 2d30 2f30 2020 2020 2020 2020 2020 03-0/0 │ │ │ │ +0024b280: 6f34 203d 202f 746d 702f 4d32 2d31 3237 o4 = /tmp/M2-127 │ │ │ │ +0024b290: 3933 2d30 2f30 2020 2020 2020 2020 2020 93-0/0 │ │ │ │ 0024b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b2c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0024b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -150466,16 +150466,16 @@ │ │ │ │ 0024bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bc20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0024bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bc70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0024bc80: 6f39 203d 202f 746d 702f 4d32 2d31 3138 o9 = /tmp/M2-118 │ │ │ │ -0024bc90: 3033 2d30 2f30 2020 2020 2020 2020 2020 03-0/0 │ │ │ │ +0024bc80: 6f39 203d 202f 746d 702f 4d32 2d31 3237 o9 = /tmp/M2-127 │ │ │ │ +0024bc90: 3933 2d30 2f30 2020 2020 2020 2020 2020 93-0/0 │ │ │ │ 0024bca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bcc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0024bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -150526,16 +150526,16 @@ │ │ │ │ 0024bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024bfe0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0024bff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c030: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0024c040: 6f31 3120 3d20 2f74 6d70 2f4d 322d 3131 o11 = /tmp/M2-11 │ │ │ │ -0024c050: 3830 332d 302f 3020 2020 2020 2020 2020 803-0/0 │ │ │ │ +0024c040: 6f31 3120 3d20 2f74 6d70 2f4d 322d 3132 o11 = /tmp/M2-12 │ │ │ │ +0024c050: 3739 332d 302f 3020 2020 2020 2020 2020 793-0/0 │ │ │ │ 0024c060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c080: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0024c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024c0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -150932,15 +150932,15 @@ │ │ │ │ 0024d930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024d940: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0024d950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024d960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024d970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024d980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024d990: 2020 2020 2020 207c 0a7c 6f31 203d 202f |.|o1 = / │ │ │ │ -0024d9a0: 746d 702f 4d32 2d31 3139 3738 2d30 2f30 tmp/M2-11978-0/0 │ │ │ │ +0024d9a0: 746d 702f 4d32 2d31 3331 3438 2d30 2f30 tmp/M2-13148-0/0 │ │ │ │ 0024d9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024d9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024d9e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0024d9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0024da00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0024da10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -150952,15 +150952,15 @@ │ │ │ │ 0024da70: 322a 795e 322a 7a5e 322b 785e 332b 362a 2*y^2*z^2+x^3+6* │ │ │ │ 0024da80: 785e 322a 792b 207c 0a7c 2020 2020 2020 x^2*y+ |.| │ │ │ │ 0024da90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024daa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024dac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024dad0: 2020 2020 2020 207c 0a7c 6f32 203d 202f |.|o2 = / │ │ │ │ -0024dae0: 746d 702f 4d32 2d31 3139 3738 2d30 2f30 tmp/M2-11978-0/0 │ │ │ │ +0024dae0: 746d 702f 4d32 2d31 3331 3438 2d30 2f30 tmp/M2-13148-0/0 │ │ │ │ 0024daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024db20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0024db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -151168,15 +151168,15 @@ │ │ │ │ 0024e7f0: 6c65 203d 2032 5e31 3030 7c0a 7c20 2020 le = 2^100|.| │ │ │ │ 0024e800: 2020 7072 696e 7420 7361 6d70 6c65 2020 print sample │ │ │ │ 0024e810: 2020 2020 2020 207c 0a7c 2020 2020 2022 |.| " │ │ │ │ 0024e820: 203c 3c20 636c 6f73 6520 2020 2020 2020 << close │ │ │ │ 0024e830: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0024e840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0024e850: 207c 0a7c 6f37 203d 202f 746d 702f 4d32 |.|o7 = /tmp/M2 │ │ │ │ -0024e860: 2d31 3139 3738 2d30 2f30 2020 2020 7c0a -11978-0/0 |. │ │ │ │ +0024e860: 2d31 3331 3438 2d30 2f30 2020 2020 7c0a -13148-0/0 |. │ │ │ │ 0024e870: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0024e880: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ 0024e890: 203a 2046 696c 6520 2020 2020 2020 2020 : File │ │ │ │ 0024e8a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 0024e8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0024e8c0: 2d2d 2d2d 2d2b 0a4e 6f77 2076 6572 6966 -----+.Now verif │ │ │ │ 0024e8d0: 7920 7468 6174 2069 7420 636f 6e74 6169 y that it contai │ │ │ │ @@ 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2020 2020 2020 2020 2020 │ │ │ │ 00250fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -152743,23 +152743,23 @@ │ │ │ │ 00254a60: 2e0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 00254a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254a80: 2d2b 0a7c 6931 203a 2066 6e20 3d20 7465 -+.|i1 : fn = te │ │ │ │ 00254a90: 6d70 6f72 6172 7946 696c 654e 616d 6528 mporaryFileName( │ │ │ │ 00254aa0: 2920 7c0a 7c20 2020 2020 2020 2020 2020 ) |.| │ │ │ │ 00254ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254ac0: 2020 207c 0a7c 6f31 203d 202f 746d 702f |.|o1 = /tmp/ │ │ │ │ -00254ad0: 4d32 2d31 3331 3136 2d30 2f30 2020 2020 M2-13116-0/0 │ │ │ │ +00254ad0: 4d32 2d31 3534 3736 2d30 2f30 2020 2020 M2-15476-0/0 │ │ │ │ 00254ae0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00254af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254b00: 2d2d 2d2d 2d2b 0a7c 6932 203a 2066 6e20 -----+.|i2 : fn │ │ │ │ 00254b10: 3c3c 2022 6869 2074 6865 7265 2220 3c3c << "hi there" << │ │ │ │ 00254b20: 2063 6c6f 7365 7c0a 7c20 2020 2020 2020 close|.| │ │ │ │ 00254b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254b40: 2020 2020 2020 207c 0a7c 6f32 203d 202f |.|o2 = / │ │ │ │ -00254b50: 746d 702f 4d32 2d31 3331 3136 2d30 2f30 tmp/M2-13116-0/0 │ │ │ │ +00254b50: 746d 702f 4d32 2d31 3534 3736 2d30 2f30 tmp/M2-15476-0/0 │ │ │ │ 00254b60: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00254b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00254b80: 2020 2020 2020 2020 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ 00254b90: 2046 696c 6520 2020 2020 2020 2020 2020 File │ │ │ │ 00254ba0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00254bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00254bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ @@ -152846,23 +152846,23 @@ │ │ │ │ 002550d0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 002550e0: 2d2d 2d2d 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3d20 2f74 |.|o3 = /t │ │ │ │ -002551c0: 6d70 2f4d 322d 3131 3231 332d 302f 3020 mp/M2-11213-0/0 │ │ │ │ +002551c0: 6d70 2f4d 322d 3131 3630 322d 302f 3020 mp/M2-11602-0/0 │ │ │ │ 002551d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 002551e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002551f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ 00255200: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ 00255210: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 00255220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255230: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ @@ -153042,24 +153042,24 @@ │ │ │ │ 00255d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255d20: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2074 -------+.|i1 : t │ │ │ │ 00255d30: 656d 706f 7261 7279 4669 6c65 4e61 6d65 emporaryFileName │ │ │ │ 00255d40: 2028 2920 7c20 222e 7465 7822 207c 0a7c () | ".tex" |.| │ │ │ │ 00255d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255d70: 2020 207c 0a7c 6f31 203d 202f 746d 702f |.|o1 = /tmp/ │ │ │ │ -00255d80: 4d32 2d31 3330 3937 2d30 2f30 2e74 6578 M2-13097-0/0.tex │ │ │ │ +00255d80: 4d32 2d31 3534 3337 2d30 2f30 2e74 6578 M2-15437-0/0.tex │ │ │ │ 00255d90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 00255da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00255dc0: 0a7c 6932 203a 2074 656d 706f 7261 7279 .|i2 : temporary │ │ │ │ 00255dd0: 4669 6c65 4e61 6d65 2028 2920 7c20 222e FileName () | ". │ │ │ │ 00255de0: 6874 6d6c 227c 0a7c 2020 2020 2020 2020 html"|.| │ │ │ │ 00255df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00255e00: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00255e10: 203d 202f 746d 702f 4d32 2d31 3330 3937 = /tmp/M2-13097 │ │ │ │ +00255e10: 203d 202f 746d 702f 4d32 2d31 3534 3337 = /tmp/M2-15437 │ │ │ │ 00255e20: 2d30 2f31 2e68 746d 6c20 2020 2020 2020 -0/1.html │ │ │ │ 00255e30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00255e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00255e50: 2d2d 2d2d 2d2d 2d2b 0a0a 5468 6973 2066 -------+..This f │ │ │ │ 00255e60: 756e 6374 696f 6e20 7769 6c6c 2077 6f72 unction will wor │ │ │ │ 00255e70: 6b20 756e 6465 7220 556e 6978 2c20 616e k under Unix, an │ │ │ │ 00255e80: 6420 616c 736f 2075 6e64 6572 2057 696e d also under Win │ │ │ │ @@ -153223,23 +153223,23 @@ │ │ │ │ 00256860: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d ===========..+-- │ │ │ │ 00256870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ 00256890: 203a 2066 6e20 3d20 7465 6d70 6f72 6172 : fn = temporar │ │ │ │ 002568a0: 7946 696c 654e 616d 6528 297c 0a7c 2020 yFileName()|.| │ │ │ │ 002568b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002568c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -002568d0: 203d 202f 746d 702f 4d32 2d31 3232 3834 = /tmp/M2-12284 │ │ │ │ +002568d0: 203d 202f 746d 702f 4d32 2d31 3337 3734 = /tmp/M2-13774 │ │ │ │ 002568e0: 2d30 2f30 2020 2020 2020 207c 0a2b 2d2d -0/0 |.+-- │ │ │ │ 002568f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ 00256910: 203a 2066 203d 2066 6e20 3c3c 2022 6869 : f = fn << "hi │ │ │ │ 00256920: 2074 6865 7265 2220 2020 207c 0a7c 2020 there" |.| │ │ │ │ 00256930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256940: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00256950: 203d 202f 746d 702f 4d32 2d31 3232 3834 = /tmp/M2-12284 │ │ │ │ +00256950: 203d 202f 746d 702f 4d32 2d31 3337 3734 = /tmp/M2-13774 │ │ │ │ 00256960: 2d30 2f30 2020 2020 2020 207c 0a7c 2020 -0/0 |.| │ │ │ │ 00256970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256980: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ 00256990: 203a 2046 696c 6520 2020 2020 2020 2020 : File │ │ │ │ 002569a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 002569b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002569c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ @@ -153251,15 +153251,15 @@ │ │ │ │ 00256a20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00256a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ 00256a50: 203a 2063 6c6f 7365 2066 2020 2020 2020 : close f │ │ │ │ 00256a60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00256a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256a80: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -00256a90: 203d 202f 746d 702f 4d32 2d31 3232 3834 = /tmp/M2-12284 │ │ │ │ +00256a90: 203d 202f 746d 702f 4d32 2d31 3337 3734 = /tmp/M2-13774 │ │ │ │ 00256aa0: 2d30 2f30 2020 2020 2020 207c 0a7c 2020 -0/0 |.| │ │ │ │ 00256ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256ac0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ 00256ad0: 203a 2046 696c 6520 2020 2020 2020 2020 : File │ │ │ │ 00256ae0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00256af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ @@ -153318,23 +153318,23 @@ │ │ │ │ 00256e50: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ==..+----------- │ │ │ │ 00256e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256e70: 2d2d 2b0a 7c69 3120 3a20 666e 203d 2074 --+.|i1 : fn = t │ │ │ │ 00256e80: 656d 706f 7261 7279 4669 6c65 4e61 6d65 emporaryFileName │ │ │ │ 00256e90: 2829 7c0a 7c20 2020 2020 2020 2020 2020 ()|.| │ │ │ │ 00256ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00256eb0: 2020 7c0a 7c6f 3120 3d20 2f74 6d70 2f4d |.|o1 = /tmp/M │ │ │ │ -00256ec0: 322d 3131 3732 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00256f90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00256fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00256fb0: 2d2d 2b0a 7c69 3320 3a20 6d20 3d20 3720 --+.|i3 : m = 7 │ │ │ │ @@ -153358,15 +153358,15 @@ │ │ │ │ 002570d0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 002570e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002570f0: 2d2d 2b0a 7c69 3620 3a20 636c 6f73 6520 --+.|i6 : close │ │ │ │ 00257100: 6620 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ 00257110: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00257120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257130: 2020 7c0a 7c6f 3620 3d20 2f74 6d70 2f4d |.|o6 = /tmp/M │ │ │ │ -00257140: 322d 3131 3732 352d 302f 3020 2020 2020 2-11725-0/0 │ │ │ │ +00257140: 322d 3132 3633 352d 302f 3020 2020 2020 2-12635-0/0 │ │ │ │ 00257150: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00257160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257170: 2020 7c0a 7c6f 3620 3a20 4669 6c65 2020 |.|o6 : File │ │ │ │ 00257180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257190: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 002571a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002571b0: 2d2d 2b0a 7c69 3720 3a20 6669 6c65 4d6f --+.|i7 : fileMo │ │ │ │ @@ -153434,23 +153434,23 @@ │ │ │ │ 00257590: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 002575a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002575b0: 2b0a 7c69 3120 3a20 666e 203d 2074 656d +.|i1 : fn = tem │ │ │ │ 002575c0: 706f 7261 7279 4669 6c65 4e61 6d65 2829 poraryFileName() │ │ │ │ 002575d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 002575e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002575f0: 2020 7c0a 7c6f 3120 3d20 2f74 6d70 2f4d |.|o1 = /tmp/M │ │ │ │ -00257600: 322d 3131 3836 302d 302f 3020 2020 2020 2-11860-0/0 │ │ │ │ +00257600: 322d 3132 3931 302d 302f 3020 2020 2020 2-12910-0/0 │ │ │ │ 00257610: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00257620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00257630: 2d2d 2d2d 2b0a 7c69 3220 3a20 666e 203c ----+.|i2 : fn < │ │ │ │ 00257640: 3c20 2268 6920 7468 6572 6522 203c 3c20 < "hi there" << │ │ │ │ 00257650: 636c 6f73 657c 0a7c 2020 2020 2020 2020 close|.| │ │ │ │ 00257660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257670: 2020 2020 2020 7c0a 7c6f 3220 3d20 2f74 |.|o2 = /t │ │ │ │ -00257680: 6d70 2f4d 322d 3131 3836 302d 302f 3020 mp/M2-11860-0/0 │ │ │ │ +00257680: 6d70 2f4d 322d 3132 3931 302d 302f 3020 mp/M2-12910-0/0 │ │ │ │ 00257690: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 002576a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002576b0: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ 002576c0: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ 002576d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 002576e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002576f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ @@ -153518,24 +153518,24 @@ │ │ │ │ 00257ad0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d =========..+---- │ │ │ │ 00257ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00257af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ 00257b00: 3a20 666e 203d 2074 656d 706f 7261 7279 : fn = temporary │ │ │ │ 00257b10: 4669 6c65 4e61 6d65 2829 207c 0a7c 2020 FileName() |.| │ │ │ │ 00257b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257b30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00257b40: 3120 3d20 2f74 6d70 2f4d 322d 3132 3930 1 = /tmp/M2-1290 │ │ │ │ +00257b40: 3120 3d20 2f74 6d70 2f4d 322d 3135 3034 1 = /tmp/M2-1504 │ │ │ │ 00257b50: 352d 302f 3020 2020 2020 2020 207c 0a2b 5-0/0 |.+ │ │ │ │ 00257b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00257b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00257b80: 7c69 3220 3a20 666e 203c 3c20 2268 6920 |i2 : fn << "hi │ │ │ │ 00257b90: 7468 6572 6522 203c 3c20 636c 6f73 657c there" << close| │ │ │ │ 00257ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00257bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257bc0: 7c0a 7c6f 3220 3d20 2f74 6d70 2f4d 322d |.|o2 = /tmp/M2- │ │ │ │ -00257bd0: 3132 3930 352d 302f 3020 2020 2020 2020 12905-0/0 │ │ │ │ +00257bd0: 3135 3034 352d 302f 3020 2020 2020 2020 15045-0/0 │ │ │ │ 00257be0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00257bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257c00: 2020 7c0a 7c6f 3220 3a20 4669 6c65 2020 |.|o2 : File │ │ │ │ 00257c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00257c20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00257c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00257c40: 2d2d 2d2d 2b0a 7c69 3320 3a20 6d20 3d20 ----+.|i3 : m = │ │ │ │ @@ -153612,15 +153612,15 @@ │ │ │ │ 002580b0: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d =======..+------ │ │ │ │ 002580c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002580d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ 002580e0: 666e 203d 2074 656d 706f 7261 7279 4669 fn = temporaryFi │ │ │ │ 002580f0: 6c65 4e61 6d65 2829 207c 0a7c 2020 2020 leName() |.| │ │ │ │ 00258100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258110: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -00258120: 3d20 2f74 6d70 2f4d 322d 3131 3339 312d = /tmp/M2-11391- │ │ │ │ +00258120: 3d20 2f74 6d70 2f4d 322d 3131 3936 302d = /tmp/M2-11960- │ │ │ │ 00258130: 302f 3020 2020 2020 2020 207c 0a2b 2d2d 0/0 |.+-- │ │ │ │ 00258140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00258150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00258160: 3220 3a20 6669 6c65 4578 6973 7473 2066 2 : fileExists f │ │ │ │ 00258170: 6e20 2020 2020 2020 2020 2020 207c 0a7c n |.| │ │ │ │ 00258180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258190: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ @@ -153629,15 +153629,15 @@ │ │ │ │ 002581c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 002581d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002581e0: 2b0a 7c69 3320 3a20 666e 203c 3c20 2268 +.|i3 : fn << "h │ │ │ │ 002581f0: 6920 7468 6572 6522 203c 3c20 636c 6f73 i there" << clos │ │ │ │ 00258200: 657c 0a7c 2020 2020 2020 2020 2020 2020 e|.| │ │ │ │ 00258210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258220: 2020 7c0a 7c6f 3320 3d20 2f74 6d70 2f4d |.|o3 = /tmp/M │ │ │ │ -00258230: 322d 3131 3339 312d 302f 3020 2020 2020 2-11391-0/0 │ │ │ │ +00258230: 322d 3131 3936 302d 302f 3020 2020 2020 2-11960-0/0 │ │ │ │ 00258240: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00258250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258260: 2020 2020 7c0a 7c6f 3320 3a20 4669 6c65 |.|o3 : File │ │ │ │ 00258270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00258280: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 00258290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002582a0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6669 ------+.|i4 : fi │ │ │ │ @@ -153930,15 +153930,15 @@ │ │ │ │ 00259490: 696e 6720 636f 6465 2e0a 2b2d 2d2d 2d2d ing code..+----- │ │ │ │ 002594a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002594b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 002594c0: 3120 3a20 6375 7272 656e 7454 696d 6528 1 : currentTime( │ │ │ │ 002594d0: 2920 2d20 6669 6c65 5469 6d65 2022 2e22 ) - fileTime "." │ │ │ │ 002594e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 002594f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00259500: 2020 2020 7c0a 7c6f 3120 3d20 3131 3120 |.|o1 = 111 │ │ │ │ +00259500: 2020 2020 7c0a 7c6f 3120 3d20 3739 2020 |.|o1 = 79 │ │ │ │ 00259510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259520: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 00259530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ 00259550: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ 00259560: 0a0a 2020 2a20 2a6e 6f74 6520 6375 7272 .. * *note curr │ │ │ │ 00259570: 656e 7454 696d 653a 2063 7572 7265 6e74 entTime: current │ │ │ │ @@ -154040,15 +154040,15 @@ │ │ │ │ 00259b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259b80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00259b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259bc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00259bd0: 0a7c 6f31 203d 202f 746d 702f 4d32 2d31 .|o1 = /tmp/M2-1 │ │ │ │ -00259be0: 3138 3431 2d30 2f30 2020 2020 2020 2020 1841-0/0 │ │ │ │ +00259be0: 3238 3731 2d30 2f30 2020 2020 2020 2020 2871-0/0 │ │ │ │ 00259bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259c10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00259c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259c50: 2d2d 2d2d 2d2b 0a7c 6932 203a 2064 7374 -----+.|i2 : dst │ │ │ │ @@ -154056,15 +154056,15 @@ │ │ │ │ 00259c70: 4e61 6d65 2829 2020 2020 2020 2020 2020 Name() │ │ │ │ 00259c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259c90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00259ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259cd0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00259ce0: 203d 202f 746d 702f 4d32 2d31 3138 3431 = /tmp/M2-11841 │ │ │ │ +00259ce0: 203d 202f 746d 702f 4d32 2d31 3238 3731 = /tmp/M2-12871 │ │ │ │ 00259cf0: 2d30 2f31 2020 2020 2020 2020 2020 2020 -0/1 │ │ │ │ 00259d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259d10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00259d20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00259d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -154073,15 +154073,15 @@ │ │ │ │ 00259d80: 6f73 6520 2020 2020 2020 2020 2020 2020 ose │ │ │ │ 00259d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259da0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00259db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259de0: 2020 2020 2020 207c 0a7c 6f33 203d 202f |.|o3 = / │ │ │ │ -00259df0: 746d 702f 4d32 2d31 3138 3431 2d30 2f30 tmp/M2-11841-0/0 │ │ │ │ +00259df0: 746d 702f 4d32 2d31 3238 3731 2d30 2f30 tmp/M2-12871-0/0 │ │ │ │ 00259e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259e20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00259e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259e60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -154094,17 +154094,17 @@ │ │ │ │ 00259ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259ef0: 2d2d 2d2b 0a7c 6934 203a 2063 6f70 7946 ---+.|i4 : copyF │ │ │ │ 00259f00: 696c 6528 7372 632c 6473 742c 5665 7262 ile(src,dst,Verb │ │ │ │ 00259f10: 6f73 653d 3e74 7275 6529 2020 2020 2020 ose=>true) │ │ │ │ 00259f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00259f30: 2020 2020 2020 7c0a 7c20 2d2d 2063 6f70 |.| -- cop │ │ │ │ -00259f40: 7969 6e67 3a20 2f74 6d70 2f4d 322d 3131 ying: /tmp/M2-11 │ │ │ │ -00259f50: 3834 312d 302f 3020 2d3e 202f 746d 702f 841-0/0 -> /tmp/ │ │ │ │ -00259f60: 4d32 2d31 3138 3431 2d30 2f31 2020 2020 M2-11841-0/1 │ │ │ │ +00259f40: 7969 6e67 3a20 2f74 6d70 2f4d 322d 3132 ying: /tmp/M2-12 │ │ │ │ +00259f50: 3837 312d 302f 3020 2d3e 202f 746d 702f 871-0/0 -> /tmp/ │ │ │ │ +00259f60: 4d32 2d31 3238 3731 2d30 2f31 2020 2020 M2-12871-0/1 │ │ │ │ 00259f70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 00259f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00259fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00259fc0: 3520 3a20 6765 7420 6473 7420 2020 2020 5 : get dst │ │ │ │ 00259fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154124,31 +154124,31 @@ │ │ │ │ 0025a0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a0c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 0025a0d0: 636f 7079 4669 6c65 2873 7263 2c64 7374 copyFile(src,dst │ │ │ │ 0025a0e0: 2c56 6572 626f 7365 3d3e 7472 7565 2c55 ,Verbose=>true,U │ │ │ │ 0025a0f0: 7064 6174 654f 6e6c 7920 3d3e 2074 7275 pdateOnly => tru │ │ │ │ 0025a100: 6529 2020 2020 2020 2020 207c 0a7c 202d e) |.| - │ │ │ │ 0025a110: 2d20 736b 6970 7069 6e67 3a20 2f74 6d70 - skipping: /tmp │ │ │ │ -0025a120: 2f4d 322d 3131 3834 312d 302f 3020 6e6f /M2-11841-0/0 no │ │ │ │ +0025a120: 2f4d 322d 3132 3837 312d 302f 3020 6e6f /M2-12871-0/0 no │ │ │ │ 0025a130: 7420 6e65 7765 7220 7468 616e 202f 746d t newer than /tm │ │ │ │ -0025a140: 702f 4d32 2d31 3138 3431 2d30 2f31 7c0a p/M2-11841-0/1|. │ │ │ │ +0025a140: 702f 4d32 2d31 3238 3731 2d30 2f31 7c0a p/M2-12871-0/1|. │ │ │ │ 0025a150: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0025a160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a190: 2d2b 0a7c 6937 203a 2073 7263 203c 3c20 -+.|i7 : src << │ │ │ │ 0025a1a0: 2268 6f20 7468 6572 6522 203c 3c20 636c "ho there" << cl │ │ │ │ 0025a1b0: 6f73 6520 2020 2020 2020 2020 2020 2020 ose │ │ │ │ 0025a1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a1d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0025a1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a210: 2020 2020 2020 207c 0a7c 6f37 203d 202f |.|o7 = / │ │ │ │ -0025a220: 746d 702f 4d32 2d31 3138 3431 2d30 2f30 tmp/M2-11841-0/0 │ │ │ │ +0025a220: 746d 702f 4d32 2d31 3238 3731 2d30 2f30 tmp/M2-12871-0/0 │ │ │ │ 0025a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a250: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0025a260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a290: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ @@ -154162,17 +154162,17 @@ │ │ │ │ 0025a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a320: 2d2d 2d2b 0a7c 6938 203a 2063 6f70 7946 ---+.|i8 : copyF │ │ │ │ 0025a330: 696c 6528 7372 632c 6473 742c 5665 7262 ile(src,dst,Verb │ │ │ │ 0025a340: 6f73 653d 3e74 7275 652c 5570 6461 7465 ose=>true,Update │ │ │ │ 0025a350: 4f6e 6c79 203d 3e20 7472 7565 2920 2020 Only => true) │ │ │ │ 0025a360: 2020 2020 2020 7c0a 7c20 2d2d 2073 6b69 |.| -- ski │ │ │ │ 0025a370: 7070 696e 673a 202f 746d 702f 4d32 2d31 pping: /tmp/M2-1 │ │ │ │ -0025a380: 3138 3431 2d30 2f30 206e 6f74 206e 6577 1841-0/0 not new │ │ │ │ +0025a380: 3238 3731 2d30 2f30 206e 6f74 206e 6577 2871-0/0 not new │ │ │ │ 0025a390: 6572 2074 6861 6e20 2f74 6d70 2f4d 322d er than /tmp/M2- │ │ │ │ -0025a3a0: 3131 3834 312d 302f 317c 0a2b 2d2d 2d2d 11841-0/1|.+---- │ │ │ │ +0025a3a0: 3132 3837 312d 302f 317c 0a2b 2d2d 2d2d 12871-0/1|.+---- │ │ │ │ 0025a3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025a3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0025a3f0: 3920 3a20 6765 7420 6473 7420 2020 2020 9 : get dst │ │ │ │ 0025a400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025a410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154310,40 +154310,40 @@ │ │ │ │ 0025ac50: 2073 7263 203d 2074 656d 706f 7261 7279 src = temporary │ │ │ │ 0025ac60: 4669 6c65 4e61 6d65 2829 2020 2020 2020 FileName() │ │ │ │ 0025ac70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0025ac80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ac90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025aca0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0025acb0: 0a7c 6f31 203d 202f 746d 702f 4d32 2d31 .|o1 = /tmp/M2-1 │ │ │ │ -0025acc0: 3134 3438 2d30 2f30 2020 2020 2020 2020 1448-0/0 │ │ │ │ +0025acc0: 3230 3737 2d30 2f30 2020 2020 2020 2020 2077-0/0 │ │ │ │ 0025acd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ace0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0025acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025ad10: 2d2d 2d2d 2d2b 0a7c 6932 203a 2064 7374 -----+.|i2 : dst │ │ │ │ 0025ad20: 203d 2074 656d 706f 7261 7279 4669 6c65 = temporaryFile │ │ │ │ 0025ad30: 4e61 6d65 2829 2020 2020 2020 2020 2020 Name() │ │ │ │ 0025ad40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0025ad50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ad70: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0025ad80: 203d 202f 746d 702f 4d32 2d31 3134 3438 = /tmp/M2-11448 │ │ │ │ +0025ad80: 203d 202f 746d 702f 4d32 2d31 3230 3737 = /tmp/M2-12077 │ │ │ │ 0025ad90: 2d30 2f31 2020 2020 2020 2020 2020 2020 -0/1 │ │ │ │ 0025ada0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0025adb0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0025adc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025add0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025ade0: 2d2b 0a7c 6933 203a 2073 7263 203c 3c20 -+.|i3 : src << │ │ │ │ 0025adf0: 2268 6920 7468 6572 6522 203c 3c20 636c "hi there" << cl │ │ │ │ 0025ae00: 6f73 6520 2020 2020 2020 2020 2020 2020 ose │ │ │ │ 0025ae10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0025ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ae40: 2020 2020 2020 207c 0a7c 6f33 203d 202f |.|o3 = / │ │ │ │ -0025ae50: 746d 702f 4d32 2d31 3134 3438 2d30 2f30 tmp/M2-11448-0/0 │ │ │ │ +0025ae50: 746d 702f 4d32 2d31 3230 3737 2d30 2f30 tmp/M2-12077-0/0 │ │ │ │ 0025ae60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ae70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0025ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ae90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025aea0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0025aeb0: 6f33 203a 2046 696c 6520 2020 2020 2020 o3 : File │ │ │ │ 0025aec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154351,17 +154351,17 @@ │ │ │ │ 0025aee0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0025aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025af10: 2d2d 2d2b 0a7c 6934 203a 206d 6f76 6546 ---+.|i4 : moveF │ │ │ │ 0025af20: 696c 6528 7372 632c 6473 742c 5665 7262 ile(src,dst,Verb │ │ │ │ 0025af30: 6f73 653d 3e74 7275 6529 2020 2020 2020 ose=>true) │ │ │ │ 0025af40: 2020 2020 2020 7c0a 7c2d 2d6d 6f76 696e |.|--movin │ │ │ │ -0025af50: 673a 202f 746d 702f 4d32 2d31 3134 3438 g: /tmp/M2-11448 │ │ │ │ +0025af50: 673a 202f 746d 702f 4d32 2d31 3230 3737 g: /tmp/M2-12077 │ │ │ │ 0025af60: 2d30 2f30 202d 3e20 2f74 6d70 2f4d 322d -0/0 -> /tmp/M2- │ │ │ │ -0025af70: 3131 3434 382d 302f 317c 0a2b 2d2d 2d2d 11448-0/1|.+---- │ │ │ │ +0025af70: 3132 3037 372d 302f 317c 0a2b 2d2d 2d2d 12077-0/1|.+---- │ │ │ │ 0025af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025afa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0025afb0: 3520 3a20 6765 7420 6473 7420 2020 2020 5 : get dst │ │ │ │ 0025afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025afd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0025afe0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ @@ -154374,21 +154374,21 @@ │ │ │ │ 0025b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b070: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ 0025b080: 6261 6b20 3d20 6d6f 7665 4669 6c65 2864 bak = moveFile(d │ │ │ │ 0025b090: 7374 2c56 6572 626f 7365 3d3e 7472 7565 st,Verbose=>true │ │ │ │ 0025b0a0: 2920 2020 2020 2020 2020 207c 0a7c 2d2d ) |.|-- │ │ │ │ 0025b0b0: 6261 636b 7570 2066 696c 6520 6372 6561 backup file crea │ │ │ │ -0025b0c0: 7465 643a 202f 746d 702f 4d32 2d31 3134 ted: /tmp/M2-114 │ │ │ │ -0025b0d0: 3438 2d30 2f31 2e62 616b 2020 2020 7c0a 48-0/1.bak |. │ │ │ │ +0025b0c0: 7465 643a 202f 746d 702f 4d32 2d31 3230 ted: /tmp/M2-120 │ │ │ │ +0025b0d0: 3737 2d30 2f31 2e62 616b 2020 2020 7c0a 77-0/1.bak |. │ │ │ │ 0025b0e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0025b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b110: 207c 0a7c 6f36 203d 202f 746d 702f 4d32 |.|o6 = /tmp/M2 │ │ │ │ -0025b120: 2d31 3134 3438 2d30 2f31 2e62 616b 2020 -11448-0/1.bak │ │ │ │ +0025b120: 2d31 3230 3737 2d30 2f31 2e62 616b 2020 -12077-0/1.bak │ │ │ │ 0025b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025b140: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0025b150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025b170: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2072 -------+.|i7 : r │ │ │ │ 0025b180: 656d 6f76 6546 696c 6520 6261 6b20 2020 emoveFile bak │ │ │ │ 0025b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -154603,15 +154603,15 @@ │ │ │ │ 0025bea0: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d =======..+------ │ │ │ │ 0025beb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025bec0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2066 -------+.|i1 : f │ │ │ │ 0025bed0: 6e20 3d20 7465 6d70 6f72 6172 7946 696c n = temporaryFil │ │ │ │ 0025bee0: 654e 616d 6528 297c 0a7c 2020 2020 2020 eName()|.| │ │ │ │ 0025bef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025bf00: 2020 2020 2020 207c 0a7c 6f31 203d 202f |.|o1 = / │ │ │ │ -0025bf10: 746d 702f 4d32 2d31 3230 3934 2d30 2f30 tmp/M2-12094-0/0 │ │ │ │ +0025bf10: 746d 702f 4d32 2d31 3333 3834 2d30 2f30 tmp/M2-13384-0/0 │ │ │ │ 0025bf20: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0025bf30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025bf40: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2073 -------+.|i2 : s │ │ │ │ 0025bf50: 796d 6c69 6e6b 4669 6c65 2822 7177 6572 ymlinkFile("qwer │ │ │ │ 0025bf60: 7422 2c20 666e 297c 0a2b 2d2d 2d2d 2d2d t", fn)|.+------ │ │ │ │ 0025bf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025bf80: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 -------+.|i3 : f │ │ │ │ @@ -154765,15 +154765,15 @@ │ │ │ │ 0025c8c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0025c8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0025c8e0: 0a7c 6931 203a 2070 203d 2074 656d 706f .|i1 : p = tempo │ │ │ │ 0025c8f0: 7261 7279 4669 6c65 4e61 6d65 2028 297c raryFileName ()| │ │ │ │ 0025c900: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0025c910: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0025c920: 0a7c 6f31 203d 202f 746d 702f 4d32 2d31 .|o1 = /tmp/M2-1 │ │ │ │ -0025c930: 3237 3334 2d30 2f30 2020 2020 2020 207c 2734-0/0 | │ │ │ │ +0025c930: 3436 3834 2d30 2f30 2020 2020 2020 207c 4684-0/0 | │ │ │ │ 0025c940: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0025c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0025c960: 0a7c 6932 203a 2073 796d 6c69 6e6b 4669 .|i2 : symlinkFi │ │ │ │ 0025c970: 6c65 2028 2266 6f6f 222c 2070 2920 207c le ("foo", p) | │ │ │ │ 0025c980: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0025c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0025c9a0: 0a7c 6933 203a 2072 6561 646c 696e 6b20 .|i3 : readlink │ │ │ │ @@ -154855,66 +154855,66 @@ │ │ │ │ 0025ce60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025ce70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0025ce80: 3120 3a20 7265 616c 7061 7468 2022 2e22 1 : realpath "." │ │ │ │ 0025ce90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0025cea0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0025ceb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025cec0: 7c0a 7c6f 3120 3d20 2f74 6d70 2f4d 322d |.|o1 = /tmp/M2- │ │ │ │ -0025ced0: 3131 3032 322d 302f 3931 2d72 756e 6469 11022-0/91-rundi │ │ │ │ +0025ced0: 3131 3234 322d 302f 3931 2d72 756e 6469 11242-0/91-rundi │ │ │ │ 0025cee0: 722f 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d r/|.+----------- │ │ │ │ 0025cef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025cf00: 2d2d 2d2d 2b0a 7c69 3220 3a20 7020 3d20 ----+.|i2 : p = │ │ │ │ 0025cf10: 7465 6d70 6f72 6172 7946 696c 654e 616d temporaryFileNam │ │ │ │ 0025cf20: 6528 2920 2020 7c0a 7c20 2020 2020 2020 e() |.| │ │ │ │ 0025cf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025cf40: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -0025cf50: 2f74 6d70 2f4d 322d 3132 3735 332d 302f /tmp/M2-12753-0/ │ │ │ │ +0025cf50: 2f74 6d70 2f4d 322d 3134 3732 332d 302f /tmp/M2-14723-0/ │ │ │ │ 0025cf60: 3020 2020 2020 2020 2020 7c0a 2b2d 2d2d 0 |.+--- │ │ │ │ 0025cf70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025cf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0025cf90: 3320 3a20 7120 3d20 7465 6d70 6f72 6172 3 : q = temporar │ │ │ │ 0025cfa0: 7946 696c 654e 616d 6528 2920 2020 7c0a yFileName() |. │ │ │ │ 0025cfb0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0025cfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025cfd0: 7c0a 7c6f 3320 3d20 2f74 6d70 2f4d 322d |.|o3 = /tmp/M2- │ │ │ │ -0025cfe0: 3132 3735 332d 302f 3120 2020 2020 2020 12753-0/1 │ │ │ │ +0025cfe0: 3134 3732 332d 302f 3120 2020 2020 2020 14723-0/1 │ │ │ │ 0025cff0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0025d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025d010: 2d2d 2d2d 2b0a 7c69 3420 3a20 7379 6d6c ----+.|i4 : syml │ │ │ │ 0025d020: 696e 6b46 696c 6528 702c 7129 2020 2020 inkFile(p,q) │ │ │ │ 0025d030: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0025d040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025d050: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ 0025d060: 7020 3c3c 2063 6c6f 7365 2020 2020 2020 p << close │ │ │ │ 0025d070: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0025d080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025d090: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0025d0a0: 3520 3d20 2f74 6d70 2f4d 322d 3132 3735 5 = /tmp/M2-1275 │ │ │ │ +0025d0a0: 3520 3d20 2f74 6d70 2f4d 322d 3134 3732 5 = /tmp/M2-1472 │ │ │ │ 0025d0b0: 332d 302f 3020 2020 2020 2020 2020 7c0a 3-0/0 |. │ │ │ │ 0025d0c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0025d0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025d0e0: 7c0a 7c6f 3520 3a20 4669 6c65 2020 2020 |.|o5 : File │ │ │ │ 0025d0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025d100: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0025d110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025d120: 2d2d 2d2d 2b0a 7c69 3620 3a20 7265 6164 ----+.|i6 : read │ │ │ │ 0025d130: 6c69 6e6b 2071 2020 2020 2020 2020 2020 link q │ │ │ │ 0025d140: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0025d150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025d160: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -0025d170: 2f74 6d70 2f4d 322d 3132 3735 332d 302f /tmp/M2-12753-0/ │ │ │ │ +0025d170: 2f74 6d70 2f4d 322d 3134 3732 332d 302f /tmp/M2-14723-0/ │ │ │ │ 0025d180: 3020 2020 2020 2020 2020 7c0a 2b2d 2d2d 0 |.+--- │ │ │ │ 0025d190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025d1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0025d1b0: 3720 3a20 7265 616c 7061 7468 2071 2020 7 : realpath q │ │ │ │ 0025d1c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0025d1d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0025d1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025d1f0: 7c0a 7c6f 3720 3d20 2f74 6d70 2f4d 322d |.|o7 = /tmp/M2- │ │ │ │ -0025d200: 3132 3735 332d 302f 3020 2020 2020 2020 12753-0/0 │ │ │ │ +0025d200: 3134 3732 332d 302f 3020 2020 2020 2020 14723-0/0 │ │ │ │ 0025d210: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0025d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025d230: 2d2d 2d2d 2b0a 7c69 3820 3a20 7265 6d6f ----+.|i8 : remo │ │ │ │ 0025d240: 7665 4669 6c65 2070 2020 2020 2020 2020 veFile p │ │ │ │ 0025d250: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 0025d260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025d270: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 --------+.|i9 : │ │ │ │ @@ -154930,15 +154930,15 @@ │ │ │ │ 0025d310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025d320: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 ----------+.|i10 │ │ │ │ 0025d330: 203a 2072 6561 6c70 6174 6820 2222 2020 : realpath "" │ │ │ │ 0025d340: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0025d350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025d360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025d370: 7c0a 7c6f 3130 203d 202f 746d 702f 4d32 |.|o10 = /tmp/M2 │ │ │ │ -0025d380: 2d31 3130 3232 2d30 2f39 312d 7275 6e64 -11022-0/91-rund │ │ │ │ +0025d380: 2d31 3132 3432 2d30 2f39 312d 7275 6e64 -11242-0/91-rund │ │ │ │ 0025d390: 6972 2f7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ir/|.+---------- │ │ │ │ 0025d3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025d3b0: 2d2d 2d2d 2d2d 2b0a 0a43 6176 6561 740a ------+..Caveat. │ │ │ │ 0025d3c0: 3d3d 3d3d 3d3d 0a0a 4576 6572 7920 636f ======..Every co │ │ │ │ 0025d3d0: 6d70 6f6e 656e 7420 6f66 2074 6865 2070 mponent of the p │ │ │ │ 0025d3e0: 6174 6820 6d75 7374 2065 7869 7374 2069 ath must exist i │ │ │ │ 0025d3f0: 6e20 7468 6520 6669 6c65 2073 7973 7465 n the file syste │ │ │ │ @@ -155095,43 +155095,43 @@ │ │ │ │ 0025dd60: 2b0a 7c69 3120 3a20 6469 7220 3d20 7465 +.|i1 : dir = te │ │ │ │ 0025dd70: 6d70 6f72 6172 7946 696c 654e 616d 6528 mporaryFileName( │ │ │ │ 0025dd80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0025dd90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0025dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ddc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0025ddd0: 7c6f 3120 3d20 2f74 6d70 2f4d 322d 3132 |o1 = /tmp/M2-12 │ │ │ │ -0025dde0: 3437 342d 302f 3020 2020 2020 2020 2020 474-0/0 │ │ │ │ +0025ddd0: 7c6f 3120 3d20 2f74 6d70 2f4d 322d 3134 |o1 = /tmp/M2-14 │ │ │ │ +0025dde0: 3136 342d 302f 3020 2020 2020 2020 2020 164-0/0 │ │ │ │ 0025ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025de00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ 0025de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0025de40: 3220 3a20 6d61 6b65 4469 7265 6374 6f72 2 : makeDirector │ │ │ │ 0025de50: 7920 6469 7220 2020 2020 2020 2020 2020 y dir │ │ │ │ 0025de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025de70: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0025de80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025de90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dea0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -0025deb0: 3d20 2f74 6d70 2f4d 322d 3132 3437 342d = /tmp/M2-12474- │ │ │ │ +0025deb0: 3d20 2f74 6d70 2f4d 322d 3134 3136 342d = /tmp/M2-14164- │ │ │ │ 0025dec0: 302f 3020 2020 2020 2020 2020 2020 2020 0/0 │ │ │ │ 0025ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dee0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0025def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025df00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025df10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ 0025df20: 2866 6e20 3d20 6469 7220 7c20 222f 2220 (fn = dir | "/" │ │ │ │ 0025df30: 7c20 2266 6f6f 2229 203c 3c20 2268 6920 | "foo") << "hi │ │ │ │ 0025df40: 7468 6572 6522 203c 3c20 636c 6f73 657c there" << close| │ │ │ │ 0025df50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0025df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025df80: 2020 2020 2020 7c0a 7c6f 3320 3d20 2f74 |.|o3 = /t │ │ │ │ -0025df90: 6d70 2f4d 322d 3132 3437 342d 302f 302f mp/M2-12474-0/0/ │ │ │ │ +0025df90: 6d70 2f4d 322d 3134 3136 342d 302f 302f mp/M2-14164-0/0/ │ │ │ │ 0025dfa0: 666f 6f20 2020 2020 2020 2020 2020 2020 foo │ │ │ │ 0025dfb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 0025dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025dff0: 2020 2020 7c0a 7c6f 3320 3a20 4669 6c65 |.|o3 : File │ │ │ │ 0025e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -155143,15 +155143,15 @@ │ │ │ │ 0025e060: 2d2d 2b0a 7c69 3420 3a20 7265 6164 4469 --+.|i4 : readDi │ │ │ │ 0025e070: 7265 6374 6f72 7920 6469 7220 2020 2020 rectory dir │ │ │ │ 0025e080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e090: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0025e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0025e0d0: 7c0a 7c6f 3420 3d20 7b2e 2c20 2e2e 2c20 |.|o4 = {., .., │ │ │ │ +0025e0d0: 7c0a 7c6f 3420 3d20 7b2e 2e2c 202e 2c20 |.|o4 = {.., ., │ │ │ │ 0025e0e0: 666f 6f7d 2020 2020 2020 2020 2020 2020 foo} │ │ │ │ 0025e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e100: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0025e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025e130: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0025e140: 7c6f 3420 3a20 4c69 7374 2020 2020 2020 |o4 : List │ │ │ │ @@ -155251,25 +155251,25 @@ │ │ │ │ 0025e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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696c 654e 616d 6528 mporaryFileName( │ │ │ │ 0025ed30: 2920 7c20 222f 2220 2020 2020 2020 2020 ) | "/" │ │ │ │ 0025ed40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0025ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ed70: 7c0a 7c6f 3120 3d20 2f74 6d70 2f4d 322d |.|o1 = /tmp/M2- │ │ │ │ -0025ed80: 3131 3631 322d 302f 302f 2020 2020 2020 11612-0/0/ │ │ │ │ +0025ed80: 3132 3430 322d 302f 302f 2020 2020 2020 12402-0/0/ │ │ │ │ 0025ed90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0025eda0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0025edb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025edc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 0025edd0: 7c69 3220 3a20 6d6b 6469 7220 7020 2020 |i2 : mkdir p │ │ │ │ 0025ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025edf0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ @@ -155375,15 +155375,15 @@ │ │ │ │ 0025eee0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ 0025eef0: 2866 6e20 3d20 7020 7c20 2266 6f6f 2229 (fn = p | "foo") │ │ │ │ 0025ef00: 203c 3c20 2268 6920 7468 6572 6522 203c << "hi there" < │ │ │ │ 0025ef10: 3c20 636c 6f73 657c 0a7c 2020 2020 2020 < close|.| │ │ │ │ 0025ef20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ef30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ef40: 2020 2020 2020 7c0a 7c6f 3420 3d20 2f74 |.|o4 = /t │ │ │ │ -0025ef50: 6d70 2f4d 322d 3131 3631 322d 302f 302f mp/M2-11612-0/0/ │ │ │ │ +0025ef50: 6d70 2f4d 322d 3132 3430 322d 302f 302f mp/M2-12402-0/0/ │ │ │ │ 0025ef60: 666f 6f20 2020 2020 2020 2020 2020 2020 foo │ │ │ │ 0025ef70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0025ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025efa0: 2020 2020 7c0a 7c6f 3420 3a20 4669 6c65 |.|o4 : File │ │ │ │ 0025efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -155478,40 +155478,40 @@ │ │ │ │ 0025f550: 2074 6f20 6469 722e 0a0a 2b2d 2d2d 2d2d to dir...+----- │ │ │ │ 0025f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f570: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ 0025f580: 2064 6972 203d 2074 656d 706f 7261 7279 dir = temporary │ │ │ │ 0025f590: 4669 6c65 4e61 6d65 2829 7c0a 7c20 2020 FileName()|.| │ │ │ │ 0025f5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f5b0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0025f5c0: 203d 202f 746d 702f 4d32 2d31 3132 3936 = /tmp/M2-11296 │ │ │ │ +0025f5c0: 203d 202f 746d 702f 4d32 2d31 3137 3635 = /tmp/M2-11765 │ │ │ │ 0025f5d0: 2d30 2f30 2020 2020 2020 2020 7c0a 2b2d -0/0 |.+- │ │ │ │ 0025f5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 0025f600: 6932 203a 206d 616b 6544 6972 6563 746f i2 : makeDirecto │ │ │ │ 0025f610: 7279 2064 6972 2020 2020 2020 2020 7c0a ry dir |. │ │ │ │ 0025f620: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0025f630: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0025f640: 0a7c 6f32 203d 202f 746d 702f 4d32 2d31 .|o2 = /tmp/M2-1 │ │ │ │ -0025f650: 3132 3936 2d30 2f30 2020 2020 2020 2020 1296-0/0 │ │ │ │ +0025f650: 3137 3635 2d30 2f30 2020 2020 2020 2020 1765-0/0 │ │ │ │ 0025f660: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0025f670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f680: 2d2b 0a7c 6933 203a 2063 6861 6e67 6544 -+.|i3 : changeD │ │ │ │ 0025f690: 6972 6563 746f 7279 2064 6972 2020 2020 irectory dir │ │ │ │ 0025f6a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0025f6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f6c0: 2020 207c 0a7c 6f33 203d 202f 746d 702f |.|o3 = /tmp/ │ │ │ │ -0025f6d0: 4d32 2d31 3132 3936 2d30 2f30 2f20 2020 M2-11296-0/0/ │ │ │ │ +0025f6d0: 4d32 2d31 3137 3635 2d30 2f30 2f20 2020 M2-11765-0/0/ │ │ │ │ 0025f6e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0025f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f700: 2d2d 2d2d 2d2b 0a7c 6934 203a 2063 7572 -----+.|i4 : cur │ │ │ │ 0025f710: 7265 6e74 4469 7265 6374 6f72 7928 2920 rentDirectory() │ │ │ │ 0025f720: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 0025f730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025f740: 2020 2020 2020 207c 0a7c 6f34 203d 202f |.|o4 = / │ │ │ │ -0025f750: 746d 702f 4d32 2d31 3132 3936 2d30 2f30 tmp/M2-11296-0/0 │ │ │ │ +0025f750: 746d 702f 4d32 2d31 3137 3635 2d30 2f30 tmp/M2-11765-0/0 │ │ │ │ 0025f760: 2f20 2020 2020 2020 7c0a 2b2d 2d2d 2d2d / |.+----- │ │ │ │ 0025f770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0025f780: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4966 2064 ---------+..If d │ │ │ │ 0025f790: 6972 2069 7320 6f6d 6974 7465 642c 2074 ir is omitted, t │ │ │ │ 0025f7a0: 6865 6e20 7468 6520 6375 7272 656e 7420 hen the current │ │ │ │ 0025f7b0: 776f 726b 696e 6720 6469 7265 6374 6f72 working director │ │ │ │ 0025f7c0: 7920 6973 2063 6861 6e67 6564 2074 6f20 y is changed to │ │ │ │ @@ -155639,15 +155639,15 @@ │ │ │ │ 0025ff60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ff70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 0025ff80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ff90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ffa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ffb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0025ffc0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -0025ffd0: 2f74 6d70 2f4d 322d 3132 3037 352d 302f /tmp/M2-12075-0/ │ │ │ │ +0025ffd0: 2f74 6d70 2f4d 322d 3133 3334 352d 302f /tmp/M2-13345-0/ │ │ │ │ 0025ffe0: 302f 2020 2020 2020 2020 2020 2020 2020 0/ │ │ │ │ 0025fff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260010: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 00260020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00260030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00260040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -155659,15 +155659,15 @@ │ │ │ │ 002600a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002600b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 002600c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002600d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002600e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002600f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260100: 2020 2020 7c0a 7c6f 3220 3d20 2f74 6d70 |.|o2 = /tmp │ │ │ │ -00260110: 2f4d 322d 3132 3037 352d 302f 312f 2020 /M2-12075-0/1/ │ │ │ │ +00260110: 2f4d 322d 3133 3334 352d 302f 312f 2020 /M2-13345-0/1/ │ │ │ │ 00260120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260150: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00260160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00260170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00260180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -155679,15 +155679,15 @@ │ │ │ │ 002601e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002601f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00260200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00260240: 7c0a 7c6f 3320 3d20 2f74 6d70 2f4d 322d |.|o3 = /tmp/M2- │ │ │ │ -00260250: 3132 3037 352d 302f 302f 612f 2020 2020 12075-0/0/a/ │ │ │ │ +00260250: 3133 3334 352d 302f 302f 612f 2020 2020 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│ 00261cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00261ce0: 207c 0a7c 6f31 203d 202f 746d 702f 4d32 |.|o1 = /tmp/M2 │ │ │ │ -00261cf0: 2d31 3136 3530 2d30 2f30 2020 2020 2020 -11650-0/0 │ │ │ │ +00261cf0: 2d31 3234 3830 2d30 2f30 2020 2020 2020 -12480-0/0 │ │ │ │ 00261d00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00261d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00261d20: 2d2d 2d2b 0a7c 6932 203a 206d 616b 6544 ---+.|i2 : makeD │ │ │ │ 00261d30: 6972 6563 746f 7279 2064 6972 2020 2020 irectory dir │ │ │ │ 00261d40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 00261d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00261d60: 2020 2020 207c 0a7c 6f32 203d 202f 746d |.|o2 = /tm │ │ │ │ -00261d70: 702f 4d32 2d31 3136 3530 2d30 2f30 2020 p/M2-11650-0/0 │ │ │ │ +00261d70: 702f 4d32 2d31 3234 3830 2d30 2f30 2020 p/M2-12480-0/0 │ │ │ │ 00261d80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 00261d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00261da0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2072 -------+.|i3 : r │ │ │ │ 00261db0: 6561 6444 6972 6563 746f 7279 2064 6972 eadDirectory dir │ │ │ │ 00261dc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00261dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00261de0: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ -00261df0: 207b 2e2c 202e 2e7d 2020 2020 2020 2020 {., ..} │ │ │ │ +00261df0: 207b 2e2e 2c20 2e7d 2020 2020 2020 2020 {.., .} │ │ │ │ 00261e00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00261e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00261e20: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ 00261e30: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ 00261e40: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00261e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00261e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ @@ -156295,15 +156295,15 @@ │ │ │ │ 00262860: 6d70 6f72 6172 7946 696c 654e 616d 6528 mporaryFileName( │ │ │ │ 00262870: 2920 7c20 222f 2220 2020 2020 2020 2020 ) | "/" │ │ │ │ 00262880: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00262890: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 002628a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002628b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002628c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -002628d0: 3120 3d20 2f74 6d70 2f4d 322d 3132 3033 1 = /tmp/M2-1203 │ │ │ │ +002628d0: 3120 3d20 2f74 6d70 2f4d 322d 3133 3236 1 = /tmp/M2-1326 │ │ │ │ 002628e0: 372d 302f 302f 2020 2020 2020 2020 2020 7-0/0/ │ │ │ │ 002628f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262900: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00262910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262940: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ @@ -156311,30 +156311,30 @@ │ │ │ │ 00262960: 696c 654e 616d 6528 2920 7c20 222f 2220 ileName() | "/" │ │ │ │ 00262970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262980: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00262990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002629a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002629b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002629c0: 2020 2020 7c0a 7c6f 3220 3d20 2f74 6d70 |.|o2 = /tmp │ │ │ │ -002629d0: 2f4d 322d 3132 3033 372d 302f 312f 2020 /M2-12037-0/1/ │ │ │ │ +002629d0: 2f4d 322d 3133 3236 372d 302f 312f 2020 /M2-13267-0/1/ │ │ │ │ 002629e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002629f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262a00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00262a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262a40: 2b0a 7c69 3320 3a20 6d61 6b65 4469 7265 +.|i3 : makeDire │ │ │ │ 00262a50: 6374 6f72 7920 2873 7263 7c22 612f 2229 ctory (src|"a/") │ │ │ │ 00262a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262a70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00262a80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00262a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262ab0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00262ac0: 3320 3d20 2f74 6d70 2f4d 322d 3132 3033 3 = /tmp/M2-1203 │ │ │ │ +00262ac0: 3320 3d20 2f74 6d70 2f4d 322d 3133 3236 3 = /tmp/M2-1326 │ │ │ │ 00262ad0: 372d 302f 302f 612f 2020 2020 2020 2020 7-0/0/a/ │ │ │ │ 00262ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262af0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00262b00: 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2020 2020 2020 2020 │ │ │ │ 00262bf0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00262c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262c30: 2b0a 7c69 3520 3a20 6d61 6b65 4469 7265 +.|i5 : makeDire │ │ │ │ 00262c40: 6374 6f72 7920 2873 7263 7c22 622f 632f ctory (src|"b/c/ │ │ │ │ 00262c50: 2229 2020 2020 2020 2020 2020 2020 2020 ") │ │ │ │ 00262c60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00262c70: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00262c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00262cb0: 3520 3d20 2f74 6d70 2f4d 322d 3132 3033 5 = /tmp/M2-1203 │ │ │ │ +00262cb0: 3520 3d20 2f74 6d70 2f4d 322d 3133 3236 5 = /tmp/M2-1326 │ │ │ │ 00262cc0: 372d 302f 302f 622f 632f 2020 2020 2020 7-0/0/b/c/ │ │ │ │ 00262cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262ce0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00262cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00262d20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ @@ -156373,15 +156373,15 @@ │ │ │ │ 00262d40: 2074 6865 7265 2220 3c3c 2063 6c6f 7365 there" << close │ │ │ │ 00262d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262d60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00262d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262da0: 2020 2020 7c0a 7c6f 3620 3d20 2f74 6d70 |.|o6 = /tmp │ │ │ │ -00262db0: 2f4d 322d 3132 3033 372d 302f 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= │ │ │ │ -00262f20: 2f74 6d70 2f4d 322d 3132 3033 372d 302f /tmp/M2-12037-0/ │ │ │ │ +00262f20: 2f74 6d70 2f4d 322d 3133 3236 372d 302f /tmp/M2-13267-0/ │ │ │ │ 00262f30: 302f 612f 6720 2020 2020 2020 2020 2020 0/a/g │ │ │ │ 00262f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262f50: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00262f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00262f90: 2020 2020 7c0a 7c6f 3720 3a20 4669 6c65 |.|o7 : File │ │ │ │ @@ -156419,15 +156419,15 @@ │ │ │ │ 00263020: 2f67 2220 3c3c 2022 686f 2074 6865 7265 /g" << "ho there │ │ │ │ 00263030: 2220 3c3c 2063 6c6f 7365 2020 2020 2020 " << close │ │ │ │ 00263040: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00263050: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00263060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263080: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00263090: 3820 3d20 2f74 6d70 2f4d 322d 3132 3033 8 = /tmp/M2-1203 │ │ │ │ +00263090: 3820 3d20 2f74 6d70 2f4d 322d 3133 3236 8 = /tmp/M2-1326 │ │ │ │ 002630a0: 372d 302f 302f 622f 632f 6720 2020 2020 7-0/0/b/c/g │ │ │ │ 002630b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002630c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 002630d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002630e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002630f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263100: 2020 2020 2020 2020 7c0a 7c6f 3820 3a20 |.|o8 : │ │ │ │ @@ -156439,25 +156439,25 @@ │ │ │ │ 00263160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263180: 2d2d 2d2d 2b0a 7c69 3920 3a20 7379 6d6c ----+.|i9 : syml │ │ │ │ 00263190: 696e 6b44 6972 6563 746f 7279 2873 7263 inkDirectory(src │ │ │ │ 002631a0: 2c64 7374 2c56 6572 626f 7365 3d3e 7472 ,dst,Verbose=>tr │ │ │ │ 002631b0: 7565 2920 2020 2020 2020 2020 2020 2020 ue) │ │ │ │ 002631c0: 2020 7c0a 7c2d 2d73 796d 6c69 6e6b 696e |.|--symlinkin │ │ │ │ -002631d0: 673a 202e 2e2f 2e2e 2f2e 2e2f 302f 622f g: ../../../0/b/ │ │ │ │ -002631e0: 632f 6720 2d3e 202f 746d 702f 4d32 2d31 c/g -> /tmp/M2-1 │ │ │ │ -002631f0: 3230 3337 2d30 2f31 2f62 2f63 2f67 2020 2037-0/1/b/c/g │ │ │ │ +002631d0: 673a 202e 2e2f 2e2e 2f30 2f61 2f67 202d g: ../../0/a/g - │ │ │ │ +002631e0: 3e20 2f74 6d70 2f4d 322d 3133 3236 372d > /tmp/M2-13267- │ │ │ │ +002631f0: 302f 312f 612f 6720 2020 2020 2020 2020 0/1/a/g │ │ │ │ 00263200: 7c0a 7c2d 2d73 796d 6c69 6e6b 696e 673a |.|--symlinking: │ │ │ │ -00263210: 202e 2e2f 2e2e 2f30 2f61 2f67 202d 3e20 ../../0/a/g -> │ │ │ │ -00263220: 2f74 6d70 2f4d 322d 3132 3033 372d 302f /tmp/M2-12037-0/ │ │ │ │ -00263230: 312f 612f 6720 2020 2020 2020 2020 7c0a 1/a/g |. │ │ │ │ +00263210: 202e 2e2f 2e2e 2f30 2f61 2f66 202d 3e20 ../../0/a/f -> │ │ │ │ +00263220: 2f74 6d70 2f4d 322d 3133 3236 372d 302f /tmp/M2-13267-0/ │ │ │ │ +00263230: 312f 612f 6620 2020 2020 2020 2020 7c0a 1/a/f |. │ │ │ │ 00263240: 7c2d 2d73 796d 6c69 6e6b 696e 673a 202e |--symlinking: . │ │ │ │ -00263250: 2e2f 2e2e 2f30 2f61 2f66 202d 3e20 2f74 ./../0/a/f -> /t │ │ │ │ -00263260: 6d70 2f4d 322d 3132 3033 372d 302f 312f mp/M2-12037-0/1/ │ │ │ │ -00263270: 612f 6620 2020 2020 2020 2020 7c0a 2b2d a/f |.+- │ │ │ │ +00263250: 2e2f 2e2e 2f2e 2e2f 302f 622f 632f 6720 ./../../0/b/c/g │ │ │ │ +00263260: 2d3e 202f 746d 702f 4d32 2d31 3332 3637 -> /tmp/M2-13267 │ │ │ │ +00263270: 2d30 2f31 2f62 2f63 2f67 2020 7c0a 2b2d -0/1/b/c/g |.+- │ │ │ │ 00263280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002632a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002632b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 ----------+.|i10 │ │ │ │ 002632c0: 203a 2067 6574 2028 6473 747c 2262 2f63 : get (dst|"b/c │ │ │ │ 002632d0: 2f67 2229 2020 2020 2020 2020 2020 2020 /g") │ │ │ │ 002632e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -156474,25 +156474,25 @@ │ │ │ │ 00263390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002633a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002633b0: 2d2d 2b0a 7c69 3131 203a 2073 796d 6c69 --+.|i11 : symli │ │ │ │ 002633c0: 6e6b 4469 7265 6374 6f72 7928 7372 632c nkDirectory(src, │ │ │ │ 002633d0: 6473 742c 5665 7262 6f73 653d 3e74 7275 dst,Verbose=>tru │ │ │ │ 002633e0: 652c 556e 646f 3d3e 7472 7565 2920 2020 e,Undo=>true) │ │ │ │ 002633f0: 7c0a 7c2d 2d75 6e73 796d 6c69 6e6b 696e |.|--unsymlinkin │ │ │ │ -00263400: 673a 202e 2e2f 2e2e 2f2e 2e2f 302f 622f g: ../../../0/b/ │ │ │ │ -00263410: 632f 6720 2d3e 202f 746d 702f 4d32 2d31 c/g -> /tmp/M2-1 │ │ │ │ -00263420: 3230 3337 2d30 2f31 2f62 2f63 2f67 7c0a 2037-0/1/b/c/g|. │ │ │ │ +00263400: 673a 202e 2e2f 2e2e 2f30 2f61 2f67 202d g: ../../0/a/g - │ │ │ │ +00263410: 3e20 2f74 6d70 2f4d 322d 3133 3236 372d > /tmp/M2-13267- │ │ │ │ +00263420: 302f 312f 612f 6720 2020 2020 2020 7c0a 0/1/a/g |. │ │ │ │ 00263430: 7c2d 2d75 6e73 796d 6c69 6e6b 696e 673a |--unsymlinking: │ │ │ │ -00263440: 202e 2e2f 2e2e 2f30 2f61 2f67 202d 3e20 ../../0/a/g -> │ │ │ │ -00263450: 2f74 6d70 2f4d 322d 3132 3033 372d 302f /tmp/M2-12037-0/ │ │ │ │ -00263460: 312f 612f 6720 2020 2020 2020 7c0a 7c2d 1/a/g |.|- │ │ │ │ +00263440: 202e 2e2f 2e2e 2f30 2f61 2f66 202d 3e20 ../../0/a/f -> │ │ │ │ +00263450: 2f74 6d70 2f4d 322d 3133 3236 372d 302f /tmp/M2-13267-0/ │ │ │ │ +00263460: 312f 612f 6620 2020 2020 2020 7c0a 7c2d 1/a/f |.|- │ │ │ │ 00263470: 2d75 6e73 796d 6c69 6e6b 696e 673a 202e -unsymlinking: . │ │ │ │ -00263480: 2e2f 2e2e 2f30 2f61 2f66 202d 3e20 2f74 ./../0/a/f -> /t │ │ │ │ -00263490: 6d70 2f4d 322d 3132 3033 372d 302f 312f mp/M2-12037-0/1/ │ │ │ │ -002634a0: 612f 6620 2020 2020 2020 7c0a 2b2d 2d2d a/f |.+--- │ │ │ │ +00263480: 2e2f 2e2e 2f2e 2e2f 302f 622f 632f 6720 ./../../0/b/c/g │ │ │ │ +00263490: 2d3e 202f 746d 702f 4d32 2d31 3332 3637 -> /tmp/M2-13267 │ │ │ │ +002634a0: 2d30 2f31 2f62 2f63 2f67 7c0a 2b2d 2d2d -0/1/b/c/g|.+--- │ │ │ │ 002634b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002634c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002634d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002634e0: 2d2d 2d2d 2d2d 2d2d 2b0a 4e6f 7720 7765 --------+.Now we │ │ │ │ 002634f0: 2072 656d 6f76 6520 7468 6520 6669 6c65 remove the file │ │ │ │ 00263500: 7320 616e 6420 6469 7265 6374 6f72 6965 s and directorie │ │ │ │ 00263510: 7320 7765 2063 7265 6174 6564 2e0a 2b2d s we created..+- │ │ │ │ @@ -156614,18 +156614,18 @@ │ │ │ │ 00263c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263c60: 2d2d 2d2b 0a7c 6931 203a 2072 756e 2022 ---+.|i1 : run " │ │ │ │ 00263c70: 756e 616d 6520 2d61 2220 2020 2020 2020 uname -a" │ │ │ │ 00263c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263cb0: 2020 207c 0a7c 4c69 6e75 7820 7362 7569 |.|Linux sbui │ │ │ │ -00263cc0: 6c64 2036 2e31 322e 3131 332b 6465 6231 ld 6.12.113+deb1 │ │ │ │ -00263cd0: 332d 616d 6436 3420 2331 2053 4d50 2050 3-amd64 #1 SMP P │ │ │ │ -00263ce0: 5245 454d 5054 5f44 594e 414d 4943 2044 REEMPT_DYNAMIC D │ │ │ │ -00263cf0: 6562 6961 6e20 362e 3132 2e31 3133 2d31 ebian 6.12.113-1 │ │ │ │ +00263cc0: 6c64 2036 2e31 322e 3130 302b 6465 6231 ld 6.12.100+deb1 │ │ │ │ +00263cd0: 332d 636c 6f75 642d 616d 6436 3420 2331 3-cloud-amd64 #1 │ │ │ │ +00263ce0: 2053 4d50 2050 5245 454d 5054 5f44 594e SMP PREEMPT_DYN │ │ │ │ +00263cf0: 414d 4943 2044 6562 6961 6e20 2020 2020 AMIC Debian │ │ │ │ 00263d00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00263d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263d50: 2020 207c 0a7c 6f31 203d 2030 2020 2020 |.|o1 = 0 │ │ │ │ 00263d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -156633,17 +156633,17 @@ │ │ │ │ 00263d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263da0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 00263db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00263df0: 2d2d 2d7c 0a7c 2832 3032 362d 3130 2d31 ---|.|(2026-10-1 │ │ │ │ -00263e00: 3529 2078 3836 5f36 3420 474e 552f 4c69 5) x86_64 GNU/Li │ │ │ │ -00263e10: 6e75 7820 2020 2020 2020 2020 2020 2020 nux │ │ │ │ +00263df0: 2d2d 2d7c 0a7c 362e 3132 2e31 3030 2d31 ---|.|6.12.100-1 │ │ │ │ +00263e00: 2028 3230 3236 2d30 372d 3330 2920 7838 (2026-07-30) x8 │ │ │ │ +00263e10: 365f 3634 2047 4e55 2f4c 696e 7578 2020 6_64 GNU/Linux │ │ │ │ 00263e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00263e40: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00263e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00263e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -156715,88126 +156715,88126 @@ │ │ │ │ 002642a0: 2073 7472 696e 6720 746f 2073 6565 0a77 string to see.w │ │ │ │ 002642b0: 6865 7468 6572 2069 7420 696e 636c 7564 hether it includ │ │ │ │ 002642c0: 6573 2061 206e 6577 6c69 6e65 2063 6861 es a newline cha │ │ │ │ 002642d0: 7261 6374 6572 2e0a 2b2d 2d2d 2d2d 2d2d racter..+------- │ │ │ │ 002642e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002642f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00264300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00264320: 7c69 3320 3a20 7065 656b 2067 6574 2022 |i3 : peek get " │ │ │ │ -00264330: 2175 6e61 6d65 202d 6122 2020 2020 2020 !uname -a" │ │ │ │ +00264310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264320: 2d2d 2d2d 2b0a 7c69 3320 3a20 7065 656b ----+.|i3 : peek │ │ │ │ +00264330: 2067 6574 2022 2175 6e61 6d65 202d 6122 get "!uname -a" │ │ │ │ 00264340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00264350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00264360: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00264370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00264360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00264370: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00264380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00264390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002643a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -002643b0: 7c6f 3320 3d20 224c 696e 7578 2073 6275 |o3 = "Linux sbu │ │ │ │ -002643c0: 696c 6420 362e 3132 2e31 3133 2b64 6562 ild 6.12.113+deb │ │ │ │ -002643d0: 3133 2d61 6d64 3634 2023 3120 534d 5020 13-amd64 #1 SMP │ │ │ │ -002643e0: 5052 4545 4d50 545f 4459 4e41 4d49 4320 PREEMPT_DYNAMIC │ │ │ │ -002643f0: 4465 6269 616e 7c0a 7c20 2020 2020 362e Debian|.| 6. │ │ │ │ -00264400: 3132 2e31 3133 2d31 2028 3230 3236 2d31 12.113-1 (2026-1 │ │ │ │ -00264410: 302d 3135 2920 7838 365f 3634 2047 4e55 0-15) x86_64 GNU │ │ │ │ -00264420: 2f4c 696e 7578 5c6e 2220 2020 2020 2020 /Linux\n" │ │ │ │ -00264430: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00264440: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00264450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002643a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002643b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002643c0: 7c0a 7c6f 3320 3d20 224c 696e 7578 2073 |.|o3 = "Linux s │ │ │ │ +002643d0: 6275 696c 6420 362e 3132 2e31 3030 2b64 build 6.12.100+d │ │ │ │ +002643e0: 6562 3133 2d63 6c6f 7564 2d61 6d64 3634 eb13-cloud-amd64 │ │ │ │ +002643f0: 2023 3120 534d 5020 5052 4545 4d50 545f #1 SMP PREEMPT_ │ │ │ │ +00264400: 4459 4e41 4d49 4320 4465 6269 616e 7c0a DYNAMIC Debian|. │ │ │ │ +00264410: 7c20 2020 2020 362e 3132 2e31 3030 2d31 | 6.12.100-1 │ │ │ │ +00264420: 2028 3230 3236 2d30 372d 3330 2920 7838 (2026-07-30) x8 │ │ │ │ +00264430: 365f 3634 2047 4e55 2f4c 696e 7578 5c6e 6_64 GNU/Linux\n │ │ │ │ +00264440: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ +00264450: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00264460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00264470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264480: 2d2d 2d2d 2d2d 2b0a 4269 6469 7265 6374 ------+.Bidirect │ │ │ │ -00264490: 696f 6e61 6c20 636f 6d6d 756e 6963 6174 ional communicat │ │ │ │ -002644a0: 696f 6e20 7769 7468 2061 2070 726f 6772 ion with a progr │ │ │ │ -002644b0: 616d 2069 7320 616c 736f 2070 6f73 7369 am is also possi │ │ │ │ -002644c0: 626c 652e 2020 5765 2075 7365 202a 6e6f ble. We use *no │ │ │ │ -002644d0: 7465 0a6f 7065 6e49 6e4f 7574 3a20 6f70 te.openInOut: op │ │ │ │ -002644e0: 656e 496e 4f75 742c 2074 6f20 6372 6561 enInOut, to crea │ │ │ │ -002644f0: 7465 2061 2066 696c 6520 7468 6174 2073 te a file that s │ │ │ │ -00264500: 6572 7665 7320 6173 2061 2062 6964 6972 erves as a bidir │ │ │ │ -00264510: 6563 7469 6f6e 616c 0a63 6f6e 6e65 6374 ectional.connect │ │ │ │ -00264520: 696f 6e20 746f 2061 2070 726f 6772 616d ion to a program │ │ │ │ -00264530: 2e20 2054 6861 7420 6669 6c65 2069 7320 . That file is │ │ │ │ -00264540: 6361 6c6c 6564 2061 6e20 696e 7075 7420 called an input │ │ │ │ -00264550: 6f75 7470 7574 2066 696c 652e 2020 496e output file. In │ │ │ │ -00264560: 2074 6869 730a 6578 616d 706c 6520 7765 this.example we │ │ │ │ -00264570: 206f 7065 6e20 6120 636f 6e6e 6563 7469 open a connecti │ │ │ │ -00264580: 6f6e 2074 6f20 7468 6520 756e 6978 2075 on to the unix u │ │ │ │ -00264590: 7469 6c69 7479 2067 7265 7020 616e 6420 tility grep and │ │ │ │ -002645a0: 7573 6520 6974 2074 6f20 6c6f 6361 7465 use it to locate │ │ │ │ -002645b0: 2074 6865 0a73 796d 626f 6c20 6e61 6d65 the.symbol name │ │ │ │ -002645c0: 7320 696e 204d 6163 6175 6c61 7932 2074 s in Macaulay2 t │ │ │ │ -002645d0: 6861 7420 6265 6769 6e20 7769 7468 2069 hat begin with i │ │ │ │ -002645e0: 6e2e 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d n..+------------ │ │ │ │ -002645f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00264620: 3420 3a20 6620 3d20 6f70 656e 496e 4f75 4 : f = openInOu │ │ │ │ -00264630: 7420 2221 6772 6570 202d 4520 275e 696e t "!grep -E '^in │ │ │ │ -00264640: 2722 2020 2020 2020 2020 2020 2020 2020 '" │ │ │ │ -00264650: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00264660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00264670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00264480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002644a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 4269 6469 ----------+.Bidi │ │ │ │ +002644b0: 7265 6374 696f 6e61 6c20 636f 6d6d 756e rectional commun │ │ │ │ +002644c0: 6963 6174 696f 6e20 7769 7468 2061 2070 ication with a p │ │ │ │ +002644d0: 726f 6772 616d 2069 7320 616c 736f 2070 rogram is also p │ │ │ │ +002644e0: 6f73 7369 626c 652e 2020 5765 2075 7365 ossible. We use │ │ │ │ +002644f0: 202a 6e6f 7465 0a6f 7065 6e49 6e4f 7574 *note.openInOut │ │ │ │ +00264500: 3a20 6f70 656e 496e 4f75 742c 2074 6f20 : openInOut, to │ │ │ │ +00264510: 6372 6561 7465 2061 2066 696c 6520 7468 create a file th │ │ │ │ +00264520: 6174 2073 6572 7665 7320 6173 2061 2062 at serves as a b │ │ │ │ +00264530: 6964 6972 6563 7469 6f6e 616c 0a63 6f6e idirectional.con │ │ │ │ +00264540: 6e65 6374 696f 6e20 746f 2061 2070 726f nection to a pro │ │ │ │ +00264550: 6772 616d 2e20 2054 6861 7420 6669 6c65 gram. That file │ │ │ │ +00264560: 2069 7320 6361 6c6c 6564 2061 6e20 696e is called an in │ │ │ │ +00264570: 7075 7420 6f75 7470 7574 2066 696c 652e put output file. │ │ │ │ +00264580: 2020 496e 2074 6869 730a 6578 616d 706c In this.exampl │ │ │ │ +00264590: 6520 7765 206f 7065 6e20 6120 636f 6e6e e we open a conn │ │ │ │ +002645a0: 6563 7469 6f6e 2074 6f20 7468 6520 756e ection to the un │ │ │ │ +002645b0: 6978 2075 7469 6c69 7479 2067 7265 7020 ix utility grep │ │ │ │ +002645c0: 616e 6420 7573 6520 6974 2074 6f20 6c6f and use it to lo │ │ │ │ +002645d0: 6361 7465 2074 6865 0a73 796d 626f 6c20 cate the.symbol │ │ │ │ +002645e0: 6e61 6d65 7320 696e 204d 6163 6175 6c61 names in Macaula │ │ │ │ +002645f0: 7932 2074 6861 7420 6265 6769 6e20 7769 y2 that begin wi │ │ │ │ +00264600: 7468 2069 6e2e 0a2b 2d2d 2d2d 2d2d 2d2d th in..+-------- │ │ │ │ +00264610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00264640: 2b0a 7c69 3420 3a20 6620 3d20 6f70 656e +.|i4 : f = open │ │ │ │ +00264650: 496e 4f75 7420 2221 6772 6570 202d 4520 InOut "!grep -E │ │ │ │ +00264660: 275e 696e 2722 2020 2020 2020 2020 2020 '^in'" │ │ │ │ +00264670: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 00264680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00264690: 2020 7c0a 7c6f 3420 3d20 2167 7265 7020 |.|o4 = !grep │ │ │ │ -002646a0: 2d45 2027 5e69 6e27 2020 2020 2020 2020 -E '^in' │ │ │ │ -002646b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002646c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00264690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002646a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002646b0: 2020 2020 2020 7c0a 7c6f 3420 3d20 2167 |.|o4 = !g │ │ │ │ +002646c0: 7265 7020 2d45 2027 5e69 6e27 2020 2020 rep -E '^in' │ │ │ │ 002646d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002646e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002646f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00264700: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ -00264710: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ -00264720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00264730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00264740: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00264750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00264770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00264780: 7c69 3520 3a20 7363 616e 286b 6579 7320 |i5 : scan(keys │ │ │ │ -00264790: 436f 7265 2e44 6963 7469 6f6e 6172 792c Core.Dictionary, │ │ │ │ -002647a0: 206b 6579 202d 3e20 6620 3c3c 206b 6579 key -> f << key │ │ │ │ -002647b0: 203c 3c20 656e 646c 297c 0a2b 2d2d 2d2d << endl)|.+---- │ │ │ │ -002647c0: 2d2d 2d2d 2d2d 2d2d 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6c77 6179 7320 6120 6461 e is always a da │ │ │ │ -00265170: 6e67 6572 206f 660a 626c 6f63 6b69 6e67 nger of.blocking │ │ │ │ -00265180: 2c20 6265 6361 7573 6520 7468 6520 6275 , because the bu │ │ │ │ -00265190: 6666 6572 7320 6173 736f 6369 6174 6564 ffers associated │ │ │ │ -002651a0: 2077 6974 6820 7468 6520 636f 6d6d 756e with the commun │ │ │ │ -002651b0: 6963 6174 696f 6e20 6368 616e 6e65 6c73 ication channels │ │ │ │ -002651c0: 0a28 7069 7065 7329 2074 7970 6963 616c .(pipes) typical │ │ │ │ -002651d0: 6c79 2068 6f6c 6420 6f6e 6c79 2034 3039 ly hold only 409 │ │ │ │ -002651e0: 3620 6279 7465 732e 2020 496e 2074 6869 6 bytes. 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In gene │ │ │ │ -00265250: 7261 6c2c 206f 6e65 0a73 686f 756c 6420 ral, one.should │ │ │ │ -00265260: 6265 2063 6172 6566 756c 2074 6f20 6172 be careful to ar │ │ │ │ -00265270: 7261 6e67 6520 7468 696e 6773 2073 6f20 range things so │ │ │ │ -00265280: 7468 6174 2074 6865 2074 776f 2070 726f that the two pro │ │ │ │ -00265290: 6772 616d 7320 7461 6b65 2074 7572 6e73 grams take turns │ │ │ │ -002652a0: 2075 7369 6e67 0a74 6865 2063 6f6d 6d75 using.the commu │ │ │ │ -002652b0: 6e69 6361 7469 6f6e 2063 6861 6e6e 656c nication channel │ │ │ │ -002652c0: 2c20 736f 2074 6861 7420 7768 656e 206f , so that when o │ │ │ │ -002652d0: 6e65 2069 7320 7772 6974 696e 6720 6461 ne is writing da │ │ │ │ -002652e0: 7461 2c20 7468 6520 6f74 6865 7220 6973 ta, the other is │ │ │ │ -002652f0: 0a72 6561 6469 6e67 2069 742e 0a0a 0a41 .reading it....A │ │ │ │ -00265300: 2075 7365 6675 6c20 6675 6e63 7469 6f6e useful function │ │ │ │ -00265310: 2069 6e20 7468 6973 2063 6f6e 6e65 6374 in this connect │ │ │ │ -00265320: 696f 6e20 6973 202a 6e6f 7465 2069 7352 ion is *note isR │ │ │ │ -00265330: 6561 6479 3a20 6973 5265 6164 795f 6c70 eady: isReady_lp │ │ │ │ -00265340: 4669 6c65 5f72 702c 2c0a 7768 6963 6820 File_rp,,.which │ │ │ │ -00265350: 7769 6c6c 2074 656c 6c20 796f 7520 7768 will tell you wh │ │ │ │ -00265360: 6574 6865 7220 616e 2069 6e70 7574 2066 ether an input f │ │ │ │ -00265370: 696c 6520 6861 7320 616e 7920 696e 7075 ile has any inpu │ │ │ │ -00265380: 7420 6176 6169 6c61 626c 6520 666f 7220 t available for │ │ │ │ -00265390: 7265 6164 696e 672c 0a6f 7220 7768 6574 reading,.or whet │ │ │ │ -002653a0: 6865 7220 6974 2068 6173 2061 7272 6976 her it has arriv │ │ │ │ -002653b0: 6564 2061 7420 7468 6520 656e 642e 2020 ed at the end. │ │ │ │ -002653c0: 5765 2069 6c6c 7573 7472 6174 6520 6974 We illustrate it │ │ │ │ -002653d0: 2069 6e20 7468 6520 666f 6c6c 6f77 696e in the followin │ │ │ │ -002653e0: 670a 6578 616d 706c 6520 6279 2073 696d g.example by sim │ │ │ │ -002653f0: 756c 6174 696e 6720 6120 636f 6d70 7574 ulating a comput │ │ │ │ -00265400: 6174 696f 6e20 7468 6174 2074 616b 6573 ation that takes │ │ │ │ -00265410: 2035 2073 6563 6f6e 6473 2074 6f20 636f 5 seconds to co │ │ │ │ -00265420: 6d70 6c65 7465 2c20 7072 696e 7469 6e67 mplete, printing │ │ │ │ -00265430: 0a6f 6e65 2064 6f74 2070 6572 2073 6563 .one dot per sec │ │ │ │ -00265440: 6f6e 6420 7768 696c 6520 7761 6974 696e ond while waitin │ │ │ │ -00265450: 672e 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d g..+------------ │ │ │ │ -00265460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ -00265490: 203a 2066 203d 206f 7065 6e49 6e20 2221 : f = openIn "! │ │ │ │ -002654a0: 736c 6565 7020 353b 2065 6368 6f20 2d6e sleep 5; echo -n │ │ │ │ -002654b0: 2074 6865 2061 6e73 7765 7220 6973 2034 the answer is 4 │ │ │ │ -002654c0: 2220 2020 207c 0a7c 2020 2020 2020 2020 " |.| │ │ │ │ -002654d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002654e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002654f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00265500: 0a7c 6f38 203d 2021 736c 6565 7020 353b .|o8 = !sleep 5; │ │ │ │ -00265510: 2065 6368 6f20 2d6e 2074 6865 2061 6e73 echo -n the ans │ │ │ │ -00265520: 7765 7220 6973 2034 2020 2020 2020 2020 wer is 4 │ │ │ │ -00265530: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00265540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265060: 2020 2020 7c0a 7c20 2020 2020 696e 7665 |.| inve │ │ │ │ +00265070: 7273 6520 2020 2020 2020 2020 2020 2020 rse │ │ │ │ +00265080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265090: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +002650a0: 0a7c 2020 2020 2069 6e73 7461 6c6c 4869 .| installHi │ │ │ │ +002650b0: 6c62 6572 7446 756e 6374 696f 6e20 2020 lbertFunction │ │ │ │ +002650c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002650d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +002650e0: 2020 696e 7374 616e 6365 7320 2020 2020 instances │ │ │ │ +002650f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265110: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00265120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265150: 2b0a 5769 7468 2074 6869 7320 666f 726d +.With this form │ │ │ │ +00265160: 206f 6620 6269 6469 7265 6374 696f 6e61 of bidirectiona │ │ │ │ +00265170: 6c20 636f 6d6d 756e 6963 6174 696f 6e20 l communication │ │ │ │ +00265180: 7468 6572 6520 6973 2061 6c77 6179 7320 there is always │ │ │ │ +00265190: 6120 6461 6e67 6572 206f 660a 626c 6f63 a danger of.bloc │ │ │ │ +002651a0: 6b69 6e67 2c20 6265 6361 7573 6520 7468 king, because th │ │ │ │ +002651b0: 6520 6275 6666 6572 7320 6173 736f 6369 e buffers associ │ │ │ │ +002651c0: 6174 6564 2077 6974 6820 7468 6520 636f ated with the co │ │ │ │ +002651d0: 6d6d 756e 6963 6174 696f 6e20 6368 616e mmunication chan │ │ │ │ +002651e0: 6e65 6c73 0a28 7069 7065 7329 2074 7970 nels.(pipes) typ │ │ │ │ +002651f0: 6963 616c 6c79 2068 6f6c 6420 6f6e 6c79 ically hold only │ │ │ │ +00265200: 2034 3039 3620 6279 7465 732e 2020 496e 4096 bytes. In │ │ │ │ +00265210: 2074 6869 7320 6578 616d 706c 6520 7765 this example we │ │ │ │ +00265220: 2073 7563 6365 6564 6564 2062 6563 6175 succeeded becau │ │ │ │ +00265230: 7365 0a74 6865 2065 6e74 6972 6520 6f75 se.the entire ou │ │ │ │ +00265240: 7470 7574 2066 726f 6d20 6772 6570 2077 tput from grep w │ │ │ │ +00265250: 6173 2073 6d61 6c6c 6572 2074 6861 6e20 as smaller than │ │ │ │ +00265260: 3430 3936 2062 7974 6573 2e20 2049 6e20 4096 bytes. In │ │ │ │ +00265270: 6765 6e65 7261 6c2c 206f 6e65 0a73 686f general, one.sho │ │ │ │ +00265280: 756c 6420 6265 2063 6172 6566 756c 2074 uld be careful t │ │ │ │ +00265290: 6f20 6172 7261 6e67 6520 7468 696e 6773 o arrange things │ │ │ │ +002652a0: 2073 6f20 7468 6174 2074 6865 2074 776f so that the two │ │ │ │ +002652b0: 2070 726f 6772 616d 7320 7461 6b65 2074 programs take t │ │ │ │ +002652c0: 7572 6e73 2075 7369 6e67 0a74 6865 2063 urns using.the c │ │ │ │ +002652d0: 6f6d 6d75 6e69 6361 7469 6f6e 2063 6861 ommunication cha │ │ │ │ +002652e0: 6e6e 656c 2c20 736f 2074 6861 7420 7768 nnel, so that wh │ │ │ │ +002652f0: 656e 206f 6e65 2069 7320 7772 6974 696e en one is writin │ │ │ │ +00265300: 6720 6461 7461 2c20 7468 6520 6f74 6865 g data, the othe │ │ │ │ +00265310: 7220 6973 0a72 6561 6469 6e67 2069 742e r is.reading it. │ │ │ │ +00265320: 0a0a 0a41 2075 7365 6675 6c20 6675 6e63 ...A useful func │ │ │ │ +00265330: 7469 6f6e 2069 6e20 7468 6973 2063 6f6e tion in this con │ │ │ │ +00265340: 6e65 6374 696f 6e20 6973 202a 6e6f 7465 nection is *note │ │ │ │ +00265350: 2069 7352 6561 6479 3a20 6973 5265 6164 isReady: isRead │ │ │ │ +00265360: 795f 6c70 4669 6c65 5f72 702c 2c0a 7768 y_lpFile_rp,,.wh │ │ │ │ +00265370: 6963 6820 7769 6c6c 2074 656c 6c20 796f ich will tell yo │ │ │ │ +00265380: 7520 7768 6574 6865 7220 616e 2069 6e70 u whether an inp │ │ │ │ +00265390: 7574 2066 696c 6520 6861 7320 616e 7920 ut file has any │ │ │ │ +002653a0: 696e 7075 7420 6176 6169 6c61 626c 6520 input available │ │ │ │ +002653b0: 666f 7220 7265 6164 696e 672c 0a6f 7220 for reading,.or │ │ │ │ +002653c0: 7768 6574 6865 7220 6974 2068 6173 2061 whether it has a │ │ │ │ +002653d0: 7272 6976 6564 2061 7420 7468 6520 656e rrived at the en │ │ │ │ +002653e0: 642e 2020 5765 2069 6c6c 7573 7472 6174 d. We illustrat │ │ │ │ +002653f0: 6520 6974 2069 6e20 7468 6520 666f 6c6c e it in the foll │ │ │ │ +00265400: 6f77 696e 670a 6578 616d 706c 6520 6279 owing.example by │ │ │ │ +00265410: 2073 696d 756c 6174 696e 6720 6120 636f simulating a co │ │ │ │ +00265420: 6d70 7574 6174 696f 6e20 7468 6174 2074 mputation that t │ │ │ │ +00265430: 616b 6573 2035 2073 6563 6f6e 6473 2074 akes 5 seconds t │ │ │ │ +00265440: 6f20 636f 6d70 6c65 7465 2c20 7072 696e o complete, prin │ │ │ │ +00265450: 7469 6e67 0a6f 6e65 2064 6f74 2070 6572 ting.one dot per │ │ │ │ +00265460: 2073 6563 6f6e 6420 7768 696c 6520 7761 second while wa │ │ │ │ +00265470: 6974 696e 672e 0a2b 2d2d 2d2d 2d2d 2d2d iting..+-------- │ │ │ │ +00265480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002654a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +002654b0: 0a7c 6938 203a 2066 203d 206f 7065 6e49 .|i8 : f = openI │ │ │ │ +002654c0: 6e20 2221 736c 6565 7020 353b 2065 6368 n "!sleep 5; ech │ │ │ │ +002654d0: 6f20 2d6e 2074 6865 2061 6e73 7765 7220 o -n the answer │ │ │ │ +002654e0: 6973 2034 2220 2020 207c 0a7c 2020 2020 is 4" |.| │ │ │ │ +002654f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265520: 2020 207c 0a7c 6f38 203d 2021 736c 6565 |.|o8 = !slee │ │ │ │ +00265530: 7020 353b 2065 6368 6f20 2d6e 2074 6865 p 5; echo -n the │ │ │ │ +00265540: 2061 6e73 7765 7220 6973 2034 2020 2020 answer is 4 │ │ │ │ +00265550: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00265560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265570: 2020 207c 0a7c 6f38 203a 2046 696c 6520 |.|o8 : File │ │ │ │ +00265570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002655a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -002655b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002655c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002655d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002655e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2069 -------+.|i9 : i │ │ │ │ -002655f0: 7352 6561 6479 2066 2020 2020 2020 2020 sReady f │ │ │ │ -00265600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265620: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00265590: 2020 2020 2020 207c 0a7c 6f38 203a 2046 |.|o8 : F │ │ │ │ +002655a0: 696c 6520 2020 2020 2020 2020 2020 2020 ile │ │ │ │ +002655b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002655c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002655d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +002655e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002655f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 -----------+.|i9 │ │ │ │ +00265610: 203a 2069 7352 6561 6479 2066 2020 2020 : isReady f │ │ │ │ +00265620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265650: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ -00265660: 203d 2066 616c 7365 2020 2020 2020 2020 = false │ │ │ │ -00265670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265690: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -002656a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002656b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002656c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -002656d0: 0a7c 6931 3020 3a20 7768 696c 6520 6e6f .|i10 : while no │ │ │ │ -002656e0: 7420 6973 5265 6164 7920 6620 646f 2028 t isReady f do ( │ │ │ │ -002656f0: 736c 6565 7020 313b 203c 3c20 222e 2220 sleep 1; << "." │ │ │ │ -00265700: 3c3c 2066 6c75 7368 297c 0a7c 2e2e 2e2e << flush)|.|.... │ │ │ │ -00265710: 2e2e 2020 2020 2020 2020 2020 2020 2020 .. │ │ │ │ -00265720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265740: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00265750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00265780: 6931 3120 3a20 7265 6164 2066 2020 2020 i11 : read f │ │ │ │ -00265790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002657a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002657b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00265640: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00265650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00265680: 0a7c 6f39 203d 2066 616c 7365 2020 2020 .|o9 = false │ │ │ │ +00265690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002656a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002656b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +002656c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002656d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002656e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002656f0: 2d2d 2d2b 0a7c 6931 3020 3a20 7768 696c ---+.|i10 : whil │ │ │ │ +00265700: 6520 6e6f 7420 6973 5265 6164 7920 6620 e not isReady f │ │ │ │ +00265710: 646f 2028 736c 6565 7020 313b 203c 3c20 do (sleep 1; << │ │ │ │ +00265720: 222e 2220 3c3c 2066 6c75 7368 297c 0a7c "." << flush)|.| │ │ │ │ +00265730: 2e2e 2e2e 2e20 2020 2020 2020 2020 2020 ..... │ │ │ │ +00265740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265760: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00265770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002657a0: 2d2b 0a7c 6931 3120 3a20 7265 6164 2066 -+.|i11 : read f │ │ │ │ +002657b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002657c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002657d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002657d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 002657e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002657f0: 207c 0a7c 6f31 3120 3d20 7468 6520 616e |.|o11 = the an │ │ │ │ -00265800: 7377 6572 2069 7320 3420 2020 2020 2020 swer is 4 │ │ │ │ -00265810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265820: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00265830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265860: 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 6973 -----+.|i12 : is │ │ │ │ -00265870: 5265 6164 7920 6620 2020 2020 2020 2020 Ready f │ │ │ │ -00265880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265890: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -002658a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +002657f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265810: 2020 2020 207c 0a7c 6f31 3120 3d20 7468 |.|o11 = th │ │ │ │ +00265820: 6520 616e 7377 6572 2069 7320 3420 2020 e answer is 4 │ │ │ │ +00265830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265840: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00265850: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00265860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265880: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ +00265890: 3a20 6973 5265 6164 7920 6620 2020 2020 : isReady f │ │ │ │ +002658a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002658b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002658c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002658d0: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -002658e0: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ -002658f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265910: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00265920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00265950: 6931 3320 3a20 6174 456e 644f 6646 696c i13 : atEndOfFil │ │ │ │ -00265960: 6520 6620 2020 2020 2020 2020 2020 2020 e f │ │ │ │ -00265970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265980: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +002658c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +002658d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002658e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002658f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00265900: 6f31 3220 3d20 7472 7565 2020 2020 2020 o12 = true │ │ │ │ +00265910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265930: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00265940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265970: 2d2b 0a7c 6931 3320 3a20 6174 456e 644f -+.|i13 : atEndO │ │ │ │ +00265980: 6646 696c 6520 6620 2020 2020 2020 2020 fFile f │ │ │ │ 00265990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002659a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002659a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 002659b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002659c0: 207c 0a7c 6f31 3320 3d20 7472 7565 2020 |.|o13 = true │ │ │ │ +002659c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002659d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002659e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002659f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00265a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265a30: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 636c -----+.|i14 : cl │ │ │ │ -00265a40: 6f73 6520 6620 2020 2020 2020 2020 2020 ose f │ │ │ │ -00265a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265a60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00265a70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +002659e0: 2020 2020 207c 0a7c 6f31 3320 3d20 7472 |.|o13 = tr │ │ │ │ +002659f0: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ +00265a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265a10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00265a20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00265a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265a50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 ---------+.|i14 │ │ │ │ +00265a60: 3a20 636c 6f73 6520 6620 2020 2020 2020 : close f │ │ │ │ +00265a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265aa0: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ -00265ab0: 3d20 2173 6c65 6570 2035 3b20 6563 686f = !sleep 5; echo │ │ │ │ -00265ac0: 202d 6e20 7468 6520 616e 7377 6572 2069 -n the answer i │ │ │ │ -00265ad0: 7320 3420 2020 2020 2020 2020 2020 2020 s 4 │ │ │ │ -00265ae0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00265af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265b10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00265b20: 6f31 3420 3a20 4669 6c65 2020 2020 2020 o14 : File │ │ │ │ +00265a90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00265aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00265ad0: 6f31 3420 3d20 2173 6c65 6570 2035 3b20 o14 = !sleep 5; │ │ │ │ +00265ae0: 6563 686f 202d 6e20 7468 6520 616e 7377 echo -n the answ │ │ │ │ +00265af0: 6572 2069 7320 3420 2020 2020 2020 2020 er is 4 │ │ │ │ +00265b00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00265b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00265b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00265b50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00265b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265b40: 207c 0a7c 6f31 3420 3a20 4669 6c65 2020 |.|o14 : File │ │ │ │ +00265b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265b70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00265b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265b90: 2d2b 0a57 6520 616c 736f 2061 6c6c 6f77 -+.We also allow │ │ │ │ -00265ba0: 2066 6f72 2062 6964 6972 6563 7469 6f6e for bidirection │ │ │ │ -00265bb0: 616c 2063 6f6d 6d75 6e69 6361 7469 6f6e al communication │ │ │ │ -00265bc0: 2074 6872 6f75 6768 2073 6f63 6b65 7473 through sockets │ │ │ │ -00265bd0: 206f 7665 7220 7468 650a 696e 7465 726e over the.intern │ │ │ │ -00265be0: 6574 2e20 2053 6565 202a 6e6f 7465 206f et. See *note o │ │ │ │ -00265bf0: 7065 6e49 6e4f 7574 3a20 6f70 656e 496e penInOut: openIn │ │ │ │ -00265c00: 4f75 742c 2061 6e64 202a 6e6f 7465 206f Out, and *note o │ │ │ │ -00265c10: 7065 6e4c 6973 7465 6e65 723a 0a6f 7065 penListener:.ope │ │ │ │ -00265c20: 6e4c 6973 7465 6e65 725f 6c70 5374 7269 nListener_lpStri │ │ │ │ -00265c30: 6e67 5f72 702c 2c20 6f72 2074 6865 206e ng_rp,, or the n │ │ │ │ -00265c40: 6578 7420 7365 6374 696f 6e2e 0a0a 0a41 ext section....A │ │ │ │ -00265c50: 6e6f 7468 6572 2075 7365 6675 6c20 6675 nother useful fu │ │ │ │ -00265c60: 6e63 7469 6f6e 2069 7320 2a6e 6f74 6520 nction is *note │ │ │ │ -00265c70: 7761 6974 3a20 7761 6974 2c2c 2077 6869 wait: wait,, whi │ │ │ │ -00265c80: 6368 2063 616e 2062 6520 7573 6564 2074 ch can be used t │ │ │ │ -00265c90: 6f20 7761 6974 2066 6f72 0a69 6e70 7574 o wait for.input │ │ │ │ -00265ca0: 2074 6f20 6265 2061 7661 696c 6162 6c65 to be available │ │ │ │ -00265cb0: 2066 726f 6d20 616e 7920 6f66 2061 206c from any of a l │ │ │ │ -00265cc0: 6973 7420 6f66 2069 6e70 7574 2066 696c ist of input fil │ │ │ │ -00265cd0: 6573 2e0a 2a20 4d65 6e75 3a0a 0a2a 2072 es..* Menu:..* r │ │ │ │ -00265ce0: 756e 3a3a 2020 2020 2020 2020 2020 2020 un:: │ │ │ │ -00265cf0: 2020 2020 2020 2020 2020 2020 2072 756e run │ │ │ │ -00265d00: 2061 6e20 6578 7465 726e 616c 2063 6f6d an external com │ │ │ │ -00265d10: 6d61 6e64 0a2a 2066 696e 6450 726f 6772 mand.* findProgr │ │ │ │ -00265d20: 616d 3a3a 2020 2020 2020 2020 2020 2020 am:: │ │ │ │ -00265d30: 2020 2020 206c 6f61 6420 6578 7465 726e load extern │ │ │ │ -00265d40: 616c 2070 726f 6772 616d 0a2a 2072 756e al program.* run │ │ │ │ -00265d50: 5072 6f67 7261 6d3a 3a20 2020 2020 2020 Program:: │ │ │ │ -00265d60: 2020 2020 2020 2020 2020 2072 756e 2061 run a │ │ │ │ -00265d70: 6e20 6578 7465 726e 616c 2070 726f 6772 n external progr │ │ │ │ -00265d80: 616d 0a2a 2050 726f 6772 616d 203c 3c20 am.* Program << │ │ │ │ -00265d90: 5468 696e 673a 3a20 2020 2020 2020 2020 Thing:: │ │ │ │ -00265da0: 2020 2072 756e 2070 726f 6772 616d 2077 run program w │ │ │ │ -00265db0: 6974 6820 696e 7075 7420 7265 6469 7265 ith input redire │ │ │ │ -00265dc0: 6374 696f 6e0a 2d2d 2d2d 2d2d 2d2d 2d2d ction.---------- │ │ │ │ -00265dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265bb0: 2d2d 2d2d 2d2b 0a57 6520 616c 736f 2061 -----+.We also a │ │ │ │ +00265bc0: 6c6c 6f77 2066 6f72 2062 6964 6972 6563 llow for bidirec │ │ │ │ +00265bd0: 7469 6f6e 616c 2063 6f6d 6d75 6e69 6361 tional communica │ │ │ │ +00265be0: 7469 6f6e 2074 6872 6f75 6768 2073 6f63 tion through soc │ │ │ │ +00265bf0: 6b65 7473 206f 7665 7220 7468 650a 696e kets over the.in │ │ │ │ +00265c00: 7465 726e 6574 2e20 2053 6565 202a 6e6f ternet. See *no │ │ │ │ +00265c10: 7465 206f 7065 6e49 6e4f 7574 3a20 6f70 te openInOut: op │ │ │ │ +00265c20: 656e 496e 4f75 742c 2061 6e64 202a 6e6f enInOut, and *no │ │ │ │ +00265c30: 7465 206f 7065 6e4c 6973 7465 6e65 723a te openListener: │ │ │ │ +00265c40: 0a6f 7065 6e4c 6973 7465 6e65 725f 6c70 .openListener_lp │ │ │ │ +00265c50: 5374 7269 6e67 5f72 702c 2c20 6f72 2074 String_rp,, or t │ │ │ │ +00265c60: 6865 206e 6578 7420 7365 6374 696f 6e2e he next section. │ │ │ │ +00265c70: 0a0a 0a41 6e6f 7468 6572 2075 7365 6675 ...Another usefu │ │ │ │ +00265c80: 6c20 6675 6e63 7469 6f6e 2069 7320 2a6e l function is *n │ │ │ │ +00265c90: 6f74 6520 7761 6974 3a20 7761 6974 2c2c ote wait: wait,, │ │ │ │ +00265ca0: 2077 6869 6368 2063 616e 2062 6520 7573 which can be us │ │ │ │ +00265cb0: 6564 2074 6f20 7761 6974 2066 6f72 0a69 ed to wait for.i │ │ │ │ +00265cc0: 6e70 7574 2074 6f20 6265 2061 7661 696c nput to be avail │ │ │ │ +00265cd0: 6162 6c65 2066 726f 6d20 616e 7920 6f66 able from any of │ │ │ │ +00265ce0: 2061 206c 6973 7420 6f66 2069 6e70 7574 a list of input │ │ │ │ +00265cf0: 2066 696c 6573 2e0a 2a20 4d65 6e75 3a0a files..* Menu:. │ │ │ │ +00265d00: 0a2a 2072 756e 3a3a 2020 2020 2020 2020 .* run:: │ │ │ │ +00265d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00265d20: 2072 756e 2061 6e20 6578 7465 726e 616c run an external │ │ │ │ +00265d30: 2063 6f6d 6d61 6e64 0a2a 2066 696e 6450 command.* findP │ │ │ │ +00265d40: 726f 6772 616d 3a3a 2020 2020 2020 2020 rogram:: │ │ │ │ +00265d50: 2020 2020 2020 2020 206c 6f61 6420 6578 load ex │ │ │ │ +00265d60: 7465 726e 616c 2070 726f 6772 616d 0a2a ternal program.* │ │ │ │ +00265d70: 2072 756e 5072 6f67 7261 6d3a 3a20 2020 runProgram:: │ │ │ │ +00265d80: 2020 2020 2020 2020 2020 2020 2020 2072 r │ │ │ │ +00265d90: 756e 2061 6e20 6578 7465 726e 616c 2070 un an external p │ │ │ │ +00265da0: 726f 6772 616d 0a2a 2050 726f 6772 616d rogram.* Program │ │ │ │ +00265db0: 203c 3c20 5468 696e 673a 3a20 2020 2020 << Thing:: │ │ │ │ +00265dc0: 2020 2020 2020 2072 756e 2070 726f 6772 run progr │ │ │ │ +00265dd0: 616d 2077 6974 6820 696e 7075 7420 7265 am with input re │ │ │ │ +00265de0: 6469 7265 6374 696f 6e0a 2d2d 2d2d 2d2d direction.------ │ │ │ │ 00265df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00265e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00265e10: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -00265e20: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -00265e30: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -00265e40: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -00265e50: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -00265e60: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -00265e70: 6179 322f 7061 636b 6167 6573 2f0a 4d61 ay2/packages/.Ma │ │ │ │ -00265e80: 6361 756c 6179 3244 6f63 2f6f 765f 6669 caulay2Doc/ov_fi │ │ │ │ -00265e90: 6c65 732e 6d32 3a34 3435 3a30 2e0a 1f0a les.m2:445:0.... │ │ │ │ -00265ea0: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -00265eb0: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2072 oc.info, Node: r │ │ │ │ -00265ec0: 756e 2c20 4e65 7874 3a20 6669 6e64 5072 un, Next: findPr │ │ │ │ -00265ed0: 6f67 7261 6d2c 2055 703a 2063 6f6d 6d75 ogram, Up: commu │ │ │ │ -00265ee0: 6e69 6361 7469 6e67 2077 6974 6820 7072 nicating with pr │ │ │ │ -00265ef0: 6f67 7261 6d73 0a0a 7275 6e20 2d2d 2072 ograms..run -- r │ │ │ │ -00265f00: 756e 2061 6e20 6578 7465 726e 616c 2063 un an external c │ │ │ │ -00265f10: 6f6d 6d61 6e64 0a2a 2a2a 2a2a 2a2a 2a2a ommand.********* │ │ │ │ -00265f20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00265f30: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -00265f40: 3a20 0a20 2020 2020 2020 2072 756e 2073 : . run s │ │ │ │ -00265f50: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -00265f60: 2020 202a 2073 2c20 6120 2a6e 6f74 6520 * s, a *note │ │ │ │ -00265f70: 7374 7269 6e67 3a20 5374 7269 6e67 2c2c string: String,, │ │ │ │ -00265f80: 2061 2063 6f6d 6d61 6e64 2075 6e64 6572 a command under │ │ │ │ -00265f90: 7374 616e 6461 626c 6520 6279 2074 6865 standable by the │ │ │ │ -00265fa0: 206f 7065 7261 7469 6e67 0a20 2020 2020 operating. │ │ │ │ -00265fb0: 2020 2073 7973 7465 6d0a 2020 2a20 4f75 system. * Ou │ │ │ │ -00265fc0: 7470 7574 733a 0a20 2020 2020 202a 2061 tputs:. * a │ │ │ │ -00265fd0: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ -00265fe0: 205a 5a2c 2c20 7468 6520 7265 7475 726e ZZ,, the return │ │ │ │ -00265ff0: 2063 6f64 650a 0a44 6573 6372 6970 7469 code..Descripti │ │ │ │ -00266000: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00266010: 5468 6520 7072 6f63 6573 7320 6973 2072 The process is r │ │ │ │ -00266020: 756e 2069 6e20 7468 6520 7361 6d65 2070 un in the same p │ │ │ │ -00266030: 726f 6365 7373 2067 726f 7570 2c20 736f rocess group, so │ │ │ │ -00266040: 2073 6967 6e61 6c73 2069 6e76 6f6b 6564 signals invoked │ │ │ │ -00266050: 2062 7920 636f 6e74 726f 6c0a 6368 6172 by control.char │ │ │ │ -00266060: 6163 7465 7273 2061 7420 7468 6520 7465 acters at the te │ │ │ │ -00266070: 726d 696e 616c 2077 696c 6c20 676f 2062 rminal will go b │ │ │ │ -00266080: 6f74 6820 746f 2069 7420 616e 6420 746f oth to it and to │ │ │ │ -00266090: 204d 6163 6175 6c61 7932 2e0a 0a55 6e64 Macaulay2...Und │ │ │ │ -002660a0: 6572 204c 696e 7578 2061 6e64 204d 6163 er Linux and Mac │ │ │ │ -002660b0: 204f 532c 2074 6865 2072 6574 7572 6e20 OS, the return │ │ │ │ -002660c0: 636f 6465 2069 7320 3235 3620 7469 6d65 code is 256 time │ │ │ │ -002660d0: 7320 7468 6520 6578 6974 2073 7461 7475 s the exit statu │ │ │ │ -002660e0: 7320 636f 6465 206f 660a 7468 6520 636f s code of.the co │ │ │ │ -002660f0: 6d6d 616e 642c 2069 6620 7468 6520 636f mmand, if the co │ │ │ │ -00266100: 6d6d 616e 6420 6578 6974 6564 206e 6f72 mmand exited nor │ │ │ │ -00266110: 6d61 6c6c 793b 2062 7920 636f 6e76 656e mally; by conven │ │ │ │ -00266120: 7469 6f6e 2c20 616e 2065 7869 7420 7374 tion, an exit st │ │ │ │ -00266130: 6174 7573 2063 6f64 650a 6f66 2030 2069 atus code.of 0 i │ │ │ │ -00266140: 6e64 6963 6174 6573 2065 7272 6f72 2066 ndicates error f │ │ │ │ -00266150: 7265 6520 6578 6563 7574 696f 6e20 6f72 ree execution or │ │ │ │ -00266160: 2074 6865 2042 6f6f 6c65 616e 2076 616c the Boolean val │ │ │ │ -00266170: 7565 2074 7275 652c 2061 6e20 6578 6974 ue true, an exit │ │ │ │ -00266180: 2073 7461 7475 730a 636f 6465 206f 6620 status.code of │ │ │ │ -00266190: 3120 696e 6469 6361 7465 7320 616e 2065 1 indicates an e │ │ │ │ -002661a0: 7272 6f72 206f 7220 7468 6520 426f 6f6c rror or the Bool │ │ │ │ -002661b0: 6561 6e20 7661 6c75 6520 6661 6c73 653b ean value false; │ │ │ │ -002661c0: 2069 6620 7468 6520 636f 6d6d 616e 640a if the command. │ │ │ │ -002661d0: 7465 726d 696e 6174 6564 2069 6e20 7265 terminated in re │ │ │ │ -002661e0: 7370 6f6e 7365 2074 6f20 6120 7369 676e sponse to a sign │ │ │ │ -002661f0: 616c 206f 7220 6661 756c 742c 2074 6865 al or fault, the │ │ │ │ -00266200: 2073 6967 6e61 6c20 6e75 6d62 6572 2028 signal number ( │ │ │ │ -00266210: 696e 2074 6865 2072 616e 6765 0a31 2d31 in the range.1-1 │ │ │ │ -00266220: 3236 2920 6973 2072 6574 7572 6e65 642c 26) is returned, │ │ │ │ -00266230: 2061 6464 6564 2074 6f20 3132 3820 6966 added to 128 if │ │ │ │ -00266240: 2061 2063 6f72 6520 6475 6d70 2077 6173 a core dump was │ │ │ │ -00266250: 2063 7265 6174 6564 3b20 6966 2074 6865 created; if the │ │ │ │ -00266260: 2073 6865 6c6c 0a28 636f 6d6d 616e 6420 shell.(command │ │ │ │ -00266270: 696e 7465 7270 7265 7465 7229 2063 6f75 interpreter) cou │ │ │ │ -00266280: 6c64 206e 6f74 2062 6520 6578 6563 7574 ld not be execut │ │ │ │ -00266290: 6564 2c20 7468 656e 2031 3237 2069 7320 ed, then 127 is │ │ │ │ -002662a0: 7265 7475 726e 6564 2e20 2053 6967 6e61 returned. Signa │ │ │ │ -002662b0: 6c0a 6e75 6d62 6572 7320 7479 7069 6361 l.numbers typica │ │ │ │ -002662c0: 6c6c 7920 696e 636c 7564 6520 3220 666f lly include 2 fo │ │ │ │ -002662d0: 7220 696e 7465 7272 7570 742c 2033 2066 r interrupt, 3 f │ │ │ │ -002662e0: 6f72 2071 7569 742c 2036 2066 6f72 2061 or quit, 6 for a │ │ │ │ -002662f0: 626f 7274 2c20 3920 666f 7220 6b69 6c6c bort, 9 for kill │ │ │ │ -00266300: 2c0a 3131 2066 6f72 2073 6567 6d65 6e74 ,.11 for segment │ │ │ │ -00266310: 6174 696f 6e20 6661 756c 742c 2061 6e64 ation fault, and │ │ │ │ -00266320: 2031 3520 666f 7220 7465 726d 696e 6174 15 for terminat │ │ │ │ -00266330: 696f 6e2e 2020 466f 7220 6465 7461 696c ion. For detail │ │ │ │ -00266340: 732c 2073 6565 2074 6865 206d 616e 0a70 s, see the man.p │ │ │ │ -00266350: 6167 6520 6f66 2074 6865 206c 6962 6320 age of the libc │ │ │ │ -00266360: 726f 7574 696e 6520 7379 7374 656d 2829 routine system() │ │ │ │ -00266370: 2e0a 0a46 6f72 2074 6865 2070 726f 6772 ...For the progr │ │ │ │ -00266380: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -00266390: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -002663a0: 6a65 6374 202a 6e6f 7465 2072 756e 3a20 ject *note run: │ │ │ │ -002663b0: 7275 6e2c 2069 7320 6120 2a6e 6f74 6520 run, is a *note │ │ │ │ -002663c0: 636f 6d70 696c 6564 2066 756e 6374 696f compiled functio │ │ │ │ -002663d0: 6e3a 2043 6f6d 7069 6c65 6446 756e 6374 n: CompiledFunct │ │ │ │ -002663e0: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ -002663f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00265e30: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00265e40: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00265e50: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00265e60: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00265e70: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00265e80: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ +00265e90: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00265ea0: 2f0a 4d61 6361 756c 6179 3244 6f63 2f6f /.Macaulay2Doc/o │ │ │ │ +00265eb0: 765f 6669 6c65 732e 6d32 3a34 3435 3a30 v_files.m2:445:0 │ │ │ │ +00265ec0: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ +00265ed0: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ +00265ee0: 653a 2072 756e 2c20 4e65 7874 3a20 6669 e: run, Next: fi │ │ │ │ +00265ef0: 6e64 5072 6f67 7261 6d2c 2055 703a 2063 ndProgram, Up: c │ │ │ │ +00265f00: 6f6d 6d75 6e69 6361 7469 6e67 2077 6974 ommunicating wit │ │ │ │ +00265f10: 6820 7072 6f67 7261 6d73 0a0a 7275 6e20 h programs..run │ │ │ │ +00265f20: 2d2d 2072 756e 2061 6e20 6578 7465 726e -- run an extern │ │ │ │ +00265f30: 616c 2063 6f6d 6d61 6e64 0a2a 2a2a 2a2a al command.***** │ │ │ │ +00265f40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00265f50: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00265f60: 7361 6765 3a20 0a20 2020 2020 2020 2072 sage: . r │ │ │ │ +00265f70: 756e 2073 0a20 202a 2049 6e70 7574 733a un s. * Inputs: │ │ │ │ +00265f80: 0a20 2020 2020 202a 2073 2c20 6120 2a6e . * s, a *n │ │ │ │ +00265f90: 6f74 6520 7374 7269 6e67 3a20 5374 7269 ote string: Stri │ │ │ │ +00265fa0: 6e67 2c2c 2061 2063 6f6d 6d61 6e64 2075 ng,, a command u │ │ │ │ +00265fb0: 6e64 6572 7374 616e 6461 626c 6520 6279 nderstandable by │ │ │ │ +00265fc0: 2074 6865 206f 7065 7261 7469 6e67 0a20 the operating. │ │ │ │ +00265fd0: 2020 2020 2020 2073 7973 7465 6d0a 2020 system. │ │ │ │ +00265fe0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00265ff0: 202a 2061 6e20 2a6e 6f74 6520 696e 7465 * an *note inte │ │ │ │ +00266000: 6765 723a 205a 5a2c 2c20 7468 6520 7265 ger: ZZ,, the re │ │ │ │ +00266010: 7475 726e 2063 6f64 650a 0a44 6573 6372 turn code..Descr │ │ │ │ +00266020: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00266030: 3d3d 0a0a 5468 6520 7072 6f63 6573 7320 ==..The process │ │ │ │ +00266040: 6973 2072 756e 2069 6e20 7468 6520 7361 is run in the sa │ │ │ │ +00266050: 6d65 2070 726f 6365 7373 2067 726f 7570 me process group │ │ │ │ +00266060: 2c20 736f 2073 6967 6e61 6c73 2069 6e76 , so signals inv │ │ │ │ +00266070: 6f6b 6564 2062 7920 636f 6e74 726f 6c0a oked by control. │ │ │ │ +00266080: 6368 6172 6163 7465 7273 2061 7420 7468 characters at th │ │ │ │ +00266090: 6520 7465 726d 696e 616c 2077 696c 6c20 e terminal will │ │ │ │ +002660a0: 676f 2062 6f74 6820 746f 2069 7420 616e go both to it an │ │ │ │ +002660b0: 6420 746f 204d 6163 6175 6c61 7932 2e0a d to Macaulay2.. │ │ │ │ +002660c0: 0a55 6e64 6572 204c 696e 7578 2061 6e64 .Under Linux and │ │ │ │ +002660d0: 204d 6163 204f 532c 2074 6865 2072 6574 Mac OS, the ret │ │ │ │ +002660e0: 7572 6e20 636f 6465 2069 7320 3235 3620 urn code is 256 │ │ │ │ +002660f0: 7469 6d65 7320 7468 6520 6578 6974 2073 times the exit s │ │ │ │ +00266100: 7461 7475 7320 636f 6465 206f 660a 7468 tatus code of.th │ │ │ │ +00266110: 6520 636f 6d6d 616e 642c 2069 6620 7468 e command, if th │ │ │ │ +00266120: 6520 636f 6d6d 616e 6420 6578 6974 6564 e command exited │ │ │ │ +00266130: 206e 6f72 6d61 6c6c 793b 2062 7920 636f normally; by co │ │ │ │ +00266140: 6e76 656e 7469 6f6e 2c20 616e 2065 7869 nvention, an exi │ │ │ │ +00266150: 7420 7374 6174 7573 2063 6f64 650a 6f66 t status code.of │ │ │ │ +00266160: 2030 2069 6e64 6963 6174 6573 2065 7272 0 indicates err │ │ │ │ +00266170: 6f72 2066 7265 6520 6578 6563 7574 696f or free executio │ │ │ │ +00266180: 6e20 6f72 2074 6865 2042 6f6f 6c65 616e n or the Boolean │ │ │ │ +00266190: 2076 616c 7565 2074 7275 652c 2061 6e20 value true, an │ │ │ │ +002661a0: 6578 6974 2073 7461 7475 730a 636f 6465 exit status.code │ │ │ │ +002661b0: 206f 6620 3120 696e 6469 6361 7465 7320 of 1 indicates │ │ │ │ +002661c0: 616e 2065 7272 6f72 206f 7220 7468 6520 an error or the │ │ │ │ +002661d0: 426f 6f6c 6561 6e20 7661 6c75 6520 6661 Boolean value fa │ │ │ │ +002661e0: 6c73 653b 2069 6620 7468 6520 636f 6d6d lse; if the comm │ │ │ │ +002661f0: 616e 640a 7465 726d 696e 6174 6564 2069 and.terminated i │ │ │ │ +00266200: 6e20 7265 7370 6f6e 7365 2074 6f20 6120 n response to a │ │ │ │ +00266210: 7369 676e 616c 206f 7220 6661 756c 742c signal or fault, │ │ │ │ +00266220: 2074 6865 2073 6967 6e61 6c20 6e75 6d62 the signal numb │ │ │ │ +00266230: 6572 2028 696e 2074 6865 2072 616e 6765 er (in the range │ │ │ │ +00266240: 0a31 2d31 3236 2920 6973 2072 6574 7572 .1-126) is retur │ │ │ │ +00266250: 6e65 642c 2061 6464 6564 2074 6f20 3132 ned, added to 12 │ │ │ │ +00266260: 3820 6966 2061 2063 6f72 6520 6475 6d70 8 if a core dump │ │ │ │ +00266270: 2077 6173 2063 7265 6174 6564 3b20 6966 was created; if │ │ │ │ +00266280: 2074 6865 2073 6865 6c6c 0a28 636f 6d6d the shell.(comm │ │ │ │ +00266290: 616e 6420 696e 7465 7270 7265 7465 7229 and interpreter) │ │ │ │ +002662a0: 2063 6f75 6c64 206e 6f74 2062 6520 6578 could not be ex │ │ │ │ +002662b0: 6563 7574 6564 2c20 7468 656e 2031 3237 ecuted, then 127 │ │ │ │ +002662c0: 2069 7320 7265 7475 726e 6564 2e20 2053 is returned. S │ │ │ │ +002662d0: 6967 6e61 6c0a 6e75 6d62 6572 7320 7479 ignal.numbers ty │ │ │ │ +002662e0: 7069 6361 6c6c 7920 696e 636c 7564 6520 pically include │ │ │ │ +002662f0: 3220 666f 7220 696e 7465 7272 7570 742c 2 for interrupt, │ │ │ │ +00266300: 2033 2066 6f72 2071 7569 742c 2036 2066 3 for quit, 6 f │ │ │ │ +00266310: 6f72 2061 626f 7274 2c20 3920 666f 7220 or abort, 9 for │ │ │ │ +00266320: 6b69 6c6c 2c0a 3131 2066 6f72 2073 6567 kill,.11 for seg │ │ │ │ +00266330: 6d65 6e74 6174 696f 6e20 6661 756c 742c mentation fault, │ │ │ │ +00266340: 2061 6e64 2031 3520 666f 7220 7465 726d and 15 for term │ │ │ │ +00266350: 696e 6174 696f 6e2e 2020 466f 7220 6465 ination. For de │ │ │ │ +00266360: 7461 696c 732c 2073 6565 2074 6865 206d tails, see the m │ │ │ │ +00266370: 616e 0a70 6167 6520 6f66 2074 6865 206c an.page of the l │ │ │ │ +00266380: 6962 6320 726f 7574 696e 6520 7379 7374 ibc routine syst │ │ │ │ +00266390: 656d 2829 2e0a 0a46 6f72 2074 6865 2070 em()...For the p │ │ │ │ +002663a0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +002663b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +002663c0: 6520 6f62 6a65 6374 202a 6e6f 7465 2072 e object *note r │ │ │ │ +002663d0: 756e 3a20 7275 6e2c 2069 7320 6120 2a6e un: run, is a *n │ │ │ │ +002663e0: 6f74 6520 636f 6d70 696c 6564 2066 756e ote compiled fun │ │ │ │ +002663f0: 6374 696f 6e3a 2043 6f6d 7069 6c65 6446 ction: CompiledF │ │ │ │ +00266400: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ 00266410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00266420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00266430: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00266440: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00266450: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00266460: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00266470: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00266480: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ -00266490: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ -002664a0: 6163 6175 6c61 7932 446f 632f 6f76 5f73 acaulay2Doc/ov_s │ │ │ │ -002664b0: 7973 7465 6d2e 6d32 3a38 3730 3a30 2e0a ystem.m2:870:0.. │ │ │ │ -002664c0: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ -002664d0: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ -002664e0: 2066 696e 6450 726f 6772 616d 2c20 4e65 findProgram, Ne │ │ │ │ -002664f0: 7874 3a20 7275 6e50 726f 6772 616d 2c20 xt: runProgram, │ │ │ │ -00266500: 5072 6576 3a20 7275 6e2c 2055 703a 2063 Prev: run, Up: c │ │ │ │ -00266510: 6f6d 6d75 6e69 6361 7469 6e67 2077 6974 ommunicating wit │ │ │ │ -00266520: 6820 7072 6f67 7261 6d73 0a0a 6669 6e64 h programs..find │ │ │ │ -00266530: 5072 6f67 7261 6d20 2d2d 206c 6f61 6420 Program -- load │ │ │ │ -00266540: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ -00266550: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -00266560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00266570: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -00266580: 3a20 0a20 2020 2020 2020 2066 696e 6450 : . findP │ │ │ │ -00266590: 726f 6772 616d 206e 616d 650a 2020 2020 rogram name. │ │ │ │ -002665a0: 2020 2020 6669 6e64 5072 6f67 7261 6d28 findProgram( │ │ │ │ -002665b0: 6e61 6d65 2c20 636d 6429 0a20 2020 2020 name, cmd). │ │ │ │ -002665c0: 2020 2066 696e 6450 726f 6772 616d 286e findProgram(n │ │ │ │ -002665d0: 616d 652c 2063 6d64 7329 0a20 202a 2049 ame, cmds). * I │ │ │ │ -002665e0: 6e70 7574 733a 0a20 2020 2020 202a 206e nputs:. * n │ │ │ │ -002665f0: 616d 652c 2061 202a 6e6f 7465 2073 7472 ame, a *note str │ │ │ │ -00266600: 696e 673a 2053 7472 696e 672c 2c20 7468 ing: String,, th │ │ │ │ -00266610: 6520 6e61 6d65 206f 6620 7468 6520 7072 e name of the pr │ │ │ │ -00266620: 6f67 7261 6d20 746f 206c 6f61 642e 2020 ogram to load. │ │ │ │ -00266630: 5468 6973 0a20 2020 2020 2020 2073 686f This. sho │ │ │ │ -00266640: 756c 6420 6d61 7463 6820 7468 6520 636f uld match the co │ │ │ │ -00266650: 7272 6573 706f 6e64 696e 6720 6b65 7920 rresponding key │ │ │ │ -00266660: 696e 202a 6e6f 7465 2070 726f 6772 616d in *note program │ │ │ │ -00266670: 5061 7468 733a 0a20 2020 2020 2020 2070 Paths:. p │ │ │ │ -00266680: 726f 6772 616d 5061 7468 732c 2e0a 2020 rogramPaths,.. │ │ │ │ -00266690: 2020 2020 2a20 636d 642c 2061 202a 6e6f * cmd, a *no │ │ │ │ -002666a0: 7465 2073 7472 696e 673a 2053 7472 696e te string: Strin │ │ │ │ -002666b0: 672c 2c20 6120 636f 6d6d 616e 6420 746f g,, a command to │ │ │ │ -002666c0: 2072 756e 2074 6861 7420 7368 6f75 6c64 run that should │ │ │ │ -002666d0: 2072 6574 7572 6e20 3020 6966 0a20 2020 return 0 if. │ │ │ │ -002666e0: 2020 2020 2074 6865 2070 726f 6772 616d the program │ │ │ │ -002666f0: 2069 7320 7072 6573 656e 742e 0a20 2020 is present.. │ │ │ │ -00266700: 2020 202a 2063 6d64 732c 2061 202a 6e6f * cmds, a *no │ │ │ │ -00266710: 7465 206c 6973 743a 204c 6973 742c 2c20 te list: List,, │ │ │ │ -00266720: 6120 6c69 7374 206f 6620 636f 6d6d 616e a list of comman │ │ │ │ -00266730: 6473 2074 6f20 7275 6e20 7468 6174 2073 ds to run that s │ │ │ │ -00266740: 686f 756c 6420 616c 6c0a 2020 2020 2020 hould all. │ │ │ │ -00266750: 2020 7265 7475 726e 2030 2069 6620 7468 return 0 if th │ │ │ │ -00266760: 6520 7072 6f67 7261 6d20 6973 2070 7265 e program is pre │ │ │ │ -00266770: 7365 6e74 2e0a 2020 2a20 2a6e 6f74 6520 sent.. * *note │ │ │ │ -00266780: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ -00266790: 2075 7369 6e67 2066 756e 6374 696f 6e73 using functions │ │ │ │ -002667a0: 2077 6974 6820 6f70 7469 6f6e 616c 2069 with optional i │ │ │ │ -002667b0: 6e70 7574 732c 3a0a 2020 2020 2020 2a20 nputs,:. * │ │ │ │ -002667c0: 5261 6973 6545 7272 6f72 203d 3e20 6120 RaiseError => a │ │ │ │ -002667d0: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661 *note Boolean va │ │ │ │ -002667e0: 6c75 653a 2042 6f6f 6c65 616e 2c2c 2064 lue: Boolean,, d │ │ │ │ -002667f0: 6566 6175 6c74 2076 616c 7565 2074 7275 efault value tru │ │ │ │ -00266800: 652c 0a20 2020 2020 2020 2077 6865 7468 e,. wheth │ │ │ │ -00266810: 6572 2074 6f20 7261 6973 6520 616e 2065 er to raise an e │ │ │ │ -00266820: 7272 6f72 2069 6620 7468 6520 7072 6f67 rror if the prog │ │ │ │ -00266830: 7261 6d20 6973 206e 6f74 2066 6f75 6e64 ram is not found │ │ │ │ -00266840: 2e0a 2020 2020 2020 2a20 5665 7262 6f73 .. * Verbos │ │ │ │ -00266850: 6520 3d3e 2061 202a 6e6f 7465 2042 6f6f e => a *note Boo │ │ │ │ -00266860: 6c65 616e 2076 616c 7565 3a20 426f 6f6c lean value: Bool │ │ │ │ -00266870: 6561 6e2c 2c20 6465 6661 756c 7420 7661 ean,, default va │ │ │ │ -00266880: 6c75 6520 6661 6c73 652c 0a20 2020 2020 lue false,. │ │ │ │ -00266890: 2020 2077 6865 7468 6572 2074 6f20 696e whether to in │ │ │ │ -002668a0: 666f 726d 2074 6865 2075 7365 7220 6f66 form the user of │ │ │ │ -002668b0: 2065 6163 6820 7061 7468 2074 6861 7420 each path that │ │ │ │ -002668c0: 6973 2063 6865 636b 6564 2e0a 2020 2020 is checked.. │ │ │ │ -002668d0: 2020 2a20 5072 6566 6978 203d 3e20 6120 * Prefix => a │ │ │ │ -002668e0: 2a6e 6f74 6520 6c69 7374 3a20 4c69 7374 *note list: List │ │ │ │ -002668f0: 2c2c 2064 6566 6175 6c74 2076 616c 7565 ,, default value │ │ │ │ -00266900: 207b 7d2c 2061 206c 6973 7420 6f66 2073 {}, a list of s │ │ │ │ -00266910: 6571 7565 6e63 6573 0a20 2020 2020 2020 equences. │ │ │ │ -00266920: 2063 6f6e 7461 696e 696e 6720 7477 6f20 containing two │ │ │ │ -00266930: 7374 7269 6e67 7320 6964 656e 7469 6679 strings identify │ │ │ │ -00266940: 696e 6720 6120 7072 6566 6978 2074 6861 ing a prefix tha │ │ │ │ -00266950: 7420 6973 2061 6464 6564 2074 6f20 7468 t is added to th │ │ │ │ -00266960: 650a 2020 2020 2020 2020 6578 6563 7574 e. execut │ │ │ │ -00266970: 6162 6c65 2062 696e 6172 6965 7320 6265 able binaries be │ │ │ │ -00266980: 6c6f 6e67 696e 6720 746f 2074 6865 2070 longing to the p │ │ │ │ -00266990: 726f 6772 616d 2062 7920 736f 6d65 2064 rogram by some d │ │ │ │ -002669a0: 6973 7472 6962 7574 696f 6e73 2e0a 2020 istributions.. │ │ │ │ -002669b0: 2020 2020 2020 5468 6573 6520 7365 7175 These sequ │ │ │ │ -002669c0: 656e 6365 7320 7368 6f75 6c64 2062 6520 ences should be │ │ │ │ -002669d0: 6f66 2074 6865 2066 6f72 6d20 2872 6567 of the form (reg │ │ │ │ -002669e0: 6578 2c20 7072 6566 6978 292c 2077 6865 ex, prefix), whe │ │ │ │ -002669f0: 7265 2072 6567 6578 2069 7320 610a 2020 re regex is a. │ │ │ │ -00266a00: 2020 2020 2020 2a6e 6f74 6520 7265 6775 *note regu │ │ │ │ -00266a10: 6c61 7220 6578 7072 6573 7369 6f6e 3a20 lar expression: │ │ │ │ -00266a20: 7265 6775 6c61 7220 6578 7072 6573 7369 regular expressi │ │ │ │ -00266a30: 6f6e 732c 2074 6861 7420 7368 6f75 6c64 ons, that should │ │ │ │ -00266a40: 206d 6174 6368 2061 6c6c 0a20 2020 2020 match all. │ │ │ │ -00266a50: 2020 2062 696e 6172 7920 6578 6563 7574 binary execut │ │ │ │ -00266a60: 6162 6c65 7320 7468 6174 206e 6565 6420 ables that need │ │ │ │ -00266a70: 7468 6520 7072 6566 6978 2061 6e64 2070 the prefix and p │ │ │ │ -00266a80: 7265 6669 7820 6973 2074 6865 2070 7265 refix is the pre │ │ │ │ -00266a90: 6669 780a 2020 2020 2020 2020 6974 7365 fix. itse │ │ │ │ -00266aa0: 6c66 2e0a 2020 2020 2020 2a20 4164 6469 lf.. * Addi │ │ │ │ -00266ab0: 7469 6f6e 616c 5061 7468 7320 3d3e 2061 tionalPaths => a │ │ │ │ -00266ac0: 202a 6e6f 7465 206c 6973 743a 204c 6973 *note list: Lis │ │ │ │ -00266ad0: 742c 2c20 6465 6661 756c 7420 7661 6c75 t,, default valu │ │ │ │ -00266ae0: 6520 7b7d 2c20 6120 6c69 7374 206f 660a e {}, a list of. │ │ │ │ -00266af0: 2020 2020 2020 2020 7374 7269 6e67 7320 strings │ │ │ │ -00266b00: 636f 6e74 6169 6e69 6e67 2061 6e79 2070 containing any p │ │ │ │ -00266b10: 6174 6873 2074 6f20 6368 6563 6b20 666f aths to check fo │ │ │ │ -00266b20: 7220 7468 6520 7072 6f67 7261 6d20 696e r the program in │ │ │ │ -00266b30: 2061 6464 6974 696f 6e20 746f 0a20 2020 addition to. │ │ │ │ -00266b40: 2020 2020 2074 6865 2064 6566 6175 6c74 the default │ │ │ │ -00266b50: 206f 6e65 732e 0a20 2020 2020 202a 204d ones.. * M │ │ │ │ -00266b60: 696e 696d 756d 5665 7273 696f 6e20 3d3e inimumVersion => │ │ │ │ -00266b70: 2061 202a 6e6f 7465 2073 6571 7565 6e63 a *note sequenc │ │ │ │ -00266b80: 653a 2053 6571 7565 6e63 652c 2c20 6465 e: Sequence,, de │ │ │ │ -00266b90: 6661 756c 7420 7661 6c75 6520 6e75 6c6c fault value null │ │ │ │ -00266ba0: 2c0a 2020 2020 2020 2020 636f 6e74 6169 ,. contai │ │ │ │ -00266bb0: 6e69 6e67 2074 776f 2073 7472 696e 6773 ning two strings │ │ │ │ -00266bc0: 2074 6865 2066 6f72 6d20 286d 696e 5665 the form (minVe │ │ │ │ -00266bd0: 7273 696f 6e2c 2076 6572 7369 6f6e 436f rsion, versionCo │ │ │ │ -00266be0: 6d6d 616e 6429 2077 6865 7265 0a20 2020 mmand) where. │ │ │ │ -00266bf0: 2020 2020 206d 696e 5665 7273 696f 6e20 minVersion │ │ │ │ -00266c00: 6973 2074 6865 206d 696e 696d 756d 2072 is the minimum r │ │ │ │ -00266c10: 6571 7569 7265 6420 7665 7273 696f 6e20 equired version │ │ │ │ -00266c20: 6f66 2074 6865 2070 726f 6772 616d 2061 of the program a │ │ │ │ -00266c30: 6e64 0a20 2020 2020 2020 2076 6572 7369 nd. versi │ │ │ │ -00266c40: 6f6e 436f 6d6d 616e 6420 6973 2061 2073 onCommand is a s │ │ │ │ -00266c50: 6865 6c6c 2063 6f6d 6d61 6e64 2074 6f20 hell command to │ │ │ │ -00266c60: 6f62 7461 696e 2074 6865 2076 6572 7369 obtain the versi │ │ │ │ -00266c70: 6f6e 206e 756d 6265 7220 6f66 2061 6e0a on number of an. │ │ │ │ -00266c80: 2020 2020 2020 2020 696e 7374 616c 6c65 installe │ │ │ │ -00266c90: 6420 7072 6f67 7261 6d2e 0a20 202a 204f d program.. * O │ │ │ │ -00266ca0: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -00266cb0: 616e 2069 6e73 7461 6e63 6520 6f66 2074 an instance of t │ │ │ │ -00266cc0: 6865 2074 7970 6520 2a6e 6f74 6520 5072 he type *note Pr │ │ │ │ -00266cd0: 6f67 7261 6d3a 2050 726f 6772 616d 2c2c ogram: Program,, │ │ │ │ -00266ce0: 2074 6865 2070 726f 6772 616d 2074 6861 the program tha │ │ │ │ -00266cf0: 7420 7761 730a 2020 2020 2020 2020 6c6f t was. lo │ │ │ │ -00266d00: 6164 6564 2e20 2049 6620 7468 6520 7072 aded. If the pr │ │ │ │ -00266d10: 6f67 7261 6d20 6973 206e 6f74 2066 6f75 ogram is not fou │ │ │ │ -00266d20: 6e64 2061 6e64 2052 6169 7365 4572 726f nd and RaiseErro │ │ │ │ -00266d30: 7220 6973 2073 6574 2074 6f20 2a6e 6f74 r is set to *not │ │ │ │ -00266d40: 650a 2020 2020 2020 2020 6661 6c73 653a e. false: │ │ │ │ -00266d50: 2066 616c 7365 2c2c 2074 6865 6e20 2a6e false,, then *n │ │ │ │ -00266d60: 6f74 6520 6e75 6c6c 3a20 6e75 6c6c 2c20 ote null: null, │ │ │ │ -00266d70: 6973 2072 6574 7572 6e65 642e 0a0a 4465 is returned...De │ │ │ │ -00266d80: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -00266d90: 3d3d 3d3d 3d0a 0a54 6869 7320 6675 6e63 =====..This func │ │ │ │ -00266da0: 7469 6f6e 2063 6865 636b 7320 666f 7220 tion checks for │ │ │ │ -00266db0: 7468 6520 6578 6973 7465 6e63 6520 6f66 the existence of │ │ │ │ -00266dc0: 2061 6e20 6578 7465 726e 616c 2070 726f an external pro │ │ │ │ -00266dd0: 6772 616d 2062 7920 7275 6e6e 696e 6720 gram by running │ │ │ │ -00266de0: 636d 640a 286f 7220 6576 6572 7920 656c cmd.(or every el │ │ │ │ -00266df0: 656d 656e 7420 6f66 2063 6d64 7329 2c20 ement of cmds), │ │ │ │ -00266e00: 7072 6570 656e 6465 6420 7769 7468 2076 prepended with v │ │ │ │ -00266e10: 6172 696f 7573 2070 6174 6873 2069 6e20 arious paths in │ │ │ │ -00266e20: 7468 6520 666f 6c6c 6f77 696e 670a 6f72 the following.or │ │ │ │ -00266e30: 6465 723a 0a0a 2020 2a20 5468 6520 7573 der:.. * The us │ │ │ │ -00266e40: 6572 2d64 6566 696e 6564 2070 6174 6820 er-defined path │ │ │ │ -00266e50: 7370 6563 6966 6965 6420 6279 2070 726f specified by pro │ │ │ │ -00266e60: 6772 616d 5061 7468 7323 6e61 6d65 2c20 gramPaths#name, │ │ │ │ -00266e70: 6966 2069 7420 6578 6973 7473 2e0a 2020 if it exists.. │ │ │ │ -00266e80: 2a20 5468 6520 7061 7468 2073 7065 6369 * The path speci │ │ │ │ -00266e90: 6669 6564 2062 7920 7072 6566 6978 4469 fied by prefixDi │ │ │ │ -00266ea0: 7265 6374 6f72 7920 7c20 6375 7272 656e rectory | curren │ │ │ │ -00266eb0: 744c 6179 6f75 7423 2270 726f 6772 616d tLayout#"program │ │ │ │ -00266ec0: 7322 2c20 7768 6572 6520 7468 650a 2020 s", where the. │ │ │ │ -00266ed0: 2020 7072 6f67 7261 6d73 2073 6869 7070 programs shipp │ │ │ │ -00266ee0: 6564 2077 6974 6820 4d61 6361 756c 6179 ed with Macaulay │ │ │ │ -00266ef0: 3220 6172 6520 696e 7374 616c 6c65 642e 2 are installed. │ │ │ │ -00266f00: 0a20 202a 2045 6163 6820 7061 7468 2073 . * Each path s │ │ │ │ -00266f10: 7065 6369 6669 6564 2062 7920 7468 6520 pecified by the │ │ │ │ -00266f20: 4164 6469 7469 6f6e 616c 5061 7468 7320 AdditionalPaths │ │ │ │ -00266f30: 6f70 7469 6f6e 2e0a 2020 2a20 4561 6368 option.. * Each │ │ │ │ -00266f40: 2070 6174 6820 7370 6563 6966 6965 6420 path specified │ │ │ │ -00266f50: 6279 2074 6865 2075 7365 7227 7320 5041 by the user's PA │ │ │ │ -00266f60: 5448 2065 6e76 6972 6f6e 6d65 6e74 2076 TH environment v │ │ │ │ -00266f70: 6172 6961 626c 652e 0a20 202a 2054 6865 ariable.. * The │ │ │ │ -00266f80: 2070 6174 6820 746f 204d 322d 6269 6e61 path to M2-bina │ │ │ │ -00266f90: 7279 2e0a 0a46 6f72 2065 6163 6820 7061 ry...For each pa │ │ │ │ -00266fa0: 7468 2c20 616e 7920 7072 6566 6978 6573 th, any prefixes │ │ │ │ -00266fb0: 2073 7065 6369 6669 6564 2062 7920 7468 specified by th │ │ │ │ -00266fc0: 6520 5072 6566 6978 206f 7074 696f 6e20 e Prefix option │ │ │ │ -00266fd0: 6172 6520 6368 6563 6b65 642e 0a0a 4f6e are checked...On │ │ │ │ -00266fe0: 6365 2074 6869 7320 6973 2073 7563 6365 ce this is succe │ │ │ │ -00266ff0: 7373 6675 6c20 2869 2e65 2e2c 2063 6d64 ssful (i.e., cmd │ │ │ │ -00267000: 206f 7220 6561 6368 2065 6c65 6d65 6e74 or each element │ │ │ │ -00267010: 206f 6620 636d 6473 2072 6574 7572 6e73 of cmds returns │ │ │ │ -00267020: 2061 2076 616c 7565 206f 660a 3029 2074 a value of.0) t │ │ │ │ -00267030: 6865 6e20 6120 2a6e 6f74 6520 5072 6f67 hen a *note Prog │ │ │ │ -00267040: 7261 6d3a 2050 726f 6772 616d 2c20 6f62 ram: Program, ob │ │ │ │ -00267050: 6a65 6374 2069 7320 7265 7475 726e 6564 ject is returned │ │ │ │ -00267060: 2e20 2049 6620 6974 2069 7320 756e 7375 . If it is unsu │ │ │ │ -00267070: 6363 6573 7366 756c 2c0a 7468 656e 2065 ccessful,.then e │ │ │ │ -00267080: 6974 6865 7220 616e 2065 7272 6f72 2069 ither an error i │ │ │ │ -00267090: 7320 7261 6973 6564 206f 7220 2a6e 6f74 s raised or *not │ │ │ │ -002670a0: 6520 6e75 6c6c 3a20 6e75 6c6c 2c20 6973 e null: null, is │ │ │ │ -002670b0: 2072 6574 7572 6e65 642c 2064 6570 656e returned, depen │ │ │ │ -002670c0: 6469 6e67 206f 6e0a 7468 6520 7661 6c75 ding on.the valu │ │ │ │ -002670d0: 6520 6f66 2052 6169 7365 4572 726f 722e e of RaiseError. │ │ │ │ -002670e0: 0a0a 4e6f 7465 2074 6861 7420 6966 2061 ..Note that if a │ │ │ │ -002670f0: 2070 726f 6772 616d 2063 6f6e 7369 7374 program consist │ │ │ │ -00267100: 7320 6f66 2061 2073 696e 676c 6520 6578 s of a single ex │ │ │ │ -00267110: 6563 7574 6162 6c65 2062 696e 6172 7920 ecutable binary │ │ │ │ -00267120: 6669 6c65 2c20 7468 656e 206e 616d 650a file, then name. │ │ │ │ -00267130: 7368 6f75 6c64 2063 6f69 6e63 6964 6520 should coincide │ │ │ │ -00267140: 7769 7468 2074 6865 206e 616d 6520 6f66 with the name of │ │ │ │ -00267150: 2074 6869 7320 6669 6c65 2e0a 0a2b 2d2d this file...+-- │ │ │ │ -00267160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00266450: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +00266460: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +00266470: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +00266480: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00266490: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +002664a0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ +002664b0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +002664c0: 732f 0a4d 6163 6175 6c61 7932 446f 632f s/.Macaulay2Doc/ │ │ │ │ +002664d0: 6f76 5f73 7973 7465 6d2e 6d32 3a38 3730 ov_system.m2:870 │ │ │ │ +002664e0: 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 6361 :0....File: Maca │ │ │ │ +002664f0: 756c 6179 3244 6f63 2e69 6e66 6f2c 204e ulay2Doc.info, N │ │ │ │ +00266500: 6f64 653a 2066 696e 6450 726f 6772 616d ode: findProgram │ │ │ │ +00266510: 2c20 4e65 7874 3a20 7275 6e50 726f 6772 , Next: runProgr │ │ │ │ +00266520: 616d 2c20 5072 6576 3a20 7275 6e2c 2055 am, Prev: run, U │ │ │ │ +00266530: 703a 2063 6f6d 6d75 6e69 6361 7469 6e67 p: communicating │ │ │ │ +00266540: 2077 6974 6820 7072 6f67 7261 6d73 0a0a with programs.. │ │ │ │ +00266550: 6669 6e64 5072 6f67 7261 6d20 2d2d 206c findProgram -- l │ │ │ │ +00266560: 6f61 6420 6578 7465 726e 616c 2070 726f oad external pro │ │ │ │ +00266570: 6772 616d 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a gram.*********** │ │ │ │ +00266580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00266590: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +002665a0: 7361 6765 3a20 0a20 2020 2020 2020 2066 sage: . f │ │ │ │ +002665b0: 696e 6450 726f 6772 616d 206e 616d 650a indProgram name. │ │ │ │ +002665c0: 2020 2020 2020 2020 6669 6e64 5072 6f67 findProg │ │ │ │ +002665d0: 7261 6d28 6e61 6d65 2c20 636d 6429 0a20 ram(name, cmd). │ │ │ │ +002665e0: 2020 2020 2020 2066 696e 6450 726f 6772 findProgr │ │ │ │ +002665f0: 616d 286e 616d 652c 2063 6d64 7329 0a20 am(name, cmds). │ │ │ │ +00266600: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +00266610: 202a 206e 616d 652c 2061 202a 6e6f 7465 * name, a *note │ │ │ │ +00266620: 2073 7472 696e 673a 2053 7472 696e 672c string: String, │ │ │ │ +00266630: 2c20 7468 6520 6e61 6d65 206f 6620 7468 , the name of th │ │ │ │ +00266640: 6520 7072 6f67 7261 6d20 746f 206c 6f61 e program to loa │ │ │ │ +00266650: 642e 2020 5468 6973 0a20 2020 2020 2020 d. This. │ │ │ │ +00266660: 2073 686f 756c 6420 6d61 7463 6820 7468 should match th │ │ │ │ +00266670: 6520 636f 7272 6573 706f 6e64 696e 6720 e corresponding │ │ │ │ +00266680: 6b65 7920 696e 202a 6e6f 7465 2070 726f key in *note pro │ │ │ │ +00266690: 6772 616d 5061 7468 733a 0a20 2020 2020 gramPaths:. │ │ │ │ +002666a0: 2020 2070 726f 6772 616d 5061 7468 732c programPaths, │ │ │ │ +002666b0: 2e0a 2020 2020 2020 2a20 636d 642c 2061 .. * cmd, a │ │ │ │ +002666c0: 202a 6e6f 7465 2073 7472 696e 673a 2053 *note string: S │ │ │ │ +002666d0: 7472 696e 672c 2c20 6120 636f 6d6d 616e tring,, a comman │ │ │ │ +002666e0: 6420 746f 2072 756e 2074 6861 7420 7368 d to run that sh │ │ │ │ +002666f0: 6f75 6c64 2072 6574 7572 6e20 3020 6966 ould return 0 if │ │ │ │ +00266700: 0a20 2020 2020 2020 2074 6865 2070 726f . the pro │ │ │ │ +00266710: 6772 616d 2069 7320 7072 6573 656e 742e gram is present. │ │ │ │ +00266720: 0a20 2020 2020 202a 2063 6d64 732c 2061 . * cmds, a │ │ │ │ +00266730: 202a 6e6f 7465 206c 6973 743a 204c 6973 *note list: Lis │ │ │ │ +00266740: 742c 2c20 6120 6c69 7374 206f 6620 636f t,, a list of co │ │ │ │ +00266750: 6d6d 616e 6473 2074 6f20 7275 6e20 7468 mmands to run th │ │ │ │ +00266760: 6174 2073 686f 756c 6420 616c 6c0a 2020 at should all. │ │ │ │ +00266770: 2020 2020 2020 7265 7475 726e 2030 2069 return 0 i │ │ │ │ +00266780: 6620 7468 6520 7072 6f67 7261 6d20 6973 f the program is │ │ │ │ +00266790: 2070 7265 7365 6e74 2e0a 2020 2a20 2a6e present.. * *n │ │ │ │ +002667a0: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70 ote Optional inp │ │ │ │ +002667b0: 7574 733a 2075 7369 6e67 2066 756e 6374 uts: using funct │ │ │ │ +002667c0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ +002667d0: 616c 2069 6e70 7574 732c 3a0a 2020 2020 al inputs,:. │ │ │ │ +002667e0: 2020 2a20 5261 6973 6545 7272 6f72 203d * RaiseError = │ │ │ │ +002667f0: 3e20 6120 2a6e 6f74 6520 426f 6f6c 6561 > a *note Boolea │ │ │ │ +00266800: 6e20 7661 6c75 653a 2042 6f6f 6c65 616e n value: Boolean │ │ │ │ +00266810: 2c2c 2064 6566 6175 6c74 2076 616c 7565 ,, default value │ │ │ │ +00266820: 2074 7275 652c 0a20 2020 2020 2020 2077 true,. w │ │ │ │ +00266830: 6865 7468 6572 2074 6f20 7261 6973 6520 hether to raise │ │ │ │ +00266840: 616e 2065 7272 6f72 2069 6620 7468 6520 an error if the │ │ │ │ +00266850: 7072 6f67 7261 6d20 6973 206e 6f74 2066 program is not f │ │ │ │ +00266860: 6f75 6e64 2e0a 2020 2020 2020 2a20 5665 ound.. * Ve │ │ │ │ +00266870: 7262 6f73 6520 3d3e 2061 202a 6e6f 7465 rbose => a *note │ │ │ │ +00266880: 2042 6f6f 6c65 616e 2076 616c 7565 3a20 Boolean value: │ │ │ │ +00266890: 426f 6f6c 6561 6e2c 2c20 6465 6661 756c Boolean,, defaul │ │ │ │ +002668a0: 7420 7661 6c75 6520 6661 6c73 652c 0a20 t value false,. │ │ │ │ +002668b0: 2020 2020 2020 2077 6865 7468 6572 2074 whether t │ │ │ │ +002668c0: 6f20 696e 666f 726d 2074 6865 2075 7365 o inform the use │ │ │ │ +002668d0: 7220 6f66 2065 6163 6820 7061 7468 2074 r of each path t │ │ │ │ +002668e0: 6861 7420 6973 2063 6865 636b 6564 2e0a hat is checked.. │ │ │ │ +002668f0: 2020 2020 2020 2a20 5072 6566 6978 203d * Prefix = │ │ │ │ +00266900: 3e20 6120 2a6e 6f74 6520 6c69 7374 3a20 > a *note list: │ │ │ │ +00266910: 4c69 7374 2c2c 2064 6566 6175 6c74 2076 List,, default v │ │ │ │ +00266920: 616c 7565 207b 7d2c 2061 206c 6973 7420 alue {}, a list │ │ │ │ +00266930: 6f66 2073 6571 7565 6e63 6573 0a20 2020 of sequences. │ │ │ │ +00266940: 2020 2020 2063 6f6e 7461 696e 696e 6720 containing │ │ │ │ +00266950: 7477 6f20 7374 7269 6e67 7320 6964 656e two strings iden │ │ │ │ +00266960: 7469 6679 696e 6720 6120 7072 6566 6978 tifying a prefix │ │ │ │ +00266970: 2074 6861 7420 6973 2061 6464 6564 2074 that is added t │ │ │ │ +00266980: 6f20 7468 650a 2020 2020 2020 2020 6578 o the. ex │ │ │ │ +00266990: 6563 7574 6162 6c65 2062 696e 6172 6965 ecutable binarie │ │ │ │ +002669a0: 7320 6265 6c6f 6e67 696e 6720 746f 2074 s belonging to t │ │ │ │ +002669b0: 6865 2070 726f 6772 616d 2062 7920 736f he program by so │ │ │ │ +002669c0: 6d65 2064 6973 7472 6962 7574 696f 6e73 me distributions │ │ │ │ +002669d0: 2e0a 2020 2020 2020 2020 5468 6573 6520 .. These │ │ │ │ +002669e0: 7365 7175 656e 6365 7320 7368 6f75 6c64 sequences should │ │ │ │ +002669f0: 2062 6520 6f66 2074 6865 2066 6f72 6d20 be of the form │ │ │ │ +00266a00: 2872 6567 6578 2c20 7072 6566 6978 292c (regex, prefix), │ │ │ │ +00266a10: 2077 6865 7265 2072 6567 6578 2069 7320 where regex is │ │ │ │ +00266a20: 610a 2020 2020 2020 2020 2a6e 6f74 6520 a. *note │ │ │ │ +00266a30: 7265 6775 6c61 7220 6578 7072 6573 7369 regular expressi │ │ │ │ +00266a40: 6f6e 3a20 7265 6775 6c61 7220 6578 7072 on: regular expr │ │ │ │ +00266a50: 6573 7369 6f6e 732c 2074 6861 7420 7368 essions, that sh │ │ │ │ +00266a60: 6f75 6c64 206d 6174 6368 2061 6c6c 0a20 ould match all. │ │ │ │ +00266a70: 2020 2020 2020 2062 696e 6172 7920 6578 binary ex │ │ │ │ +00266a80: 6563 7574 6162 6c65 7320 7468 6174 206e ecutables that n │ │ │ │ +00266a90: 6565 6420 7468 6520 7072 6566 6978 2061 eed the prefix a │ │ │ │ +00266aa0: 6e64 2070 7265 6669 7820 6973 2074 6865 nd prefix is the │ │ │ │ +00266ab0: 2070 7265 6669 780a 2020 2020 2020 2020 prefix. │ │ │ │ +00266ac0: 6974 7365 6c66 2e0a 2020 2020 2020 2a20 itself.. * │ │ │ │ +00266ad0: 4164 6469 7469 6f6e 616c 5061 7468 7320 AdditionalPaths │ │ │ │ +00266ae0: 3d3e 2061 202a 6e6f 7465 206c 6973 743a => a *note list: │ │ │ │ +00266af0: 204c 6973 742c 2c20 6465 6661 756c 7420 List,, default │ │ │ │ +00266b00: 7661 6c75 6520 7b7d 2c20 6120 6c69 7374 value {}, a list │ │ │ │ +00266b10: 206f 660a 2020 2020 2020 2020 7374 7269 of. stri │ │ │ │ +00266b20: 6e67 7320 636f 6e74 6169 6e69 6e67 2061 ngs containing a │ │ │ │ +00266b30: 6e79 2070 6174 6873 2074 6f20 6368 6563 ny paths to chec │ │ │ │ +00266b40: 6b20 666f 7220 7468 6520 7072 6f67 7261 k for the progra │ │ │ │ +00266b50: 6d20 696e 2061 6464 6974 696f 6e20 746f m in addition to │ │ │ │ +00266b60: 0a20 2020 2020 2020 2074 6865 2064 6566 . the def │ │ │ │ +00266b70: 6175 6c74 206f 6e65 732e 0a20 2020 2020 ault ones.. │ │ │ │ +00266b80: 202a 204d 696e 696d 756d 5665 7273 696f * MinimumVersio │ │ │ │ +00266b90: 6e20 3d3e 2061 202a 6e6f 7465 2073 6571 n => a *note seq │ │ │ │ +00266ba0: 7565 6e63 653a 2053 6571 7565 6e63 652c uence: Sequence, │ │ │ │ +00266bb0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +00266bc0: 6e75 6c6c 2c0a 2020 2020 2020 2020 636f null,. co │ │ │ │ +00266bd0: 6e74 6169 6e69 6e67 2074 776f 2073 7472 ntaining two str │ │ │ │ +00266be0: 696e 6773 2074 6865 2066 6f72 6d20 286d ings the form (m │ │ │ │ +00266bf0: 696e 5665 7273 696f 6e2c 2076 6572 7369 inVersion, versi │ │ │ │ +00266c00: 6f6e 436f 6d6d 616e 6429 2077 6865 7265 onCommand) where │ │ │ │ +00266c10: 0a20 2020 2020 2020 206d 696e 5665 7273 . minVers │ │ │ │ +00266c20: 696f 6e20 6973 2074 6865 206d 696e 696d ion is the minim │ │ │ │ +00266c30: 756d 2072 6571 7569 7265 6420 7665 7273 um required vers │ │ │ │ +00266c40: 696f 6e20 6f66 2074 6865 2070 726f 6772 ion of the progr │ │ │ │ +00266c50: 616d 2061 6e64 0a20 2020 2020 2020 2076 am and. v │ │ │ │ +00266c60: 6572 7369 6f6e 436f 6d6d 616e 6420 6973 ersionCommand is │ │ │ │ +00266c70: 2061 2073 6865 6c6c 2063 6f6d 6d61 6e64 a shell command │ │ │ │ +00266c80: 2074 6f20 6f62 7461 696e 2074 6865 2076 to obtain the v │ │ │ │ +00266c90: 6572 7369 6f6e 206e 756d 6265 7220 6f66 ersion number of │ │ │ │ +00266ca0: 2061 6e0a 2020 2020 2020 2020 696e 7374 an. inst │ │ │ │ +00266cb0: 616c 6c65 6420 7072 6f67 7261 6d2e 0a20 alled program.. │ │ │ │ +00266cc0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +00266cd0: 2020 2a20 616e 2069 6e73 7461 6e63 6520 * an instance │ │ │ │ +00266ce0: 6f66 2074 6865 2074 7970 6520 2a6e 6f74 of the type *not │ │ │ │ +00266cf0: 6520 5072 6f67 7261 6d3a 2050 726f 6772 e Program: Progr │ │ │ │ +00266d00: 616d 2c2c 2074 6865 2070 726f 6772 616d am,, the program │ │ │ │ +00266d10: 2074 6861 7420 7761 730a 2020 2020 2020 that was. │ │ │ │ +00266d20: 2020 6c6f 6164 6564 2e20 2049 6620 7468 loaded. If th │ │ │ │ +00266d30: 6520 7072 6f67 7261 6d20 6973 206e 6f74 e program is not │ │ │ │ +00266d40: 2066 6f75 6e64 2061 6e64 2052 6169 7365 found and Raise │ │ │ │ +00266d50: 4572 726f 7220 6973 2073 6574 2074 6f20 Error is set to │ │ │ │ +00266d60: 2a6e 6f74 650a 2020 2020 2020 2020 6661 *note. fa │ │ │ │ +00266d70: 6c73 653a 2066 616c 7365 2c2c 2074 6865 lse: false,, the │ │ │ │ +00266d80: 6e20 2a6e 6f74 6520 6e75 6c6c 3a20 6e75 n *note null: nu │ │ │ │ +00266d90: 6c6c 2c20 6973 2072 6574 7572 6e65 642e ll, is returned. │ │ │ │ +00266da0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00266db0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 =========..This │ │ │ │ +00266dc0: 6675 6e63 7469 6f6e 2063 6865 636b 7320 function checks │ │ │ │ +00266dd0: 666f 7220 7468 6520 6578 6973 7465 6e63 for the existenc │ │ │ │ +00266de0: 6520 6f66 2061 6e20 6578 7465 726e 616c e of an external │ │ │ │ +00266df0: 2070 726f 6772 616d 2062 7920 7275 6e6e program by runn │ │ │ │ +00266e00: 696e 6720 636d 640a 286f 7220 6576 6572 ing cmd.(or ever │ │ │ │ +00266e10: 7920 656c 656d 656e 7420 6f66 2063 6d64 y element of cmd │ │ │ │ +00266e20: 7329 2c20 7072 6570 656e 6465 6420 7769 s), prepended wi │ │ │ │ +00266e30: 7468 2076 6172 696f 7573 2070 6174 6873 th various paths │ │ │ │ +00266e40: 2069 6e20 7468 6520 666f 6c6c 6f77 696e in the followin │ │ │ │ +00266e50: 670a 6f72 6465 723a 0a0a 2020 2a20 5468 g.order:.. * Th │ │ │ │ +00266e60: 6520 7573 6572 2d64 6566 696e 6564 2070 e user-defined p │ │ │ │ +00266e70: 6174 6820 7370 6563 6966 6965 6420 6279 ath specified by │ │ │ │ +00266e80: 2070 726f 6772 616d 5061 7468 7323 6e61 programPaths#na │ │ │ │ +00266e90: 6d65 2c20 6966 2069 7420 6578 6973 7473 me, if it exists │ │ │ │ +00266ea0: 2e0a 2020 2a20 5468 6520 7061 7468 2073 .. * The path s │ │ │ │ +00266eb0: 7065 6369 6669 6564 2062 7920 7072 6566 pecified by pref │ │ │ │ +00266ec0: 6978 4469 7265 6374 6f72 7920 7c20 6375 ixDirectory | cu │ │ │ │ +00266ed0: 7272 656e 744c 6179 6f75 7423 2270 726f rrentLayout#"pro │ │ │ │ +00266ee0: 6772 616d 7322 2c20 7768 6572 6520 7468 grams", where th │ │ │ │ +00266ef0: 650a 2020 2020 7072 6f67 7261 6d73 2073 e. programs s │ │ │ │ +00266f00: 6869 7070 6564 2077 6974 6820 4d61 6361 hipped with Maca │ │ │ │ +00266f10: 756c 6179 3220 6172 6520 696e 7374 616c ulay2 are instal │ │ │ │ +00266f20: 6c65 642e 0a20 202a 2045 6163 6820 7061 led.. * Each pa │ │ │ │ +00266f30: 7468 2073 7065 6369 6669 6564 2062 7920 th specified by │ │ │ │ +00266f40: 7468 6520 4164 6469 7469 6f6e 616c 5061 the AdditionalPa │ │ │ │ +00266f50: 7468 7320 6f70 7469 6f6e 2e0a 2020 2a20 ths option.. * │ │ │ │ +00266f60: 4561 6368 2070 6174 6820 7370 6563 6966 Each path specif │ │ │ │ +00266f70: 6965 6420 6279 2074 6865 2075 7365 7227 ied by the user' │ │ │ │ +00266f80: 7320 5041 5448 2065 6e76 6972 6f6e 6d65 s PATH environme │ │ │ │ +00266f90: 6e74 2076 6172 6961 626c 652e 0a20 202a nt variable.. * │ │ │ │ +00266fa0: 2054 6865 2070 6174 6820 746f 204d 322d The path to M2- │ │ │ │ +00266fb0: 6269 6e61 7279 2e0a 0a46 6f72 2065 6163 binary...For eac │ │ │ │ +00266fc0: 6820 7061 7468 2c20 616e 7920 7072 6566 h path, any pref │ │ │ │ +00266fd0: 6978 6573 2073 7065 6369 6669 6564 2062 ixes specified b │ │ │ │ +00266fe0: 7920 7468 6520 5072 6566 6978 206f 7074 y the Prefix opt │ │ │ │ +00266ff0: 696f 6e20 6172 6520 6368 6563 6b65 642e ion are checked. │ │ │ │ +00267000: 0a0a 4f6e 6365 2074 6869 7320 6973 2073 ..Once this is s │ │ │ │ +00267010: 7563 6365 7373 6675 6c20 2869 2e65 2e2c uccessful (i.e., │ │ │ │ +00267020: 2063 6d64 206f 7220 6561 6368 2065 6c65 cmd or each ele │ │ │ │ +00267030: 6d65 6e74 206f 6620 636d 6473 2072 6574 ment of cmds ret │ │ │ │ +00267040: 7572 6e73 2061 2076 616c 7565 206f 660a urns a value of. │ │ │ │ +00267050: 3029 2074 6865 6e20 6120 2a6e 6f74 6520 0) then a *note │ │ │ │ +00267060: 5072 6f67 7261 6d3a 2050 726f 6772 616d Program: Program │ │ │ │ +00267070: 2c20 6f62 6a65 6374 2069 7320 7265 7475 , object is retu │ │ │ │ +00267080: 726e 6564 2e20 2049 6620 6974 2069 7320 rned. If it is │ │ │ │ +00267090: 756e 7375 6363 6573 7366 756c 2c0a 7468 unsuccessful,.th │ │ │ │ +002670a0: 656e 2065 6974 6865 7220 616e 2065 7272 en either an err │ │ │ │ +002670b0: 6f72 2069 7320 7261 6973 6564 206f 7220 or is raised or │ │ │ │ +002670c0: 2a6e 6f74 6520 6e75 6c6c 3a20 6e75 6c6c *note null: null │ │ │ │ +002670d0: 2c20 6973 2072 6574 7572 6e65 642c 2064 , is returned, d │ │ │ │ +002670e0: 6570 656e 6469 6e67 206f 6e0a 7468 6520 epending on.the │ │ │ │ +002670f0: 7661 6c75 6520 6f66 2052 6169 7365 4572 value of RaiseEr │ │ │ │ +00267100: 726f 722e 0a0a 4e6f 7465 2074 6861 7420 ror...Note that │ │ │ │ +00267110: 6966 2061 2070 726f 6772 616d 2063 6f6e if a program con │ │ │ │ +00267120: 7369 7374 7320 6f66 2061 2073 696e 676c sists of a singl │ │ │ │ +00267130: 6520 6578 6563 7574 6162 6c65 2062 696e e executable bin │ │ │ │ +00267140: 6172 7920 6669 6c65 2c20 7468 656e 206e ary file, then n │ │ │ │ +00267150: 616d 650a 7368 6f75 6c64 2063 6f69 6e63 ame.should coinc │ │ │ │ +00267160: 6964 6520 7769 7468 2074 6865 206e 616d ide with the nam │ │ │ │ +00267170: 6520 6f66 2074 6869 7320 6669 6c65 2e0a e of this file.. │ │ │ │ +00267180: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00267190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002671a0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7072 ------+.|i1 : pr │ │ │ │ -002671b0: 6f67 7261 6d50 6174 6873 2322 6766 616e ogramPaths#"gfan │ │ │ │ -002671c0: 2220 3d20 222f 7061 7468 2f74 6f2f 6766 " = "/path/to/gf │ │ │ │ -002671d0: 616e 2f22 2020 2020 2020 2020 2020 2020 an/" │ │ │ │ -002671e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002671f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +002671a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002671b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002671c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +002671d0: 3a20 7072 6f67 7261 6d50 6174 6873 2322 : programPaths#" │ │ │ │ +002671e0: 6766 616e 2220 3d20 222f 7061 7468 2f74 gfan" = "/path/t │ │ │ │ +002671f0: 6f2f 6766 616e 2f22 2020 2020 2020 2020 o/gfan/" │ │ │ │ 00267200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267210: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00267220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00267240: 3120 3d20 2f70 6174 682f 746f 2f67 6661 1 = /path/to/gfa │ │ │ │ -00267250: 6e2f 2020 2020 2020 2020 2020 2020 2020 n/ │ │ │ │ -00267260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267280: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00267290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002672a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267260: 7c0a 7c6f 3120 3d20 2f70 6174 682f 746f |.|o1 = /path/to │ │ │ │ +00267270: 2f67 6661 6e2f 2020 2020 2020 2020 2020 /gfan/ │ │ │ │ +00267280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002672a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 002672b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002672c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002672d0: 2d2d 2b0a 7c69 3220 3a20 6766 616e 203d --+.|i2 : gfan = │ │ │ │ -002672e0: 2066 696e 6450 726f 6772 616d 2822 6766 findProgram("gf │ │ │ │ -002672f0: 616e 222c 2022 6766 616e 205f 7665 7273 an", "gfan _vers │ │ │ │ -00267300: 696f 6e20 2d2d 6865 6c70 222c 2056 6572 ion --help", Ver │ │ │ │ -00267310: 626f 7365 203d 3e20 7472 7565 297c 0a7c bose => true)|.| │ │ │ │ -00267320: 202d 2d20 2f70 6174 682f 746f 2f67 6661 -- /path/to/gfa │ │ │ │ -00267330: 6e2f 6766 616e 2064 6f65 7320 6e6f 7420 n/gfan does not │ │ │ │ -00267340: 6578 6973 7420 2020 2020 2020 2020 2020 exist │ │ │ │ -00267350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267360: 2020 2020 2020 2020 7c0a 7c20 2d2d 202f |.| -- / │ │ │ │ -00267370: 7573 722f 6c69 6265 7865 632f 4d61 6361 usr/libexec/Maca │ │ │ │ -00267380: 756c 6179 322f 6269 6e2f 6766 616e 2064 ulay2/bin/gfan d │ │ │ │ -00267390: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -002673a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002673b0: 2020 207c 0a7c 202d 2d20 2f75 7372 2f6c |.| -- /usr/l │ │ │ │ -002673c0: 6f63 616c 2f73 6269 6e2f 6766 616e 2064 ocal/sbin/gfan d │ │ │ │ -002673d0: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -002673e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002673f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00267400: 7c20 2d2d 202f 7573 722f 6c6f 6361 6c2f | -- /usr/local/ │ │ │ │ -00267410: 6269 6e2f 6766 616e 2064 6f65 7320 6e6f bin/gfan does no │ │ │ │ -00267420: 7420 6578 6973 7420 2020 2020 2020 2020 t exist │ │ │ │ -00267430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267440: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -00267450: 2f75 7372 2f73 6269 6e2f 6766 616e 2064 /usr/sbin/gfan d │ │ │ │ -00267460: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -00267470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267490: 2020 2020 7c0a 7c20 2d2d 202f 7573 722f |.| -- /usr/ │ │ │ │ -002674a0: 6269 6e2f 6766 616e 2065 7869 7374 7320 bin/gfan exists │ │ │ │ -002674b0: 616e 6420 6973 2065 7865 6375 7461 626c and is executabl │ │ │ │ -002674c0: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -002674d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -002674e0: 0a7c 202d 2d20 7275 6e6e 696e 6720 222f .| -- running "/ │ │ │ │ -002674f0: 7573 722f 6269 6e2f 6766 616e 205f 7665 usr/bin/gfan _ve │ │ │ │ -00267500: 7273 696f 6e20 2d2d 6865 6c70 223a 2020 rsion --help": │ │ │ │ -00267510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267520: 2020 2020 2020 2020 2020 7c0a 7c54 6869 |.|Thi │ │ │ │ -00267530: 7320 7072 6f67 7261 6d20 7772 6974 6573 s program writes │ │ │ │ -00267540: 206f 7574 2076 6572 7369 6f6e 2069 6e66 out version inf │ │ │ │ -00267550: 6f72 6d61 7469 6f6e 206f 6620 7468 6520 ormation of the │ │ │ │ -00267560: 4766 616e 2069 6e73 7461 6c6c 6174 696f Gfan installatio │ │ │ │ -00267570: 6e2e 2020 207c 0a7c 202d 2d20 7265 7475 n. |.| -- retu │ │ │ │ -00267580: 726e 2076 616c 7565 3a20 3020 2020 2020 rn value: 0 │ │ │ │ -00267590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002675a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002672d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002672e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002672f0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 6766 ------+.|i2 : gf │ │ │ │ +00267300: 616e 203d 2066 696e 6450 726f 6772 616d an = findProgram │ │ │ │ +00267310: 2822 6766 616e 222c 2022 6766 616e 205f ("gfan", "gfan _ │ │ │ │ +00267320: 7665 7273 696f 6e20 2d2d 6865 6c70 222c version --help", │ │ │ │ +00267330: 2056 6572 626f 7365 203d 3e20 7472 7565 Verbose => true │ │ │ │ +00267340: 297c 0a7c 202d 2d20 2f70 6174 682f 746f )|.| -- /path/to │ │ │ │ +00267350: 2f67 6661 6e2f 6766 616e 2064 6f65 7320 /gfan/gfan does │ │ │ │ +00267360: 6e6f 7420 6578 6973 7420 2020 2020 2020 not exist │ │ │ │ +00267370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267380: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00267390: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ +002673a0: 4d61 6361 756c 6179 322f 6269 6e2f 6766 Macaulay2/bin/gf │ │ │ │ +002673b0: 616e 2064 6f65 7320 6e6f 7420 6578 6973 an does not exis │ │ │ │ +002673c0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +002673d0: 2020 2020 2020 207c 0a7c 202d 2d20 2f75 |.| -- /u │ │ │ │ +002673e0: 7372 2f6c 6f63 616c 2f73 6269 6e2f 6766 sr/local/sbin/gf │ │ │ │ +002673f0: 616e 2064 6f65 7320 6e6f 7420 6578 6973 an does not exis │ │ │ │ +00267400: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00267410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267420: 2020 7c0a 7c20 2d2d 202f 7573 722f 6c6f |.| -- /usr/lo │ │ │ │ +00267430: 6361 6c2f 6269 6e2f 6766 616e 2064 6f65 cal/bin/gfan doe │ │ │ │ +00267440: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +00267450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267460: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00267470: 202d 2d20 2f75 7372 2f73 6269 6e2f 6766 -- /usr/sbin/gf │ │ │ │ +00267480: 616e 2064 6f65 7320 6e6f 7420 6578 6973 an does not exis │ │ │ │ +00267490: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +002674a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002674b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 202f |.| -- / │ │ │ │ +002674c0: 7573 722f 6269 6e2f 6766 616e 2065 7869 usr/bin/gfan exi │ │ │ │ +002674d0: 7374 7320 616e 6420 6973 2065 7865 6375 sts and is execu │ │ │ │ +002674e0: 7461 626c 6520 2020 2020 2020 2020 2020 table │ │ │ │ +002674f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267500: 2020 207c 0a7c 202d 2d20 7275 6e6e 696e |.| -- runnin │ │ │ │ +00267510: 6720 222f 7573 722f 6269 6e2f 6766 616e g "/usr/bin/gfan │ │ │ │ +00267520: 205f 7665 7273 696f 6e20 2d2d 6865 6c70 _version --help │ │ │ │ +00267530: 223a 2020 2020 2020 2020 2020 2020 2020 ": │ │ │ │ +00267540: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00267550: 7c54 6869 7320 7072 6f67 7261 6d20 7772 |This program wr │ │ │ │ +00267560: 6974 6573 206f 7574 2076 6572 7369 6f6e ites out version │ │ │ │ +00267570: 2069 6e66 6f72 6d61 7469 6f6e 206f 6620 information of │ │ │ │ +00267580: 7468 6520 4766 616e 2069 6e73 7461 6c6c the Gfan install │ │ │ │ +00267590: 6174 696f 6e2e 2020 207c 0a7c 202d 2d20 ation. |.| -- │ │ │ │ +002675a0: 7265 7475 726e 2076 616c 7565 3a20 3020 return value: 0 │ │ │ │ 002675b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002675c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +002675c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002675d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002675e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002675e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 002675f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267600: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00267610: 203d 2067 6661 6e20 2020 2020 2020 2020 = gfan │ │ │ │ -00267620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267620: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00267630: 0a7c 6f32 203d 2067 6661 6e20 2020 2020 .|o2 = gfan │ │ │ │ 00267640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267650: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00267650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00267660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267670: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00267680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00267690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002676a0: 207c 0a7c 6f32 203a 2050 726f 6772 616d |.|o2 : Program │ │ │ │ +002676a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002676b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002676c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002676d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002676e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -002676f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002676c0: 2020 2020 207c 0a7c 6f32 203a 2050 726f |.|o2 : Pro │ │ │ │ +002676d0: 6772 616d 2020 2020 2020 2020 2020 2020 gram │ │ │ │ +002676e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002676f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267710: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00267720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267730: 2d2d 2d2d 2d2d 2d2b 0a0a 4966 2063 6d64 -------+..If cmd │ │ │ │ -00267740: 2069 7320 6e6f 7420 7072 6f76 6964 6564 is not provided │ │ │ │ -00267750: 2c20 7468 656e 206e 616d 6520 6973 2072 , then name is r │ │ │ │ -00267760: 756e 2077 6974 6820 7468 6520 636f 6d6d un with the comm │ │ │ │ -00267770: 6f6e 202d 2d76 6572 7369 6f6e 2063 6f6d on --version com │ │ │ │ -00267780: 6d61 6e64 206c 696e 650a 6f70 7469 6f6e mand line.option │ │ │ │ -00267790: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ -002677a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -002677b0: 0a7c 6933 203a 2066 696e 6450 726f 6772 .|i3 : findProgr │ │ │ │ -002677c0: 616d 2022 6e6f 726d 616c 697a 227c 0a7c am "normaliz"|.| │ │ │ │ -002677d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002677e0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ -002677f0: 203d 206e 6f72 6d61 6c69 7a20 2020 2020 = normaliz │ │ │ │ -00267800: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00267810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267820: 2020 2020 2020 207c 0a7c 6f33 203a 2050 |.|o3 : P │ │ │ │ -00267830: 726f 6772 616d 2020 2020 2020 2020 2020 rogram │ │ │ │ -00267840: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00267850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267860: 2d2d 2d2b 0a0a 4f6e 6520 7072 6f67 7261 ---+..One progra │ │ │ │ -00267870: 6d20 7468 6174 2069 7320 7368 6970 7065 m that is shippe │ │ │ │ -00267880: 6420 7769 7468 2061 2076 6172 6965 7479 d with a variety │ │ │ │ -00267890: 206f 6620 7072 6566 6978 6573 2069 6e20 of prefixes in │ │ │ │ -002678a0: 6469 6666 6572 656e 740a 6469 7374 7269 different.distri │ │ │ │ -002678b0: 6275 7469 6f6e 7320 616e 6420 666f 7220 butions and for │ │ │ │ -002678c0: 7768 6963 6820 7468 6520 5072 6566 6978 which the Prefix │ │ │ │ -002678d0: 206f 7074 696f 6e20 6973 2075 7365 6675 option is usefu │ │ │ │ -002678e0: 6c20 6973 2054 4f50 434f 4d3a 0a0a 2b2d l is TOPCOM:..+- │ │ │ │ -002678f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4966 -----------+..If │ │ │ │ +00267760: 2063 6d64 2069 7320 6e6f 7420 7072 6f76 cmd is not prov │ │ │ │ +00267770: 6964 6564 2c20 7468 656e 206e 616d 6520 ided, then name │ │ │ │ +00267780: 6973 2072 756e 2077 6974 6820 7468 6520 is run with the │ │ │ │ +00267790: 636f 6d6d 6f6e 202d 2d76 6572 7369 6f6e common --version │ │ │ │ +002677a0: 2063 6f6d 6d61 6e64 206c 696e 650a 6f70 command line.op │ │ │ │ +002677b0: 7469 6f6e 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d tion...+-------- │ │ │ │ +002677c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002677d0: 2d2d 2d2b 0a7c 6933 203a 2066 696e 6450 ---+.|i3 : findP │ │ │ │ +002677e0: 726f 6772 616d 2022 6e6f 726d 616c 697a rogram "normaliz │ │ │ │ +002677f0: 227c 0a7c 2020 2020 2020 2020 2020 2020 "|.| │ │ │ │ +00267800: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00267810: 0a7c 6f33 203d 206e 6f72 6d61 6c69 7a20 .|o3 = normaliz │ │ │ │ +00267820: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00267830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267840: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +00267850: 203a 2050 726f 6772 616d 2020 2020 2020 : Program │ │ │ │ +00267860: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00267870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267880: 2d2d 2d2d 2d2d 2d2b 0a0a 4f6e 6520 7072 -------+..One pr │ │ │ │ +00267890: 6f67 7261 6d20 7468 6174 2069 7320 7368 ogram that is sh │ │ │ │ +002678a0: 6970 7065 6420 7769 7468 2061 2076 6172 ipped with a var │ │ │ │ +002678b0: 6965 7479 206f 6620 7072 6566 6978 6573 iety of prefixes │ │ │ │ +002678c0: 2069 6e20 6469 6666 6572 656e 740a 6469 in different.di │ │ │ │ +002678d0: 7374 7269 6275 7469 6f6e 7320 616e 6420 stributions and │ │ │ │ +002678e0: 666f 7220 7768 6963 6820 7468 6520 5072 for which the Pr │ │ │ │ +002678f0: 6566 6978 206f 7074 696f 6e20 6973 2075 efix option is u │ │ │ │ +00267900: 7365 6675 6c20 6973 2054 4f50 434f 4d3a seful is TOPCOM: │ │ │ │ +00267910: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 00267920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00267930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00267940: 3420 3a20 6669 6e64 5072 6f67 7261 6d28 4 : findProgram( │ │ │ │ -00267950: 2274 6f70 636f 6d22 2c20 2263 7562 6520 "topcom", "cube │ │ │ │ -00267960: 3322 2c20 5665 7262 6f73 6520 3d3e 2074 3", Verbose => t │ │ │ │ -00267970: 7275 652c 2050 7265 6669 7820 3d3e 207b rue, Prefix => { │ │ │ │ -00267980: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267990: 2020 2020 2020 2822 2e2a 222c 2022 746f (".*", "to │ │ │ │ -002679a0: 7063 6f6d 2d22 292c 2020 2020 2020 2020 pcom-"), │ │ │ │ -002679b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002679c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002679d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -002679e0: 2020 2020 2020 2822 5e28 6372 6f73 737c ("^(cross| │ │ │ │ -002679f0: 6375 6265 7c63 7963 6c69 637c 6879 7065 cube|cyclic|hype │ │ │ │ -00267a00: 7273 696d 706c 6578 7c6c 6174 7469 6365 rsimplex|lattice │ │ │ │ -00267a10: 2924 222c 2022 544f 5043 4f4d 2d22 292c )$", "TOPCOM-"), │ │ │ │ -00267a20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267a30: 2020 2020 2020 2822 5e63 7562 6524 222c ("^cube$", │ │ │ │ -00267a40: 2022 746f 7063 6f6d 5f22 297d 2920 2020 "topcom_")}) │ │ │ │ -00267a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267a70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267a80: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ -00267a90: 4d61 6361 756c 6179 322f 6269 6e2f 6375 Macaulay2/bin/cu │ │ │ │ -00267aa0: 6265 2064 6f65 7320 6e6f 7420 6578 6973 be does not exis │ │ │ │ -00267ab0: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00267ac0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267ad0: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ -00267ae0: 4d61 6361 756c 6179 322f 6269 6e2f 746f Macaulay2/bin/to │ │ │ │ -00267af0: 7063 6f6d 2d63 7562 6520 646f 6573 206e pcom-cube does n │ │ │ │ -00267b00: 6f74 2065 7869 7374 2020 2020 2020 2020 ot exist │ │ │ │ -00267b10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267b20: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ -00267b30: 4d61 6361 756c 6179 322f 6269 6e2f 544f Macaulay2/bin/TO │ │ │ │ -00267b40: 5043 4f4d 2d63 7562 6520 646f 6573 206e PCOM-cube does n │ │ │ │ -00267b50: 6f74 2065 7869 7374 2020 2020 2020 2020 ot exist │ │ │ │ -00267b60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267b70: 2d2d 202f 7573 722f 6c69 6265 7865 632f -- /usr/libexec/ │ │ │ │ -00267b80: 4d61 6361 756c 6179 322f 6269 6e2f 746f Macaulay2/bin/to │ │ │ │ -00267b90: 7063 6f6d 5f63 7562 6520 646f 6573 206e pcom_cube does n │ │ │ │ -00267ba0: 6f74 2065 7869 7374 2020 2020 2020 2020 ot exist │ │ │ │ -00267bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267bc0: 2d2d 202f 7573 722f 6c6f 6361 6c2f 7362 -- /usr/local/sb │ │ │ │ -00267bd0: 696e 2f63 7562 6520 646f 6573 206e 6f74 in/cube does not │ │ │ │ -00267be0: 2065 7869 7374 2020 2020 2020 2020 2020 exist │ │ │ │ -00267bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267c00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267c10: 2d2d 202f 7573 722f 6c6f 6361 6c2f 7362 -- /usr/local/sb │ │ │ │ -00267c20: 696e 2f74 6f70 636f 6d2d 6375 6265 2064 in/topcom-cube d │ │ │ │ -00267c30: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -00267c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267c50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267c60: 2d2d 202f 7573 722f 6c6f 6361 6c2f 7362 -- /usr/local/sb │ │ │ │ -00267c70: 696e 2f54 4f50 434f 4d2d 6375 6265 2064 in/TOPCOM-cube d │ │ │ │ -00267c80: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -00267c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267cb0: 2d2d 202f 7573 722f 6c6f 6361 6c2f 7362 -- /usr/local/sb │ │ │ │ -00267cc0: 696e 2f74 6f70 636f 6d5f 6375 6265 2064 in/topcom_cube d │ │ │ │ -00267cd0: 6f65 7320 6e6f 7420 6578 6973 7420 2020 oes not exist │ │ │ │ -00267ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267d00: 2d2d 202f 7573 722f 6c6f 6361 6c2f 6269 -- /usr/local/bi │ │ │ │ -00267d10: 6e2f 6375 6265 2064 6f65 7320 6e6f 7420 n/cube does not │ │ │ │ -00267d20: 6578 6973 7420 2020 2020 2020 2020 2020 exist │ │ │ │ -00267d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267d40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267d50: 2d2d 202f 7573 722f 6c6f 6361 6c2f 6269 -- /usr/local/bi │ │ │ │ -00267d60: 6e2f 746f 7063 6f6d 2d63 7562 6520 646f n/topcom-cube do │ │ │ │ -00267d70: 6573 206e 6f74 2065 7869 7374 2020 2020 es not exist │ │ │ │ -00267d80: 2020 2020 2020 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-00267e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267e80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267e90: 2d2d 202f 7573 722f 7362 696e 2f74 6f70 -- /usr/sbin/top │ │ │ │ -00267ea0: 636f 6d2d 6375 6265 2064 6f65 7320 6e6f com-cube does no │ │ │ │ -00267eb0: 7420 6578 6973 7420 2020 2020 2020 2020 t exist │ │ │ │ -00267ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267ed0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267ee0: 2d2d 202f 7573 722f 7362 696e 2f54 4f50 -- /usr/sbin/TOP │ │ │ │ -00267ef0: 434f 4d2d 6375 6265 2064 6f65 7320 6e6f COM-cube does no │ │ │ │ -00267f00: 7420 6578 6973 7420 2020 2020 2020 2020 t exist │ │ │ │ -00267f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267f20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267f30: 2d2d 202f 7573 722f 7362 696e 2f74 6f70 -- /usr/sbin/top │ │ │ │ -00267f40: 636f 6d5f 6375 6265 2064 6f65 7320 6e6f com_cube does no │ │ │ │ -00267f50: 7420 6578 6973 7420 2020 2020 2020 2020 t exist │ │ │ │ -00267f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267f70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267f80: 2d2d 202f 7573 722f 6269 6e2f 6375 6265 -- /usr/bin/cube │ │ │ │ -00267f90: 2064 6f65 7320 6e6f 7420 6578 6973 7420 does not exist │ │ │ │ -00267fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00267fc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00267fd0: 2d2d 202f 7573 722f 6269 6e2f 746f 7063 -- /usr/bin/topc │ │ │ │ -00267fe0: 6f6d 2d63 7562 6520 6578 6973 7473 2061 om-cube exists a │ │ │ │ -00267ff0: 6e64 2069 7320 6578 6563 7574 6162 6c65 nd is executable │ │ │ │ -00268000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00268020: 2d2d 2072 756e 6e69 6e67 2022 2f75 7372 -- running "/usr │ │ │ │ -00268030: 2f62 696e 2f74 6f70 636f 6d2d 6375 6265 /bin/topcom-cube │ │ │ │ -00268040: 2033 223a 2020 2020 2020 2020 2020 2020 3": │ │ │ │ -00268050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268060: 2020 2020 2020 2020 2020 2020 7c0a 7c5b |.|[ │ │ │ │ -00268070: 5b30 2c30 2c30 2c31 5d2c 5b31 2c30 2c30 [0,0,0,1],[1,0,0 │ │ │ │ -00268080: 2c31 5d2c 5b30 2c31 2c30 2c31 5d2c 5b31 ,1],[0,1,0,1],[1 │ │ │ │ -00268090: 2c31 2c30 2c31 5d2c 5b30 2c30 2c31 2c31 ,1,0,1],[0,0,1,1 │ │ │ │ -002680a0: 5d2c 5b31 2c30 2c31 2c31 5d2c 5b30 2c31 ],[1,0,1,1],[0,1 │ │ │ │ -002680b0: 2c31 2c31 5d2c 5b31 2c31 2c31 7c0a 7c5b ,1,1],[1,1,1|.|[ │ │ │ │ -002680c0: 5b37 2c36 2c35 2c34 2c33 2c32 2c31 2c30 [7,6,5,4,3,2,1,0 │ │ │ │ -002680d0: 5d2c 5b36 2c37 2c34 2c35 2c32 2c33 2c30 ],[6,7,4,5,2,3,0 │ │ │ │ -002680e0: 2c31 5d2c 5b34 2c36 2c35 2c37 2c30 2c32 ,1],[4,6,5,7,0,2 │ │ │ │ -002680f0: 2c31 2c33 5d2c 5b30 2c34 2c32 2c36 2c31 ,1,3],[0,4,2,6,1 │ │ │ │ -00268100: 2c35 2c33 2c37 5d5d 2020 2020 7c0a 7c20 ,5,3,7]] |.| │ │ │ │ -00268110: 2d2d 2072 6574 7572 6e20 7661 6c75 653a -- return value: │ │ │ │ -00268120: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -00268130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268150: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00267930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00267960: 2b0a 7c69 3420 3a20 6669 6e64 5072 6f67 +.|i4 : findProg │ │ │ │ +00267970: 7261 6d28 2274 6f70 636f 6d22 2c20 2263 ram("topcom", "c │ │ │ │ +00267980: 7562 6520 3322 2c20 5665 7262 6f73 6520 ube 3", Verbose │ │ │ │ +00267990: 3d3e 2074 7275 652c 2050 7265 6669 7820 => true, Prefix │ │ │ │ +002679a0: 3d3e 207b 2020 2020 2020 2020 2020 2020 => { │ │ │ │ +002679b0: 7c0a 7c20 2020 2020 2020 2822 2e2a 222c |.| (".*", │ │ │ │ +002679c0: 2022 746f 7063 6f6d 2d22 292c 2020 2020 "topcom-"), │ │ │ │ +002679d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002679e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002679f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267a00: 7c0a 7c20 2020 2020 2020 2822 5e28 6372 |.| ("^(cr │ │ │ │ +00267a10: 6f73 737c 6375 6265 7c63 7963 6c69 637c oss|cube|cyclic| │ │ │ │ +00267a20: 6879 7065 7273 696d 706c 6578 7c6c 6174 hypersimplex|lat │ │ │ │ +00267a30: 7469 6365 2924 222c 2022 544f 5043 4f4d tice)$", "TOPCOM │ │ │ │ +00267a40: 2d22 292c 2020 2020 2020 2020 2020 2020 -"), │ │ │ │ +00267a50: 7c0a 7c20 2020 2020 2020 2822 5e63 7562 |.| ("^cub │ │ │ │ +00267a60: 6524 222c 2022 746f 7063 6f6d 5f22 297d e$", "topcom_")} │ │ │ │ +00267a70: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00267a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267aa0: 7c0a 7c20 2d2d 202f 7573 722f 6c69 6265 |.| -- /usr/libe │ │ │ │ +00267ab0: 7865 632f 4d61 6361 756c 6179 322f 6269 xec/Macaulay2/bi │ │ │ │ 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│ +00267d70: 7c0a 7c20 2d2d 202f 7573 722f 6c6f 6361 |.| -- /usr/loca │ │ │ │ +00267d80: 6c2f 6269 6e2f 746f 7063 6f6d 2d63 7562 l/bin/topcom-cub │ │ │ │ +00267d90: 6520 646f 6573 206e 6f74 2065 7869 7374 e does not exist │ │ │ │ +00267da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267dc0: 7c0a 7c20 2d2d 202f 7573 722f 6c6f 6361 |.| -- /usr/loca │ │ │ │ +00267dd0: 6c2f 6269 6e2f 544f 5043 4f4d 2d63 7562 l/bin/TOPCOM-cub │ │ │ │ +00267de0: 6520 646f 6573 206e 6f74 2065 7869 7374 e does not exist │ │ │ │ +00267df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267e10: 7c0a 7c20 2d2d 202f 7573 722f 6c6f 6361 |.| -- /usr/loca │ │ │ │ +00267e20: 6c2f 6269 6e2f 746f 7063 6f6d 5f63 7562 l/bin/topcom_cub │ │ │ │ +00267e30: 6520 646f 6573 206e 6f74 2065 7869 7374 e does not exist │ │ │ │ +00267e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267e60: 7c0a 7c20 2d2d 202f 7573 722f 7362 696e |.| -- /usr/sbin │ │ │ │ +00267e70: 2f63 7562 6520 646f 6573 206e 6f74 2065 /cube does not e │ │ │ │ +00267e80: 7869 7374 2020 2020 2020 2020 2020 2020 xist │ │ │ │ +00267e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267eb0: 7c0a 7c20 2d2d 202f 7573 722f 7362 696e |.| -- /usr/sbin │ │ │ │ +00267ec0: 2f74 6f70 636f 6d2d 6375 6265 2064 6f65 /topcom-cube doe │ │ │ │ +00267ed0: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +00267ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267f00: 7c0a 7c20 2d2d 202f 7573 722f 7362 696e |.| -- /usr/sbin │ │ │ │ +00267f10: 2f54 4f50 434f 4d2d 6375 6265 2064 6f65 /TOPCOM-cube doe │ │ │ │ +00267f20: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +00267f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267f50: 7c0a 7c20 2d2d 202f 7573 722f 7362 696e |.| -- /usr/sbin │ │ │ │ +00267f60: 2f74 6f70 636f 6d5f 6375 6265 2064 6f65 /topcom_cube doe │ │ │ │ +00267f70: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +00267f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267fa0: 7c0a 7c20 2d2d 202f 7573 722f 6269 6e2f |.| -- /usr/bin/ │ │ │ │ +00267fb0: 6375 6265 2064 6f65 7320 6e6f 7420 6578 cube does not ex │ │ │ │ +00267fc0: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +00267fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00267ff0: 7c0a 7c20 2d2d 202f 7573 722f 6269 6e2f |.| -- /usr/bin/ │ │ │ │ +00268000: 746f 7063 6f6d 2d63 7562 6520 6578 6973 topcom-cube exis │ │ │ │ +00268010: 7473 2061 6e64 2069 7320 6578 6563 7574 ts and is execut │ │ │ │ +00268020: 6162 6c65 2020 2020 2020 2020 2020 2020 able │ │ │ │ +00268030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268040: 7c0a 7c20 2d2d 2072 756e 6e69 6e67 2022 |.| -- running " │ │ │ │ +00268050: 2f75 7372 2f62 696e 2f74 6f70 636f 6d2d /usr/bin/topcom- │ │ │ │ +00268060: 6375 6265 2033 223a 2020 2020 2020 2020 cube 3": │ │ │ │ +00268070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268090: 7c0a 7c5b 5b30 2c30 2c30 2c31 5d2c 5b31 |.|[[0,0,0,1],[1 │ │ │ │ +002680a0: 2c30 2c30 2c31 5d2c 5b30 2c31 2c30 2c31 ,0,0,1],[0,1,0,1 │ │ │ │ +002680b0: 5d2c 5b31 2c31 2c30 2c31 5d2c 5b30 2c30 ],[1,1,0,1],[0,0 │ │ │ │ +002680c0: 2c31 2c31 5d2c 5b31 2c30 2c31 2c31 5d2c ,1,1],[1,0,1,1], │ │ │ │ +002680d0: 5b30 2c31 2c31 2c31 5d2c 5b31 2c31 2c31 [0,1,1,1],[1,1,1 │ │ │ │ +002680e0: 7c0a 7c5b 5b37 2c36 2c35 2c34 2c33 2c32 |.|[[7,6,5,4,3,2 │ │ │ │ +002680f0: 2c31 2c30 5d2c 5b36 2c37 2c34 2c35 2c32 ,1,0],[6,7,4,5,2 │ │ │ │ +00268100: 2c33 2c30 2c31 5d2c 5b34 2c36 2c35 2c37 ,3,0,1],[4,6,5,7 │ │ │ │ +00268110: 2c30 2c32 2c31 2c33 5d2c 5b30 2c34 2c32 ,0,2,1,3],[0,4,2 │ │ │ │ +00268120: 2c36 2c31 2c35 2c33 2c37 5d5d 2020 2020 ,6,1,5,3,7]] │ │ │ │ +00268130: 7c0a 7c20 2d2d 2072 6574 7572 6e20 7661 |.| -- return va │ │ │ │ +00268140: 6c75 653a 2030 2020 2020 2020 2020 2020 lue: 0 │ │ │ │ +00268150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00268160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00268170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268180: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00268190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002681a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -002681b0: 3420 3d20 746f 7063 6f6d 2020 2020 2020 4 = topcom │ │ │ │ +002681a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002681b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002681c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002681d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002681d0: 7c0a 7c6f 3420 3d20 746f 7063 6f6d 2020 |.|o4 = topcom │ │ │ │ 002681e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002681f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +002681f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00268200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00268210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268220: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00268230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00268250: 3420 3a20 5072 6f67 7261 6d20 2020 2020 4 : Program │ │ │ │ +00268240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00268260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268270: 7c0a 7c6f 3420 3a20 5072 6f67 7261 6d20 |.|o4 : Program │ │ │ │ 00268280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268290: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ -002682a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002682b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002682c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002682a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002682b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002682c0: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------- │ │ │ │ 002682d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002682e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c2c ------------|.|, │ │ │ │ -002682f0: 315d 5d20 2020 2020 2020 2020 2020 2020 1]] │ │ │ │ -00268300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002682e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002682f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268310: 7c0a 7c2c 315d 5d20 2020 2020 2020 2020 |.|,1]] │ │ │ │ 00268320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268330: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00268340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268360: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00268370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e ------------+..N │ │ │ │ -00268390: 6f74 6520 7468 6174 2077 6865 6e20 7573 ote that when us │ │ │ │ -002683a0: 696e 6720 7468 6520 4d69 6e69 6d75 6d56 ing the MinimumV │ │ │ │ -002683b0: 6572 7369 6f6e 206f 7074 696f 6e2c 2074 ersion option, t │ │ │ │ -002683c0: 6865 2063 6f6d 6d61 6e64 2075 7365 6420 he command used │ │ │ │ -002683d0: 746f 206f 6274 6169 6e20 7468 650a 6375 to obtain the.cu │ │ │ │ -002683e0: 7272 656e 7420 7665 7273 696f 6e20 6e75 rrent version nu │ │ │ │ -002683f0: 6d62 6572 206d 7573 7420 7265 6d6f 7665 mber must remove │ │ │ │ -00268400: 2065 7665 7279 7468 696e 6720 6578 6365 everything exce │ │ │ │ -00268410: 7074 2074 6865 2076 6572 7369 6f6e 206e pt the version n │ │ │ │ -00268420: 756d 6265 7220 6974 7365 6c66 0a61 6e64 umber itself.and │ │ │ │ -00268430: 2061 6e79 206c 6561 6469 6e67 206f 7220 any leading or │ │ │ │ -00268440: 7472 6169 6c69 6e67 2077 6869 7465 7370 trailing whitesp │ │ │ │ -00268450: 6163 652e 2020 5069 7069 6e67 2077 6974 ace. Piping wit │ │ │ │ -00268460: 6820 7374 616e 6461 7264 2055 4e49 5820 h standard UNIX │ │ │ │ -00268470: 7574 696c 6974 6965 730a 7375 6368 2061 utilities.such a │ │ │ │ -00268480: 7320 7365 642c 2068 6561 642c 2074 6169 s sed, head, tai │ │ │ │ -00268490: 6c2c 2063 7574 2c20 616e 6420 7472 206d l, cut, and tr m │ │ │ │ -002684a0: 6179 2062 6520 7573 6566 756c 2e0a 0a2b ay be useful...+ │ │ │ │ -002684b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002684c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002684d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002683a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002683b0: 2b0a 0a4e 6f74 6520 7468 6174 2077 6865 +..Note that whe │ │ │ │ +002683c0: 6e20 7573 696e 6720 7468 6520 4d69 6e69 n using the Mini │ │ │ │ +002683d0: 6d75 6d56 6572 7369 6f6e 206f 7074 696f mumVersion optio │ │ │ │ +002683e0: 6e2c 2074 6865 2063 6f6d 6d61 6e64 2075 n, the command u │ │ │ │ +002683f0: 7365 6420 746f 206f 6274 6169 6e20 7468 sed to obtain th │ │ │ │ +00268400: 650a 6375 7272 656e 7420 7665 7273 696f e.current versio │ │ │ │ +00268410: 6e20 6e75 6d62 6572 206d 7573 7420 7265 n number must re │ │ │ │ +00268420: 6d6f 7665 2065 7665 7279 7468 696e 6720 move everything │ │ │ │ +00268430: 6578 6365 7074 2074 6865 2076 6572 7369 except the versi │ │ │ │ +00268440: 6f6e 206e 756d 6265 7220 6974 7365 6c66 on number itself │ │ │ │ +00268450: 0a61 6e64 2061 6e79 206c 6561 6469 6e67 .and any leading │ │ │ │ +00268460: 206f 7220 7472 6169 6c69 6e67 2077 6869 or trailing whi │ │ │ │ +00268470: 7465 7370 6163 652e 2020 5069 7069 6e67 tespace. Piping │ │ │ │ +00268480: 2077 6974 6820 7374 616e 6461 7264 2055 with standard U │ │ │ │ +00268490: 4e49 5820 7574 696c 6974 6965 730a 7375 NIX utilities.su │ │ │ │ +002684a0: 6368 2061 7320 7365 642c 2068 6561 642c ch as sed, head, │ │ │ │ +002684b0: 2074 6169 6c2c 2063 7574 2c20 616e 6420 tail, cut, and │ │ │ │ +002684c0: 7472 206d 6179 2062 6520 7573 6566 756c tr may be useful │ │ │ │ +002684d0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 002684e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002684f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2066 696e -----+.|i5 : fin │ │ │ │ -00268500: 6450 726f 6772 616d 2822 6766 616e 222c dProgram("gfan", │ │ │ │ -00268510: 2022 6766 616e 205f 7665 7273 696f 6e20 "gfan _version │ │ │ │ -00268520: 2d2d 6865 6c70 222c 2056 6572 626f 7365 --help", Verbose │ │ │ │ -00268530: 203d 3e20 7472 7565 2c20 2020 207c 0a7c => true, |.| │ │ │ │ -00268540: 2020 2020 2020 204d 696e 696d 756d 5665 MinimumVe │ │ │ │ -00268550: 7273 696f 6e20 3d3e 2028 2230 2e35 222c rsion => ("0.5", │ │ │ │ -00268560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268580: 2020 2020 207c 0a7c 2020 2020 2020 2267 |.| "g │ │ │ │ -00268590: 6661 6e20 5f76 6572 7369 6f6e 207c 2068 fan _version | h │ │ │ │ -002685a0: 6561 6420 2d32 207c 2074 6169 6c20 2d31 ead -2 | tail -1 │ │ │ │ -002685b0: 207c 2073 6564 2027 732f 6766 616e 2f2f | sed 's/gfan// │ │ │ │ -002685c0: 2722 2929 2020 2020 2020 2020 207c 0a7c '")) |.| │ │ │ │ -002685d0: 202d 2d20 2f70 6174 682f 746f 2f67 6661 -- /path/to/gfa │ │ │ │ -002685e0: 6e2f 6766 616e 2064 6f65 7320 6e6f 7420 n/gfan does not │ │ │ │ -002685f0: 6578 6973 7420 2020 2020 2020 2020 2020 exist │ │ │ │ -00268600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268610: 2020 2020 207c 0a7c 202d 2d20 2f75 7372 |.| -- /usr │ │ │ │ -00268620: 2f6c 6962 6578 6563 2f4d 6163 6175 6c61 /libexec/Macaula │ │ │ │ -00268630: 7932 2f62 696e 2f67 6661 6e20 646f 6573 y2/bin/gfan does │ │ │ │ -00268640: 206e 6f74 2065 7869 7374 2020 2020 2020 not exist │ │ │ │ -00268650: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00268660: 202d 2d20 2f75 7372 2f6c 6f63 616c 2f73 -- /usr/local/s │ │ │ │ -00268670: 6269 6e2f 6766 616e 2064 6f65 7320 6e6f bin/gfan does no │ │ │ │ -00268680: 7420 6578 6973 7420 2020 2020 2020 2020 t exist │ │ │ │ -00268690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002686a0: 2020 2020 207c 0a7c 202d 2d20 2f75 7372 |.| -- /usr │ │ │ │ -002686b0: 2f6c 6f63 616c 2f62 696e 2f67 6661 6e20 /local/bin/gfan │ │ │ │ -002686c0: 646f 6573 206e 6f74 2065 7869 7374 2020 does not exist │ │ │ │ -002686d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002686e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -002686f0: 202d 2d20 2f75 7372 2f73 6269 6e2f 6766 -- /usr/sbin/gf │ │ │ │ -00268700: 616e 2064 6f65 7320 6e6f 7420 6578 6973 an does not exis │ │ │ │ -00268710: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ -00268720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268730: 2020 2020 207c 0a7c 202d 2d20 2f75 7372 |.| -- /usr │ │ │ │ -00268740: 2f62 696e 2f67 6661 6e20 6578 6973 7473 /bin/gfan exists │ │ │ │ -00268750: 2061 6e64 2069 7320 6578 6563 7574 6162 and is executab │ │ │ │ -00268760: 6c65 2020 2020 2020 2020 2020 2020 2020 le │ │ │ │ -00268770: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00268780: 202d 2d20 7275 6e6e 696e 6720 222f 7573 -- running "/us │ │ │ │ -00268790: 722f 6269 6e2f 6766 616e 205f 7665 7273 r/bin/gfan _vers │ │ │ │ -002687a0: 696f 6e20 2d2d 6865 6c70 223a 2020 2020 ion --help": │ │ │ │ -002687b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002687c0: 2020 2020 207c 0a7c 5468 6973 2070 726f |.|This pro │ │ │ │ -002687d0: 6772 616d 2077 7269 7465 7320 6f75 7420 gram writes out │ │ │ │ -002687e0: 7665 7273 696f 6e20 696e 666f 726d 6174 version informat │ │ │ │ -002687f0: 696f 6e20 6f66 2074 6865 2047 6661 6e20 ion of the Gfan │ │ │ │ -00268800: 696e 7374 616c 6c61 7469 6f6e 2e7c 0a7c installation.|.| │ │ │ │ -00268810: 202d 2d20 7265 7475 726e 2076 616c 7565 -- return value │ │ │ │ -00268820: 3a20 3020 2020 2020 2020 2020 2020 2020 : 0 │ │ │ │ -00268830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268850: 2020 2020 207c 0a7c 202d 2d20 666f 756e |.| -- foun │ │ │ │ -00268860: 6420 7665 7273 696f 6e20 302e 3862 6574 d version 0.8bet │ │ │ │ -00268870: 6120 3e3d 2030 2e35 2020 2020 2020 2020 a >= 0.5 │ │ │ │ -00268880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268890: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +002684f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268510: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ +00268520: 2066 696e 6450 726f 6772 616d 2822 6766 findProgram("gf │ │ │ │ +00268530: 616e 222c 2022 6766 616e 205f 7665 7273 an", "gfan _vers │ │ │ │ +00268540: 696f 6e20 2d2d 6865 6c70 222c 2056 6572 ion --help", Ver │ │ │ │ +00268550: 626f 7365 203d 3e20 7472 7565 2c20 2020 bose => true, │ │ │ │ +00268560: 207c 0a7c 2020 2020 2020 204d 696e 696d |.| Minim │ │ │ │ +00268570: 756d 5665 7273 696f 6e20 3d3e 2028 2230 umVersion => ("0 │ │ │ │ +00268580: 2e35 222c 2020 2020 2020 2020 2020 2020 .5", │ │ │ │ +00268590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002685a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +002685b0: 2020 2267 6661 6e20 5f76 6572 7369 6f6e "gfan _version │ │ │ │ +002685c0: 207c 2068 6561 6420 2d32 207c 2074 6169 | head -2 | tai │ │ │ │ +002685d0: 6c20 2d31 207c 2073 6564 2027 732f 6766 l -1 | sed 's/gf │ │ │ │ +002685e0: 616e 2f2f 2722 2929 2020 2020 2020 2020 an//'")) │ │ │ │ +002685f0: 207c 0a7c 202d 2d20 2f70 6174 682f 746f |.| -- /path/to │ │ │ │ +00268600: 2f67 6661 6e2f 6766 616e 2064 6f65 7320 /gfan/gfan does │ │ │ │ +00268610: 6e6f 7420 6578 6973 7420 2020 2020 2020 not exist │ │ │ │ +00268620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268630: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +00268640: 2f75 7372 2f6c 6962 6578 6563 2f4d 6163 /usr/libexec/Mac │ │ │ │ +00268650: 6175 6c61 7932 2f62 696e 2f67 6661 6e20 aulay2/bin/gfan │ │ │ │ +00268660: 646f 6573 206e 6f74 2065 7869 7374 2020 does not exist │ │ │ │ +00268670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268680: 207c 0a7c 202d 2d20 2f75 7372 2f6c 6f63 |.| -- /usr/loc │ │ │ │ +00268690: 616c 2f73 6269 6e2f 6766 616e 2064 6f65 al/sbin/gfan doe │ │ │ │ +002686a0: 7320 6e6f 7420 6578 6973 7420 2020 2020 s not exist │ │ │ │ +002686b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002686c0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +002686d0: 2f75 7372 2f6c 6f63 616c 2f62 696e 2f67 /usr/local/bin/g │ │ │ │ +002686e0: 6661 6e20 646f 6573 206e 6f74 2065 7869 fan does not exi │ │ │ │ +002686f0: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ +00268700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268710: 207c 0a7c 202d 2d20 2f75 7372 2f73 6269 |.| -- /usr/sbi │ │ │ │ +00268720: 6e2f 6766 616e 2064 6f65 7320 6e6f 7420 n/gfan does not │ │ │ │ +00268730: 6578 6973 7420 2020 2020 2020 2020 2020 exist │ │ │ │ +00268740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268750: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +00268760: 2f75 7372 2f62 696e 2f67 6661 6e20 6578 /usr/bin/gfan ex │ │ │ │ +00268770: 6973 7473 2061 6e64 2069 7320 6578 6563 ists and is exec │ │ │ │ +00268780: 7574 6162 6c65 2020 2020 2020 2020 2020 utable │ │ │ │ +00268790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002687a0: 207c 0a7c 202d 2d20 7275 6e6e 696e 6720 |.| -- running │ │ │ │ +002687b0: 222f 7573 722f 6269 6e2f 6766 616e 205f "/usr/bin/gfan _ │ │ │ │ +002687c0: 7665 7273 696f 6e20 2d2d 6865 6c70 223a version --help": │ │ │ │ +002687d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002687e0: 2020 2020 2020 2020 207c 0a7c 5468 6973 |.|This │ │ │ │ +002687f0: 2070 726f 6772 616d 2077 7269 7465 7320 program writes │ │ │ │ +00268800: 6f75 7420 7665 7273 696f 6e20 696e 666f out version info │ │ │ │ +00268810: 726d 6174 696f 6e20 6f66 2074 6865 2047 rmation of the G │ │ │ │ +00268820: 6661 6e20 696e 7374 616c 6c61 7469 6f6e fan installation │ │ │ │ +00268830: 2e7c 0a7c 202d 2d20 7265 7475 726e 2076 .|.| -- return v │ │ │ │ +00268840: 616c 7565 3a20 3020 2020 2020 2020 2020 alue: 0 │ │ │ │ +00268850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268870: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +00268880: 666f 756e 6420 7665 7273 696f 6e20 302e found version 0. │ │ │ │ +00268890: 3862 6574 6120 3e3d 2030 2e35 2020 2020 8beta >= 0.5 │ │ │ │ 002688a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002688b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002688c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002688c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 002688d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002688e0: 2020 2020 207c 0a7c 6f35 203d 2067 6661 |.|o5 = gfa │ │ │ │ -002688f0: 6e20 2020 2020 2020 2020 2020 2020 2020 n │ │ │ │ -00268900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268920: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +002688e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002688f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268900: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +00268910: 2067 6661 6e20 2020 2020 2020 2020 2020 gfan │ │ │ │ +00268920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00268930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00268940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268950: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00268960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00268970: 2020 2020 207c 0a7c 6f35 203a 2050 726f |.|o5 : Pro │ │ │ │ -00268980: 6772 616d 2020 2020 2020 2020 2020 2020 gram │ │ │ │ -00268990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002689a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002689b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -002689c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002689d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002689e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00268990: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ +002689a0: 2050 726f 6772 616d 2020 2020 2020 2020 Program │ │ │ │ +002689b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002689c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002689d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002689e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 002689f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268a00: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ -00268a10: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -00268a20: 6e6f 7465 2072 756e 5072 6f67 7261 6d3a note runProgram: │ │ │ │ -00268a30: 2072 756e 5072 6f67 7261 6d2c 202d 2d20 runProgram, -- │ │ │ │ -00268a40: 7275 6e20 616e 2065 7874 6572 6e61 6c20 run an external │ │ │ │ -00268a50: 7072 6f67 7261 6d0a 2020 2a20 2a6e 6f74 program. * *not │ │ │ │ -00268a60: 6520 7365 6172 6368 5061 7468 3a20 7365 e searchPath: se │ │ │ │ -00268a70: 6172 6368 5061 7468 5f6c 704c 6973 745f archPath_lpList_ │ │ │ │ -00268a80: 636d 5374 7269 6e67 5f72 702c 202d 2d20 cmString_rp, -- │ │ │ │ -00268a90: 7365 6172 6368 2061 2070 6174 6820 666f search a path fo │ │ │ │ -00268aa0: 7220 610a 2020 2020 6669 6c65 0a2a 204d r a. file.* M │ │ │ │ -00268ab0: 656e 753a 0a0a 2a20 5072 6f67 7261 6d3a enu:..* Program: │ │ │ │ -00268ac0: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ -00268ad0: 2020 2020 2020 6578 7465 726e 616c 2070 external p │ │ │ │ -00268ae0: 726f 6772 616d 206f 626a 6563 740a 2a20 rogram object.* │ │ │ │ -00268af0: 7072 6f67 7261 6d50 6174 6873 3a3a 2020 programPaths:: │ │ │ │ -00268b00: 2020 2020 2020 2020 2020 2020 2020 7573 us │ │ │ │ -00268b10: 6572 2d64 6566 696e 6564 2065 7874 6572 er-defined exter │ │ │ │ -00268b20: 6e61 6c20 7072 6f67 7261 6d20 7061 7468 nal program path │ │ │ │ -00268b30: 730a 0a57 6179 7320 746f 2075 7365 2066 s..Ways to use f │ │ │ │ -00268b40: 696e 6450 726f 6772 616d 3a0a 3d3d 3d3d indProgram:.==== │ │ │ │ -00268b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00268b60: 3d3d 3d3d 0a0a 2020 2a20 2266 696e 6450 ====.. * "findP │ │ │ │ -00268b70: 726f 6772 616d 2853 7472 696e 6729 220a rogram(String)". │ │ │ │ -00268b80: 2020 2a20 2266 696e 6450 726f 6772 616d * "findProgram │ │ │ │ -00268b90: 2853 7472 696e 672c 4c69 7374 2922 0a20 (String,List)". │ │ │ │ -00268ba0: 202a 2022 6669 6e64 5072 6f67 7261 6d28 * "findProgram( │ │ │ │ -00268bb0: 5374 7269 6e67 2c53 7472 696e 6729 220a String,String)". │ │ │ │ -00268bc0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00268bd0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00268be0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00268bf0: 6374 202a 6e6f 7465 2066 696e 6450 726f ct *note findPro │ │ │ │ -00268c00: 6772 616d 3a20 6669 6e64 5072 6f67 7261 gram: findProgra │ │ │ │ -00268c10: 6d2c 2069 7320 6120 2a6e 6f74 6520 6d65 m, is a *note me │ │ │ │ -00268c20: 7468 6f64 2066 756e 6374 696f 6e20 7769 thod function wi │ │ │ │ -00268c30: 7468 0a6f 7074 696f 6e73 3a20 4d65 7468 th.options: Meth │ │ │ │ -00268c40: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70 odFunctionWithOp │ │ │ │ -00268c50: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d tions,...------- │ │ │ │ -00268c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268a20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ +00268a30: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ +00268a40: 202a 202a 6e6f 7465 2072 756e 5072 6f67 * *note runProg │ │ │ │ +00268a50: 7261 6d3a 2072 756e 5072 6f67 7261 6d2c ram: runProgram, │ │ │ │ +00268a60: 202d 2d20 7275 6e20 616e 2065 7874 6572 -- run an exter │ │ │ │ +00268a70: 6e61 6c20 7072 6f67 7261 6d0a 2020 2a20 nal program. * │ │ │ │ +00268a80: 2a6e 6f74 6520 7365 6172 6368 5061 7468 *note searchPath │ │ │ │ +00268a90: 3a20 7365 6172 6368 5061 7468 5f6c 704c : searchPath_lpL │ │ │ │ +00268aa0: 6973 745f 636d 5374 7269 6e67 5f72 702c ist_cmString_rp, │ │ │ │ +00268ab0: 202d 2d20 7365 6172 6368 2061 2070 6174 -- search a pat │ │ │ │ +00268ac0: 6820 666f 7220 610a 2020 2020 6669 6c65 h for a. file │ │ │ │ +00268ad0: 0a2a 204d 656e 753a 0a0a 2a20 5072 6f67 .* Menu:..* Prog │ │ │ │ +00268ae0: 7261 6d3a 3a20 2020 2020 2020 2020 2020 ram:: │ │ │ │ +00268af0: 2020 2020 2020 2020 2020 6578 7465 726e extern │ │ │ │ +00268b00: 616c 2070 726f 6772 616d 206f 626a 6563 al program objec │ │ │ │ +00268b10: 740a 2a20 7072 6f67 7261 6d50 6174 6873 t.* programPaths │ │ │ │ +00268b20: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: │ │ │ │ +00268b30: 2020 7573 6572 2d64 6566 696e 6564 2065 user-defined e │ │ │ │ +00268b40: 7874 6572 6e61 6c20 7072 6f67 7261 6d20 xternal program │ │ │ │ +00268b50: 7061 7468 730a 0a57 6179 7320 746f 2075 paths..Ways to u │ │ │ │ +00268b60: 7365 2066 696e 6450 726f 6772 616d 3a0a se findProgram:. │ │ │ │ +00268b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00268b80: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2266 ========.. * "f │ │ │ │ +00268b90: 696e 6450 726f 6772 616d 2853 7472 696e indProgram(Strin │ │ │ │ +00268ba0: 6729 220a 2020 2a20 2266 696e 6450 726f g)". * "findPro │ │ │ │ +00268bb0: 6772 616d 2853 7472 696e 672c 4c69 7374 gram(String,List │ │ │ │ +00268bc0: 2922 0a20 202a 2022 6669 6e64 5072 6f67 )". * "findProg │ │ │ │ +00268bd0: 7261 6d28 5374 7269 6e67 2c53 7472 696e ram(String,Strin │ │ │ │ +00268be0: 6729 220a 0a46 6f72 2074 6865 2070 726f g)"..For the pro │ │ │ │ +00268bf0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00268c00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00268c10: 6f62 6a65 6374 202a 6e6f 7465 2066 696e object *note fin │ │ │ │ +00268c20: 6450 726f 6772 616d 3a20 6669 6e64 5072 dProgram: findPr │ │ │ │ +00268c30: 6f67 7261 6d2c 2069 7320 6120 2a6e 6f74 ogram, is a *not │ │ │ │ +00268c40: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +00268c50: 6e20 7769 7468 0a6f 7074 696f 6e73 3a20 n with.options: │ │ │ │ +00268c60: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769 MethodFunctionWi │ │ │ │ +00268c70: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d thOptions,...--- │ │ │ │ 00268c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00268c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00268ca0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00268cb0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00268cc0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00268cd0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00268ce0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00268cf0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -00268d00: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00268d10: 0a4d 6163 6175 6c61 7932 446f 632f 6675 .Macaulay2Doc/fu │ │ │ │ -00268d20: 6e63 7469 6f6e 732f 6669 6e64 5072 6f67 nctions/findProg │ │ │ │ -00268d30: 7261 6d2d 646f 632e 6d32 3a31 3734 3a30 ram-doc.m2:174:0 │ │ │ │ -00268d40: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ -00268d50: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ -00268d60: 653a 2050 726f 6772 616d 2c20 4e65 7874 e: Program, Next │ │ │ │ -00268d70: 3a20 7072 6f67 7261 6d50 6174 6873 2c20 : programPaths, │ │ │ │ -00268d80: 5570 3a20 6669 6e64 5072 6f67 7261 6d0a Up: findProgram. │ │ │ │ -00268d90: 0a50 726f 6772 616d 202d 2d20 6578 7465 .Program -- exte │ │ │ │ -00268da0: 726e 616c 2070 726f 6772 616d 206f 626a rnal program obj │ │ │ │ -00268db0: 6563 740a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ect.************ │ │ │ │ -00268dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00268dd0: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074 ******..Descript │ │ │ │ -00268de0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00268df0: 0a41 2068 6173 6820 7461 626c 6520 7265 .A hash table re │ │ │ │ -00268e00: 7475 726e 6564 2062 7920 2a6e 6f74 6520 turned by *note │ │ │ │ -00268e10: 6669 6e64 5072 6f67 7261 6d3a 2066 696e findProgram: fin │ │ │ │ -00268e20: 6450 726f 6772 616d 2c20 7769 7468 2074 dProgram, with t │ │ │ │ -00268e30: 6865 2066 6f6c 6c6f 7769 6e67 0a73 7472 he following.str │ │ │ │ -00268e40: 696e 6773 2061 7320 6b65 7973 3a0a 0a20 ings as keys:.. │ │ │ │ -00268e50: 202a 2022 6e61 6d65 222c 2074 6865 206e * "name", the n │ │ │ │ -00268e60: 616d 6520 6f66 2074 6865 206c 6f61 6465 ame of the loade │ │ │ │ -00268e70: 6420 7072 6f67 7261 6d2e 2020 5468 6973 d program. This │ │ │ │ -00268e80: 2063 6f6d 6573 2066 726f 6d20 7468 6520 comes from the │ │ │ │ -00268e90: 6669 7273 7420 6172 6775 6d65 6e74 0a20 first argument. │ │ │ │ -00268ea0: 2020 2070 6173 7365 6420 746f 202a 6e6f passed to *no │ │ │ │ -00268eb0: 7465 2066 696e 6450 726f 6772 616d 3a20 te findProgram: │ │ │ │ -00268ec0: 6669 6e64 5072 6f67 7261 6d2c 2e20 2049 findProgram,. I │ │ │ │ -00268ed0: 7420 6973 2061 6c73 6f20 7768 6174 2069 t is also what i │ │ │ │ -00268ee0: 7320 6469 7370 6c61 7965 640a 2020 2020 s displayed. │ │ │ │ -00268ef0: 7768 656e 2070 7269 6e74 696e 6720 6120 when printing a │ │ │ │ -00268f00: 5072 6f67 7261 6d2e 0a20 202a 2022 7061 Program.. * "pa │ │ │ │ -00268f10: 7468 222c 2074 6865 2070 6174 6820 746f th", the path to │ │ │ │ -00268f20: 2074 6865 2070 726f 6772 616d 2061 7320 the program as │ │ │ │ -00268f30: 6465 7465 726d 696e 6564 2062 7920 2a6e determined by *n │ │ │ │ -00268f40: 6f74 6520 6669 6e64 5072 6f67 7261 6d3a ote findProgram: │ │ │ │ -00268f50: 0a20 2020 2066 696e 6450 726f 6772 616d . findProgram │ │ │ │ -00268f60: 2c2e 0a20 202a 2022 7072 6566 6978 222c ,.. * "prefix", │ │ │ │ -00268f70: 2061 2073 6571 7565 6e63 6520 6f66 2074 a sequence of t │ │ │ │ -00268f80: 776f 2073 7472 696e 6773 2069 6465 6e74 wo strings ident │ │ │ │ -00268f90: 6966 7969 6e67 2074 6865 2070 7265 6669 ifying the prefi │ │ │ │ -00268fa0: 7820 7072 6570 656e 6465 6420 746f 2074 x prepended to t │ │ │ │ -00268fb0: 6865 0a20 2020 2062 696e 6172 7920 6578 he. binary ex │ │ │ │ -00268fc0: 6563 7574 6162 6c65 732e 2020 5365 6520 ecutables. See │ │ │ │ -00268fd0: 2a6e 6f74 6520 6669 6e64 5072 6f67 7261 *note findProgra │ │ │ │ -00268fe0: 6d3a 2066 696e 6450 726f 6772 616d 2c2c m: findProgram,, │ │ │ │ -00268ff0: 2073 7065 6369 6669 6361 6c6c 7920 7468 specifically th │ │ │ │ -00269000: 650a 2020 2020 6465 7363 7269 7074 696f e. descriptio │ │ │ │ -00269010: 6e20 6f66 2074 6865 2050 7265 6669 7820 n of the Prefix │ │ │ │ -00269020: 6f70 7469 6f6e 2c20 666f 7220 6d6f 7265 option, for more │ │ │ │ -00269030: 2e0a 2020 2a20 2276 6572 7369 6f6e 222c .. * "version", │ │ │ │ -00269040: 2061 2073 7472 696e 6720 636f 6e74 6169 a string contai │ │ │ │ -00269050: 6e69 6e67 2074 6865 2076 6572 7369 6f6e ning the version │ │ │ │ -00269060: 206e 756d 6265 7220 6f66 2074 6865 2070 number of the p │ │ │ │ -00269070: 726f 6772 616d 2e20 204f 6e6c 790a 2020 rogram. Only. │ │ │ │ -00269080: 2020 7072 6573 656e 7420 6966 202a 6e6f present if *no │ │ │ │ -00269090: 7465 2066 696e 6450 726f 6772 616d 3a20 te findProgram: │ │ │ │ -002690a0: 6669 6e64 5072 6f67 7261 6d2c 2077 6173 findProgram, was │ │ │ │ -002690b0: 2063 616c 6c65 6420 7769 7468 2074 6865 called with the │ │ │ │ -002690c0: 0a20 2020 204d 696e 696d 756d 5665 7273 . MinimumVers │ │ │ │ -002690d0: 696f 6e20 6f70 7469 6f6e 2e0a 0a53 6565 ion option...See │ │ │ │ -002690e0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -002690f0: 2020 2a20 2a6e 6f74 6520 6669 6e64 5072 * *note findPr │ │ │ │ -00269100: 6f67 7261 6d3a 2066 696e 6450 726f 6772 ogram: findProgr │ │ │ │ -00269110: 616d 2c20 2d2d 206c 6f61 6420 6578 7465 am, -- load exte │ │ │ │ -00269120: 726e 616c 2070 726f 6772 616d 0a20 202a rnal program. * │ │ │ │ -00269130: 202a 6e6f 7465 2072 756e 5072 6f67 7261 *note runProgra │ │ │ │ -00269140: 6d3a 2072 756e 5072 6f67 7261 6d2c 202d m: runProgram, - │ │ │ │ -00269150: 2d20 7275 6e20 616e 2065 7874 6572 6e61 - run an externa │ │ │ │ -00269160: 6c20 7072 6f67 7261 6d0a 2020 2a20 2a6e l program. * *n │ │ │ │ -00269170: 6f74 6520 7072 6f67 7261 6d50 6174 6873 ote programPaths │ │ │ │ -00269180: 3a20 7072 6f67 7261 6d50 6174 6873 2c20 : programPaths, │ │ │ │ -00269190: 2d2d 2075 7365 722d 6465 6669 6e65 6420 -- user-defined │ │ │ │ -002691a0: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ -002691b0: 2070 6174 6873 0a0a 4675 6e63 7469 6f6e paths..Function │ │ │ │ -002691c0: 7320 616e 6420 6d65 7468 6f64 7320 7265 s and methods re │ │ │ │ -002691d0: 7475 726e 696e 6720 616e 206f 626a 6563 turning an objec │ │ │ │ -002691e0: 7420 6f66 2063 6c61 7373 2050 726f 6772 t of class Progr │ │ │ │ -002691f0: 616d 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d am:.============ │ │ │ │ -00269200: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00269210: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00269220: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00269230: 0a20 202a 202a 6e6f 7465 2066 696e 6450 . * *note findP │ │ │ │ -00269240: 726f 6772 616d 3a20 6669 6e64 5072 6f67 rogram: findProg │ │ │ │ -00269250: 7261 6d2c 202d 2d20 6c6f 6164 2065 7874 ram, -- load ext │ │ │ │ -00269260: 6572 6e61 6c20 7072 6f67 7261 6d0a 0a4d ernal program..M │ │ │ │ -00269270: 6574 686f 6473 2074 6861 7420 7573 6520 ethods that use │ │ │ │ -00269280: 616e 206f 626a 6563 7420 6f66 2063 6c61 an object of cla │ │ │ │ -00269290: 7373 2050 726f 6772 616d 3a0a 3d3d 3d3d ss Program:.==== │ │ │ │ -002692a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -002692b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -002692c0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -002692d0: 6f74 6520 5072 6f67 7261 6d20 3c3c 2054 ote Program << T │ │ │ │ -002692e0: 6869 6e67 3a20 5072 6f67 7261 6d20 3c3c hing: Program << │ │ │ │ -002692f0: 2054 6869 6e67 2c20 2d2d 2072 756e 2070 Thing, -- run p │ │ │ │ -00269300: 726f 6772 616d 2077 6974 6820 696e 7075 rogram with inpu │ │ │ │ -00269310: 740a 2020 2020 7265 6469 7265 6374 696f t. redirectio │ │ │ │ -00269320: 6e0a 2020 2a20 2272 756e 5072 6f67 7261 n. * "runProgra │ │ │ │ -00269330: 6d28 5072 6f67 7261 6d2c 5374 7269 6e67 m(Program,String │ │ │ │ -00269340: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -00269350: 7275 6e50 726f 6772 616d 3a20 7275 6e50 runProgram: runP │ │ │ │ -00269360: 726f 6772 616d 2c20 2d2d 2072 756e 2061 rogram, -- run a │ │ │ │ -00269370: 6e0a 2020 2020 6578 7465 726e 616c 2070 n. external p │ │ │ │ -00269380: 726f 6772 616d 0a20 202a 2022 7275 6e50 rogram. * "runP │ │ │ │ -00269390: 726f 6772 616d 2850 726f 6772 616d 2c53 rogram(Program,S │ │ │ │ -002693a0: 7472 696e 672c 5374 7269 6e67 2922 202d tring,String)" - │ │ │ │ -002693b0: 2d20 7365 6520 2a6e 6f74 6520 7275 6e50 - see *note runP │ │ │ │ -002693c0: 726f 6772 616d 3a20 7275 6e50 726f 6772 rogram: runProgr │ │ │ │ -002693d0: 616d 2c20 2d2d 0a20 2020 2072 756e 2061 am, --. run a │ │ │ │ -002693e0: 6e20 6578 7465 726e 616c 2070 726f 6772 n external progr │ │ │ │ -002693f0: 616d 0a0a 466f 7220 7468 6520 7072 6f67 am..For the prog │ │ │ │ -00269400: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -00269410: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -00269420: 626a 6563 7420 2a6e 6f74 6520 5072 6f67 bject *note Prog │ │ │ │ -00269430: 7261 6d3a 2050 726f 6772 616d 2c20 6973 ram: Program, is │ │ │ │ -00269440: 2061 202a 6e6f 7465 2074 7970 653a 2054 a *note type: T │ │ │ │ -00269450: 7970 652c 2c20 7769 7468 2061 6e63 6573 ype,, with ances │ │ │ │ -00269460: 746f 720a 636c 6173 7365 7320 2a6e 6f74 tor.classes *not │ │ │ │ -00269470: 6520 4861 7368 5461 626c 653a 2048 6173 e HashTable: Has │ │ │ │ -00269480: 6854 6162 6c65 2c20 3c20 2a6e 6f74 6520 hTable, < *note │ │ │ │ -00269490: 5468 696e 673a 2054 6869 6e67 2c2e 0a0a Thing: Thing,... │ │ │ │ -002694a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002694b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002694c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00268cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00268cd0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00268ce0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00268cf0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00268d00: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00268d10: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ +00268d20: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00268d30: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ +00268d40: 632f 6675 6e63 7469 6f6e 732f 6669 6e64 c/functions/find │ │ │ │ +00268d50: 5072 6f67 7261 6d2d 646f 632e 6d32 3a31 Program-doc.m2:1 │ │ │ │ +00268d60: 3734 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 74:0....File: Ma │ │ │ │ +00268d70: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ +00268d80: 204e 6f64 653a 2050 726f 6772 616d 2c20 Node: Program, │ │ │ │ +00268d90: 4e65 7874 3a20 7072 6f67 7261 6d50 6174 Next: programPat │ │ │ │ +00268da0: 6873 2c20 5570 3a20 6669 6e64 5072 6f67 hs, Up: findProg │ │ │ │ +00268db0: 7261 6d0a 0a50 726f 6772 616d 202d 2d20 ram..Program -- │ │ │ │ +00268dc0: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ +00268dd0: 206f 626a 6563 740a 2a2a 2a2a 2a2a 2a2a object.******** │ │ │ │ +00268de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00268df0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 **********..Desc │ │ │ │ +00268e00: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00268e10: 3d3d 3d0a 0a41 2068 6173 6820 7461 626c ===..A hash tabl │ │ │ │ +00268e20: 6520 7265 7475 726e 6564 2062 7920 2a6e e returned by *n │ │ │ │ +00268e30: 6f74 6520 6669 6e64 5072 6f67 7261 6d3a ote findProgram: │ │ │ │ +00268e40: 2066 696e 6450 726f 6772 616d 2c20 7769 findProgram, wi │ │ │ │ +00268e50: 7468 2074 6865 2066 6f6c 6c6f 7769 6e67 th the following │ │ │ │ +00268e60: 0a73 7472 696e 6773 2061 7320 6b65 7973 .strings as keys │ │ │ │ +00268e70: 3a0a 0a20 202a 2022 6e61 6d65 222c 2074 :.. * "name", t │ │ │ │ +00268e80: 6865 206e 616d 6520 6f66 2074 6865 206c he name of the l │ │ │ │ +00268e90: 6f61 6465 6420 7072 6f67 7261 6d2e 2020 oaded program. │ │ │ │ +00268ea0: 5468 6973 2063 6f6d 6573 2066 726f 6d20 This comes from │ │ │ │ +00268eb0: 7468 6520 6669 7273 7420 6172 6775 6d65 the first argume │ │ │ │ +00268ec0: 6e74 0a20 2020 2070 6173 7365 6420 746f nt. passed to │ │ │ │ +00268ed0: 202a 6e6f 7465 2066 696e 6450 726f 6772 *note findProgr │ │ │ │ +00268ee0: 616d 3a20 6669 6e64 5072 6f67 7261 6d2c am: findProgram, │ │ │ │ +00268ef0: 2e20 2049 7420 6973 2061 6c73 6f20 7768 . It is also wh │ │ │ │ +00268f00: 6174 2069 7320 6469 7370 6c61 7965 640a at is displayed. │ │ │ │ +00268f10: 2020 2020 7768 656e 2070 7269 6e74 696e when printin │ │ │ │ +00268f20: 6720 6120 5072 6f67 7261 6d2e 0a20 202a g a Program.. * │ │ │ │ +00268f30: 2022 7061 7468 222c 2074 6865 2070 6174 "path", the pat │ │ │ │ +00268f40: 6820 746f 2074 6865 2070 726f 6772 616d h to the program │ │ │ │ +00268f50: 2061 7320 6465 7465 726d 696e 6564 2062 as determined b │ │ │ │ +00268f60: 7920 2a6e 6f74 6520 6669 6e64 5072 6f67 y *note findProg │ │ │ │ +00268f70: 7261 6d3a 0a20 2020 2066 696e 6450 726f ram:. findPro │ │ │ │ +00268f80: 6772 616d 2c2e 0a20 202a 2022 7072 6566 gram,.. * "pref │ │ │ │ +00268f90: 6978 222c 2061 2073 6571 7565 6e63 6520 ix", a sequence │ │ │ │ +00268fa0: 6f66 2074 776f 2073 7472 696e 6773 2069 of two strings i │ │ │ │ +00268fb0: 6465 6e74 6966 7969 6e67 2074 6865 2070 dentifying the p │ │ │ │ +00268fc0: 7265 6669 7820 7072 6570 656e 6465 6420 refix prepended │ │ │ │ +00268fd0: 746f 2074 6865 0a20 2020 2062 696e 6172 to the. binar │ │ │ │ +00268fe0: 7920 6578 6563 7574 6162 6c65 732e 2020 y executables. │ │ │ │ +00268ff0: 5365 6520 2a6e 6f74 6520 6669 6e64 5072 See *note findPr │ │ │ │ +00269000: 6f67 7261 6d3a 2066 696e 6450 726f 6772 ogram: findProgr │ │ │ │ +00269010: 616d 2c2c 2073 7065 6369 6669 6361 6c6c am,, specificall │ │ │ │ +00269020: 7920 7468 650a 2020 2020 6465 7363 7269 y the. descri │ │ │ │ +00269030: 7074 696f 6e20 6f66 2074 6865 2050 7265 ption of the Pre │ │ │ │ +00269040: 6669 7820 6f70 7469 6f6e 2c20 666f 7220 fix option, for │ │ │ │ +00269050: 6d6f 7265 2e0a 2020 2a20 2276 6572 7369 more.. * "versi │ │ │ │ +00269060: 6f6e 222c 2061 2073 7472 696e 6720 636f on", a string co │ │ │ │ +00269070: 6e74 6169 6e69 6e67 2074 6865 2076 6572 ntaining the ver │ │ │ │ +00269080: 7369 6f6e 206e 756d 6265 7220 6f66 2074 sion number of t │ │ │ │ +00269090: 6865 2070 726f 6772 616d 2e20 204f 6e6c he program. Onl │ │ │ │ +002690a0: 790a 2020 2020 7072 6573 656e 7420 6966 y. present if │ │ │ │ +002690b0: 202a 6e6f 7465 2066 696e 6450 726f 6772 *note findProgr │ │ │ │ +002690c0: 616d 3a20 6669 6e64 5072 6f67 7261 6d2c am: findProgram, │ │ │ │ +002690d0: 2077 6173 2063 616c 6c65 6420 7769 7468 was called with │ │ │ │ +002690e0: 2074 6865 0a20 2020 204d 696e 696d 756d the. Minimum │ │ │ │ +002690f0: 5665 7273 696f 6e20 6f70 7469 6f6e 2e0a Version option.. │ │ │ │ +00269100: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +00269110: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6669 ==.. * *note fi │ │ │ │ +00269120: 6e64 5072 6f67 7261 6d3a 2066 696e 6450 ndProgram: findP │ │ │ │ +00269130: 726f 6772 616d 2c20 2d2d 206c 6f61 6420 rogram, -- load │ │ │ │ +00269140: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ +00269150: 0a20 202a 202a 6e6f 7465 2072 756e 5072 . * *note runPr │ │ │ │ +00269160: 6f67 7261 6d3a 2072 756e 5072 6f67 7261 ogram: runProgra │ │ │ │ +00269170: 6d2c 202d 2d20 7275 6e20 616e 2065 7874 m, -- run an ext │ │ │ │ +00269180: 6572 6e61 6c20 7072 6f67 7261 6d0a 2020 ernal program. │ │ │ │ +00269190: 2a20 2a6e 6f74 6520 7072 6f67 7261 6d50 * *note programP │ │ │ │ +002691a0: 6174 6873 3a20 7072 6f67 7261 6d50 6174 aths: programPat │ │ │ │ +002691b0: 6873 2c20 2d2d 2075 7365 722d 6465 6669 hs, -- user-defi │ │ │ │ +002691c0: 6e65 6420 6578 7465 726e 616c 2070 726f ned external pro │ │ │ │ +002691d0: 6772 616d 2070 6174 6873 0a0a 4675 6e63 gram paths..Func │ │ │ │ +002691e0: 7469 6f6e 7320 616e 6420 6d65 7468 6f64 tions and method │ │ │ │ +002691f0: 7320 7265 7475 726e 696e 6720 616e 206f s returning an o │ │ │ │ +00269200: 626a 6563 7420 6f66 2063 6c61 7373 2050 bject of class P │ │ │ │ +00269210: 726f 6772 616d 3a0a 3d3d 3d3d 3d3d 3d3d rogram:.======== │ │ │ │ +00269220: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00269230: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00269240: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00269250: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2066 ===.. * *note f │ │ │ │ +00269260: 696e 6450 726f 6772 616d 3a20 6669 6e64 indProgram: find │ │ │ │ +00269270: 5072 6f67 7261 6d2c 202d 2d20 6c6f 6164 Program, -- load │ │ │ │ +00269280: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ +00269290: 6d0a 0a4d 6574 686f 6473 2074 6861 7420 m..Methods that │ │ │ │ +002692a0: 7573 6520 616e 206f 626a 6563 7420 6f66 use an object of │ │ │ │ +002692b0: 2063 6c61 7373 2050 726f 6772 616d 3a0a class Program:. │ │ │ │ +002692c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002692d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002692e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +002692f0: 2a20 2a6e 6f74 6520 5072 6f67 7261 6d20 * *note Program │ │ │ │ +00269300: 3c3c 2054 6869 6e67 3a20 5072 6f67 7261 << Thing: Progra │ │ │ │ +00269310: 6d20 3c3c 2054 6869 6e67 2c20 2d2d 2072 m << Thing, -- r │ │ │ │ +00269320: 756e 2070 726f 6772 616d 2077 6974 6820 un program with │ │ │ │ +00269330: 696e 7075 740a 2020 2020 7265 6469 7265 input. redire │ │ │ │ +00269340: 6374 696f 6e0a 2020 2a20 2272 756e 5072 ction. * "runPr │ │ │ │ +00269350: 6f67 7261 6d28 5072 6f67 7261 6d2c 5374 ogram(Program,St │ │ │ │ +00269360: 7269 6e67 2922 202d 2d20 7365 6520 2a6e ring)" -- see *n │ │ │ │ +00269370: 6f74 6520 7275 6e50 726f 6772 616d 3a20 ote runProgram: │ │ │ │ +00269380: 7275 6e50 726f 6772 616d 2c20 2d2d 2072 runProgram, -- r │ │ │ │ +00269390: 756e 2061 6e0a 2020 2020 6578 7465 726e un an. extern │ │ │ │ +002693a0: 616c 2070 726f 6772 616d 0a20 202a 2022 al program. * " │ │ │ │ +002693b0: 7275 6e50 726f 6772 616d 2850 726f 6772 runProgram(Progr │ │ │ │ +002693c0: 616d 2c53 7472 696e 672c 5374 7269 6e67 am,String,String │ │ │ │ +002693d0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ +002693e0: 7275 6e50 726f 6772 616d 3a20 7275 6e50 runProgram: runP │ │ │ │ +002693f0: 726f 6772 616d 2c20 2d2d 0a20 2020 2072 rogram, --. r │ │ │ │ +00269400: 756e 2061 6e20 6578 7465 726e 616c 2070 un an external p │ │ │ │ +00269410: 726f 6772 616d 0a0a 466f 7220 7468 6520 rogram..For the │ │ │ │ +00269420: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +00269430: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +00269440: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +00269450: 5072 6f67 7261 6d3a 2050 726f 6772 616d Program: Program │ │ │ │ +00269460: 2c20 6973 2061 202a 6e6f 7465 2074 7970 , is a *note typ │ │ │ │ +00269470: 653a 2054 7970 652c 2c20 7769 7468 2061 e: Type,, with a │ │ │ │ +00269480: 6e63 6573 746f 720a 636c 6173 7365 7320 ncestor.classes │ │ │ │ +00269490: 2a6e 6f74 6520 4861 7368 5461 626c 653a *note HashTable: │ │ │ │ +002694a0: 2048 6173 6854 6162 6c65 2c20 3c20 2a6e HashTable, < *n │ │ │ │ +002694b0: 6f74 6520 5468 696e 673a 2054 6869 6e67 ote Thing: Thing │ │ │ │ +002694c0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ 002694d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002694e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -002694f0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00269500: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00269510: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00269520: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -00269530: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -00269540: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -00269550: 636b 6167 6573 2f0a 4d61 6361 756c 6179 ckages/.Macaulay │ │ │ │ -00269560: 3244 6f63 2f66 756e 6374 696f 6e73 2f66 2Doc/functions/f │ │ │ │ -00269570: 696e 6450 726f 6772 616d 2d64 6f63 2e6d indProgram-doc.m │ │ │ │ -00269580: 323a 3330 3a30 2e0a 1f0a 4669 6c65 3a20 2:30:0....File: │ │ │ │ -00269590: 4d61 6361 756c 6179 3244 6f63 2e69 6e66 Macaulay2Doc.inf │ │ │ │ -002695a0: 6f2c 204e 6f64 653a 2070 726f 6772 616d o, Node: program │ │ │ │ -002695b0: 5061 7468 732c 2050 7265 763a 2050 726f Paths, Prev: Pro │ │ │ │ -002695c0: 6772 616d 2c20 5570 3a20 6669 6e64 5072 gram, Up: findPr │ │ │ │ -002695d0: 6f67 7261 6d0a 0a70 726f 6772 616d 5061 ogram..programPa │ │ │ │ -002695e0: 7468 7320 2d2d 2075 7365 722d 6465 6669 ths -- user-defi │ │ │ │ -002695f0: 6e65 6420 6578 7465 726e 616c 2070 726f ned external pro │ │ │ │ -00269600: 6772 616d 2070 6174 6873 0a2a 2a2a 2a2a gram paths.***** │ │ │ │ -00269610: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00269620: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00269630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00269640: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00269650: 3d3d 3d3d 3d3d 3d0a 0a41 206d 7574 6162 =======..A mutab │ │ │ │ -00269660: 6c65 2068 6173 6820 7461 626c 6520 636f le hash table co │ │ │ │ -00269670: 6e74 6169 6e69 6e67 2075 7365 722d 6465 ntaining user-de │ │ │ │ -00269680: 6669 6e65 6420 7061 7468 7320 746f 2065 fined paths to e │ │ │ │ -00269690: 7874 6572 6e61 6c20 7072 6f67 7261 6d73 xternal programs │ │ │ │ -002696a0: 2075 7365 6420 6279 0a4d 6163 6175 6c61 used by.Macaula │ │ │ │ -002696b0: 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Its keys ar │ │ │ │ -002696c0: 6520 7374 7269 6e67 7320 636f 6e74 6169 e strings contai │ │ │ │ -002696d0: 6e69 6e67 2074 6865 206e 616d 6573 206f ning the names o │ │ │ │ -002696e0: 6620 7072 6f67 7261 6d73 2061 6e64 206d f programs and m │ │ │ │ -002696f0: 7573 740a 636f 696e 6369 6465 2077 6974 ust.coincide wit │ │ │ │ -00269700: 6820 7468 6520 6669 7273 7420 6172 6775 h the first argu │ │ │ │ -00269710: 6d65 6e74 206f 6620 2a6e 6f74 6520 6669 ment of *note fi │ │ │ │ -00269720: 6e64 5072 6f67 7261 6d3a 2066 696e 6450 ndProgram: findP │ │ │ │ -00269730: 726f 6772 616d 2c2e 0a0a 4974 2069 7320 rogram,...It is │ │ │ │ -00269740: 6f6e 6c79 206e 6563 6573 7361 7279 2074 only necessary t │ │ │ │ -00269750: 6f20 6465 6669 6e65 2061 2070 6174 6820 o define a path │ │ │ │ -00269760: 696e 2074 6869 7320 7761 7920 6966 2061 in this way if a │ │ │ │ -00269770: 2070 726f 6772 616d 2069 7320 696e 7374 program is inst │ │ │ │ -00269780: 616c 6c65 6420 696e 0a61 206e 6f6e 2d73 alled in.a non-s │ │ │ │ -00269790: 7461 6e64 6172 6420 6c6f 6361 7469 6f6e tandard location │ │ │ │ -002697a0: 2e20 2049 6e20 7061 7274 6963 756c 6172 . In particular │ │ │ │ -002697b0: 2c20 2a6e 6f74 6520 6669 6e64 5072 6f67 , *note findProg │ │ │ │ -002697c0: 7261 6d3a 2066 696e 6450 726f 6772 616d ram: findProgram │ │ │ │ -002697d0: 2c0a 616c 7265 6164 7920 6368 6563 6b73 ,.already checks │ │ │ │ -002697e0: 2070 7265 6669 7844 6972 6563 746f 7279 prefixDirectory │ │ │ │ -002697f0: 207c 2063 7572 7265 6e74 4c61 796f 7574 | currentLayout │ │ │ │ -00269800: 2322 7072 6f67 7261 6d73 222c 2028 7768 #"programs", (wh │ │ │ │ -00269810: 6572 6520 7468 6520 7072 6f67 7261 6d73 ere the programs │ │ │ │ -00269820: 0a73 6869 7070 6564 2077 6974 6820 4d61 .shipped with Ma │ │ │ │ -00269830: 6361 756c 6179 3220 6172 6520 696e 7374 caulay2 are inst │ │ │ │ -00269840: 616c 6c65 6429 2061 6e64 2061 6c6c 206f alled) and all o │ │ │ │ -00269850: 6620 7468 6520 6469 7265 6374 6f72 6965 f the directorie │ │ │ │ -00269860: 7320 696e 2074 6865 2075 7365 7227 730a s in the user's. │ │ │ │ -00269870: 5041 5448 2065 6e76 6972 6f6e 6d65 6e74 PATH environment │ │ │ │ -00269880: 2076 6172 6961 626c 652e 0a0a 4966 2079 variable...If y │ │ │ │ -00269890: 6f75 2075 7365 2061 2070 6172 7469 6375 ou use a particu │ │ │ │ -002698a0: 6c61 7220 7072 6f67 7261 6d20 6672 6571 lar program freq │ │ │ │ -002698b0: 7565 6e74 6c79 2061 6e64 2069 7420 6973 uently and it is │ │ │ │ -002698c0: 2069 6e73 7461 6c6c 6564 2069 6e20 610a installed in a. │ │ │ │ -002698d0: 6e6f 6e2d 7374 616e 6461 7264 206c 6f63 non-standard loc │ │ │ │ -002698e0: 6174 696f 6e2c 2074 6865 6e20 6974 206d ation, then it m │ │ │ │ -002698f0: 6179 2062 6520 7573 6566 756c 2074 6f20 ay be useful to │ │ │ │ -00269900: 6164 6420 6120 6c69 6e65 2074 6f20 796f add a line to yo │ │ │ │ -00269910: 7572 202a 6e6f 7465 0a69 6e69 7469 616c ur *note.initial │ │ │ │ -00269920: 697a 6174 696f 6e20 6669 6c65 3a20 696e ization file: in │ │ │ │ -00269930: 6974 6961 6c69 7a61 7469 6f6e 2066 696c itialization fil │ │ │ │ -00269940: 652c 2064 6566 696e 696e 6720 6974 7320 e, defining its │ │ │ │ -00269950: 7061 7468 2069 6e20 7468 6973 2077 6179 path in this way │ │ │ │ -00269960: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -00269970: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -00269980: 5072 6f67 7261 6d3a 2050 726f 6772 616d Program: Program │ │ │ │ -00269990: 2c20 2d2d 2065 7874 6572 6e61 6c20 7072 , -- external pr │ │ │ │ -002699a0: 6f67 7261 6d20 6f62 6a65 6374 0a20 202a ogram object. * │ │ │ │ -002699b0: 202a 6e6f 7465 2070 6174 683a 2070 6174 *note path: pat │ │ │ │ -002699c0: 682c 202d 2d20 6c69 7374 206f 6620 6469 h, -- list of di │ │ │ │ -002699d0: 7265 6374 6f72 6965 7320 746f 206c 6f6f rectories to loo │ │ │ │ -002699e0: 6b20 696e 0a0a 466f 7220 7468 6520 7072 k in..For the pr │ │ │ │ -002699f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -00269a00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -00269a10: 206f 626a 6563 7420 2a6e 6f74 6520 7072 object *note pr │ │ │ │ -00269a20: 6f67 7261 6d50 6174 6873 3a20 7072 6f67 ogramPaths: prog │ │ │ │ -00269a30: 7261 6d50 6174 6873 2c20 6973 2061 202a ramPaths, is a * │ │ │ │ -00269a40: 6e6f 7465 206d 7574 6162 6c65 2068 6173 note mutable has │ │ │ │ -00269a50: 6820 7461 626c 653a 0a4d 7574 6162 6c65 h table:.Mutable │ │ │ │ -00269a60: 4861 7368 5461 626c 652c 2e0a 0a2d 2d2d HashTable,...--- │ │ │ │ -00269a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002694e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002694f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269510: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +00269520: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +00269530: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +00269540: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +00269550: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +00269560: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +00269570: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ +00269580: 756c 6179 3244 6f63 2f66 756e 6374 696f ulay2Doc/functio │ │ │ │ +00269590: 6e73 2f66 696e 6450 726f 6772 616d 2d64 ns/findProgram-d │ │ │ │ +002695a0: 6f63 2e6d 323a 3330 3a30 2e0a 1f0a 4669 oc.m2:30:0....Fi │ │ │ │ +002695b0: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ +002695c0: 2e69 6e66 6f2c 204e 6f64 653a 2070 726f .info, Node: pro │ │ │ │ +002695d0: 6772 616d 5061 7468 732c 2050 7265 763a gramPaths, Prev: │ │ │ │ +002695e0: 2050 726f 6772 616d 2c20 5570 3a20 6669 Program, Up: fi │ │ │ │ +002695f0: 6e64 5072 6f67 7261 6d0a 0a70 726f 6772 ndProgram..progr │ │ │ │ +00269600: 616d 5061 7468 7320 2d2d 2075 7365 722d amPaths -- user- │ │ │ │ +00269610: 6465 6669 6e65 6420 6578 7465 726e 616c defined external │ │ │ │ +00269620: 2070 726f 6772 616d 2070 6174 6873 0a2a program paths.* │ │ │ │ +00269630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00269640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00269650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00269660: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a **..Description. │ │ │ │ +00269670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 206d ===========..A m │ │ │ │ +00269680: 7574 6162 6c65 2068 6173 6820 7461 626c utable hash tabl │ │ │ │ +00269690: 6520 636f 6e74 6169 6e69 6e67 2075 7365 e containing use │ │ │ │ +002696a0: 722d 6465 6669 6e65 6420 7061 7468 7320 r-defined paths │ │ │ │ +002696b0: 746f 2065 7874 6572 6e61 6c20 7072 6f67 to external prog │ │ │ │ +002696c0: 7261 6d73 2075 7365 6420 6279 0a4d 6163 rams used by.Mac │ │ │ │ +002696d0: 6175 6c61 7932 2e20 2049 7473 206b 6579 aulay2. Its key │ │ │ │ +002696e0: 7320 6172 6520 7374 7269 6e67 7320 636f s are strings co │ │ │ │ +002696f0: 6e74 6169 6e69 6e67 2074 6865 206e 616d ntaining the nam │ │ │ │ +00269700: 6573 206f 6620 7072 6f67 7261 6d73 2061 es of programs a │ │ │ │ +00269710: 6e64 206d 7573 740a 636f 696e 6369 6465 nd must.coincide │ │ │ │ +00269720: 2077 6974 6820 7468 6520 6669 7273 7420 with the first │ │ │ │ +00269730: 6172 6775 6d65 6e74 206f 6620 2a6e 6f74 argument of *not │ │ │ │ +00269740: 6520 6669 6e64 5072 6f67 7261 6d3a 2066 e findProgram: f │ │ │ │ +00269750: 696e 6450 726f 6772 616d 2c2e 0a0a 4974 indProgram,...It │ │ │ │ +00269760: 2069 7320 6f6e 6c79 206e 6563 6573 7361 is only necessa │ │ │ │ +00269770: 7279 2074 6f20 6465 6669 6e65 2061 2070 ry to define a p │ │ │ │ +00269780: 6174 6820 696e 2074 6869 7320 7761 7920 ath in this way │ │ │ │ +00269790: 6966 2061 2070 726f 6772 616d 2069 7320 if a program is │ │ │ │ +002697a0: 696e 7374 616c 6c65 6420 696e 0a61 206e installed in.a n │ │ │ │ +002697b0: 6f6e 2d73 7461 6e64 6172 6420 6c6f 6361 on-standard loca │ │ │ │ +002697c0: 7469 6f6e 2e20 2049 6e20 7061 7274 6963 tion. In partic │ │ │ │ +002697d0: 756c 6172 2c20 2a6e 6f74 6520 6669 6e64 ular, *note find │ │ │ │ +002697e0: 5072 6f67 7261 6d3a 2066 696e 6450 726f Program: findPro │ │ │ │ +002697f0: 6772 616d 2c0a 616c 7265 6164 7920 6368 gram,.already ch │ │ │ │ +00269800: 6563 6b73 2070 7265 6669 7844 6972 6563 ecks prefixDirec │ │ │ │ +00269810: 746f 7279 207c 2063 7572 7265 6e74 4c61 tory | currentLa │ │ │ │ +00269820: 796f 7574 2322 7072 6f67 7261 6d73 222c yout#"programs", │ │ │ │ +00269830: 2028 7768 6572 6520 7468 6520 7072 6f67 (where the prog │ │ │ │ +00269840: 7261 6d73 0a73 6869 7070 6564 2077 6974 rams.shipped wit │ │ │ │ +00269850: 6820 4d61 6361 756c 6179 3220 6172 6520 h Macaulay2 are │ │ │ │ +00269860: 696e 7374 616c 6c65 6429 2061 6e64 2061 installed) and a │ │ │ │ +00269870: 6c6c 206f 6620 7468 6520 6469 7265 6374 ll of the direct │ │ │ │ +00269880: 6f72 6965 7320 696e 2074 6865 2075 7365 ories in the use │ │ │ │ +00269890: 7227 730a 5041 5448 2065 6e76 6972 6f6e r's.PATH environ │ │ │ │ +002698a0: 6d65 6e74 2076 6172 6961 626c 652e 0a0a ment variable... │ │ │ │ +002698b0: 4966 2079 6f75 2075 7365 2061 2070 6172 If you use a par │ │ │ │ +002698c0: 7469 6375 6c61 7220 7072 6f67 7261 6d20 ticular program │ │ │ │ +002698d0: 6672 6571 7565 6e74 6c79 2061 6e64 2069 frequently and i │ │ │ │ +002698e0: 7420 6973 2069 6e73 7461 6c6c 6564 2069 t is installed i │ │ │ │ +002698f0: 6e20 610a 6e6f 6e2d 7374 616e 6461 7264 n a.non-standard │ │ │ │ +00269900: 206c 6f63 6174 696f 6e2c 2074 6865 6e20 location, then │ │ │ │ +00269910: 6974 206d 6179 2062 6520 7573 6566 756c it may be useful │ │ │ │ +00269920: 2074 6f20 6164 6420 6120 6c69 6e65 2074 to add a line t │ │ │ │ +00269930: 6f20 796f 7572 202a 6e6f 7465 0a69 6e69 o your *note.ini │ │ │ │ +00269940: 7469 616c 697a 6174 696f 6e20 6669 6c65 tialization file │ │ │ │ +00269950: 3a20 696e 6974 6961 6c69 7a61 7469 6f6e : initialization │ │ │ │ +00269960: 2066 696c 652c 2064 6566 696e 696e 6720 file, defining │ │ │ │ +00269970: 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7261 6d50 6174 6873 3a20 e programPaths: │ │ │ │ +00269a50: 7072 6f67 7261 6d50 6174 6873 2c20 6973 programPaths, is │ │ │ │ +00269a60: 2061 202a 6e6f 7465 206d 7574 6162 6c65 a *note mutable │ │ │ │ +00269a70: 2068 6173 6820 7461 626c 653a 0a4d 7574 hash table:.Mut │ │ │ │ +00269a80: 6162 6c65 4861 7368 5461 626c 652c 2e0a ableHashTable,.. │ │ │ │ +00269a90: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 00269aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00269ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00269ac0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00269ad0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00269ae0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00269af0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00269b00: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00269b10: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00269b20: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ -00269b30: 632f 6675 6e63 7469 6f6e 732f 6669 6e64 c/functions/find │ │ │ │ -00269b40: 5072 6f67 7261 6d2d 646f 632e 6d32 3a35 Program-doc.m2:5 │ │ │ │ -00269b50: 383a 302e 0a1f 0a46 696c 653a 204d 6163 8:0....File: Mac │ │ │ │ -00269b60: 6175 6c61 7932 446f 632e 696e 666f 2c20 aulay2Doc.info, │ │ │ │ -00269b70: 4e6f 6465 3a20 7275 6e50 726f 6772 616d Node: runProgram │ │ │ │ -00269b80: 2c20 4e65 7874 3a20 5072 6f67 7261 6d20 , Next: Program │ │ │ │ -00269b90: 3c3c 2054 6869 6e67 2c20 5072 6576 3a20 << Thing, Prev: │ │ │ │ -00269ba0: 6669 6e64 5072 6f67 7261 6d2c 2055 703a findProgram, Up: │ │ │ │ -00269bb0: 2063 6f6d 6d75 6e69 6361 7469 6e67 2077 communicating w │ │ │ │ -00269bc0: 6974 6820 7072 6f67 7261 6d73 0a0a 7275 ith programs..ru │ │ │ │ -00269bd0: 6e50 726f 6772 616d 202d 2d20 7275 6e20 nProgram -- run │ │ │ │ -00269be0: 616e 2065 7874 6572 6e61 6c20 7072 6f67 an external prog │ │ │ │ -00269bf0: 7261 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ram.************ │ │ │ │ -00269c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00269c10: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00269c20: 7361 6765 3a20 0a20 2020 2020 2020 2072 sage: . r │ │ │ │ -00269c30: 756e 5072 6f67 7261 6d28 7072 6f67 7261 unProgram(progra │ │ │ │ -00269c40: 6d2c 2061 7267 7329 0a20 2020 2020 2020 m, args). │ │ │ │ -00269c50: 2072 756e 5072 6f67 7261 6d28 7072 6f67 runProgram(prog │ │ │ │ -00269c60: 7261 6d2c 2065 7865 2c20 6172 6773 290a ram, exe, args). │ │ │ │ -00269c70: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -00269c80: 2020 2a20 7072 6f67 7261 6d2c 2061 6e20 * program, an │ │ │ │ -00269c90: 696e 7374 616e 6365 206f 6620 7468 6520 instance of the │ │ │ │ -00269ca0: 7479 7065 202a 6e6f 7465 2050 726f 6772 type *note Progr │ │ │ │ -00269cb0: 616d 3a20 5072 6f67 7261 6d2c 2c20 7468 am: Program,, th │ │ │ │ -00269cc0: 6520 7072 6f67 7261 6d0a 2020 2020 2020 e program. │ │ │ │ -00269cd0: 2020 746f 2072 756e 2c20 6765 6e65 7261 to run, genera │ │ │ │ -00269ce0: 7465 6420 6279 202a 6e6f 7465 2066 696e ted by *note fin │ │ │ │ -00269cf0: 6450 726f 6772 616d 3a20 6669 6e64 5072 dProgram: findPr │ │ │ │ -00269d00: 6f67 7261 6d2c 2e0a 2020 2020 2020 2a20 ogram,.. * │ │ │ │ -00269d10: 6578 652c 2061 202a 6e6f 7465 2073 7472 exe, a *note str │ │ │ │ -00269d20: 696e 673a 2053 7472 696e 672c 2c20 7468 ing: String,, th │ │ │ │ -00269d30: 6520 7370 6563 6966 6963 2065 7865 6375 e specific execu │ │ │ │ -00269d40: 7461 626c 6520 6669 6c65 2074 6f20 7275 table file to ru │ │ │ │ -00269d50: 6e2e 2054 6869 730a 2020 2020 2020 2020 n. This. │ │ │ │ -00269d60: 6973 206f 6e6c 7920 6e65 6365 7373 6172 is only necessar │ │ │ │ -00269d70: 7920 6966 2074 6865 2070 726f 6772 616d y if the program │ │ │ │ -00269d80: 2063 6f6e 7369 7374 7320 6f66 206d 756c consists of mul │ │ │ │ -00269d90: 7469 706c 6520 7375 6368 2066 696c 6573 tiple such files │ │ │ │ -00269da0: 2e20 2049 660a 2020 2020 2020 2020 6e6f . If. no │ │ │ │ -00269db0: 7420 6769 7665 6e2c 2074 6865 6e20 7072 t given, then pr │ │ │ │ -00269dc0: 6f67 7261 6d23 226e 616d 6522 2069 7320 ogram#"name" is │ │ │ │ -00269dd0: 7573 6564 2e0a 2020 2020 2020 2a20 6172 used.. * ar │ │ │ │ -00269de0: 6773 2c20 6120 2a6e 6f74 6520 7374 7269 gs, a *note stri │ │ │ │ -00269df0: 6e67 3a20 5374 7269 6e67 2c2c 2074 6865 ng: String,, the │ │ │ │ -00269e00: 2063 6f6d 6d61 6e64 206c 696e 6520 6172 command line ar │ │ │ │ -00269e10: 6775 6d65 6e74 7320 7061 7373 6564 2074 guments passed t │ │ │ │ -00269e20: 6f20 7468 650a 2020 2020 2020 2020 7072 o the. pr │ │ │ │ -00269e30: 6f67 7261 6d2e 0a20 202a 202a 6e6f 7465 ogram.. * *note │ │ │ │ -00269e40: 204f 7074 696f 6e61 6c20 696e 7075 7473 Optional inputs │ │ │ │ -00269e50: 3a20 7573 696e 6720 6675 6e63 7469 6f6e : using function │ │ │ │ -00269e60: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ -00269e70: 696e 7075 7473 2c3a 0a20 2020 2020 202a inputs,:. * │ │ │ │ -00269e80: 204b 6565 7046 696c 6573 203d 3e20 6120 KeepFiles => a │ │ │ │ -00269e90: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661 *note Boolean va │ │ │ │ -00269ea0: 6c75 653a 2042 6f6f 6c65 616e 2c2c 2064 lue: Boolean,, d │ │ │ │ -00269eb0: 6566 6175 6c74 2076 616c 7565 2066 616c efault value fal │ │ │ │ -00269ec0: 7365 2c0a 2020 2020 2020 2020 7768 6574 se,. whet │ │ │ │ -00269ed0: 6865 7220 746f 206b 6565 7020 7468 6520 her to keep the │ │ │ │ -00269ee0: 7465 6d70 6f72 6172 7920 6669 6c65 7320 temporary files │ │ │ │ -00269ef0: 636f 6e74 6169 6e69 6e67 2074 6865 2070 containing the p │ │ │ │ -00269f00: 726f 6772 616d 2773 206f 7574 7075 742e rogram's output. │ │ │ │ -00269f10: 0a20 2020 2020 202a 2052 6169 7365 4572 . * RaiseEr │ │ │ │ -00269f20: 726f 7220 3d3e 2061 202a 6e6f 7465 2042 ror => a *note B │ │ │ │ -00269f30: 6f6f 6c65 616e 2076 616c 7565 3a20 426f oolean value: Bo │ │ │ │ -00269f40: 6f6c 6561 6e2c 2c20 6465 6661 756c 7420 olean,, default │ │ │ │ -00269f50: 7661 6c75 6520 7472 7565 2c0a 2020 2020 value true,. │ │ │ │ -00269f60: 2020 2020 7768 6574 6865 7220 746f 2072 whether to r │ │ │ │ -00269f70: 6169 7365 2061 6e20 6572 726f 7220 6966 aise an error if │ │ │ │ -00269f80: 2074 6865 2070 726f 6772 616d 2072 6574 the program ret │ │ │ │ -00269f90: 7572 6e73 2061 206e 6f6e 7a65 726f 2076 urns a nonzero v │ │ │ │ -00269fa0: 616c 7565 2e0a 2020 2020 2020 2a20 5275 alue.. * Ru │ │ │ │ -00269fb0: 6e44 6972 6563 746f 7279 203d 3e20 6120 nDirectory => a │ │ │ │ -00269fc0: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374 *note string: St │ │ │ │ -00269fd0: 7269 6e67 2c2c 2064 6566 6175 6c74 2076 ring,, default v │ │ │ │ -00269fe0: 616c 7565 206e 756c 6c2c 2074 6865 0a20 alue null, the. │ │ │ │ -00269ff0: 2020 2020 2020 2064 6972 6563 746f 7279 directory │ │ │ │ -0026a000: 2066 726f 6d20 7768 6963 6820 746f 2072 from which to r │ │ │ │ -0026a010: 756e 2074 6865 2070 726f 6772 616d 2e20 un the program. │ │ │ │ -0026a020: 2049 6620 6974 2064 6f65 7320 6e6f 7420 If it does not │ │ │ │ -0026a030: 6578 6973 742c 2074 6865 6e20 6974 0a20 exist, then it. │ │ │ │ -0026a040: 2020 2020 2020 2077 696c 6c20 6265 2063 will be c │ │ │ │ -0026a050: 7265 6174 6564 2e20 2049 6620 2a6e 6f74 reated. If *not │ │ │ │ -0026a060: 6520 6e75 6c6c 3a20 6e75 6c6c 2c2c 2074 e null: null,, t │ │ │ │ -0026a070: 6865 6e20 7468 6520 7072 6f67 7261 6d20 hen the program │ │ │ │ -0026a080: 7769 6c6c 2062 6520 7275 6e0a 2020 2020 will be run. │ │ │ │ -0026a090: 2020 2020 6672 6f6d 2074 6865 2063 7572 from the cur │ │ │ │ -0026a0a0: 7265 6e74 2077 6f72 6b69 6e67 2064 6972 rent working dir │ │ │ │ -0026a0b0: 6563 746f 7279 2028 7365 6520 2a6e 6f74 ectory (see *not │ │ │ │ -0026a0c0: 6520 6375 7272 656e 7444 6972 6563 746f e currentDirecto │ │ │ │ -0026a0d0: 7279 3a0a 2020 2020 2020 2020 6375 7272 ry:. curr │ │ │ │ -0026a0e0: 656e 7444 6972 6563 746f 7279 2c29 2e0a entDirectory,).. │ │ │ │ -0026a0f0: 2020 2020 2020 2a20 5665 7262 6f73 6520 * Verbose │ │ │ │ -0026a100: 3d3e 2061 202a 6e6f 7465 2042 6f6f 6c65 => a *note Boole │ │ │ │ -0026a110: 616e 2076 616c 7565 3a20 426f 6f6c 6561 an value: Boolea │ │ │ │ -0026a120: 6e2c 2c20 6465 6661 756c 7420 7661 6c75 n,, default valu │ │ │ │ -0026a130: 6520 6661 6c73 652c 0a20 2020 2020 2020 e false,. │ │ │ │ -0026a140: 2077 6865 7468 6572 2074 6f20 7072 696e whether to prin │ │ │ │ -0026a150: 7420 7468 6520 636f 6d6d 616e 6420 6c69 t the command li │ │ │ │ -0026a160: 6e65 2069 6e70 7574 2061 6e64 2074 6865 ne input and the │ │ │ │ -0026a170: 2070 726f 6772 616d 2773 206f 7574 7075 program's outpu │ │ │ │ -0026a180: 742e 0a0a 4465 7363 7269 7074 696f 6e0a t...Description. │ │ │ │ -0026a190: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -0026a1a0: 7320 6d65 7468 6f64 2072 756e 7320 616e s method runs an │ │ │ │ -0026a1b0: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ -0026a1c0: 6d20 7768 6963 6820 6861 7320 616c 7265 m which has alre │ │ │ │ -0026a1d0: 6164 7920 6265 656e 206c 6f61 6465 6420 ady been loaded │ │ │ │ -0026a1e0: 7573 696e 6720 2a6e 6f74 650a 6669 6e64 using *note.find │ │ │ │ -0026a1f0: 5072 6f67 7261 6d3a 2066 696e 6450 726f Program: findPro │ │ │ │ -0026a200: 6772 616d 2c2e 2020 5468 6520 7265 7375 gram,. The resu │ │ │ │ -0026a210: 6c74 7320 6f66 2074 6869 7320 7275 6e20 lts of this run │ │ │ │ -0026a220: 6172 6520 6176 6169 6c61 626c 6520 696e are available in │ │ │ │ -0026a230: 2061 202a 6e6f 7465 0a50 726f 6772 616d a *note.Program │ │ │ │ -0026a240: 5275 6e3a 2050 726f 6772 616d 5275 6e2c Run: ProgramRun, │ │ │ │ -0026a250: 206f 626a 6563 742e 0a0a 2b2d 2d2d 2d2d object...+----- │ │ │ │ -0026a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00269ae0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00269af0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +00269b00: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +00269b10: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00269b20: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +00269b30: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00269b40: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ +00269b50: 7932 446f 632f 6675 6e63 7469 6f6e 732f y2Doc/functions/ │ │ │ │ +00269b60: 6669 6e64 5072 6f67 7261 6d2d 646f 632e findProgram-doc. │ │ │ │ +00269b70: 6d32 3a35 383a 302e 0a1f 0a46 696c 653a m2:58:0....File: │ │ │ │ +00269b80: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ +00269b90: 666f 2c20 4e6f 6465 3a20 7275 6e50 726f fo, Node: runPro │ │ │ │ +00269ba0: 6772 616d 2c20 4e65 7874 3a20 5072 6f67 gram, Next: Prog │ │ │ │ +00269bb0: 7261 6d20 3c3c 2054 6869 6e67 2c20 5072 ram << Thing, Pr │ │ │ │ +00269bc0: 6576 3a20 6669 6e64 5072 6f67 7261 6d2c ev: findProgram, │ │ │ │ +00269bd0: 2055 703a 2063 6f6d 6d75 6e69 6361 7469 Up: communicati │ │ │ │ +00269be0: 6e67 2077 6974 6820 7072 6f67 7261 6d73 ng with programs │ │ │ │ +00269bf0: 0a0a 7275 6e50 726f 6772 616d 202d 2d20 ..runProgram -- │ │ │ │ +00269c00: 7275 6e20 616e 2065 7874 6572 6e61 6c20 run an external │ │ │ │ +00269c10: 7072 6f67 7261 6d0a 2a2a 2a2a 2a2a 2a2a program.******** │ │ │ │ +00269c20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00269c30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ +00269c40: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ +00269c50: 2020 2072 756e 5072 6f67 7261 6d28 7072 runProgram(pr │ │ │ │ +00269c60: 6f67 7261 6d2c 2061 7267 7329 0a20 2020 ogram, args). │ │ │ │ +00269c70: 2020 2020 2072 756e 5072 6f67 7261 6d28 runProgram( │ │ │ │ +00269c80: 7072 6f67 7261 6d2c 2065 7865 2c20 6172 program, exe, ar │ │ │ │ +00269c90: 6773 290a 2020 2a20 496e 7075 7473 3a0a gs). * Inputs:. │ │ │ │ +00269ca0: 2020 2020 2020 2a20 7072 6f67 7261 6d2c * program, │ │ │ │ +00269cb0: 2061 6e20 696e 7374 616e 6365 206f 6620 an instance of │ │ │ │ +00269cc0: 7468 6520 7479 7065 202a 6e6f 7465 2050 the type *note P │ │ │ │ +00269cd0: 726f 6772 616d 3a20 5072 6f67 7261 6d2c rogram: Program, │ │ │ │ +00269ce0: 2c20 7468 6520 7072 6f67 7261 6d0a 2020 , the program. │ │ │ │ +00269cf0: 2020 2020 2020 746f 2072 756e 2c20 6765 to run, ge │ │ │ │ +00269d00: 6e65 7261 7465 6420 6279 202a 6e6f 7465 nerated by *note │ │ │ │ +00269d10: 2066 696e 6450 726f 6772 616d 3a20 6669 findProgram: fi │ │ │ │ +00269d20: 6e64 5072 6f67 7261 6d2c 2e0a 2020 2020 ndProgram,.. │ │ │ │ +00269d30: 2020 2a20 6578 652c 2061 202a 6e6f 7465 * exe, a *note │ │ │ │ +00269d40: 2073 7472 696e 673a 2053 7472 696e 672c string: String, │ │ │ │ +00269d50: 2c20 7468 6520 7370 6563 6966 6963 2065 , the specific e │ │ │ │ +00269d60: 7865 6375 7461 626c 6520 6669 6c65 2074 xecutable file t │ │ │ │ +00269d70: 6f20 7275 6e2e 2054 6869 730a 2020 2020 o run. This. │ │ │ │ +00269d80: 2020 2020 6973 206f 6e6c 7920 6e65 6365 is only nece │ │ │ │ +00269d90: 7373 6172 7920 6966 2074 6865 2070 726f ssary if the pro │ │ │ │ +00269da0: 6772 616d 2063 6f6e 7369 7374 7320 6f66 gram consists of │ │ │ │ +00269db0: 206d 756c 7469 706c 6520 7375 6368 2066 multiple such f │ │ │ │ +00269dc0: 696c 6573 2e20 2049 660a 2020 2020 2020 iles. If. │ │ │ │ +00269dd0: 2020 6e6f 7420 6769 7665 6e2c 2074 6865 not given, the │ │ │ │ +00269de0: 6e20 7072 6f67 7261 6d23 226e 616d 6522 n program#"name" │ │ │ │ +00269df0: 2069 7320 7573 6564 2e0a 2020 2020 2020 is used.. │ │ │ │ +00269e00: 2a20 6172 6773 2c20 6120 2a6e 6f74 6520 * args, a *note │ │ │ │ +00269e10: 7374 7269 6e67 3a20 5374 7269 6e67 2c2c string: String,, │ │ │ │ +00269e20: 2074 6865 2063 6f6d 6d61 6e64 206c 696e the command lin │ │ │ │ +00269e30: 6520 6172 6775 6d65 6e74 7320 7061 7373 e arguments pass │ │ │ │ +00269e40: 6564 2074 6f20 7468 650a 2020 2020 2020 ed to the. │ │ │ │ +00269e50: 2020 7072 6f67 7261 6d2e 0a20 202a 202a program.. * * │ │ │ │ +00269e60: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ +00269e70: 7075 7473 3a20 7573 696e 6720 6675 6e63 puts: using func │ │ │ │ +00269e80: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ +00269e90: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020 nal inputs,:. │ │ │ │ +00269ea0: 2020 202a 204b 6565 7046 696c 6573 203d * KeepFiles = │ │ │ │ +00269eb0: 3e20 6120 2a6e 6f74 6520 426f 6f6c 6561 > a *note Boolea │ │ │ │ +00269ec0: 6e20 7661 6c75 653a 2042 6f6f 6c65 616e n value: Boolean │ │ │ │ +00269ed0: 2c2c 2064 6566 6175 6c74 2076 616c 7565 ,, default value │ │ │ │ +00269ee0: 2066 616c 7365 2c0a 2020 2020 2020 2020 false,. │ │ │ │ +00269ef0: 7768 6574 6865 7220 746f 206b 6565 7020 whether to keep │ │ │ │ +00269f00: 7468 6520 7465 6d70 6f72 6172 7920 6669 the temporary fi │ │ │ │ +00269f10: 6c65 7320 636f 6e74 6169 6e69 6e67 2074 les containing t │ │ │ │ +00269f20: 6865 2070 726f 6772 616d 2773 206f 7574 he program's out │ │ │ │ +00269f30: 7075 742e 0a20 2020 2020 202a 2052 6169 put.. * Rai │ │ │ │ +00269f40: 7365 4572 726f 7220 3d3e 2061 202a 6e6f seError => a *no │ │ │ │ +00269f50: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ +00269f60: 3a20 426f 6f6c 6561 6e2c 2c20 6465 6661 : Boolean,, defa │ │ │ │ +00269f70: 756c 7420 7661 6c75 6520 7472 7565 2c0a ult value true,. │ │ │ │ +00269f80: 2020 2020 2020 2020 7768 6574 6865 7220 whether │ │ │ │ +00269f90: 746f 2072 6169 7365 2061 6e20 6572 726f to raise an erro │ │ │ │ +00269fa0: 7220 6966 2074 6865 2070 726f 6772 616d r if the program │ │ │ │ +00269fb0: 2072 6574 7572 6e73 2061 206e 6f6e 7a65 returns a nonze │ │ │ │ +00269fc0: 726f 2076 616c 7565 2e0a 2020 2020 2020 ro value.. │ │ │ │ +00269fd0: 2a20 5275 6e44 6972 6563 746f 7279 203d * RunDirectory = │ │ │ │ +00269fe0: 3e20 6120 2a6e 6f74 6520 7374 7269 6e67 > a *note string │ │ │ │ +00269ff0: 3a20 5374 7269 6e67 2c2c 2064 6566 6175 : String,, defau │ │ │ │ +0026a000: 6c74 2076 616c 7565 206e 756c 6c2c 2074 lt value null, t │ │ │ │ +0026a010: 6865 0a20 2020 2020 2020 2064 6972 6563 he. direc │ │ │ │ +0026a020: 746f 7279 2066 726f 6d20 7768 6963 6820 tory from which │ │ │ │ +0026a030: 746f 2072 756e 2074 6865 2070 726f 6772 to run the progr │ │ │ │ +0026a040: 616d 2e20 2049 6620 6974 2064 6f65 7320 am. If it does │ │ │ │ +0026a050: 6e6f 7420 6578 6973 742c 2074 6865 6e20 not exist, then │ │ │ │ +0026a060: 6974 0a20 2020 2020 2020 2077 696c 6c20 it. will │ │ │ │ +0026a070: 6265 2063 7265 6174 6564 2e20 2049 6620 be created. If │ │ │ │ +0026a080: 2a6e 6f74 6520 6e75 6c6c 3a20 6e75 6c6c *note null: null │ │ │ │ +0026a090: 2c2c 2074 6865 6e20 7468 6520 7072 6f67 ,, then the prog │ │ │ │ +0026a0a0: 7261 6d20 7769 6c6c 2062 6520 7275 6e0a ram will be run. │ │ │ │ +0026a0b0: 2020 2020 2020 2020 6672 6f6d 2074 6865 from the │ │ │ │ +0026a0c0: 2063 7572 7265 6e74 2077 6f72 6b69 6e67 current working │ │ │ │ +0026a0d0: 2064 6972 6563 746f 7279 2028 7365 6520 directory (see │ │ │ │ +0026a0e0: 2a6e 6f74 6520 6375 7272 656e 7444 6972 *note currentDir │ │ │ │ +0026a0f0: 6563 746f 7279 3a0a 2020 2020 2020 2020 ectory:. │ │ │ │ +0026a100: 6375 7272 656e 7444 6972 6563 746f 7279 currentDirectory │ │ │ │ +0026a110: 2c29 2e0a 2020 2020 2020 2a20 5665 7262 ,).. * Verb │ │ │ │ +0026a120: 6f73 6520 3d3e 2061 202a 6e6f 7465 2042 ose => a *note B │ │ │ │ +0026a130: 6f6f 6c65 616e 2076 616c 7565 3a20 426f oolean value: Bo │ │ │ │ +0026a140: 6f6c 6561 6e2c 2c20 6465 6661 756c 7420 olean,, default │ │ │ │ +0026a150: 7661 6c75 6520 6661 6c73 652c 0a20 2020 value false,. │ │ │ │ +0026a160: 2020 2020 2077 6865 7468 6572 2074 6f20 whether to │ │ │ │ +0026a170: 7072 696e 7420 7468 6520 636f 6d6d 616e print the comman │ │ │ │ +0026a180: 6420 6c69 6e65 2069 6e70 7574 2061 6e64 d line input and │ │ │ │ +0026a190: 2074 6865 2070 726f 6772 616d 2773 206f the program's o │ │ │ │ +0026a1a0: 7574 7075 742e 0a0a 4465 7363 7269 7074 utput...Descript │ │ │ │ +0026a1b0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0026a1c0: 0a54 6869 7320 6d65 7468 6f64 2072 756e .This method run │ │ │ │ +0026a1d0: 7320 616e 2065 7874 6572 6e61 6c20 7072 s an external pr │ │ │ │ +0026a1e0: 6f67 7261 6d20 7768 6963 6820 6861 7320 ogram which has │ │ │ │ +0026a1f0: 616c 7265 6164 7920 6265 656e 206c 6f61 already been loa │ │ │ │ +0026a200: 6465 6420 7573 696e 6720 2a6e 6f74 650a ded using *note. │ │ │ │ +0026a210: 6669 6e64 5072 6f67 7261 6d3a 2066 696e findProgram: fin │ │ │ │ +0026a220: 6450 726f 6772 616d 2c2e 2020 5468 6520 dProgram,. The │ │ │ │ +0026a230: 7265 7375 6c74 7320 6f66 2074 6869 7320 results of this │ │ │ │ +0026a240: 7275 6e20 6172 6520 6176 6169 6c61 626c run are availabl │ │ │ │ +0026a250: 6520 696e 2061 202a 6e6f 7465 0a50 726f e in a *note.Pro │ │ │ │ +0026a260: 6772 616d 5275 6e3a 2050 726f 6772 616d gramRun: Program │ │ │ │ +0026a270: 5275 6e2c 206f 626a 6563 742e 0a0a 2b2d Run, object...+- │ │ │ │ 0026a280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a290: 2d2d 2d2d 2b0a 7c69 3120 3a20 6766 616e ----+.|i1 : gfan │ │ │ │ -0026a2a0: 203d 2066 696e 6450 726f 6772 616d 2822 = findProgram(" │ │ │ │ -0026a2b0: 6766 616e 222c 2022 6766 616e 202d 2d68 gfan", "gfan --h │ │ │ │ -0026a2c0: 656c 7022 2920 2020 2020 2020 2020 2020 elp") │ │ │ │ -0026a2d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0026a2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a300: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0026a310: 3120 3d20 6766 616e 2020 2020 2020 2020 1 = gfan │ │ │ │ +0026a290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a2b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +0026a2c0: 6766 616e 203d 2066 696e 6450 726f 6772 gfan = findProgr │ │ │ │ +0026a2d0: 616d 2822 6766 616e 222c 2022 6766 616e am("gfan", "gfan │ │ │ │ +0026a2e0: 202d 2d68 656c 7022 2920 2020 2020 2020 --help") │ │ │ │ +0026a2f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0026a300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a340: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0026a330: 7c0a 7c6f 3120 3d20 6766 616e 2020 2020 |.|o1 = gfan │ │ │ │ +0026a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a360: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0026a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a380: 2020 2020 7c0a 7c6f 3120 3a20 5072 6f67 |.|o1 : Prog │ │ │ │ -0026a390: 7261 6d20 2020 2020 2020 2020 2020 2020 ram │ │ │ │ -0026a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a3c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0026a3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0026a400: 3220 3a20 7275 6e50 726f 6772 616d 2867 2 : runProgram(g │ │ │ │ -0026a410: 6661 6e2c 2022 5f76 6572 7369 6f6e 2229 fan, "_version") │ │ │ │ -0026a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a430: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0026a440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a3a0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +0026a3b0: 5072 6f67 7261 6d20 2020 2020 2020 2020 Program │ │ │ │ +0026a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a3e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0026a3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a420: 2b0a 7c69 3220 3a20 7275 6e50 726f 6772 +.|i2 : runProgr │ │ │ │ +0026a430: 616d 2867 6661 6e2c 2022 5f76 6572 7369 am(gfan, "_versi │ │ │ │ +0026a440: 6f6e 2229 2020 2020 2020 2020 2020 2020 on") │ │ │ │ +0026a450: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0026a460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a470: 2020 2020 7c0a 7c6f 3220 3d20 3020 2020 |.|o2 = 0 │ │ │ │ +0026a470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a4b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0026a490: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +0026a4a0: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +0026a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a4e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0026a4f0: 3220 3a20 5072 6f67 7261 6d52 756e 2020 2 : ProgramRun │ │ │ │ +0026a4d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0026a4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a520: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0026a530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a510: 7c0a 7c6f 3220 3a20 5072 6f67 7261 6d52 |.|o2 : ProgramR │ │ │ │ +0026a520: 756e 2020 2020 2020 2020 2020 2020 2020 un │ │ │ │ +0026a530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a540: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0026a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a560: 2d2d 2d2d 2b0a 7c69 3320 3a20 6f6f 2322 ----+.|i3 : oo#" │ │ │ │ -0026a570: 6f75 7470 7574 2220 2020 2020 2020 2020 output" │ │ │ │ -0026a580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a5a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0026a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a580: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +0026a590: 6f6f 2322 6f75 7470 7574 2220 2020 2020 oo#"output" │ │ │ │ +0026a5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a5d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0026a5e0: 3320 3d20 4766 616e 2076 6572 7369 6f6e 3 = Gfan version │ │ │ │ -0026a5f0: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ -0026a600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a610: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0026a620: 6766 616e 302e 3862 6574 6120 2020 2020 gfan0.8beta │ │ │ │ -0026a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a650: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0026a5c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0026a5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a600: 7c0a 7c6f 3320 3d20 4766 616e 2076 6572 |.|o3 = Gfan ver │ │ │ │ +0026a610: 7369 6f6e 3a20 2020 2020 2020 2020 2020 sion: │ │ │ │ +0026a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a630: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0026a640: 2020 2020 6766 616e 302e 3862 6574 6120 gfan0.8beta │ │ │ │ +0026a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a670: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0026a680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a690: 7c0a 7c20 2020 2020 466f 726b 6564 2066 |.| Forked f │ │ │ │ -0026a6a0: 726f 6d20 736f 7572 6365 2074 7265 6520 rom source tree │ │ │ │ -0026a6b0: 6f6e 3a20 2020 2020 2020 2020 2020 2020 on: │ │ │ │ -0026a6c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0026a6d0: 2020 2020 3137 3738 3138 3433 3637 2054 1778184367 T │ │ │ │ -0026a6e0: 6875 204d 6179 2030 3720 3232 3a30 363a hu May 07 22:06: │ │ │ │ -0026a6f0: 3037 2032 3032 3620 2020 2020 2020 2020 07 2026 │ │ │ │ -0026a700: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0026a710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a6b0: 2020 2020 7c0a 7c20 2020 2020 466f 726b |.| Fork │ │ │ │ +0026a6c0: 6564 2066 726f 6d20 736f 7572 6365 2074 ed from source t │ │ │ │ +0026a6d0: 7265 6520 6f6e 3a20 2020 2020 2020 2020 ree on: │ │ │ │ +0026a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a6f0: 7c0a 7c20 2020 2020 3137 3738 3138 3433 |.| 17781843 │ │ │ │ +0026a700: 3637 2054 6875 204d 6179 2030 3720 3232 67 Thu May 07 22 │ │ │ │ +0026a710: 3a30 363a 3037 2032 3032 3620 2020 2020 :06:07 2026 │ │ │ │ +0026a720: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0026a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a740: 2020 2020 7c0a 7c20 2020 2020 4c69 6e6b |.| Link │ │ │ │ -0026a750: 6564 206c 6962 7261 7269 6573 3a20 2020 ed libraries: │ │ │ │ -0026a760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a780: 7c0a 7c20 2020 2020 474d 5020 362e 332e |.| GMP 6.3. │ │ │ │ -0026a790: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -0026a7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0026a7c0: 2020 2020 4364 646c 6962 2020 2020 2020 Cddlib │ │ │ │ -0026a7d0: 2059 4553 2020 2020 2020 2020 2020 2020 YES │ │ │ │ -0026a7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a7f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0026a800: 536f 506c 6578 2020 2020 2020 2020 4e4f SoPlex NO │ │ │ │ -0026a810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a830: 2020 2020 7c0a 7c20 2020 2020 5369 6e67 |.| Sing │ │ │ │ -0026a840: 756c 6172 2020 2020 2020 4e4f 2020 2020 ular NO │ │ │ │ -0026a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a870: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0026a880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0026a8b0: 3420 3a20 7275 6e50 726f 6772 616d 2867 4 : runProgram(g │ │ │ │ -0026a8c0: 6661 6e2c 2022 5f66 6f6f 222c 2052 6169 fan, "_foo", Rai │ │ │ │ -0026a8d0: 7365 4572 726f 7220 3d3e 2066 616c 7365 seError => false │ │ │ │ -0026a8e0: 2920 2020 2020 2020 7c0a 7c20 2020 2020 ) |.| │ │ │ │ -0026a8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a760: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0026a770: 4c69 6e6b 6564 206c 6962 7261 7269 6573 Linked libraries │ │ │ │ +0026a780: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ +0026a790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a7a0: 2020 2020 7c0a 7c20 2020 2020 474d 5020 |.| GMP │ │ │ │ +0026a7b0: 362e 332e 3020 2020 2020 2020 2020 2020 6.3.0 │ │ │ │ +0026a7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a7e0: 7c0a 7c20 2020 2020 4364 646c 6962 2020 |.| Cddlib │ │ │ │ +0026a7f0: 2020 2020 2059 4553 2020 2020 2020 2020 YES │ │ │ │ +0026a800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a810: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0026a820: 2020 2020 536f 506c 6578 2020 2020 2020 SoPlex │ │ │ │ +0026a830: 2020 4e4f 2020 2020 2020 2020 2020 2020 NO │ │ │ │ +0026a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a850: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0026a860: 5369 6e67 756c 6172 2020 2020 2020 4e4f Singular NO │ │ │ │ +0026a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a890: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0026a8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a8d0: 2b0a 7c69 3420 3a20 7275 6e50 726f 6772 +.|i4 : runProgr │ │ │ │ +0026a8e0: 616d 2867 6661 6e2c 2022 5f66 6f6f 222c am(gfan, "_foo", │ │ │ │ +0026a8f0: 2052 6169 7365 4572 726f 7220 3d3e 2066 RaiseError => f │ │ │ │ +0026a900: 616c 7365 2920 2020 2020 2020 7c0a 7c20 alse) |.| │ │ │ │ 0026a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a920: 2020 2020 7c0a 7c6f 3420 3d20 3235 3620 |.|o4 = 256 │ │ │ │ +0026a920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a960: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0026a940: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ +0026a950: 3235 3620 2020 2020 2020 2020 2020 2020 256 │ │ │ │ +0026a960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a990: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0026a9a0: 3420 3a20 5072 6f67 7261 6d52 756e 2020 4 : ProgramRun │ │ │ │ +0026a980: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0026a990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026a9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026a9d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0026a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026a9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026a9c0: 7c0a 7c6f 3420 3a20 5072 6f67 7261 6d52 |.|o4 : ProgramR │ │ │ │ +0026a9d0: 756e 2020 2020 2020 2020 2020 2020 2020 un │ │ │ │ +0026a9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026a9f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0026aa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026aa10: 2d2d 2d2d 2b0a 7c69 3520 3a20 6f6f 2322 ----+.|i5 : oo#" │ │ │ │ -0026aa20: 6572 726f 7222 2020 2020 2020 2020 2020 error" │ │ │ │ -0026aa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026aa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026aa50: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0026aa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026aa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026aa30: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ +0026aa40: 6f6f 2322 6572 726f 7222 2020 2020 2020 oo#"error" │ │ │ │ +0026aa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026aa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026aa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026aa80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0026aa90: 3520 3d20 554e 4b4e 4f57 4e20 4f50 5449 5 = UNKNOWN OPTI │ │ │ │ -0026aaa0: 4f4e 3a20 5f66 6f6f 2e20 2020 2020 2020 ON: _foo. │ │ │ │ -0026aab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026aac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0026aad0: 5553 4520 2d2d 6865 6c70 2041 5320 4120 USE --help AS A │ │ │ │ -0026aae0: 5349 4e47 4c45 204f 5054 494f 4e20 544f SINGLE OPTION TO │ │ │ │ -0026aaf0: 2056 4945 5720 5448 4520 4845 4c50 2054 VIEW THE HELP T │ │ │ │ -0026ab00: 4558 542e 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d EXT.|.+--------- │ │ │ │ -0026ab10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ab20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026aa70: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0026aa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026aa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026aaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026aab0: 7c0a 7c6f 3520 3d20 554e 4b4e 4f57 4e20 |.|o5 = UNKNOWN │ │ │ │ +0026aac0: 4f50 5449 4f4e 3a20 5f66 6f6f 2e20 2020 OPTION: _foo. │ │ │ │ +0026aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026aae0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0026aaf0: 2020 2020 5553 4520 2d2d 6865 6c70 2041 USE --help A │ │ │ │ +0026ab00: 5320 4120 5349 4e47 4c45 204f 5054 494f S A SINGLE OPTIO │ │ │ │ +0026ab10: 4e20 544f 2056 4945 5720 5448 4520 4845 N TO VIEW THE HE │ │ │ │ +0026ab20: 4c50 2054 4558 542e 7c0a 2b2d 2d2d 2d2d LP TEXT.|.+----- │ │ │ │ 0026ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ab40: 2b0a 0a54 6865 2076 616c 7565 2063 6f72 +..The value cor │ │ │ │ -0026ab50: 7265 7370 6f6e 6469 6e67 2074 6f20 7468 responding to th │ │ │ │ -0026ab60: 6520 226f 7574 7075 7422 206b 6579 206d e "output" key m │ │ │ │ -0026ab70: 6179 2061 6c73 6f20 6265 206f 6274 6169 ay also be obtai │ │ │ │ -0026ab80: 6e65 6420 7573 696e 6720 2a6e 6f74 650a ned using *note. │ │ │ │ -0026ab90: 746f 5374 7269 6e67 3a20 746f 5374 7269 toString: toStri │ │ │ │ -0026aba0: 6e67 2c2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ng,...+--------- │ │ │ │ -0026abb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026abc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026abd0: 2d2b 0a7c 6936 203a 2074 6f53 7472 696e -+.|i6 : toStrin │ │ │ │ -0026abe0: 6720 7275 6e50 726f 6772 616d 2867 6661 g runProgram(gfa │ │ │ │ -0026abf0: 6e2c 2022 5f76 6572 7369 6f6e 2229 7c0a n, "_version")|. │ │ │ │ -0026ac00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0026ac10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ac20: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0026ac30: 203d 2047 6661 6e20 7665 7273 696f 6e3a = Gfan version: │ │ │ │ -0026ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ac50: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0026ac60: 6766 616e 302e 3862 6574 6120 2020 2020 gfan0.8beta │ │ │ │ -0026ac70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ac80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0026ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ab60: 2d2d 2d2d 2b0a 0a54 6865 2076 616c 7565 ----+..The value │ │ │ │ +0026ab70: 2063 6f72 7265 7370 6f6e 6469 6e67 2074 corresponding t │ │ │ │ +0026ab80: 6f20 7468 6520 226f 7574 7075 7422 206b o the "output" k │ │ │ │ +0026ab90: 6579 206d 6179 2061 6c73 6f20 6265 206f ey may also be o │ │ │ │ +0026aba0: 6274 6169 6e65 6420 7573 696e 6720 2a6e btained using *n │ │ │ │ +0026abb0: 6f74 650a 746f 5374 7269 6e67 3a20 746f ote.toString: to │ │ │ │ +0026abc0: 5374 7269 6e67 2c2e 0a0a 2b2d 2d2d 2d2d String,...+----- │ │ │ │ +0026abd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026abe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026abf0: 2d2d 2d2d 2d2b 0a7c 6936 203a 2074 6f53 -----+.|i6 : toS │ │ │ │ +0026ac00: 7472 696e 6720 7275 6e50 726f 6772 616d tring runProgram │ │ │ │ +0026ac10: 2867 6661 6e2c 2022 5f76 6572 7369 6f6e (gfan, "_version │ │ │ │ +0026ac20: 2229 7c0a 7c20 2020 2020 2020 2020 2020 ")|.| │ │ │ │ +0026ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ac40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026ac50: 0a7c 6f36 203d 2047 6661 6e20 7665 7273 .|o6 = Gfan vers │ │ │ │ +0026ac60: 696f 6e3a 2020 2020 2020 2020 2020 2020 ion: │ │ │ │ +0026ac70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0026ac80: 2020 2020 6766 616e 302e 3862 6574 6120 gfan0.8beta │ │ │ │ 0026ac90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026aca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026acb0: 2020 7c0a 7c20 2020 2020 466f 726b 6564 |.| Forked │ │ │ │ -0026acc0: 2066 726f 6d20 736f 7572 6365 2074 7265 from source tre │ │ │ │ -0026acd0: 6520 6f6e 3a20 2020 2020 2020 2020 207c e on: | │ │ │ │ -0026ace0: 0a7c 2020 2020 2031 3737 3831 3834 3336 .| 177818436 │ │ │ │ -0026acf0: 3720 5468 7520 4d61 7920 3037 2032 323a 7 Thu May 07 22: │ │ │ │ -0026ad00: 3036 3a30 3720 3230 3236 2020 7c0a 7c20 06:07 2026 |.| │ │ │ │ -0026ad10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ad20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ad30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0026ad40: 204c 696e 6b65 6420 6c69 6272 6172 6965 Linked librarie │ │ │ │ -0026ad50: 733a 2020 2020 2020 2020 2020 2020 2020 s: │ │ │ │ -0026ad60: 2020 2020 2020 7c0a 7c20 2020 2020 474d |.| GM │ │ │ │ -0026ad70: 5020 362e 332e 3020 2020 2020 2020 2020 P 6.3.0 │ │ │ │ -0026ad80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ad90: 2020 207c 0a7c 2020 2020 2043 6464 6c69 |.| Cddli │ │ │ │ -0026ada0: 6220 2020 2020 2020 5945 5320 2020 2020 b YES │ │ │ │ -0026adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026adc0: 7c0a 7c20 2020 2020 536f 506c 6578 2020 |.| SoPlex │ │ │ │ -0026add0: 2020 2020 2020 4e4f 2020 2020 2020 2020 NO │ │ │ │ -0026ade0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0026adf0: 2020 2020 2053 696e 6775 6c61 7220 2020 Singular │ │ │ │ -0026ae00: 2020 204e 4f20 2020 2020 2020 2020 2020 NO │ │ │ │ -0026ae10: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -0026ae20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ae30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ae40: 2d2d 2d2d 2d2d 2d2b 0a0a 4974 2069 7320 -------+..It is │ │ │ │ -0026ae50: 616c 736f 2070 6f73 7369 626c 6520 746f also possible to │ │ │ │ -0026ae60: 2073 6b69 7020 2a6e 6f74 6520 6669 6e64 skip *note find │ │ │ │ -0026ae70: 5072 6f67 7261 6d3a 2066 696e 6450 726f Program: findPro │ │ │ │ -0026ae80: 6772 616d 2c20 616e 6420 6a75 7374 2070 gram, and just p │ │ │ │ -0026ae90: 726f 7669 6465 0a74 776f 2073 7472 696e rovide.two strin │ │ │ │ -0026aea0: 6773 3a20 7468 6520 6e61 6d65 206f 6620 gs: the name of │ │ │ │ -0026aeb0: 7468 6520 7072 6f67 7261 6d20 616e 6420 the program and │ │ │ │ -0026aec0: 7468 6520 636f 6d6d 616e 6420 6c69 6e65 the command line │ │ │ │ -0026aed0: 2061 7267 756d 656e 7473 2e0a 0a2b 2d2d arguments...+-- │ │ │ │ -0026aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026aef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026af00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026aca0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0026acb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026acc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026acd0: 2020 2020 2020 7c0a 7c20 2020 2020 466f |.| Fo │ │ │ │ +0026ace0: 726b 6564 2066 726f 6d20 736f 7572 6365 rked from source │ │ │ │ +0026acf0: 2074 7265 6520 6f6e 3a20 2020 2020 2020 tree on: │ │ │ │ +0026ad00: 2020 207c 0a7c 2020 2020 2031 3737 3831 |.| 17781 │ │ │ │ +0026ad10: 3834 3336 3720 5468 7520 4d61 7920 3037 84367 Thu May 07 │ │ │ │ +0026ad20: 2032 323a 3036 3a30 3720 3230 3236 2020 22:06:07 2026 │ │ │ │ +0026ad30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0026ad40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ad50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0026ad60: 2020 2020 204c 696e 6b65 6420 6c69 6272 Linked libr │ │ │ │ +0026ad70: 6172 6965 733a 2020 2020 2020 2020 2020 aries: │ │ │ │ +0026ad80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0026ad90: 2020 474d 5020 362e 332e 3020 2020 2020 GMP 6.3.0 │ │ │ │ +0026ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026adb0: 2020 2020 2020 207c 0a7c 2020 2020 2043 |.| C │ │ │ │ +0026adc0: 6464 6c69 6220 2020 2020 2020 5945 5320 ddlib YES │ │ │ │ +0026add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ade0: 2020 2020 7c0a 7c20 2020 2020 536f 506c |.| SoPl │ │ │ │ +0026adf0: 6578 2020 2020 2020 2020 4e4f 2020 2020 ex NO │ │ │ │ +0026ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ae10: 207c 0a7c 2020 2020 2053 696e 6775 6c61 |.| Singula │ │ │ │ +0026ae20: 7220 2020 2020 204e 4f20 2020 2020 2020 r NO │ │ │ │ +0026ae30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0026ae40: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0026ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4974 -----------+..It │ │ │ │ +0026ae70: 2069 7320 616c 736f 2070 6f73 7369 626c is also possibl │ │ │ │ +0026ae80: 6520 746f 2073 6b69 7020 2a6e 6f74 6520 e to skip *note │ │ │ │ +0026ae90: 6669 6e64 5072 6f67 7261 6d3a 2066 696e findProgram: fin │ │ │ │ +0026aea0: 6450 726f 6772 616d 2c20 616e 6420 6a75 dProgram, and ju │ │ │ │ +0026aeb0: 7374 2070 726f 7669 6465 0a74 776f 2073 st provide.two s │ │ │ │ +0026aec0: 7472 696e 6773 3a20 7468 6520 6e61 6d65 trings: the name │ │ │ │ +0026aed0: 206f 6620 7468 6520 7072 6f67 7261 6d20 of the program │ │ │ │ +0026aee0: 616e 6420 7468 6520 636f 6d6d 616e 6420 and the command │ │ │ │ +0026aef0: 6c69 6e65 2061 7267 756d 656e 7473 2e0a line arguments.. │ │ │ │ +0026af00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0026af10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026af20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -0026af30: 203a 2072 756e 5072 6f67 7261 6d28 226e : runProgram("n │ │ │ │ -0026af40: 6f72 6d61 6c69 7a22 2c20 222d 2d76 6572 ormaliz", "--ver │ │ │ │ -0026af50: 7369 6f6e 2229 2020 2020 2020 2020 2020 sion") │ │ │ │ -0026af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026af70: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0026af20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026af30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026af40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0026af50: 0a7c 6937 203a 2072 756e 5072 6f67 7261 .|i7 : runProgra │ │ │ │ +0026af60: 6d28 226e 6f72 6d61 6c69 7a22 2c20 222d m("normaliz", "- │ │ │ │ +0026af70: 2d76 6572 7369 6f6e 2229 2020 2020 2020 -version") │ │ │ │ 0026af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026af90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026afa0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0026afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026afc0: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -0026afd0: 203d 2030 2020 2020 2020 2020 2020 2020 = 0 │ │ │ │ -0026afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026afd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026afe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026aff0: 0a7c 6f37 203d 2030 2020 2020 2020 2020 .|o7 = 0 │ │ │ │ 0026b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b010: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0026b010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026b020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b030: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b040: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0026b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b060: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -0026b070: 203a 2050 726f 6772 616d 5275 6e20 2020 : ProgramRun │ │ │ │ -0026b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b0b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0026b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b080: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b090: 0a7c 6f37 203a 2050 726f 6772 616d 5275 .|o7 : ProgramRu │ │ │ │ +0026b0a0: 6e20 2020 2020 2020 2020 2020 2020 2020 n │ │ │ │ +0026b0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b0d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b0e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0026b0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026b100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ -0026b110: 203a 2070 6565 6b20 6f6f 2020 2020 2020 : peek oo │ │ │ │ -0026b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026b110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026b120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0026b130: 0a7c 6938 203a 2070 6565 6b20 6f6f 2020 .|i8 : peek oo │ │ │ │ 0026b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b150: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0026b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b170: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b180: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0026b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b1a0: 2020 2020 2020 2020 2020 207c 0a7c 6f38 |.|o8 │ │ │ │ -0026b1b0: 203d 2050 726f 6772 616d 5275 6e7b 636f = ProgramRun{co │ │ │ │ -0026b1c0: 6d6d 616e 6420 3d3e 202f 7573 722f 6269 mmand => /usr/bi │ │ │ │ -0026b1d0: 6e2f 6e6f 726d 616c 697a 202d 2d76 6572 n/normaliz --ver │ │ │ │ -0026b1e0: 7369 6f6e 2020 2020 2020 2020 2020 2020 sion │ │ │ │ -0026b1f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0026b200: 2020 2020 2020 2020 2020 2020 2020 6572 er │ │ │ │ -0026b210: 726f 7220 3d3e 2020 2020 2020 2020 2020 ror => │ │ │ │ -0026b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b240: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0026b250: 2020 2020 2020 2020 2020 2020 2020 6f75 ou │ │ │ │ -0026b260: 7470 7574 203d 3e20 4e6f 726d 616c 697a tput => Normaliz │ │ │ │ -0026b270: 2033 2e31 312e 3120 2020 2020 2020 2020 3.11.1 │ │ │ │ -0026b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b290: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0026b1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b1c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b1d0: 0a7c 6f38 203d 2050 726f 6772 616d 5275 .|o8 = ProgramRu │ │ │ │ +0026b1e0: 6e7b 636f 6d6d 616e 6420 3d3e 202f 7573 n{command => /us │ │ │ │ +0026b1f0: 722f 6269 6e2f 6e6f 726d 616c 697a 202d r/bin/normaliz - │ │ │ │ +0026b200: 2d76 6572 7369 6f6e 2020 2020 2020 2020 -version │ │ │ │ +0026b210: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0026b230: 2020 6572 726f 7220 3d3e 2020 2020 2020 error => │ │ │ │ +0026b240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b260: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b270: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0026b280: 2020 6f75 7470 7574 203d 3e20 4e6f 726d output => Norm │ │ │ │ +0026b290: 616c 697a 2033 2e31 312e 3120 2020 2020 aliz 3.11.1 │ │ │ │ 0026b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b2b0: 2020 2020 2020 2020 2d2d 2d2d 2d2d 2d2d -------- │ │ │ │ -0026b2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026b2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026b2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -0026b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b300: 2020 2020 2020 2020 7769 7468 2070 6163 with pac │ │ │ │ -0026b310: 6b61 6765 2873 2920 466c 696e 7420 652d kage(s) Flint e- │ │ │ │ -0026b320: 616e 7469 6320 6e61 7574 7920 2020 2020 antic nauty │ │ │ │ -0026b330: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0026b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b350: 2020 2020 2020 2020 436f 7079 7269 6768 Copyrigh │ │ │ │ -0026b360: 7420 2843 2920 3230 3037 2d32 3032 3520 t (C) 2007-2025 │ │ │ │ -0026b370: 2054 6865 204e 6f72 6d61 6c69 7a20 5465 The Normaliz Te │ │ │ │ -0026b380: 616d 2c20 556e 6976 6572 737c 0a7c 2020 am, Univers|.| │ │ │ │ -0026b390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b3a0: 2020 2020 2020 2020 5468 6973 2070 726f This pro │ │ │ │ -0026b3b0: 6772 616d 2063 6f6d 6573 2077 6974 6820 gram comes with │ │ │ │ -0026b3c0: 4142 534f 4c55 5445 4c59 204e 4f20 5741 ABSOLUTELY NO WA │ │ │ │ -0026b3d0: 5252 414e 5459 3b20 5468 697c 0a7c 2020 RRANTY; Thi|.| │ │ │ │ -0026b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b3f0: 2020 2020 2020 2020 616e 6420 796f 7520 and you │ │ │ │ -0026b400: 6172 6520 7765 6c63 6f6d 6520 746f 2072 are welcome to r │ │ │ │ -0026b410: 6564 6973 7472 6962 7574 6520 6974 2075 edistribute it u │ │ │ │ -0026b420: 6e64 6572 2063 6572 7461 697c 0a7c 2020 nder certai|.| │ │ │ │ -0026b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b440: 2020 2020 2020 2020 5365 6520 434f 5059 See COPY │ │ │ │ -0026b450: 494e 4720 666f 7220 6465 7461 696c 732e ING for details. │ │ │ │ -0026b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b470: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0026b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b490: 2020 2020 2020 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2920 3230 3037 2d32 right (C) 2007-2 │ │ │ │ +0026b390: 3032 3520 2054 6865 204e 6f72 6d61 6c69 025 The Normali │ │ │ │ +0026b3a0: 7a20 5465 616d 2c20 556e 6976 6572 737c z Team, Univers| │ │ │ │ +0026b3b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0026b3c0: 2020 2020 2020 2020 2020 2020 5468 6973 This │ │ │ │ +0026b3d0: 2070 726f 6772 616d 2063 6f6d 6573 2077 program comes w │ │ │ │ +0026b3e0: 6974 6820 4142 534f 4c55 5445 4c59 204e ith ABSOLUTELY N │ │ │ │ +0026b3f0: 4f20 5741 5252 414e 5459 3b20 5468 697c O WARRANTY; Thi| │ │ │ │ +0026b400: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0026b410: 2020 2020 2020 2020 2020 2020 616e 6420 and │ │ │ │ +0026b420: 796f 7520 6172 6520 7765 6c63 6f6d 6520 you are welcome │ │ │ │ +0026b430: 746f 2072 6564 6973 7472 6962 7574 6520 to redistribute │ │ │ │ +0026b440: 6974 2075 6e64 6572 2063 6572 7461 697c it under certai| │ │ │ │ +0026b450: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0026b460: 2020 2020 2020 2020 2020 2020 5365 6520 See │ │ │ │ +0026b470: 434f 5059 494e 4720 666f 7220 6465 7461 COPYING for deta │ │ │ │ +0026b480: 696c 732e 2020 2020 2020 2020 2020 2020 ils. │ │ │ │ +0026b490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b4a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0026b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b4c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0026b4d0: 2020 2020 2020 2020 2020 2020 2020 7265 re │ │ │ │ -0026b4e0: 7475 726e 2076 616c 7565 203d 3e20 3020 turn value => 0 │ │ │ │ -0026b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b510: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ -0026b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026b4c0: 2020 2020 2020 2020 2020 2020 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6e73 3b20 2020 2020 conditions; │ │ │ │ -0026b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b710: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b720: 0a7c 6974 7920 6f66 204f 736e 6162 7275 .|ity of Osnabru │ │ │ │ +0026b730: 6563 6b2e 2020 2020 2020 2020 2020 2020 eck. │ │ │ │ +0026b740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b770: 0a7c 7320 6973 2066 7265 6520 736f 6674 .|s is free soft │ │ │ │ +0026b780: 7761 7265 2c20 2020 2020 2020 2020 2020 ware, │ │ │ │ +0026b790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026b7b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026b7c0: 0a7c 6e20 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-0026b940: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -0026b950: 6f74 6520 5072 6f67 7261 6d3a 2050 726f ote Program: Pro │ │ │ │ -0026b960: 6772 616d 2c20 2d2d 2065 7874 6572 6e61 gram, -- externa │ │ │ │ -0026b970: 6c20 7072 6f67 7261 6d20 6f62 6a65 6374 l program object │ │ │ │ -0026b980: 0a20 202a 202a 6e6f 7465 2066 696e 6450 . * *note findP │ │ │ │ -0026b990: 726f 6772 616d 3a20 6669 6e64 5072 6f67 rogram: findProg │ │ │ │ -0026b9a0: 7261 6d2c 202d 2d20 6c6f 6164 2065 7874 ram, -- load ext │ │ │ │ -0026b9b0: 6572 6e61 6c20 7072 6f67 7261 6d0a 2020 ernal program. │ │ │ │ -0026b9c0: 2a20 2a6e 6f74 6520 7374 6174 7573 2850 * *note status(P │ │ │ │ -0026b9d0: 726f 6772 616d 5275 6e29 3a20 7374 6174 rogramRun): stat │ │ │ │ -0026b9e0: 7573 5f6c 7050 726f 6772 616d 5275 6e5f us_lpProgramRun_ │ │ │ │ -0026b9f0: 7270 2c20 2d2d 2067 6574 2074 6865 2072 rp, -- get the r │ │ │ │ -0026ba00: 6574 7572 6e20 7374 6174 7573 0a20 2020 eturn status. │ │ │ │ -0026ba10: 206f 6620 6120 7072 6f67 7261 6d20 7275 of a program ru │ │ │ │ -0026ba20: 6e0a 2020 2a20 2a6e 6f74 6520 7275 6e3a n. * *note run: │ │ │ │ -0026ba30: 2072 756e 2c20 2d2d 2072 756e 2061 6e20 run, -- run an │ │ │ │ -0026ba40: 6578 7465 726e 616c 2063 6f6d 6d61 6e64 external command │ │ │ │ -0026ba50: 0a20 202a 202a 6e6f 7465 2067 6574 3a20 . * *note get: │ │ │ │ -0026ba60: 6765 742c 202d 2d20 6765 7420 7468 6520 get, -- get the │ │ │ │ -0026ba70: 636f 6e74 656e 7473 206f 6620 6120 6669 contents of a fi │ │ │ │ -0026ba80: 6c65 0a2a 204d 656e 753a 0a0a 2a20 5072 le.* Menu:..* Pr │ │ │ │ -0026ba90: 6f67 7261 6d52 756e 3a3a 2020 2020 2020 ogramRun:: │ │ │ │ -0026baa0: 2020 2020 2020 2020 2020 2020 7265 7375 resu │ │ │ │ -0026bab0: 6c74 206f 6620 7275 6e6e 696e 6720 616e lt of running an │ │ │ │ -0026bac0: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ -0026bad0: 6d0a 0a57 6179 7320 746f 2075 7365 2072 m..Ways to use r │ │ │ │ -0026bae0: 756e 5072 6f67 7261 6d3a 0a3d 3d3d 3d3d unProgram:.===== │ │ │ │ -0026baf0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026bb00: 3d3d 0a0a 2020 2a20 2272 756e 5072 6f67 ==.. * "runProg │ │ │ │ -0026bb10: 7261 6d28 5072 6f67 7261 6d2c 5374 7269 ram(Program,Stri │ │ │ │ -0026bb20: 6e67 2922 0a20 202a 2022 7275 6e50 726f ng)". * "runPro │ │ │ │ -0026bb30: 6772 616d 2850 726f 6772 616d 2c53 7472 gram(Program,Str │ │ │ │ -0026bb40: 696e 672c 5374 7269 6e67 2922 0a20 202a ing,String)". * │ │ │ │ -0026bb50: 2022 7275 6e50 726f 6772 616d 2853 7472 "runProgram(Str │ │ │ │ -0026bb60: 696e 672c 5374 7269 6e67 2922 0a0a 466f ing,String)"..Fo │ │ │ │ -0026bb70: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -0026bb80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0026bb90: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -0026bba0: 2a6e 6f74 6520 7275 6e50 726f 6772 616d *note runProgram │ │ │ │ -0026bbb0: 3a20 7275 6e50 726f 6772 616d 2c20 6973 : runProgram, is 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+0026b890: 2041 6e6f 7468 6572 2077 6179 2074 6f20 Another way to │ │ │ │ +0026b8a0: 696e 7465 7261 6374 2077 6974 680a 7072 interact with.pr │ │ │ │ +0026b8b0: 6f67 7261 6d73 2069 7320 746f 2070 6173 ograms is to pas │ │ │ │ +0026b8c0: 7320 6120 7374 7269 6e67 2062 6567 696e s a string begin │ │ │ │ +0026b8d0: 6e69 6e67 2077 6974 6820 2221 2220 746f ning with "!" to │ │ │ │ +0026b8e0: 202a 6e6f 7465 2067 6574 3a20 6765 742c *note get: get, │ │ │ │ +0026b8f0: 2c20 2a6e 6f74 650a 6f70 656e 496e 3a20 , *note.openIn: │ │ │ │ +0026b900: 6f70 656e 496e 5f6c 7053 7472 696e 675f openIn_lpString_ │ │ │ │ +0026b910: 7270 2c2c 202a 6e6f 7465 206f 7065 6e4f rp,, *note openO │ │ │ │ +0026b920: 7574 3a20 6f70 656e 4f75 745f 6c70 5374 ut: openOut_lpSt │ │ │ │ +0026b930: 7269 6e67 5f72 702c 2c20 6f72 202a 6e6f ring_rp,, or *no │ │ │ │ +0026b940: 7465 0a6f 7065 6e49 6e4f 7574 3a20 6f70 te.openInOut: op │ │ │ │ +0026b950: 656e 496e 4f75 742c 2e0a 0a53 6565 2061 enInOut,...See a │ │ │ │ +0026b960: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +0026b970: 2a20 2a6e 6f74 6520 5072 6f67 7261 6d3a * *note Program: │ │ │ │ +0026b980: 2050 726f 6772 616d 2c20 2d2d 2065 7874 Program, -- ext │ │ │ │ +0026b990: 6572 6e61 6c20 7072 6f67 7261 6d20 6f62 ernal program ob │ │ │ │ +0026b9a0: 6a65 6374 0a20 202a 202a 6e6f 7465 2066 ject. * *note f │ │ │ │ +0026b9b0: 696e 6450 726f 6772 616d 3a20 6669 6e64 indProgram: find │ │ │ │ +0026b9c0: 5072 6f67 7261 6d2c 202d 2d20 6c6f 6164 Program, -- load │ │ │ │ +0026b9d0: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ +0026b9e0: 6d0a 2020 2a20 2a6e 6f74 6520 7374 6174 m. * *note stat │ │ │ │ +0026b9f0: 7573 2850 726f 6772 616d 5275 6e29 3a20 us(ProgramRun): │ │ │ │ +0026ba00: 7374 6174 7573 5f6c 7050 726f 6772 616d status_lpProgram │ │ │ │ +0026ba10: 5275 6e5f 7270 2c20 2d2d 2067 6574 2074 Run_rp, -- get t │ │ │ │ +0026ba20: 6865 2072 6574 7572 6e20 7374 6174 7573 he return status │ │ │ │ +0026ba30: 0a20 2020 206f 6620 6120 7072 6f67 7261 . of a progra │ │ │ │ +0026ba40: 6d20 7275 6e0a 2020 2a20 2a6e 6f74 6520 m run. * *note │ │ │ │ +0026ba50: 7275 6e3a 2072 756e 2c20 2d2d 2072 756e run: run, -- run │ │ │ │ +0026ba60: 2061 6e20 6578 7465 726e 616c 2063 6f6d an external com │ │ │ │ +0026ba70: 6d61 6e64 0a20 202a 202a 6e6f 7465 2067 mand. * *note g │ │ │ │ +0026ba80: 6574 3a20 6765 742c 202d 2d20 6765 7420 et: get, -- get │ │ │ │ +0026ba90: 7468 6520 636f 6e74 656e 7473 206f 6620 the contents of │ │ │ │ +0026baa0: 6120 6669 6c65 0a2a 204d 656e 753a 0a0a a file.* Menu:.. │ │ │ │ +0026bab0: 2a20 5072 6f67 7261 6d52 756e 3a3a 2020 * ProgramRun:: │ │ │ │ +0026bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026bad0: 7265 7375 6c74 206f 6620 7275 6e6e 696e result of runnin │ │ │ │ +0026bae0: 6720 616e 2065 7874 6572 6e61 6c20 7072 g an external pr │ │ │ │ +0026baf0: 6f67 7261 6d0a 0a57 6179 7320 746f 2075 ogram..Ways to u │ │ │ │ +0026bb00: 7365 2072 756e 5072 6f67 7261 6d3a 0a3d se runProgram:.= │ │ │ │ +0026bb10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026bb20: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2272 756e ======.. * "run │ │ │ │ +0026bb30: 5072 6f67 7261 6d28 5072 6f67 7261 6d2c Program(Program, │ │ │ │ +0026bb40: 5374 7269 6e67 2922 0a20 202a 2022 7275 String)". * "ru │ │ │ │ +0026bb50: 6e50 726f 6772 616d 2850 726f 6772 616d nProgram(Program │ │ │ │ +0026bb60: 2c53 7472 696e 672c 5374 7269 6e67 2922 ,String,String)" │ │ │ │ +0026bb70: 0a20 202a 2022 7275 6e50 726f 6772 616d . * "runProgram │ │ │ │ +0026bb80: 2853 7472 696e 672c 5374 7269 6e67 2922 (String,String)" │ │ │ │ +0026bb90: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ +0026bba0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ +0026bbb0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ +0026bbc0: 6563 7420 2a6e 6f74 6520 7275 6e50 726f ect *note runPro │ │ │ │ +0026bbd0: 6772 616d 3a20 7275 6e50 726f 6772 616d gram: runProgram │ │ │ │ +0026bbe0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +0026bbf0: 686f 6420 6675 6e63 7469 6f6e 2077 6974 hod function wit │ │ │ │ +0026bc00: 680a 6f70 7469 6f6e 733a 204d 6574 686f h.options: Metho │ │ │ │ +0026bc10: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ +0026bc20: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ 0026bc30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0026bc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026bc50: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0026bc60: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0026bc70: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0026bc80: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0026bc90: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0026bca0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -0026bcb0: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -0026bcc0: 756c 6179 3244 6f63 2f66 756e 6374 696f ulay2Doc/functio │ │ │ │ -0026bcd0: 6e73 2f72 756e 5072 6f67 7261 6d2d 646f ns/runProgram-do │ │ │ │ -0026bce0: 632e 6d32 3a31 3037 3a30 2e0a 1f0a 4669 c.m2:107:0....Fi │ │ │ │ -0026bcf0: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -0026bd00: 2e69 6e66 6f2c 204e 6f64 653a 2050 726f .info, Node: Pro │ │ │ │ -0026bd10: 6772 616d 5275 6e2c 2055 703a 2072 756e gramRun, Up: run │ │ │ │ -0026bd20: 5072 6f67 7261 6d0a 0a50 726f 6772 616d Program..Program │ │ │ │ -0026bd30: 5275 6e20 2d2d 2072 6573 756c 7420 6f66 Run -- result of │ │ │ │ -0026bd40: 2072 756e 6e69 6e67 2061 6e20 6578 7465 running an exte │ │ │ │ -0026bd50: 726e 616c 2070 726f 6772 616d 0a2a 2a2a rnal program.*** │ │ │ │ -0026bd60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026bd70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026bd80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026bd90: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0026bda0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 2068 6173 =========..A has │ │ │ │ -0026bdb0: 6820 7461 626c 6520 7265 7475 726e 6564 h table returned │ │ │ │ -0026bdc0: 2062 7920 2a6e 6f74 6520 7275 6e50 726f by *note runPro │ │ │ │ -0026bdd0: 6772 616d 3a20 7275 6e50 726f 6772 616d gram: runProgram │ │ │ │ -0026bde0: 2c20 7769 7468 2074 6865 2066 6f6c 6c6f , with the follo │ │ │ │ -0026bdf0: 7769 6e67 0a73 7472 696e 6773 2061 7320 wing.strings as │ │ │ │ -0026be00: 6b65 7973 3a0a 0a20 202a 2022 636f 6d6d keys:.. * "comm │ │ │ │ -0026be10: 616e 6422 2c20 7468 6520 636f 6d6d 616e and", the comman │ │ │ │ -0026be20: 6420 7468 6174 2077 6173 2075 7365 6420 d that was used │ │ │ │ -0026be30: 746f 2072 756e 2074 6865 2070 726f 6772 to run the progr │ │ │ │ -0026be40: 616d 2e0a 2020 2a20 226f 7574 7075 7422 am.. * "output" │ │ │ │ -0026be50: 2c20 7468 6520 6f75 7470 7574 206f 6620 , the output of │ │ │ │ -0026be60: 7468 6520 7072 6f67 7261 6d20 746f 2073 the program to s │ │ │ │ -0026be70: 7464 6f75 742e 0a20 202a 2022 6572 726f tdout.. * "erro │ │ │ │ -0026be80: 7222 2c20 7468 6520 6f75 7470 7574 206f r", the output o │ │ │ │ -0026be90: 6620 7468 6520 7072 6f67 7261 6d20 746f f the program to │ │ │ │ -0026bea0: 2073 7464 6572 722e 0a20 202a 2022 7265 stderr.. * "re │ │ │ │ -0026beb0: 7475 726e 2076 616c 7565 222c 2074 6865 turn value", the │ │ │ │ -0026bec0: 2072 6574 7572 6e20 7661 6c75 6520 6f66 return value of │ │ │ │ -0026bed0: 2074 6865 2070 726f 6772 616d 2c20 706f the program, po │ │ │ │ -0026bee0: 7373 6962 6c79 206d 756c 7469 706c 6965 ssibly multiplie │ │ │ │ -0026bef0: 6420 6279 2032 3536 0a20 2020 2028 7365 d by 256. (se │ │ │ │ -0026bf00: 6520 2a6e 6f74 6520 7275 6e3a 2072 756e e *note run: run │ │ │ │ -0026bf10: 2c29 2e20 204e 6f74 6520 7468 6174 2074 ,). Note that t │ │ │ │ -0026bf20: 6869 7320 6973 2077 6861 7420 6973 2064 his is what is d │ │ │ │ -0026bf30: 6973 706c 6179 6564 2077 6865 6e20 7072 isplayed when pr │ │ │ │ -0026bf40: 696e 7469 6e67 2061 0a20 2020 2050 726f inting a. Pro │ │ │ │ -0026bf50: 6772 616d 5275 6e20 6f62 6a65 6374 2e0a gramRun object.. │ │ │ │ -0026bf60: 0a49 6e20 6164 6469 7469 6f6e 2c20 6966 .In addition, if │ │ │ │ -0026bf70: 202a 6e6f 7465 2072 756e 5072 6f67 7261 *note runProgra │ │ │ │ -0026bf80: 6d3a 2072 756e 5072 6f67 7261 6d2c 2069 m: runProgram, i │ │ │ │ -0026bf90: 7320 6361 6c6c 6564 2077 6974 6820 7468 s called with th │ │ │ │ -0026bfa0: 6520 4b65 6570 4669 6c65 730a 6f70 7469 e KeepFiles.opti │ │ │ │ -0026bfb0: 6f6e 2073 6574 2074 6f20 2a6e 6f74 6520 on set to *note │ │ │ │ -0026bfc0: 7472 7565 3a20 7472 7565 2c2c 2074 6865 true: true,, the │ │ │ │ -0026bfd0: 6e20 7468 6520 666f 6c6c 6f77 696e 6720 n the following │ │ │ │ -0026bfe0: 6b65 7973 2077 696c 6c20 6265 2070 7265 keys will be pre │ │ │ │ -0026bff0: 7365 6e74 3a0a 0a20 202a 2022 6f75 7470 sent:.. * "outp │ │ │ │ -0026c000: 7574 2066 696c 6522 2c20 7468 6520 7061 ut file", the pa │ │ │ │ -0026c010: 7468 2074 6f20 6120 6669 6c65 2063 6f6e th to a file con │ │ │ │ -0026c020: 7461 696e 696e 6720 7468 6520 6f75 7470 taining the outp │ │ │ │ -0026c030: 7574 206f 6620 7468 6520 7072 6f67 7261 ut of the progra │ │ │ │ -0026c040: 6d20 746f 0a20 2020 2073 7464 6f75 742e m to. stdout. │ │ │ │ -0026c050: 0a20 202a 2022 6572 726f 7220 6669 6c65 . * "error file │ │ │ │ -0026c060: 222c 2074 6865 2070 6174 6820 746f 2061 ", the path to a │ │ │ │ -0026c070: 2066 696c 6520 636f 6e74 6169 6e69 6e67 file containing │ │ │ │ -0026c080: 2074 6865 206f 7574 7075 7420 6f66 2074 the output of t │ │ │ │ -0026c090: 6865 2070 726f 6772 616d 2074 6f0a 2020 he program to. │ │ │ │ -0026c0a0: 2020 7374 6465 7272 2e0a 0a53 6565 2061 stderr...See a │ │ │ │ -0026c0b0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -0026c0c0: 2a20 2a6e 6f74 6520 7275 6e50 726f 6772 * *note runProgr │ │ │ │ -0026c0d0: 616d 3a20 7275 6e50 726f 6772 616d 2c20 am: runProgram, │ │ │ │ -0026c0e0: 2d2d 2072 756e 2061 6e20 6578 7465 726e -- run an extern │ │ │ │ -0026c0f0: 616c 2070 726f 6772 616d 0a20 202a 202a al program. * * │ │ │ │ -0026c100: 6e6f 7465 2050 726f 6772 616d 203c 3c20 note Program << │ │ │ │ -0026c110: 5468 696e 673a 2050 726f 6772 616d 203c Thing: Program < │ │ │ │ -0026c120: 3c20 5468 696e 672c 202d 2d20 7275 6e20 < Thing, -- run │ │ │ │ -0026c130: 7072 6f67 7261 6d20 7769 7468 2069 6e70 program with inp │ │ │ │ -0026c140: 7574 0a20 2020 2072 6564 6972 6563 7469 ut. redirecti │ │ │ │ -0026c150: 6f6e 0a20 202a 202a 6e6f 7465 2050 726f on. * *note Pro │ │ │ │ -0026c160: 6772 616d 3a20 5072 6f67 7261 6d2c 202d gram: Program, - │ │ │ │ -0026c170: 2d20 6578 7465 726e 616c 2070 726f 6772 - external progr │ │ │ │ -0026c180: 616d 206f 626a 6563 740a 2a20 4d65 6e75 am object.* Menu │ │ │ │ -0026c190: 3a0a 0a2a 2073 7461 7475 7328 5072 6f67 :..* status(Prog │ │ │ │ -0026c1a0: 7261 6d52 756e 293a 2073 7461 7475 735f ramRun): status_ │ │ │ │ -0026c1b0: 6c70 5072 6f67 7261 6d52 756e 5f72 702e lpProgramRun_rp. │ │ │ │ -0026c1c0: 2020 6765 7420 7468 6520 7265 7475 726e get the return │ │ │ │ -0026c1d0: 2073 7461 7475 7320 6f66 2061 2070 726f status of a pro │ │ │ │ -0026c1e0: 6772 616d 2072 756e 0a0a 4675 6e63 7469 gram run..Functi │ │ │ │ -0026c1f0: 6f6e 7320 616e 6420 6d65 7468 6f64 7320 ons and methods │ │ │ │ -0026c200: 7265 7475 726e 696e 6720 616e 206f 626a returning an obj │ │ │ │ -0026c210: 6563 7420 6f66 2063 6c61 7373 2050 726f ect of class Pro │ │ │ │ -0026c220: 6772 616d 5275 6e3a 0a3d 3d3d 3d3d 3d3d gramRun:.======= │ │ │ │ -0026c230: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026c240: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026bc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026bc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026bc70: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +0026bc80: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +0026bc90: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +0026bca0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +0026bcb0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +0026bcc0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +0026bcd0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +0026bce0: 4d61 6361 756c 6179 3244 6f63 2f66 756e Macaulay2Doc/fun │ │ │ │ +0026bcf0: 6374 696f 6e73 2f72 756e 5072 6f67 7261 ctions/runProgra │ │ │ │ +0026bd00: 6d2d 646f 632e 6d32 3a31 3037 3a30 2e0a m-doc.m2:107:0.. │ │ │ │ +0026bd10: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +0026bd20: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +0026bd30: 2050 726f 6772 616d 5275 6e2c 2055 703a ProgramRun, Up: │ │ │ │ +0026bd40: 2072 756e 5072 6f67 7261 6d0a 0a50 726f runProgram..Pro │ │ │ │ +0026bd50: 6772 616d 5275 6e20 2d2d 2072 6573 756c gramRun -- resul │ │ │ │ +0026bd60: 7420 6f66 2072 756e 6e69 6e67 2061 6e20 t of running an │ │ │ │ +0026bd70: 6578 7465 726e 616c 2070 726f 6772 616d external program │ │ │ │ +0026bd80: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +0026bd90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026bda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026bdb0: 2a2a 2a2a 0a0a 4465 7363 7269 7074 696f ****..Descriptio │ │ │ │ +0026bdc0: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 n.===========..A │ │ │ │ +0026bdd0: 2068 6173 6820 7461 626c 6520 7265 7475 hash table retu │ │ │ │ +0026bde0: 726e 6564 2062 7920 2a6e 6f74 6520 7275 rned by *note ru │ │ │ │ +0026bdf0: 6e50 726f 6772 616d 3a20 7275 6e50 726f nProgram: runPro │ │ │ │ +0026be00: 6772 616d 2c20 7769 7468 2074 6865 2066 gram, with the f │ │ │ │ +0026be10: 6f6c 6c6f 7769 6e67 0a73 7472 696e 6773 ollowing.strings │ │ │ │ +0026be20: 2061 7320 6b65 7973 3a0a 0a20 202a 2022 as keys:.. * " │ │ │ │ +0026be30: 636f 6d6d 616e 6422 2c20 7468 6520 636f command", the co │ │ │ │ +0026be40: 6d6d 616e 6420 7468 6174 2077 6173 2075 mmand that was u │ │ │ │ +0026be50: 7365 6420 746f 2072 756e 2074 6865 2070 sed to run the p │ │ │ │ +0026be60: 726f 6772 616d 2e0a 2020 2a20 226f 7574 rogram.. * "out │ │ │ │ +0026be70: 7075 7422 2c20 7468 6520 6f75 7470 7574 put", the output │ │ │ │ +0026be80: 206f 6620 7468 6520 7072 6f67 7261 6d20 of the program │ │ │ │ +0026be90: 746f 2073 7464 6f75 742e 0a20 202a 2022 to stdout.. * " │ │ │ │ +0026bea0: 6572 726f 7222 2c20 7468 6520 6f75 7470 error", the outp │ │ │ │ +0026beb0: 7574 206f 6620 7468 6520 7072 6f67 7261 ut of the progra │ │ │ │ +0026bec0: 6d20 746f 2073 7464 6572 722e 0a20 202a m to stderr.. * │ │ │ │ +0026bed0: 2022 7265 7475 726e 2076 616c 7565 222c "return value", │ │ │ │ +0026bee0: 2074 6865 2072 6574 7572 6e20 7661 6c75 the return valu │ │ │ │ +0026bef0: 6520 6f66 2074 6865 2070 726f 6772 616d e of the program │ │ │ │ +0026bf00: 2c20 706f 7373 6962 6c79 206d 756c 7469 , possibly multi │ │ │ │ +0026bf10: 706c 6965 6420 6279 2032 3536 0a20 2020 plied by 256. │ │ │ │ +0026bf20: 2028 7365 6520 2a6e 6f74 6520 7275 6e3a (see *note run: │ │ │ │ +0026bf30: 2072 756e 2c29 2e20 204e 6f74 6520 7468 run,). Note th │ │ │ │ +0026bf40: 6174 2074 6869 7320 6973 2077 6861 7420 at this is what │ │ │ │ +0026bf50: 6973 2064 6973 706c 6179 6564 2077 6865 is displayed whe │ │ │ │ +0026bf60: 6e20 7072 696e 7469 6e67 2061 0a20 2020 n printing a. │ │ │ │ +0026bf70: 2050 726f 6772 616d 5275 6e20 6f62 6a65 ProgramRun obje │ │ │ │ +0026bf80: 6374 2e0a 0a49 6e20 6164 6469 7469 6f6e ct...In addition │ │ │ │ +0026bf90: 2c20 6966 202a 6e6f 7465 2072 756e 5072 , if *note runPr │ │ │ │ +0026bfa0: 6f67 7261 6d3a 2072 756e 5072 6f67 7261 ogram: runProgra │ │ │ │ +0026bfb0: 6d2c 2069 7320 6361 6c6c 6564 2077 6974 m, is called wit │ │ │ │ +0026bfc0: 6820 7468 6520 4b65 6570 4669 6c65 730a h the KeepFiles. │ │ │ │ +0026bfd0: 6f70 7469 6f6e 2073 6574 2074 6f20 2a6e option set to *n │ │ │ │ +0026bfe0: 6f74 6520 7472 7565 3a20 7472 7565 2c2c ote true: true,, │ │ │ │ +0026bff0: 2074 6865 6e20 7468 6520 666f 6c6c 6f77 then the follow │ │ │ │ +0026c000: 696e 6720 6b65 7973 2077 696c 6c20 6265 ing keys will be │ │ │ │ +0026c010: 2070 7265 7365 6e74 3a0a 0a20 202a 2022 present:.. * " │ │ │ │ +0026c020: 6f75 7470 7574 2066 696c 6522 2c20 7468 output file", th │ │ │ │ +0026c030: 6520 7061 7468 2074 6f20 6120 6669 6c65 e path to a file │ │ │ │ +0026c040: 2063 6f6e 7461 696e 696e 6720 7468 6520 containing the │ │ │ │ +0026c050: 6f75 7470 7574 206f 6620 7468 6520 7072 output of the pr │ │ │ │ +0026c060: 6f67 7261 6d20 746f 0a20 2020 2073 7464 ogram to. std │ │ │ │ +0026c070: 6f75 742e 0a20 202a 2022 6572 726f 7220 out.. * "error │ │ │ │ +0026c080: 6669 6c65 222c 2074 6865 2070 6174 6820 file", the path │ │ │ │ +0026c090: 746f 2061 2066 696c 6520 636f 6e74 6169 to a file contai │ │ │ │ +0026c0a0: 6e69 6e67 2074 6865 206f 7574 7075 7420 ning the output │ │ │ │ +0026c0b0: 6f66 2074 6865 2070 726f 6772 616d 2074 of the program t │ │ │ │ +0026c0c0: 6f0a 2020 2020 7374 6465 7272 2e0a 0a53 o. stderr...S │ │ │ │ +0026c0d0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +0026c0e0: 0a0a 2020 2a20 2a6e 6f74 6520 7275 6e50 .. * *note runP │ │ │ │ +0026c0f0: 726f 6772 616d 3a20 7275 6e50 726f 6772 rogram: runProgr │ │ │ │ +0026c100: 616d 2c20 2d2d 2072 756e 2061 6e20 6578 am, -- run an ex │ │ │ │ +0026c110: 7465 726e 616c 2070 726f 6772 616d 0a20 ternal program. │ │ │ │ +0026c120: 202a 202a 6e6f 7465 2050 726f 6772 616d * *note Program │ │ │ │ +0026c130: 203c 3c20 5468 696e 673a 2050 726f 6772 << Thing: Progr │ │ │ │ +0026c140: 616d 203c 3c20 5468 696e 672c 202d 2d20 am << Thing, -- │ │ │ │ +0026c150: 7275 6e20 7072 6f67 7261 6d20 7769 7468 run program with │ │ │ │ +0026c160: 2069 6e70 7574 0a20 2020 2072 6564 6972 input. redir │ │ │ │ +0026c170: 6563 7469 6f6e 0a20 202a 202a 6e6f 7465 ection. * *note │ │ │ │ +0026c180: 2050 726f 6772 616d 3a20 5072 6f67 7261 Program: Progra │ │ │ │ +0026c190: 6d2c 202d 2d20 6578 7465 726e 616c 2070 m, -- external p │ │ │ │ +0026c1a0: 726f 6772 616d 206f 626a 6563 740a 2a20 rogram object.* │ │ │ │ +0026c1b0: 4d65 6e75 3a0a 0a2a 2073 7461 7475 7328 Menu:..* status( │ │ │ │ +0026c1c0: 5072 6f67 7261 6d52 756e 293a 2073 7461 ProgramRun): sta │ │ │ │ +0026c1d0: 7475 735f 6c70 5072 6f67 7261 6d52 756e tus_lpProgramRun │ │ │ │ +0026c1e0: 5f72 702e 2020 6765 7420 7468 6520 7265 _rp. get the re │ │ │ │ +0026c1f0: 7475 726e 2073 7461 7475 7320 6f66 2061 turn status of a │ │ │ │ +0026c200: 2070 726f 6772 616d 2072 756e 0a0a 4675 program run..Fu │ │ │ │ +0026c210: 6e63 7469 6f6e 7320 616e 6420 6d65 7468 nctions and meth │ │ │ │ +0026c220: 6f64 7320 7265 7475 726e 696e 6720 616e ods returning an │ │ │ │ +0026c230: 206f 626a 6563 7420 6f66 2063 6c61 7373 object of class │ │ │ │ +0026c240: 2050 726f 6772 616d 5275 6e3a 0a3d 3d3d ProgramRun:.=== │ │ │ │ 0026c250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026c260: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -0026c270: 7465 2072 756e 5072 6f67 7261 6d3a 2072 te runProgram: r │ │ │ │ -0026c280: 756e 5072 6f67 7261 6d2c 202d 2d20 7275 unProgram, -- ru │ │ │ │ -0026c290: 6e20 616e 2065 7874 6572 6e61 6c20 7072 n an external pr │ │ │ │ -0026c2a0: 6f67 7261 6d0a 0a4d 6574 686f 6473 2074 ogram..Methods t │ │ │ │ -0026c2b0: 6861 7420 7573 6520 616e 206f 626a 6563 hat use an objec │ │ │ │ -0026c2c0: 7420 6f66 2063 6c61 7373 2050 726f 6772 t of class Progr │ │ │ │ -0026c2d0: 616d 5275 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d amRun:.========= │ │ │ │ -0026c2e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026c2f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026c300: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ -0026c310: 6520 7374 6174 7573 2850 726f 6772 616d e status(Program │ │ │ │ -0026c320: 5275 6e29 3a20 7374 6174 7573 5f6c 7050 Run): status_lpP │ │ │ │ -0026c330: 726f 6772 616d 5275 6e5f 7270 2c20 2d2d rogramRun_rp, -- │ │ │ │ -0026c340: 2067 6574 2074 6865 2072 6574 7572 6e20 get the return │ │ │ │ -0026c350: 7374 6174 7573 0a20 2020 206f 6620 6120 status. of a │ │ │ │ -0026c360: 7072 6f67 7261 6d20 7275 6e0a 0a46 6f72 program run..For │ │ │ │ -0026c370: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -0026c380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026c390: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -0026c3a0: 6e6f 7465 2050 726f 6772 616d 5275 6e3a note ProgramRun: │ │ │ │ -0026c3b0: 2050 726f 6772 616d 5275 6e2c 2069 7320 ProgramRun, is │ │ │ │ -0026c3c0: 6120 2a6e 6f74 6520 7479 7065 3a20 5479 a *note type: Ty │ │ │ │ -0026c3d0: 7065 2c2c 2077 6974 6820 616e 6365 7374 pe,, with ancest │ │ │ │ -0026c3e0: 6f72 0a63 6c61 7373 6573 202a 6e6f 7465 or.classes *note │ │ │ │ -0026c3f0: 2048 6173 6854 6162 6c65 3a20 4861 7368 HashTable: Hash │ │ │ │ -0026c400: 5461 626c 652c 203c 202a 6e6f 7465 2054 Table, < *note T │ │ │ │ -0026c410: 6869 6e67 3a20 5468 696e 672c 2e0a 0a2d hing: Thing,...- │ │ │ │ -0026c420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026c430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026c440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026c260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026c270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026c280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +0026c290: 202a 6e6f 7465 2072 756e 5072 6f67 7261 *note runProgra │ │ │ │ +0026c2a0: 6d3a 2072 756e 5072 6f67 7261 6d2c 202d m: runProgram, - │ │ │ │ +0026c2b0: 2d20 7275 6e20 616e 2065 7874 6572 6e61 - run an externa │ │ │ │ +0026c2c0: 6c20 7072 6f67 7261 6d0a 0a4d 6574 686f l program..Metho │ │ │ │ +0026c2d0: 6473 2074 6861 7420 7573 6520 616e 206f ds that use an o │ │ │ │ +0026c2e0: 626a 6563 7420 6f66 2063 6c61 7373 2050 bject of class P │ │ │ │ +0026c2f0: 726f 6772 616d 5275 6e3a 0a3d 3d3d 3d3d rogramRun:.===== │ │ │ │ +0026c300: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026c310: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026c320: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0026c330: 2a6e 6f74 6520 7374 6174 7573 2850 726f *note status(Pro │ │ │ │ +0026c340: 6772 616d 5275 6e29 3a20 7374 6174 7573 gramRun): status │ │ │ │ +0026c350: 5f6c 7050 726f 6772 616d 5275 6e5f 7270 _lpProgramRun_rp │ │ │ │ +0026c360: 2c20 2d2d 2067 6574 2074 6865 2072 6574 , -- get the ret │ │ │ │ +0026c370: 7572 6e20 7374 6174 7573 0a20 2020 206f urn status. o │ │ │ │ +0026c380: 6620 6120 7072 6f67 7261 6d20 7275 6e0a f a program run. │ │ │ │ +0026c390: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0026c3a0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0026c3b0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0026c3c0: 6374 202a 6e6f 7465 2050 726f 6772 616d ct *note Program │ │ │ │ +0026c3d0: 5275 6e3a 2050 726f 6772 616d 5275 6e2c Run: ProgramRun, │ │ │ │ +0026c3e0: 2069 7320 6120 2a6e 6f74 6520 7479 7065 is a *note type │ │ │ │ +0026c3f0: 3a20 5479 7065 2c2c 2077 6974 6820 616e : Type,, with an │ │ │ │ +0026c400: 6365 7374 6f72 0a63 6c61 7373 6573 202a cestor.classes * │ │ │ │ +0026c410: 6e6f 7465 2048 6173 6854 6162 6c65 3a20 note HashTable: │ │ │ │ +0026c420: 4861 7368 5461 626c 652c 203c 202a 6e6f HashTable, < *no │ │ │ │ +0026c430: 7465 2054 6869 6e67 3a20 5468 696e 672c te Thing: Thing, │ │ │ │ +0026c440: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0026c450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -0026c470: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -0026c480: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -0026c490: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -0026c4a0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -0026c4b0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -0026c4c0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -0026c4d0: 6b61 6765 732f 0a4d 6163 6175 6c61 7932 kages/.Macaulay2 │ │ │ │ -0026c4e0: 446f 632f 6675 6e63 7469 6f6e 732f 7275 Doc/functions/ru │ │ │ │ -0026c4f0: 6e50 726f 6772 616d 2d64 6f63 2e6d 323a nProgram-doc.m2: │ │ │ │ -0026c500: 3335 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 35:0....File: Ma │ │ │ │ -0026c510: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ -0026c520: 204e 6f64 653a 2073 7461 7475 735f 6c70 Node: status_lp │ │ │ │ -0026c530: 5072 6f67 7261 6d52 756e 5f72 702c 2055 ProgramRun_rp, U │ │ │ │ -0026c540: 703a 2050 726f 6772 616d 5275 6e0a 0a73 p: ProgramRun..s │ │ │ │ -0026c550: 7461 7475 7328 5072 6f67 7261 6d52 756e tatus(ProgramRun │ │ │ │ -0026c560: 2920 2d2d 2067 6574 2074 6865 2072 6574 ) -- get the ret │ │ │ │ -0026c570: 7572 6e20 7374 6174 7573 206f 6620 6120 urn status of a │ │ │ │ -0026c580: 7072 6f67 7261 6d20 7275 6e0a 2a2a 2a2a program run.**** │ │ │ │ -0026c590: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026c5a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026c460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026c470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026c480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026c490: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0026c4a0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0026c4b0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0026c4c0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0026c4d0: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +0026c4e0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0026c4f0: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ +0026c500: 6c61 7932 446f 632f 6675 6e63 7469 6f6e lay2Doc/function │ │ │ │ +0026c510: 732f 7275 6e50 726f 6772 616d 2d64 6f63 s/runProgram-doc │ │ │ │ +0026c520: 2e6d 323a 3335 3a30 2e0a 1f0a 4669 6c65 .m2:35:0....File │ │ │ │ +0026c530: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ +0026c540: 6e66 6f2c 204e 6f64 653a 2073 7461 7475 nfo, Node: statu │ │ │ │ +0026c550: 735f 6c70 5072 6f67 7261 6d52 756e 5f72 s_lpProgramRun_r │ │ │ │ +0026c560: 702c 2055 703a 2050 726f 6772 616d 5275 p, Up: ProgramRu │ │ │ │ +0026c570: 6e0a 0a73 7461 7475 7328 5072 6f67 7261 n..status(Progra │ │ │ │ +0026c580: 6d52 756e 2920 2d2d 2067 6574 2074 6865 mRun) -- get the │ │ │ │ +0026c590: 2072 6574 7572 6e20 7374 6174 7573 206f return status o │ │ │ │ +0026c5a0: 6620 6120 7072 6f67 7261 6d20 7275 6e0a f a program run. │ │ │ │ 0026c5b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026c5c0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 ********.. * Fu │ │ │ │ -0026c5d0: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 7374 nction: *note st │ │ │ │ -0026c5e0: 6174 7573 3a20 7374 6174 7573 2c0a 2020 atus: status,. │ │ │ │ -0026c5f0: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -0026c600: 2020 7374 6174 7573 2070 720a 2020 2a20 status pr. * │ │ │ │ -0026c610: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -0026c620: 7072 2c20 616e 2069 6e73 7461 6e63 6520 pr, an instance │ │ │ │ -0026c630: 6f66 2074 6865 2074 7970 6520 2a6e 6f74 of the type *not │ │ │ │ -0026c640: 6520 5072 6f67 7261 6d52 756e 3a20 5072 e ProgramRun: Pr │ │ │ │ -0026c650: 6f67 7261 6d52 756e 2c2c 200a 2020 2a20 ogramRun,, . * │ │ │ │ -0026c660: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069 *note Optional i │ │ │ │ -0026c670: 6e70 7574 733a 2075 7369 6e67 2066 756e nputs: using fun │ │ │ │ -0026c680: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ -0026c690: 6f6e 616c 2069 6e70 7574 732c 3a0a 2020 onal inputs,:. │ │ │ │ -0026c6a0: 2020 2020 2a20 4d6f 6e6f 6d69 616c 7320 * Monomials │ │ │ │ -0026c6b0: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ -0026c6c0: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ -0026c6d0: 6465 6661 756c 7420 7661 6c75 6520 6661 default value fa │ │ │ │ -0026c6e0: 6c73 652c 200a 2020 2020 2020 2a20 5061 lse, . * Pa │ │ │ │ -0026c6f0: 6972 7352 656d 6169 6e69 6e67 2028 6d69 irsRemaining (mi │ │ │ │ -0026c700: 7373 696e 6720 646f 6375 6d65 6e74 6174 ssing documentat │ │ │ │ -0026c710: 696f 6e29 203d 3e20 2e2e 2e2c 2064 6566 ion) => ..., def │ │ │ │ -0026c720: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ -0026c730: 2c20 0a20 2020 2020 202a 2054 6f74 616c , . * Total │ │ │ │ -0026c740: 5061 6972 7320 286d 6973 7369 6e67 2064 Pairs (missing d │ │ │ │ -0026c750: 6f63 756d 656e 7461 7469 6f6e 2920 3d3e ocumentation) => │ │ │ │ -0026c760: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -0026c770: 6c75 6520 7472 7565 2c20 0a20 202a 204f lue true, . * O │ │ │ │ -0026c780: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -0026c790: 616e 202a 6e6f 7465 2069 6e74 6567 6572 an *note integer │ │ │ │ -0026c7a0: 3a20 5a5a 2c2c 200a 0a44 6573 6372 6970 : ZZ,, ..Descrip │ │ │ │ -0026c7b0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -0026c7c0: 0a0a 4765 7420 7468 6520 7265 7475 726e ..Get the return │ │ │ │ -0026c7d0: 2073 7461 7475 7320 6f66 2061 2070 726f status of a pro │ │ │ │ -0026c7e0: 6772 616d 2072 756e 2e20 2055 7375 616c gram run. Usual │ │ │ │ -0026c7f0: 6c79 2c20 3020 6d65 616e 7320 7468 6174 ly, 0 means that │ │ │ │ -0026c800: 2069 7420 7761 730a 7375 6363 6573 7366 it was.successf │ │ │ │ -0026c810: 756c 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ul...+---------- │ │ │ │ -0026c820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026c840: 2d2d 2d2b 0a7c 6931 203a 206e 6f72 6d61 ---+.|i1 : norma │ │ │ │ -0026c850: 6c69 7a20 3d20 6669 6e64 5072 6f67 7261 liz = findProgra │ │ │ │ -0026c860: 6d20 226e 6f72 6d61 6c69 7a22 2020 2020 m "normaliz" │ │ │ │ -0026c870: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0026c880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c8a0: 2020 207c 0a7c 6f31 203d 206e 6f72 6d61 |.|o1 = norma │ │ │ │ -0026c8b0: 6c69 7a20 2020 2020 2020 2020 2020 2020 liz │ │ │ │ -0026c8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c8d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0026c5c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026c5d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026c5e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +0026c5f0: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74 * Function: *not │ │ │ │ +0026c600: 6520 7374 6174 7573 3a20 7374 6174 7573 e status: status │ │ │ │ +0026c610: 2c0a 2020 2a20 5573 6167 653a 200a 2020 ,. * Usage: . │ │ │ │ +0026c620: 2020 2020 2020 7374 6174 7573 2070 720a status pr. │ │ │ │ +0026c630: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0026c640: 2020 2a20 7072 2c20 616e 2069 6e73 7461 * pr, an insta │ │ │ │ +0026c650: 6e63 6520 6f66 2074 6865 2074 7970 6520 nce of the type │ │ │ │ +0026c660: 2a6e 6f74 6520 5072 6f67 7261 6d52 756e *note ProgramRun │ │ │ │ +0026c670: 3a20 5072 6f67 7261 6d52 756e 2c2c 200a : ProgramRun,, . │ │ │ │ +0026c680: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ +0026c690: 616c 2069 6e70 7574 733a 2075 7369 6e67 al inputs: using │ │ │ │ +0026c6a0: 2066 756e 6374 696f 6e73 2077 6974 6820 functions with │ │ │ │ +0026c6b0: 6f70 7469 6f6e 616c 2069 6e70 7574 732c optional inputs, │ │ │ │ +0026c6c0: 3a0a 2020 2020 2020 2a20 4d6f 6e6f 6d69 :. * Monomi │ │ │ │ +0026c6d0: 616c 7320 286d 6973 7369 6e67 2064 6f63 als (missing doc │ │ │ │ +0026c6e0: 756d 656e 7461 7469 6f6e 2920 3d3e 202e umentation) => . │ │ │ │ +0026c6f0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ +0026c700: 6520 6661 6c73 652c 200a 2020 2020 2020 e false, . │ │ │ │ +0026c710: 2a20 5061 6972 7352 656d 6169 6e69 6e67 * PairsRemaining │ │ │ │ +0026c720: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +0026c730: 6e74 6174 696f 6e29 203d 3e20 2e2e 2e2c ntation) => ..., │ │ │ │ +0026c740: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ +0026c750: 616c 7365 2c20 0a20 2020 2020 202a 2054 alse, . * T │ │ │ │ +0026c760: 6f74 616c 5061 6972 7320 286d 6973 7369 otalPairs (missi │ │ │ │ +0026c770: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ +0026c780: 2920 3d3e 202e 2e2e 2c20 6465 6661 756c ) => ..., defaul │ │ │ │ +0026c790: 7420 7661 6c75 6520 7472 7565 2c20 0a20 t value true, . │ │ │ │ +0026c7a0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +0026c7b0: 2020 2a20 616e 202a 6e6f 7465 2069 6e74 * an *note int │ │ │ │ +0026c7c0: 6567 6572 3a20 5a5a 2c2c 200a 0a44 6573 eger: ZZ,, ..Des │ │ │ │ +0026c7d0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +0026c7e0: 3d3d 3d3d 0a0a 4765 7420 7468 6520 7265 ====..Get the re │ │ │ │ +0026c7f0: 7475 726e 2073 7461 7475 7320 6f66 2061 turn status of a │ │ │ │ +0026c800: 2070 726f 6772 616d 2072 756e 2e20 2055 program run. U │ │ │ │ +0026c810: 7375 616c 6c79 2c20 3020 6d65 616e 7320 sually, 0 means │ │ │ │ +0026c820: 7468 6174 2069 7420 7761 730a 7375 6363 that it was.succ │ │ │ │ +0026c830: 6573 7366 756c 2e0a 0a2b 2d2d 2d2d 2d2d essful...+------ │ │ │ │ +0026c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026c860: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206e -------+.|i1 : n │ │ │ │ +0026c870: 6f72 6d61 6c69 7a20 3d20 6669 6e64 5072 ormaliz = findPr │ │ │ │ +0026c880: 6f67 7261 6d20 226e 6f72 6d61 6c69 7a22 ogram "normaliz" │ │ │ │ +0026c890: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0026c8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026c8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026c8c0: 2020 2020 2020 207c 0a7c 6f31 203d 206e |.|o1 = n │ │ │ │ +0026c8d0: 6f72 6d61 6c69 7a20 2020 2020 2020 2020 ormaliz │ │ │ │ 0026c8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c900: 2020 207c 0a7c 6f31 203a 2050 726f 6772 |.|o1 : Progr │ │ │ │ -0026c910: 616d 2020 2020 2020 2020 2020 2020 2020 am │ │ │ │ -0026c920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c930: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0026c940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026c960: 2d2d 2d2b 0a7c 6932 203a 2073 7461 7475 ---+.|i2 : statu │ │ │ │ -0026c970: 7320 7275 6e50 726f 6772 616d 286e 6f72 s runProgram(nor │ │ │ │ -0026c980: 6d61 6c69 7a2c 2022 2d2d 7665 7273 696f maliz, "--versio │ │ │ │ -0026c990: 6e22 297c 0a7c 2020 2020 2020 2020 2020 n")|.| │ │ │ │ -0026c9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c9c0: 2020 207c 0a7c 6f32 203d 2030 2020 2020 |.|o2 = 0 │ │ │ │ +0026c8f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0026c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026c920: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ +0026c930: 726f 6772 616d 2020 2020 2020 2020 2020 rogram │ │ │ │ +0026c940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026c950: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0026c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026c980: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2073 -------+.|i2 : s │ │ │ │ +0026c990: 7461 7475 7320 7275 6e50 726f 6772 616d tatus runProgram │ │ │ │ +0026c9a0: 286e 6f72 6d61 6c69 7a2c 2022 2d2d 7665 (normaliz, "--ve │ │ │ │ +0026c9b0: 7273 696f 6e22 297c 0a7c 2020 2020 2020 rsion")|.| │ │ │ │ +0026c9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026c9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026c9f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0026ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ca20: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ -0026ca30: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -0026ca40: 7465 2066 696e 6450 726f 6772 616d 3a20 te findProgram: │ │ │ │ -0026ca50: 6669 6e64 5072 6f67 7261 6d2c 202d 2d20 findProgram, -- │ │ │ │ -0026ca60: 6c6f 6164 2065 7874 6572 6e61 6c20 7072 load external pr │ │ │ │ -0026ca70: 6f67 7261 6d0a 2020 2a20 2a6e 6f74 6520 ogram. * *note │ │ │ │ -0026ca80: 7275 6e50 726f 6772 616d 3a20 7275 6e50 runProgram: runP │ │ │ │ -0026ca90: 726f 6772 616d 2c20 2d2d 2072 756e 2061 rogram, -- run a │ │ │ │ -0026caa0: 6e20 6578 7465 726e 616c 2070 726f 6772 n external progr │ │ │ │ -0026cab0: 616d 0a0a 5761 7973 2074 6f20 7573 6520 am..Ways to use │ │ │ │ -0026cac0: 7468 6973 206d 6574 686f 643a 0a3d 3d3d this method:.=== │ │ │ │ -0026cad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026cae0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0026caf0: 2073 7461 7475 7328 5072 6f67 7261 6d52 status(ProgramR │ │ │ │ -0026cb00: 756e 293a 2073 7461 7475 735f 6c70 5072 un): status_lpPr │ │ │ │ -0026cb10: 6f67 7261 6d52 756e 5f72 702c 202d 2d20 ogramRun_rp, -- │ │ │ │ -0026cb20: 6765 7420 7468 6520 7265 7475 726e 2073 get the return s │ │ │ │ -0026cb30: 7461 7475 730a 2020 2020 6f66 2061 2070 tatus. of a p │ │ │ │ -0026cb40: 726f 6772 616d 2072 756e 0a2d 2d2d 2d2d rogram run.----- │ │ │ │ -0026cb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026cb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026c9e0: 2020 2020 2020 207c 0a7c 6f32 203d 2030 |.|o2 = 0 │ │ │ │ +0026c9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ca00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ca10: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0026ca20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ca40: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ +0026ca50: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ +0026ca60: 202a 6e6f 7465 2066 696e 6450 726f 6772 *note findProgr │ │ │ │ +0026ca70: 616d 3a20 6669 6e64 5072 6f67 7261 6d2c am: findProgram, │ │ │ │ +0026ca80: 202d 2d20 6c6f 6164 2065 7874 6572 6e61 -- load externa │ │ │ │ +0026ca90: 6c20 7072 6f67 7261 6d0a 2020 2a20 2a6e l program. * *n │ │ │ │ +0026caa0: 6f74 6520 7275 6e50 726f 6772 616d 3a20 ote runProgram: │ │ │ │ +0026cab0: 7275 6e50 726f 6772 616d 2c20 2d2d 2072 runProgram, -- r │ │ │ │ +0026cac0: 756e 2061 6e20 6578 7465 726e 616c 2070 un an external p │ │ │ │ +0026cad0: 726f 6772 616d 0a0a 5761 7973 2074 6f20 rogram..Ways to │ │ │ │ +0026cae0: 7573 6520 7468 6973 206d 6574 686f 643a use this method: │ │ │ │ +0026caf0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +0026cb00: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a =========.. * * │ │ │ │ +0026cb10: 6e6f 7465 2073 7461 7475 7328 5072 6f67 note status(Prog │ │ │ │ +0026cb20: 7261 6d52 756e 293a 2073 7461 7475 735f ramRun): status_ │ │ │ │ +0026cb30: 6c70 5072 6f67 7261 6d52 756e 5f72 702c lpProgramRun_rp, │ │ │ │ +0026cb40: 202d 2d20 6765 7420 7468 6520 7265 7475 -- get the retu │ │ │ │ +0026cb50: 726e 2073 7461 7475 730a 2020 2020 6f66 rn status. of │ │ │ │ +0026cb60: 2061 2070 726f 6772 616d 2072 756e 0a2d a program run.- │ │ │ │ 0026cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0026cb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -0026cba0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -0026cbb0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -0026cbc0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -0026cbd0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -0026cbe0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -0026cbf0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -0026cc00: 732f 0a4d 6163 6175 6c61 7932 446f 632f s/.Macaulay2Doc/ │ │ │ │ -0026cc10: 6675 6e63 7469 6f6e 732f 7275 6e50 726f functions/runPro │ │ │ │ -0026cc20: 6772 616d 2d64 6f63 2e6d 323a 3133 303a gram-doc.m2:130: │ │ │ │ -0026cc30: 302e 0a1f 0a46 696c 653a 204d 6163 6175 0....File: Macau │ │ │ │ -0026cc40: 6c61 7932 446f 632e 696e 666f 2c20 4e6f lay2Doc.info, No │ │ │ │ -0026cc50: 6465 3a20 5072 6f67 7261 6d20 3c3c 2054 de: Program << T │ │ │ │ -0026cc60: 6869 6e67 2c20 5072 6576 3a20 7275 6e50 hing, Prev: runP │ │ │ │ -0026cc70: 726f 6772 616d 2c20 5570 3a20 636f 6d6d rogram, Up: comm │ │ │ │ -0026cc80: 756e 6963 6174 696e 6720 7769 7468 2070 unicating with p │ │ │ │ -0026cc90: 726f 6772 616d 730a 0a50 726f 6772 616d rograms..Program │ │ │ │ -0026cca0: 203c 3c20 5468 696e 6720 2d2d 2072 756e << Thing -- run │ │ │ │ -0026ccb0: 2070 726f 6772 616d 2077 6974 6820 696e program with in │ │ │ │ -0026ccc0: 7075 7420 7265 6469 7265 6374 696f 6e0a put redirection. │ │ │ │ -0026ccd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026cce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026ccf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026cd00: 2a2a 2a2a 2a2a 0a0a 2020 2a20 4f70 6572 ******.. * Oper │ │ │ │ -0026cd10: 6174 6f72 3a20 2a6e 6f74 6520 3c3c 3a20 ator: *note <<: │ │ │ │ -0026cd20: 3c3c 2c0a 2020 2a20 5573 6167 653a 200a <<,. * Usage: . │ │ │ │ -0026cd30: 2020 2020 2020 2020 7072 6f67 203c 3c20 prog << │ │ │ │ -0026cd40: 780a 2020 2a20 496e 7075 7473 3a0a 2020 x. * Inputs:. │ │ │ │ -0026cd50: 2020 2020 2a20 7072 6f67 2c20 616e 2069 * prog, an i │ │ │ │ -0026cd60: 6e73 7461 6e63 6520 6f66 2074 6865 2074 nstance of the t │ │ │ │ -0026cd70: 7970 6520 2a6e 6f74 6520 5072 6f67 7261 ype *note Progra │ │ │ │ -0026cd80: 6d3a 2050 726f 6772 616d 2c2c 200a 2020 m: Program,, . │ │ │ │ -0026cd90: 2020 2020 2a20 782c 2061 202a 6e6f 7465 * x, a *note │ │ │ │ -0026cda0: 2074 6869 6e67 3a20 5468 696e 672c 2c20 thing: Thing,, │ │ │ │ -0026cdb0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -0026cdc0: 2020 2020 2a20 616e 2069 6e73 7461 6e63 * an instanc │ │ │ │ -0026cdd0: 6520 6f66 2074 6865 2074 7970 6520 2a6e e of the type *n │ │ │ │ -0026cde0: 6f74 6520 5072 6f67 7261 6d52 756e 3a20 ote ProgramRun: │ │ │ │ -0026cdf0: 5072 6f67 7261 6d52 756e 2c2c 200a 0a44 ProgramRun,, ..D │ │ │ │ -0026ce00: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0026ce10: 3d3d 3d3d 3d3d 0a0a 5772 6974 6520 7820 ======..Write x │ │ │ │ -0026ce20: 746f 2061 2074 656d 706f 7261 7279 2066 to a temporary f │ │ │ │ -0026ce30: 696c 6520 616e 6420 7275 6e20 7072 6f67 ile and run prog │ │ │ │ -0026ce40: 2077 6974 6820 7468 6973 2066 696c 6520 with this file │ │ │ │ -0026ce50: 6173 2069 6e70 7574 2075 7369 6e67 2069 as input using i │ │ │ │ -0026ce60: 6e70 7574 0a72 6564 6972 6563 7469 6f6e nput.redirection │ │ │ │ -0026ce70: 2028 7468 6520 3c20 6f70 6572 6174 6f72 (the < operator │ │ │ │ -0026ce80: 2069 6e20 6120 504f 5349 5820 7368 656c in a POSIX shel │ │ │ │ -0026ce90: 6c29 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d l)...+---------- │ │ │ │ -0026cea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0026cec0: 6931 203a 204d 3220 3d20 6669 6e64 5072 i1 : M2 = findPr │ │ │ │ -0026ced0: 6f67 7261 6d20 224d 3222 2020 2020 2020 ogram "M2" │ │ │ │ -0026cee0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0026cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026cf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026cf10: 207c 0a7c 6f31 203d 204d 3220 2020 2020 |.|o1 = M2 │ │ │ │ +0026cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0026cbc0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0026cbd0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0026cbe0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0026cbf0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0026cc00: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +0026cc10: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0026cc20: 6b61 6765 732f 0a4d 6163 6175 6c61 7932 kages/.Macaulay2 │ │ │ │ +0026cc30: 446f 632f 6675 6e63 7469 6f6e 732f 7275 Doc/functions/ru │ │ │ │ +0026cc40: 6e50 726f 6772 616d 2d64 6f63 2e6d 323a nProgram-doc.m2: │ │ │ │ +0026cc50: 3133 303a 302e 0a1f 0a46 696c 653a 204d 130:0....File: M │ │ │ │ +0026cc60: 6163 6175 6c61 7932 446f 632e 696e 666f acaulay2Doc.info │ │ │ │ +0026cc70: 2c20 4e6f 6465 3a20 5072 6f67 7261 6d20 , Node: Program │ │ │ │ +0026cc80: 3c3c 2054 6869 6e67 2c20 5072 6576 3a20 << Thing, Prev: │ │ │ │ +0026cc90: 7275 6e50 726f 6772 616d 2c20 5570 3a20 runProgram, Up: │ │ │ │ +0026cca0: 636f 6d6d 756e 6963 6174 696e 6720 7769 communicating wi │ │ │ │ +0026ccb0: 7468 2070 726f 6772 616d 730a 0a50 726f th programs..Pro │ │ │ │ +0026ccc0: 6772 616d 203c 3c20 5468 696e 6720 2d2d gram << Thing -- │ │ │ │ +0026ccd0: 2072 756e 2070 726f 6772 616d 2077 6974 run program wit │ │ │ │ +0026cce0: 6820 696e 7075 7420 7265 6469 7265 6374 h input redirect │ │ │ │ +0026ccf0: 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ion.************ │ │ │ │ +0026cd00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026cd10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026cd20: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +0026cd30: 4f70 6572 6174 6f72 3a20 2a6e 6f74 6520 Operator: *note │ │ │ │ +0026cd40: 3c3c 3a20 3c3c 2c0a 2020 2a20 5573 6167 <<: <<,. * Usag │ │ │ │ +0026cd50: 653a 200a 2020 2020 2020 2020 7072 6f67 e: . prog │ │ │ │ +0026cd60: 203c 3c20 780a 2020 2a20 496e 7075 7473 << x. * Inputs │ │ │ │ +0026cd70: 3a0a 2020 2020 2020 2a20 7072 6f67 2c20 :. * prog, │ │ │ │ +0026cd80: 616e 2069 6e73 7461 6e63 6520 6f66 2074 an instance of t │ │ │ │ +0026cd90: 6865 2074 7970 6520 2a6e 6f74 6520 5072 he type *note Pr │ │ │ │ +0026cda0: 6f67 7261 6d3a 2050 726f 6772 616d 2c2c ogram: Program,, │ │ │ │ +0026cdb0: 200a 2020 2020 2020 2a20 782c 2061 202a . * x, a * │ │ │ │ +0026cdc0: 6e6f 7465 2074 6869 6e67 3a20 5468 696e note thing: Thin │ │ │ │ +0026cdd0: 672c 2c20 0a20 202a 204f 7574 7075 7473 g,, . * Outputs │ │ │ │ +0026cde0: 3a0a 2020 2020 2020 2a20 616e 2069 6e73 :. * an ins │ │ │ │ +0026cdf0: 7461 6e63 6520 6f66 2074 6865 2074 7970 tance of the typ │ │ │ │ +0026ce00: 6520 2a6e 6f74 6520 5072 6f67 7261 6d52 e *note ProgramR │ │ │ │ +0026ce10: 756e 3a20 5072 6f67 7261 6d52 756e 2c2c un: ProgramRun,, │ │ │ │ +0026ce20: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d ..Description.= │ │ │ │ +0026ce30: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5772 6974 ==========..Writ │ │ │ │ +0026ce40: 6520 7820 746f 2061 2074 656d 706f 7261 e x to a tempora │ │ │ │ +0026ce50: 7279 2066 696c 6520 616e 6420 7275 6e20 ry file and run │ │ │ │ +0026ce60: 7072 6f67 2077 6974 6820 7468 6973 2066 prog with this f │ │ │ │ +0026ce70: 696c 6520 6173 2069 6e70 7574 2075 7369 ile as input usi │ │ │ │ +0026ce80: 6e67 2069 6e70 7574 0a72 6564 6972 6563 ng input.redirec │ │ │ │ +0026ce90: 7469 6f6e 2028 7468 6520 3c20 6f70 6572 tion (the < oper │ │ │ │ +0026cea0: 6174 6f72 2069 6e20 6120 504f 5349 5820 ator in a POSIX │ │ │ │ +0026ceb0: 7368 656c 6c29 2e0a 0a2b 2d2d 2d2d 2d2d shell)...+------ │ │ │ │ +0026cec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ced0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026cee0: 2d2b 0a7c 6931 203a 204d 3220 3d20 6669 -+.|i1 : M2 = fi │ │ │ │ +0026cef0: 6e64 5072 6f67 7261 6d20 224d 3222 2020 ndProgram "M2" │ │ │ │ +0026cf00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0026cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026cf30: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0026cf30: 2020 2020 207c 0a7c 6f31 203d 204d 3220 |.|o1 = M2 │ │ │ │ 0026cf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026cf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026cf60: 2020 2020 207c 0a7c 6f31 203a 2050 726f |.|o1 : Pro │ │ │ │ -0026cf70: 6772 616d 2020 2020 2020 2020 2020 2020 gram │ │ │ │ -0026cf80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0026cf90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0026cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026cfb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -0026cfc0: 204d 3220 3c3c 2022 3220 2b20 3222 2020 M2 << "2 + 2" │ │ │ │ -0026cfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026cfe0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0026cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d000: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0026d010: 6f32 203d 2030 2020 2020 2020 2020 2020 o2 = 0 │ │ │ │ +0026cf50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026cf60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0026cf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026cf80: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ +0026cf90: 2050 726f 6772 616d 2020 2020 2020 2020 Program │ │ │ │ +0026cfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026cfb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0026cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0026cfe0: 6932 203a 204d 3220 3c3c 2022 3220 2b20 i2 : M2 << "2 + │ │ │ │ +0026cff0: 3222 2020 2020 2020 2020 2020 2020 2020 2" │ │ │ │ +0026d000: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0026d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d030: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0026d030: 207c 0a7c 6f32 203d 2030 2020 2020 2020 |.|o2 = 0 │ │ │ │ 0026d040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d060: 207c 0a7c 6f32 203a 2050 726f 6772 616d |.|o2 : Program │ │ │ │ -0026d070: 5275 6e20 2020 2020 2020 2020 2020 2020 Run │ │ │ │ -0026d080: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0026d090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026d0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026d0b0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2074 6f53 -----+.|i3 : toS │ │ │ │ -0026d0c0: 7472 696e 6720 6f6f 2020 2020 2020 2020 tring oo │ │ │ │ -0026d0d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0026d0e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0026d050: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0026d060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026d070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026d080: 2020 2020 207c 0a7c 6f32 203a 2050 726f |.|o2 : Pro │ │ │ │ +0026d090: 6772 616d 5275 6e20 2020 2020 2020 2020 gramRun │ │ │ │ +0026d0a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026d0b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0026d0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026d0d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +0026d0e0: 2074 6f53 7472 696e 6720 6f6f 2020 2020 toString oo │ │ │ │ 0026d0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d100: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ -0026d110: 204d 6163 6175 6c61 7932 2c20 7665 7273 Macaulay2, vers │ │ │ │ -0026d120: 696f 6e20 312e 3236 2e30 3620 2020 2020 ion 1.26.06 │ │ │ │ -0026d130: 2020 207c 0a7c 2020 2020 2054 7970 6520 |.| Type │ │ │ │ -0026d140: 2268 656c 7022 2074 6f20 7365 6520 7573 "help" to see us │ │ │ │ -0026d150: 6566 756c 2063 6f6d 6d61 6e64 737c 0a7c eful commands|.| │ │ │ │ -0026d160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d180: 2020 2020 2020 207c 0a7c 2020 2020 2069 |.| i │ │ │ │ -0026d190: 3120 3a20 3220 2b20 3220 2020 2020 2020 1 : 2 + 2 │ │ │ │ -0026d1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d1b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0026d100: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0026d110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026d120: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0026d130: 6f33 203d 204d 6163 6175 6c61 7932 2c20 o3 = Macaulay2, │ │ │ │ +0026d140: 7665 7273 696f 6e20 312e 3236 2e30 3620 version 1.26.06 │ │ │ │ +0026d150: 2020 2020 2020 207c 0a7c 2020 2020 2054 |.| T │ │ │ │ +0026d160: 7970 6520 2268 656c 7022 2074 6f20 7365 ype "help" to se │ │ │ │ +0026d170: 6520 7573 6566 756c 2063 6f6d 6d61 6e64 e useful command │ │ │ │ +0026d180: 737c 0a7c 2020 2020 2020 2020 2020 2020 s|.| │ │ │ │ +0026d190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026d1a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0026d1b0: 2020 2069 3120 3a20 3220 2b20 3220 2020 i1 : 2 + 2 │ │ │ │ 0026d1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d1d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0026d1e0: 2020 206f 3120 3d20 3420 2020 2020 2020 o1 = 4 │ │ │ │ -0026d1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026d200: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0026d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0026d230: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -0026d240: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2066 ===.. * *note f │ │ │ │ -0026d250: 696e 6450 726f 6772 616d 3a20 6669 6e64 indProgram: find │ │ │ │ -0026d260: 5072 6f67 7261 6d2c 202d 2d20 6c6f 6164 Program, -- load │ │ │ │ -0026d270: 2065 7874 6572 6e61 6c20 7072 6f67 7261 external progra │ │ │ │ -0026d280: 6d0a 2020 2a20 2a6e 6f74 6520 7275 6e50 m. * *note runP │ │ │ │ -0026d290: 726f 6772 616d 3a20 7275 6e50 726f 6772 rogram: runProgr │ │ │ │ -0026d2a0: 616d 2c20 2d2d 2072 756e 2061 6e20 6578 am, -- run an ex │ │ │ │ -0026d2b0: 7465 726e 616c 2070 726f 6772 616d 0a20 ternal program. │ │ │ │ -0026d2c0: 202a 202a 6e6f 7465 2067 6574 3a20 6765 * *note get: ge │ │ │ │ -0026d2d0: 742c 202d 2d20 6765 7420 7468 6520 636f t, -- get the co │ │ │ │ -0026d2e0: 6e74 656e 7473 206f 6620 6120 6669 6c65 ntents of a file │ │ │ │ -0026d2f0: 0a0a 5761 7973 2074 6f20 7573 6520 7468 ..Ways to use th │ │ │ │ -0026d300: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d is method:.===== │ │ │ │ -0026d310: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026d320: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2050 ===.. * *note P │ │ │ │ -0026d330: 726f 6772 616d 203c 3c20 5468 696e 673a rogram << Thing: │ │ │ │ -0026d340: 2050 726f 6772 616d 203c 3c20 5468 696e Program << Thin │ │ │ │ -0026d350: 672c 202d 2d20 7275 6e20 7072 6f67 7261 g, -- run progra │ │ │ │ -0026d360: 6d20 7769 7468 2069 6e70 7574 0a20 2020 m with input. │ │ │ │ -0026d370: 2072 6564 6972 6563 7469 6f6e 0a2d 2d2d redirection.--- │ │ │ │ -0026d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026d390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026d3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026d1d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0026d1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026d1f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026d200: 0a7c 2020 2020 206f 3120 3d20 3420 2020 .| o1 = 4 │ │ │ │ +0026d210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026d220: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0026d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026d250: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ +0026d260: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +0026d270: 7465 2066 696e 6450 726f 6772 616d 3a20 te findProgram: │ │ │ │ +0026d280: 6669 6e64 5072 6f67 7261 6d2c 202d 2d20 findProgram, -- │ │ │ │ +0026d290: 6c6f 6164 2065 7874 6572 6e61 6c20 7072 load external pr │ │ │ │ +0026d2a0: 6f67 7261 6d0a 2020 2a20 2a6e 6f74 6520 ogram. * *note │ │ │ │ +0026d2b0: 7275 6e50 726f 6772 616d 3a20 7275 6e50 runProgram: runP │ │ │ │ +0026d2c0: 726f 6772 616d 2c20 2d2d 2072 756e 2061 rogram, -- run a │ │ │ │ +0026d2d0: 6e20 6578 7465 726e 616c 2070 726f 6772 n external progr │ │ │ │ +0026d2e0: 616d 0a20 202a 202a 6e6f 7465 2067 6574 am. * *note get │ │ │ │ +0026d2f0: 3a20 6765 742c 202d 2d20 6765 7420 7468 : get, -- get th │ │ │ │ +0026d300: 6520 636f 6e74 656e 7473 206f 6620 6120 e contents of a │ │ │ │ +0026d310: 6669 6c65 0a0a 5761 7973 2074 6f20 7573 file..Ways to us │ │ │ │ +0026d320: 6520 7468 6973 206d 6574 686f 643a 0a3d e this method:.= │ │ │ │ +0026d330: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026d340: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +0026d350: 7465 2050 726f 6772 616d 203c 3c20 5468 te Program << Th │ │ │ │ +0026d360: 696e 673a 2050 726f 6772 616d 203c 3c20 ing: Program << │ │ │ │ +0026d370: 5468 696e 672c 202d 2d20 7275 6e20 7072 Thing, -- run pr │ │ │ │ +0026d380: 6f67 7261 6d20 7769 7468 2069 6e70 7574 ogram with input │ │ │ │ +0026d390: 0a20 2020 2072 6564 6972 6563 7469 6f6e . redirection │ │ │ │ +0026d3a0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 0026d3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026d3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -0026d3d0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -0026d3e0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -0026d3f0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -0026d400: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -0026d410: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -0026d420: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -0026d430: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ -0026d440: 632f 6675 6e63 7469 6f6e 732f 7275 6e50 c/functions/runP │ │ │ │ -0026d450: 726f 6772 616d 2d64 6f63 2e6d 323a 3135 rogram-doc.m2:15 │ │ │ │ -0026d460: 373a 302e 0a1f 0a46 696c 653a 204d 6163 7:0....File: Mac │ │ │ │ -0026d470: 6175 6c61 7932 446f 632e 696e 666f 2c20 aulay2Doc.info, │ │ │ │ -0026d480: 4e6f 6465 3a20 7573 696e 6720 736f 636b Node: using sock │ │ │ │ -0026d490: 6574 732c 204e 6578 743a 206e 6f74 6966 ets, Next: notif │ │ │ │ -0026d4a0: 792c 2050 7265 763a 2063 6f6d 6d75 6e69 y, Prev: communi │ │ │ │ -0026d4b0: 6361 7469 6e67 2077 6974 6820 7072 6f67 cating with prog │ │ │ │ -0026d4c0: 7261 6d73 2c20 5570 3a20 7379 7374 656d rams, Up: system │ │ │ │ -0026d4d0: 2066 6163 696c 6974 6965 730a 0a75 7369 facilities..usi │ │ │ │ -0026d4e0: 6e67 2073 6f63 6b65 7473 0a2a 2a2a 2a2a ng sockets.***** │ │ │ │ -0026d4f0: 2a2a 2a2a 2a2a 2a2a 0a0a 4974 2773 2065 ********..It's e │ │ │ │ -0026d500: 6173 7920 746f 2075 7365 2073 6f63 6b65 asy to use socke │ │ │ │ -0026d510: 7473 2061 7320 7468 6f75 6768 2074 6865 ts as though the │ │ │ │ -0026d520: 7920 7765 7265 2066 696c 6573 2e20 2053 y were files. S │ │ │ │ -0026d530: 696d 706c 7920 7265 706c 6163 6520 7468 imply replace th │ │ │ │ -0026d540: 6520 6669 6c65 0a6e 616d 6520 6279 2061 e file.name by a │ │ │ │ -0026d550: 2073 7472 696e 6720 6f66 2074 6865 2066 string of the f │ │ │ │ -0026d560: 6f72 6d20 2468 6f73 743a 7365 7276 6963 orm $host:servic │ │ │ │ -0026d570: 6520 7768 6572 6520 686f 7374 2069 7320 e where host is │ │ │ │ -0026d580: 7468 6520 6e61 6d65 206f 6620 4950 206e the name of IP n │ │ │ │ -0026d590: 756d 6265 720a 6f66 2068 6f73 7420 746f umber.of host to │ │ │ │ -0026d5a0: 2063 6f6e 7461 6374 2c20 616e 6420 7365 contact, and se │ │ │ │ -0026d5b0: 7276 6963 6520 6973 2074 6865 2070 6f72 rvice is the por │ │ │ │ -0026d5c0: 7420 6e75 6d62 6572 206f 7220 6e61 6d65 t number or name │ │ │ │ -0026d5d0: 2074 6f20 7573 652e 2020 4966 2073 6572 to use. 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If hos │ │ │ │ -0026d610: 7420 6973 206f 6d69 7474 6564 2c20 7468 t is omitted, th │ │ │ │ -0026d620: 656e 2061 6e20 696e 636f 6d69 6e67 0a63 en an incoming.c │ │ │ │ -0026d630: 6f6e 6e65 6374 696f 6e20 7769 6c6c 2062 onnection will b │ │ │ │ -0026d640: 6520 6c69 7374 656e 6564 2066 6f72 2e0a e listened for.. │ │ │ │ -0026d650: 0a54 6865 2066 6f6c 6c6f 7769 6e67 2063 .The following c │ │ │ │ -0026d660: 6f64 6520 7769 6c6c 2069 6c6c 7573 7472 ode will illustr │ │ │ │ -0026d670: 6174 6520 7477 6f2d 7761 7920 636f 6d6d ate two-way comm │ │ │ │ -0026d680: 756e 6963 6174 696f 6e20 7573 696e 6720 unication using │ │ │ │ -0026d690: 736f 636b 6574 7320 7369 6d69 6c61 720a sockets similar. │ │ │ │ -0026d6a0: 746f 2074 6865 2069 6e74 6572 6163 7469 to the interacti │ │ │ │ -0026d6b0: 6f6e 2075 7365 6420 6279 2077 6562 2073 on used by web s │ │ │ │ -0026d6c0: 6572 7665 7273 2c20 616e 6420 796f 7520 ervers, and you │ │ │ │ -0026d6d0: 6d61 7920 7472 7920 6974 206f 7574 206f may try it out o │ │ │ │ -0026d6e0: 6e20 796f 7572 206d 6163 6869 6e65 2c0a n your machine,. │ │ │ │ -0026d6f0: 756e 6c65 7373 2061 2066 6972 6577 616c unless a firewal │ │ │ │ -0026d700: 6c20 7072 6576 656e 7473 2069 742e 0a0a l prevents it... │ │ │ │ -0026d710: 6966 2028 7069 6420 3d20 666f 726b 2829 if (pid = fork() │ │ │ │ -0026d720: 2920 3d3d 2030 2074 6865 6e20 280a 2020 ) == 0 then (. │ │ │ │ -0026d730: 2020 2074 7279 2022 243a 3735 3030 2220 try "$:7500" │ │ │ │ -0026d740: 3c3c 2022 6869 2074 6865 7265 2220 3c3c << "hi there" << │ │ │ │ -0026d750: 2063 6c6f 7365 3b0a 2020 2020 2065 7869 close;. exi │ │ │ │ -0026d760: 7420 303b 0a20 2020 2020 290a 736c 6565 t 0;. ).slee │ │ │ │ -0026d770: 7020 320a 6765 7420 2224 6c6f 6361 6c68 p 2.get "$localh │ │ │ │ -0026d780: 6f73 743a 3735 3030 220a 7761 6974 2070 ost:7500".wait p │ │ │ │ -0026d790: 6964 0a0a 5468 6520 636f 6465 2075 7365 id..The code use │ │ │ │ -0026d7a0: 7320 2a6e 6f74 6520 666f 726b 3a20 666f s *note fork: fo │ │ │ │ -0026d7b0: 726b 2c20 746f 2063 7265 6174 6520 6120 rk, to create a │ │ │ │ -0026d7c0: 7365 7061 7261 7465 2070 726f 6365 7373 separate process │ │ │ │ -0026d7d0: 2074 6861 7420 7769 6c6c 206c 6973 7465 that will liste │ │ │ │ -0026d7e0: 6e0a 666f 7220 6120 636f 6e6e 6563 7469 n.for a connecti │ │ │ │ -0026d7f0: 6f6e 206f 6e20 706f 7274 2037 3530 3020 on on port 7500 │ │ │ │ -0026d800: 616e 6420 7468 656e 2073 656e 6420 7573 and then send us │ │ │ │ -0026d810: 2061 206d 6573 7361 6765 2e20 2054 6865 a message. The │ │ │ │ -0026d820: 202a 6e6f 7465 2073 6c65 6570 3a0a 736c *note sleep:.sl │ │ │ │ -0026d830: 6565 702c 2063 6f6d 6d61 6e64 2070 6175 eep, command pau │ │ │ │ -0026d840: 7365 7320 666f 7220 6120 7768 696c 6520 ses for a while │ │ │ │ -0026d850: 746f 206d 616b 6520 7375 7265 2074 6865 to make sure the │ │ │ │ -0026d860: 2063 6869 6c64 2070 726f 6365 7373 2068 child process h │ │ │ │ -0026d870: 6173 2068 6164 2074 696d 650a 746f 2073 as had time.to s │ │ │ │ -0026d880: 7461 7274 206c 6973 7465 6e69 6e67 2e20 tart listening. │ │ │ │ -0026d890: 5468 656e 2077 6520 7573 6520 616e 206f Then we use an o │ │ │ │ -0026d8a0: 7264 696e 6172 7920 696e 7075 7420 636f rdinary input co │ │ │ │ -0026d8b0: 6d6d 616e 642c 206e 616d 656c 7920 2a6e mmand, namely *n │ │ │ │ -0026d8c0: 6f74 6520 6765 743a 0a67 6574 2c2c 2074 ote get:.get,, t │ │ │ │ -0026d8d0: 6f20 6f62 7461 696e 2074 6865 206d 6573 o obtain the mes │ │ │ │ -0026d8e0: 7361 6765 2e20 4669 6e61 6c6c 792c 2077 sage. Finally, w │ │ │ │ -0026d8f0: 6520 2a6e 6f74 6520 7761 6974 3a20 7761 e *note wait: wa │ │ │ │ -0026d900: 6974 2c20 666f 7220 7468 6520 6368 696c it, for the chil │ │ │ │ -0026d910: 640a 7072 6f63 6573 7320 746f 2066 696e d.process to fin │ │ │ │ -0026d920: 6973 682c 2061 7320 7765 2073 686f 756c ish, as we shoul │ │ │ │ -0026d930: 642e 0a0a 5365 6520 616c 736f 0a3d 3d3d d...See also.=== │ │ │ │ -0026d940: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0026d950: 2067 6574 5757 573a 2067 6574 5757 572c getWWW: getWWW, │ │ │ │ -0026d960: 202d 2d20 6765 7420 6120 7765 6220 7061 -- get a web pa │ │ │ │ -0026d970: 6765 0a2a 204d 656e 753a 0a0a 0a43 7265 ge.* Menu:...Cre │ │ │ │ -0026d980: 6174 696e 6720 6120 736f 636b 6574 0a2a ating a socket.* │ │ │ │ -0026d990: 206f 7065 6e4c 6973 7465 6e65 723a 206f openListener: o │ │ │ │ -0026d9a0: 7065 6e4c 6973 7465 6e65 725f 6c70 5374 penListener_lpSt │ │ │ │ -0026d9b0: 7269 6e67 5f72 702e 2020 6f70 656e 2061 ring_rp. open a │ │ │ 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696e 675f 7270 pend_lpString_rp │ │ │ │ -0026daa0: 2e20 206f 7065 6e20 616e 206f 7574 7075 . open an outpu │ │ │ │ -0026dab0: 7420 6669 6c65 2066 6f72 2061 7070 656e t file for appen │ │ │ │ -0026dac0: 6469 6e67 0a0a 4d61 6e69 7075 6c61 7469 ding..Manipulati │ │ │ │ -0026dad0: 6e67 2061 2073 6f63 6b65 740a 2a20 656e ng a socket.* en │ │ │ │ -0026dae0: 646c 3a3a 2020 2020 2020 2020 2020 2020 dl:: │ │ │ │ -0026daf0: 2020 2020 2020 2020 2020 2020 656e 6420 end │ │ │ │ -0026db00: 616e 206f 7574 7075 7420 6c69 6e65 0a2a an output line.* │ │ │ │ -0026db10: 2066 6c75 7368 3a3a 2020 2020 2020 2020 flush:: │ │ │ │ -0026db20: 2020 2020 2020 2020 2020 2020 2020 2066 f │ │ │ │ -0026db30: 6c75 7368 206f 7574 7075 7420 746f 2066 lush output to f │ │ │ │ -0026db40: 696c 650a 2a20 636c 6f73 653a 3a20 2020 ile.* close:: │ │ │ │ -0026db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026db60: 2020 2020 636c 6f73 6520 6120 6669 6c65 close a file │ │ │ │ -0026db70: 0a2a 2063 6c6f 7365 496e 3a3a 2020 2020 .* closeIn:: │ │ │ │ -0026db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026db90: 2063 6c6f 7365 2061 6e20 696e 7075 7420 close an input │ │ │ │ -0026dba0: 6669 6c65 0a2a 2063 6c6f 7365 4f75 743a file.* closeOut: │ │ │ │ -0026dbb0: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ -0026dbc0: 2020 2020 2063 6c6f 7365 2061 6e20 6f75 close an ou │ │ │ │ -0026dbd0: 7470 7574 2066 696c 650a 0a49 6e74 6572 tput file..Inter │ │ │ │ -0026dbe0: 6163 7469 6e67 2061 2073 6f63 6b65 740a acting a socket. │ │ │ │ -0026dbf0: 2a20 6765 743a 3a20 2020 2020 2020 2020 * get:: │ │ │ │ -0026dc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026dc10: 6765 7420 7468 6520 636f 6e74 656e 7473 get the contents │ │ │ │ -0026dc20: 206f 6620 6120 6669 6c65 0a2a 2067 6574 of a file.* get │ │ │ │ -0026dc30: 633a 3a20 2020 2020 2020 2020 2020 2020 c:: │ │ │ │ -0026dc40: 2020 2020 2020 2020 2020 2067 6574 2061 get a │ │ │ │ -0026dc50: 2062 7974 650a 2a20 7265 6164 3a3a 2020 byte.* read:: │ │ │ │ -0026dc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026dc70: 2020 2020 2020 7265 6164 2066 726f 6d20 read from │ │ │ │ -0026dc80: 6120 6669 6c65 0a2a 2073 6361 6e4c 696e a file.* scanLin │ │ │ │ -0026dc90: 6573 3a3a 2020 2020 2020 2020 2020 2020 es:: │ │ │ │ -0026dca0: 2020 2020 2020 2061 7070 6c79 2061 2066 apply a f │ │ │ │ -0026dcb0: 756e 6374 696f 6e20 746f 2065 6163 6820 unction to each │ │ │ │ -0026dcc0: 6c69 6e65 206f 6620 6120 6669 6c65 0a2a line of a file.* │ │ │ │ -0026dcd0: 2066 696c 654c 656e 6774 683a 3a20 2020 fileLength:: │ │ │ │ -0026dce0: 2020 2020 2020 2020 2020 2020 2020 2074 t │ │ │ │ -0026dcf0: 6865 206c 656e 6774 6820 6f66 2061 2066 he length of a f │ │ │ │ -0026dd00: 696c 650a 2a20 6865 6967 6874 2846 696c ile.* height(Fil │ │ │ │ -0026dd10: 6529 3a20 6865 6967 6874 5f6c 7046 696c e): height_lpFil │ │ │ │ -0026dd20: 655f 7270 2e20 2067 6574 2077 696e 646f e_rp. get windo │ │ │ │ -0026dd30: 7720 6865 6967 6874 0a2a 2077 6964 7468 w height.* width │ │ │ │ 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│ -0026de20: 2020 2020 2020 2020 7768 6574 6865 7220 whether │ │ │ │ -0026de30: 6120 6669 6c65 206f 7220 6461 7461 6261 a file or databa │ │ │ │ -0026de40: 7365 2069 7320 6f70 656e 0a2a 2069 7352 se is open.* isR │ │ │ │ -0026de50: 6561 6479 3a20 6973 5265 6164 795f 6c70 eady: isReady_lp │ │ │ │ -0026de60: 4669 6c65 5f72 702e 2020 2077 6865 7468 File_rp. wheth │ │ │ │ -0026de70: 6572 2061 2066 696c 6520 6861 7320 6461 er a file has da │ │ │ │ -0026de80: 7461 2061 7661 696c 6162 6c65 2066 6f72 ta available for │ │ │ │ -0026de90: 2072 6561 6469 6e67 0a2a 2069 734f 7574 reading.* isOut │ │ │ │ -0026dea0: 7075 7446 696c 653a 2069 734f 7574 7075 putFile: isOutpu │ │ │ │ -0026deb0: 7446 696c 655f 6c70 4669 6c65 5f72 702e tFile_lpFile_rp. │ │ │ │ -0026dec0: 2020 7768 6574 6865 7220 6120 6669 6c65 whether a file │ │ │ │ -0026ded0: 2069 7320 6f70 656e 2066 6f72 206f 7574 is open for out │ │ │ │ -0026dee0: 7075 740a 2a20 6973 496e 7075 7446 696c put.* isInputFil │ │ │ │ -0026def0: 653a 2069 7349 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│ -0026dfd0: 6d62 6572 206f 6620 636f 6e6e 6563 7469 mber of connecti │ │ │ │ -0026dfe0: 6f6e 730a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ons.------------ │ │ │ │ -0026dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026d3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026d3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026d3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026d3f0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0026d400: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0026d410: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0026d420: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0026d430: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +0026d440: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0026d450: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ +0026d460: 7932 446f 632f 6675 6e63 7469 6f6e 732f y2Doc/functions/ │ │ │ │ +0026d470: 7275 6e50 726f 6772 616d 2d64 6f63 2e6d runProgram-doc.m │ │ │ │ +0026d480: 323a 3135 373a 302e 0a1f 0a46 696c 653a 2:157:0....File: │ │ │ │ +0026d490: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ +0026d4a0: 666f 2c20 4e6f 6465 3a20 7573 696e 6720 fo, Node: using │ │ │ │ +0026d4b0: 736f 636b 6574 732c 204e 6578 743a 206e sockets, Next: n │ │ │ │ +0026d4c0: 6f74 6966 792c 2050 7265 763a 2063 6f6d otify, Prev: com │ │ │ │ +0026d4d0: 6d75 6e69 6361 7469 6e67 2077 6974 6820 municating with │ │ │ │ +0026d4e0: 7072 6f67 7261 6d73 2c20 5570 3a20 7379 programs, Up: sy │ │ │ │ +0026d4f0: 7374 656d 2066 6163 696c 6974 6965 730a stem facilities. │ │ │ │ +0026d500: 0a75 7369 6e67 2073 6f63 6b65 7473 0a2a .using sockets.* │ │ │ │ +0026d510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4974 ************..It │ │ │ │ +0026d520: 2773 2065 6173 7920 746f 2075 7365 2073 's easy to use s │ │ │ │ +0026d530: 6f63 6b65 7473 2061 7320 7468 6f75 6768 ockets as though │ │ │ │ +0026d540: 2074 6865 7920 7765 7265 2066 696c 6573 they were files │ │ │ │ +0026d550: 2e20 2053 696d 706c 7920 7265 706c 6163 . Simply replac │ │ │ │ +0026d560: 6520 7468 6520 6669 6c65 0a6e 616d 6520 e the file.name │ │ │ │ +0026d570: 6279 2061 2073 7472 696e 6720 6f66 2074 by a string of t │ │ │ │ +0026d580: 6865 2066 6f72 6d20 2468 6f73 743a 7365 he form $host:se │ │ │ │ +0026d590: 7276 6963 6520 7768 6572 6520 686f 7374 rvice where host │ │ │ │ +0026d5a0: 2069 7320 7468 6520 6e61 6d65 206f 6620 is the name of │ │ │ │ +0026d5b0: 4950 206e 756d 6265 720a 6f66 2068 6f73 IP number.of hos │ │ │ │ +0026d5c0: 7420 746f 2063 6f6e 7461 6374 2c20 616e t to contact, an │ │ │ │ +0026d5d0: 6420 7365 7276 6963 6520 6973 2074 6865 d service is the │ │ │ │ +0026d5e0: 2070 6f72 7420 6e75 6d62 6572 206f 7220 port number or │ │ │ │ +0026d5f0: 6e61 6d65 2074 6f20 7573 652e 2020 4966 name to use. If │ │ │ │ +0026d600: 2073 6572 7669 6365 0a69 7320 6f6d 6974 service.is omit │ │ │ │ +0026d610: 7465 642c 2074 6865 6e20 706f 7274 2032 ted, then port 2 │ │ │ │ +0026d620: 3530 3020 6973 2075 7365 642e 2020 4966 500 is used. If │ │ │ │ +0026d630: 2068 6f73 7420 6973 206f 6d69 7474 6564 host is omitted │ │ │ │ +0026d640: 2c20 7468 656e 2061 6e20 696e 636f 6d69 , then an incomi │ │ │ │ +0026d650: 6e67 0a63 6f6e 6e65 6374 696f 6e20 7769 ng.connection wi │ │ │ │ +0026d660: 6c6c 2062 6520 6c69 7374 656e 6564 2066 ll be listened f │ │ │ │ +0026d670: 6f72 2e0a 0a54 6865 2066 6f6c 6c6f 7769 or...The followi │ │ │ │ +0026d680: 6e67 2063 6f64 6520 7769 6c6c 2069 6c6c ng code will ill │ │ │ │ +0026d690: 7573 7472 6174 6520 7477 6f2d 7761 7920 ustrate two-way │ │ │ │ +0026d6a0: 636f 6d6d 756e 6963 6174 696f 6e20 7573 communication us │ │ │ │ +0026d6b0: 696e 6720 736f 636b 6574 7320 7369 6d69 ing sockets simi │ │ │ │ +0026d6c0: 6c61 720a 746f 2074 6865 2069 6e74 6572 lar.to the inter │ │ │ │ +0026d6d0: 6163 7469 6f6e 2075 7365 6420 6279 2077 action used by w │ │ │ │ +0026d6e0: 6562 2073 6572 7665 7273 2c20 616e 6420 eb servers, and │ │ │ │ +0026d6f0: 796f 7520 6d61 7920 7472 7920 6974 206f you may try it o │ │ │ │ +0026d700: 7574 206f 6e20 796f 7572 206d 6163 6869 ut on your machi │ │ │ │ +0026d710: 6e65 2c0a 756e 6c65 7373 2061 2066 6972 ne,.unless a fir │ │ │ │ +0026d720: 6577 616c 6c20 7072 6576 656e 7473 2069 ewall prevents i │ │ │ │ +0026d730: 742e 0a0a 6966 2028 7069 6420 3d20 666f t...if (pid = fo │ │ │ │ +0026d740: 726b 2829 2920 3d3d 2030 2074 6865 6e20 rk()) == 0 then │ │ │ │ +0026d750: 280a 2020 2020 2074 7279 2022 243a 3735 (. try "$:75 │ │ │ │ +0026d760: 3030 2220 3c3c 2022 6869 2074 6865 7265 00" << "hi there │ │ │ │ +0026d770: 2220 3c3c 2063 6c6f 7365 3b0a 2020 2020 " << close;. │ │ │ │ +0026d780: 2065 7869 7420 303b 0a20 2020 2020 290a exit 0;. ). │ │ │ │ +0026d790: 736c 6565 7020 320a 6765 7420 2224 6c6f sleep 2.get "$lo │ │ │ │ +0026d7a0: 6361 6c68 6f73 743a 3735 3030 220a 7761 calhost:7500".wa │ │ │ │ +0026d7b0: 6974 2070 6964 0a0a 5468 6520 636f 6465 it pid..The code │ │ │ │ +0026d7c0: 2075 7365 7320 2a6e 6f74 6520 666f 726b uses *note fork │ │ │ │ +0026d7d0: 3a20 666f 726b 2c20 746f 2063 7265 6174 : fork, to creat │ │ │ │ +0026d7e0: 6520 6120 7365 7061 7261 7465 2070 726f e a separate pro │ │ │ │ +0026d7f0: 6365 7373 2074 6861 7420 7769 6c6c 206c cess that will l │ │ │ │ +0026d800: 6973 7465 6e0a 666f 7220 6120 636f 6e6e isten.for a conn │ │ │ │ +0026d810: 6563 7469 6f6e 206f 6e20 706f 7274 2037 ection on port 7 │ │ │ │ +0026d820: 3530 3020 616e 6420 7468 656e 2073 656e 500 and then sen │ │ │ │ +0026d830: 6420 7573 2061 206d 6573 7361 6765 2e20 d us a message. │ │ │ │ +0026d840: 2054 6865 202a 6e6f 7465 2073 6c65 6570 The *note sleep │ │ │ │ +0026d850: 3a0a 736c 6565 702c 2063 6f6d 6d61 6e64 :.sleep, command │ │ │ │ +0026d860: 2070 6175 7365 7320 666f 7220 6120 7768 pauses for a wh │ │ │ │ +0026d870: 696c 6520 746f 206d 616b 6520 7375 7265 ile to make sure │ │ │ │ +0026d880: 2074 6865 2063 6869 6c64 2070 726f 6365 the child proce │ │ │ │ +0026d890: 7373 2068 6173 2068 6164 2074 696d 650a ss has had time. │ │ │ │ +0026d8a0: 746f 2073 7461 7274 206c 6973 7465 6e69 to start listeni │ │ │ │ +0026d8b0: 6e67 2e20 5468 656e 2077 6520 7573 6520 ng. Then we use │ │ │ │ +0026d8c0: 616e 206f 7264 696e 6172 7920 696e 7075 an ordinary inpu │ │ │ │ +0026d8d0: 7420 636f 6d6d 616e 642c 206e 616d 656c t command, namel │ │ │ │ +0026d8e0: 7920 2a6e 6f74 6520 6765 743a 0a67 6574 y *note get:.get │ │ │ │ +0026d8f0: 2c2c 2074 6f20 6f62 7461 696e 2074 6865 ,, to obtain the │ │ │ │ +0026d900: 206d 6573 7361 6765 2e20 4669 6e61 6c6c message. Finall │ │ │ │ +0026d910: 792c 2077 6520 2a6e 6f74 6520 7761 6974 y, we *note wait │ │ │ │ +0026d920: 3a20 7761 6974 2c20 666f 7220 7468 6520 : wait, for the │ │ │ │ +0026d930: 6368 696c 640a 7072 6f63 6573 7320 746f child.process to │ │ │ │ +0026d940: 2066 696e 6973 682c 2061 7320 7765 2073 finish, as we s │ │ │ │ +0026d950: 686f 756c 642e 0a0a 5365 6520 616c 736f hould...See also │ │ │ │ +0026d960: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0026d970: 6e6f 7465 2067 6574 5757 573a 2067 6574 note getWWW: get │ │ │ │ +0026d980: 5757 572c 202d 2d20 6765 7420 6120 7765 WWW, -- get a we │ │ │ │ +0026d990: 6220 7061 6765 0a2a 204d 656e 753a 0a0a b page.* Menu:.. │ │ │ │ +0026d9a0: 0a43 7265 6174 696e 6720 6120 736f 636b .Creating a sock │ │ │ │ +0026d9b0: 6574 0a2a 206f 7065 6e4c 6973 7465 6e65 et.* openListene │ │ │ │ +0026d9c0: 723a 206f 7065 6e4c 6973 7465 6e65 725f r: openListener_ │ │ │ │ +0026d9d0: 6c70 5374 7269 6e67 5f72 702e 2020 6f70 lpString_rp. op │ │ │ │ +0026d9e0: 656e 2061 2070 6f72 7420 666f 7220 6c69 en a port for li │ │ │ │ +0026d9f0: 7374 656e 696e 670a 2a20 6f70 656e 496e stening.* openIn │ │ │ │ +0026da00: 3a20 6f70 656e 496e 5f6c 7053 7472 696e : openIn_lpStrin │ │ │ │ +0026da10: 675f 7270 2e20 2020 6f70 656e 2061 6e20 g_rp. open an │ │ │ │ +0026da20: 696e 7075 7420 6669 6c65 0a2a 206f 7065 input file.* ope │ │ │ │ +0026da30: 6e49 6e4f 7574 3a3a 2020 2020 2020 2020 nInOut:: │ │ │ │ +0026da40: 2020 2020 2020 2020 2020 206f 7065 6e20 open │ │ │ │ +0026da50: 616e 2069 6e70 7574 206f 7574 7075 7420 an input output │ │ │ │ +0026da60: 6669 6c65 0a2a 206f 7065 6e4f 7574 3a20 file.* openOut: │ │ │ │ +0026da70: 6f70 656e 4f75 745f 6c70 5374 7269 6e67 openOut_lpString │ │ │ │ +0026da80: 5f72 702e 2020 6f70 656e 2061 6e20 6f75 _rp. open an ou │ │ │ │ +0026da90: 7470 7574 2066 696c 650a 2a20 6f70 656e tput file.* open │ │ │ │ +0026daa0: 4f75 7441 7070 656e 643a 206f 7065 6e4f OutAppend: openO │ │ │ │ +0026dab0: 7574 4170 7065 6e64 5f6c 7053 7472 696e utAppend_lpStrin │ │ │ │ +0026dac0: 675f 7270 2e20 206f 7065 6e20 616e 206f g_rp. open an o │ │ │ │ +0026dad0: 7574 7075 7420 6669 6c65 2066 6f72 2061 utput file for a │ │ │ │ +0026dae0: 7070 656e 6469 6e67 0a0a 4d61 6e69 7075 ppending..Manipu │ │ │ │ +0026daf0: 6c61 7469 6e67 2061 2073 6f63 6b65 740a lating a socket. │ │ │ │ +0026db00: 2a20 656e 646c 3a3a 2020 2020 2020 2020 * endl:: │ │ │ │ +0026db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026db20: 656e 6420 616e 206f 7574 7075 7420 6c69 end an output li │ │ │ │ +0026db30: 6e65 0a2a 2066 6c75 7368 3a3a 2020 2020 ne.* flush:: │ │ │ │ +0026db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026db50: 2020 2066 6c75 7368 206f 7574 7075 7420 flush output │ │ │ │ +0026db60: 746f 2066 696c 650a 2a20 636c 6f73 653a to file.* close: │ │ │ │ +0026db70: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ +0026db80: 2020 2020 2020 2020 636c 6f73 6520 6120 close a │ │ │ │ +0026db90: 6669 6c65 0a2a 2063 6c6f 7365 496e 3a3a file.* closeIn:: │ │ │ │ +0026dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026dbb0: 2020 2020 2063 6c6f 7365 2061 6e20 696e close an in │ │ │ │ +0026dbc0: 7075 7420 6669 6c65 0a2a 2063 6c6f 7365 put file.* close │ │ │ │ +0026dbd0: 4f75 743a 3a20 2020 2020 2020 2020 2020 Out:: │ │ │ │ +0026dbe0: 2020 2020 2020 2020 2063 6c6f 7365 2061 close a │ │ │ │ +0026dbf0: 6e20 6f75 7470 7574 2066 696c 650a 0a49 n output file..I │ │ │ │ +0026dc00: 6e74 6572 6163 7469 6e67 2061 2073 6f63 nteracting a soc │ │ │ │ +0026dc10: 6b65 740a 2a20 6765 743a 3a20 2020 2020 ket.* get:: │ │ │ │ +0026dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026dc30: 2020 2020 6765 7420 7468 6520 636f 6e74 get the cont │ │ │ │ +0026dc40: 656e 7473 206f 6620 6120 6669 6c65 0a2a ents of a file.* │ │ │ │ +0026dc50: 2067 6574 633a 3a20 2020 2020 2020 2020 getc:: │ │ │ │ +0026dc60: 2020 2020 2020 2020 2020 2020 2020 2067 g │ │ │ │ +0026dc70: 6574 2061 2062 7974 650a 2a20 7265 6164 et a byte.* read │ │ │ │ +0026dc80: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: │ │ │ │ +0026dc90: 2020 2020 2020 2020 2020 7265 6164 2066 read f │ │ │ │ +0026dca0: 726f 6d20 6120 6669 6c65 0a2a 2073 6361 rom a file.* sca │ │ │ │ +0026dcb0: 6e4c 696e 6573 3a3a 2020 2020 2020 2020 nLines:: │ │ │ │ +0026dcc0: 2020 2020 2020 2020 2020 2061 7070 6c79 apply │ │ │ │ +0026dcd0: 2061 2066 756e 6374 696f 6e20 746f 2065 a function to e │ │ │ │ +0026dce0: 6163 6820 6c69 6e65 206f 6620 6120 6669 ach line of a fi │ │ │ │ +0026dcf0: 6c65 0a2a 2066 696c 654c 656e 6774 683a le.* fileLength: │ │ │ │ +0026dd00: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ +0026dd10: 2020 2074 6865 206c 656e 6774 6820 6f66 the length of │ │ │ │ +0026dd20: 2061 2066 696c 650a 2a20 6865 6967 6874 a file.* height │ │ │ │ +0026dd30: 2846 696c 6529 3a20 6865 6967 6874 5f6c (File): height_l │ │ │ │ +0026dd40: 7046 696c 655f 7270 2e20 2067 6574 2077 pFile_rp. get w │ │ │ │ +0026dd50: 696e 646f 7720 6865 6967 6874 0a2a 2077 indow height.* w │ │ │ │ +0026dd60: 6964 7468 2846 696c 6529 3a20 7769 6474 idth(File): widt │ │ │ │ +0026dd70: 685f 6c70 4669 6c65 5f72 702e 2020 6765 h_lpFile_rp. ge │ │ │ │ +0026dd80: 7420 7769 6e64 6f77 2077 6964 7468 0a2a t window width.* │ │ │ │ +0026dd90: 2061 7445 6e64 4f66 4669 6c65 3a20 6174 atEndOfFile: at │ │ │ │ +0026dda0: 456e 644f 6646 696c 655f 6c70 4669 6c65 EndOfFile_lpFile │ │ │ │ +0026ddb0: 5f72 702e 2020 7465 7374 2066 6f72 2065 _rp. test for e │ │ │ │ +0026ddc0: 6e64 206f 6620 6669 6c65 0a2a 2065 6368 nd of file.* ech │ │ │ │ +0026ddd0: 6f4f 6e3a 3a20 2020 2020 2020 2020 2020 oOn:: │ │ │ │ +0026dde0: 2020 2020 2020 2020 2020 2074 7572 6e20 turn │ │ │ │ +0026ddf0: 6f6e 2065 6368 6f69 6e67 0a2a 2065 6368 on echoing.* ech │ │ │ │ +0026de00: 6f4f 6666 3a3a 2020 2020 2020 2020 2020 oOff:: │ │ │ │ +0026de10: 2020 2020 2020 2020 2020 2074 7572 6e20 turn │ │ │ │ +0026de20: 6f66 6620 6563 686f 696e 670a 2a20 6973 off echoing.* is │ │ │ │ +0026de30: 4f70 656e 3a3a 2020 2020 2020 2020 2020 Open:: │ │ │ │ +0026de40: 2020 2020 2020 2020 2020 2020 7768 6574 whet │ │ │ │ +0026de50: 6865 7220 6120 6669 6c65 206f 7220 6461 her a file or da │ │ │ │ +0026de60: 7461 6261 7365 2069 7320 6f70 656e 0a2a tabase is open.* │ │ │ │ +0026de70: 2069 7352 6561 6479 3a20 6973 5265 6164 isReady: isRead │ │ │ │ +0026de80: 795f 6c70 4669 6c65 5f72 702e 2020 2077 y_lpFile_rp. w │ │ │ │ +0026de90: 6865 7468 6572 2061 2066 696c 6520 6861 hether a file ha │ │ │ │ +0026dea0: 7320 6461 7461 2061 7661 696c 6162 6c65 s data available │ │ │ │ +0026deb0: 2066 6f72 2072 6561 6469 6e67 0a2a 2069 for reading.* i │ │ │ │ +0026dec0: 734f 7574 7075 7446 696c 653a 2069 734f sOutputFile: isO │ │ │ │ +0026ded0: 7574 7075 7446 696c 655f 6c70 4669 6c65 utputFile_lpFile │ │ │ │ +0026dee0: 5f72 702e 2020 7768 6574 6865 7220 6120 _rp. whether a │ │ │ │ +0026def0: 6669 6c65 2069 7320 6f70 656e 2066 6f72 file is open for │ │ │ │ +0026df00: 206f 7574 7075 740a 2a20 6973 496e 7075 output.* isInpu │ │ │ │ +0026df10: 7446 696c 653a 2069 7349 6e70 7574 4669 tFile: isInputFi │ │ │ │ +0026df20: 6c65 5f6c 7046 696c 655f 7270 2e20 2077 le_lpFile_rp. w │ │ │ │ +0026df30: 6865 7468 6572 2061 2066 696c 6520 6973 hether a file is │ │ │ │ +0026df40: 206f 7065 6e20 666f 7220 696e 7075 740a open for input. │ │ │ │ +0026df50: 2a20 6973 4c69 7374 656e 6572 3a20 6973 * isListener: is │ │ │ │ +0026df60: 4c69 7374 656e 6572 5f6c 7046 696c 655f Listener_lpFile_ │ │ │ │ +0026df70: 7270 2e20 2077 6865 7468 6572 2061 2066 rp. whether a f │ │ │ │ +0026df80: 696c 6520 6973 206f 7065 6e20 666f 7220 ile is open for │ │ │ │ +0026df90: 6c69 7374 656e 696e 670a 2a20 6f70 656e listening.* open │ │ │ │ +0026dfa0: 4669 6c65 733a 3a20 2020 2020 2020 2020 Files:: │ │ │ │ +0026dfb0: 2020 2020 2020 2020 2020 6c69 7374 2074 list t │ │ │ │ +0026dfc0: 6865 206f 7065 6e20 6669 6c65 730a 2a20 he open files.* │ │ │ │ +0026dfd0: 636f 6e6e 6563 7469 6f6e 436f 756e 743a connectionCount: │ │ │ │ +0026dfe0: 3a20 2020 2020 2020 2020 2020 2020 7468 : th │ │ │ │ +0026dff0: 6520 6e75 6d62 6572 206f 6620 636f 6e6e e number of conn │ │ │ │ +0026e000: 6563 7469 6f6e 730a 2d2d 2d2d 2d2d 2d2d ections.-------- │ │ │ │ 0026e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0026e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026e030: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0026e040: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0026e050: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0026e060: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0026e070: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0026e080: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -0026e090: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -0026e0a0: 756c 6179 3244 6f63 2f6f 765f 6669 6c65 ulay2Doc/ov_file │ │ │ │ -0026e0b0: 732e 6d32 3a35 3039 3a30 2e0a 1f0a 4669 s.m2:509:0....Fi │ │ │ │ -0026e0c0: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -0026e0d0: 2e69 6e66 6f2c 204e 6f64 653a 206f 7065 .info, Node: ope │ │ │ │ -0026e0e0: 6e4c 6973 7465 6e65 725f 6c70 5374 7269 nListener_lpStri │ │ │ │ -0026e0f0: 6e67 5f72 702c 204e 6578 743a 206f 7065 ng_rp, Next: ope │ │ │ │ -0026e100: 6e49 6e5f 6c70 5374 7269 6e67 5f72 702c nIn_lpString_rp, │ │ │ │ -0026e110: 2055 703a 2075 7369 6e67 2073 6f63 6b65 Up: using socke │ │ │ │ -0026e120: 7473 0a0a 6f70 656e 4c69 7374 656e 6572 ts..openListener │ │ │ │ -0026e130: 2853 7472 696e 6729 202d 2d20 6f70 656e (String) -- open │ │ │ │ -0026e140: 2061 2070 6f72 7420 666f 7220 6c69 7374 a port for list │ │ │ │ -0026e150: 656e 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a ening.********** │ │ │ │ -0026e160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026e170: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026e180: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e *******.. * Fun │ │ │ │ -0026e190: 6374 696f 6e3a 202a 6e6f 7465 206f 7065 ction: *note ope │ │ │ │ -0026e1a0: 6e4c 6973 7465 6e65 723a 206f 7065 6e4c nListener: openL │ │ │ │ -0026e1b0: 6973 7465 6e65 725f 6c70 5374 7269 6e67 istener_lpString │ │ │ │ -0026e1c0: 5f72 702c 0a20 202a 2055 7361 6765 3a20 _rp,. * Usage: │ │ │ │ -0026e1d0: 0a20 2020 2020 2020 2066 203d 206f 7065 . f = ope │ │ │ │ -0026e1e0: 6e4c 6973 7465 6e65 7220 730a 2020 2a20 nListener s. * │ │ │ │ -0026e1f0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -0026e200: 732c 2061 202a 6e6f 7465 2073 7472 696e s, a *note strin │ │ │ │ -0026e210: 673a 2053 7472 696e 672c 2c20 6f66 2074 g: String,, of t │ │ │ │ -0026e220: 6865 2066 6f72 6d20 2224 696e 7465 7266 he form "$interf │ │ │ │ -0026e230: 6163 653a 706f 7274 222e 2020 426f 7468 ace:port". Both │ │ │ │ -0026e240: 2070 6172 7473 0a20 2020 2020 2020 2061 parts. a │ │ │ │ -0026e250: 7265 206f 7074 696f 6e61 6c2e 2020 4966 re optional. If │ │ │ │ -0026e260: 2074 6865 2070 6f72 7420 6973 206f 6d69 the port is omi │ │ │ │ -0026e270: 7474 6564 2c20 7468 6520 636f 6c6f 6e20 tted, the colon │ │ │ │ -0026e280: 6973 206f 7074 696f 6e61 6c2e 0a20 202a is optional.. * │ │ │ │ -0026e290: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0026e2a0: 2a20 662c 2061 202a 6e6f 7465 2066 696c * f, a *note fil │ │ │ │ -0026e2b0: 653a 2046 696c 652c 2c20 616e 206f 7065 e: File,, an ope │ │ │ │ -0026e2c0: 6e20 6c69 7374 656e 6572 206f 6e20 7468 n listener on th │ │ │ │ -0026e2d0: 6520 7370 6563 6966 6965 6420 696e 7465 e specified inte │ │ │ │ -0026e2e0: 7266 6163 6520 6f66 0a20 2020 2020 2020 rface of. │ │ │ │ -0026e2f0: 2074 6865 206c 6f63 616c 2068 6f73 7420 the local host │ │ │ │ -0026e300: 6174 2074 6865 2073 7065 6369 6669 6564 at the specified │ │ │ │ -0026e310: 2073 6572 7669 6365 2070 6f72 742e 2049 service port. I │ │ │ │ -0026e320: 6620 7468 6520 706f 7274 2069 7320 6f6d f the port is om │ │ │ │ -0026e330: 6974 7465 642c 0a20 2020 2020 2020 2069 itted,. i │ │ │ │ -0026e340: 7420 6973 2074 616b 656e 2074 6f20 6265 t is taken to be │ │ │ │ -0026e350: 2070 6f72 7420 3235 3030 2e20 2049 6620 port 2500. If │ │ │ │ -0026e360: 7468 6520 696e 7465 7266 6163 6520 6973 the interface is │ │ │ │ -0026e370: 206f 6d69 7474 6564 2c20 7468 6520 6c69 omitted, the li │ │ │ │ -0026e380: 7374 656e 6572 0a20 2020 2020 2020 2061 stener. a │ │ │ │ -0026e390: 6363 6570 7473 2063 6f6e 6e65 6374 696f ccepts connectio │ │ │ │ -0026e3a0: 6e73 206f 6e20 616c 6c20 696e 7465 7266 ns on all interf │ │ │ │ -0026e3b0: 6163 6573 2e0a 0a44 6573 6372 6970 7469 aces...Descripti │ │ │ │ -0026e3c0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -0026e3d0: 5573 6520 6f70 656e 496e 4f75 7420 6620 Use openInOut f │ │ │ │ -0026e3e0: 746f 2061 6363 6570 7420 616e 2069 6e63 to accept an inc │ │ │ │ -0026e3f0: 6f6d 696e 6720 636f 6e6e 6563 7469 6f6e oming connection │ │ │ │ -0026e400: 206f 6e20 7468 6520 6c69 7374 656e 6572 on the listener │ │ │ │ -0026e410: 2c20 7265 7475 726e 696e 6720 610a 6e65 , returning a.ne │ │ │ │ -0026e420: 7720 696e 7075 7420 6f75 7470 7574 2066 w input output f │ │ │ │ -0026e430: 696c 6520 7468 6174 2073 6572 7665 7320 ile that serves │ │ │ │ -0026e440: 6173 2074 6865 2063 6f6e 6e65 6374 696f as the connectio │ │ │ │ -0026e450: 6e2e 2020 5468 6520 6675 6e63 7469 6f6e n. The function │ │ │ │ -0026e460: 202a 6e6f 7465 0a69 7352 6561 6479 3a20 *note.isReady: │ │ │ │ -0026e470: 6973 5265 6164 795f 6c70 4669 6c65 5f72 isReady_lpFile_r │ │ │ │ -0026e480: 702c 2063 616e 2062 6520 7573 6564 2074 p, can be used t │ │ │ │ -0026e490: 6f20 6465 7465 726d 696e 6520 7768 6574 o determine whet │ │ │ │ -0026e4a0: 6865 7220 616e 2069 6e63 6f6d 696e 670a her an incoming. │ │ │ │ -0026e4b0: 636f 6e6e 6563 7469 6f6e 2068 6173 2061 connection has a │ │ │ │ -0026e4c0: 7272 6976 6564 2c20 7769 7468 6f75 7420 rrived, without │ │ │ │ -0026e4d0: 626c 6f63 6b69 6e67 2e0a 0a53 6565 2061 blocking...See a │ │ │ │ -0026e4e0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -0026e4f0: 2a20 2a6e 6f74 6520 6f70 656e 496e 4f75 * *note openInOu │ │ │ │ -0026e500: 743a 206f 7065 6e49 6e4f 7574 2c20 2d2d t: openInOut, -- │ │ │ │ -0026e510: 206f 7065 6e20 616e 2069 6e70 7574 206f open an input o │ │ │ │ -0026e520: 7574 7075 7420 6669 6c65 0a20 202a 202a utput file. * * │ │ │ │ -0026e530: 6e6f 7465 2069 7352 6561 6479 3a20 6973 note isReady: is │ │ │ │ -0026e540: 5265 6164 795f 6c70 4669 6c65 5f72 702c Ready_lpFile_rp, │ │ │ │ -0026e550: 202d 2d20 7768 6574 6865 7220 6120 6669 -- whether a fi │ │ │ │ -0026e560: 6c65 2068 6173 2064 6174 6120 6176 6169 le has data avai │ │ │ │ -0026e570: 6c61 626c 6520 666f 720a 2020 2020 7265 lable for. re │ │ │ │ -0026e580: 6164 696e 670a 0a57 6179 7320 746f 2075 ading..Ways to u │ │ │ │ -0026e590: 7365 2074 6869 7320 6d65 7468 6f64 3a0a se this method:. │ │ │ │ -0026e5a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026e5b0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -0026e5c0: 6f74 6520 6f70 656e 4c69 7374 656e 6572 ote openListener │ │ │ │ -0026e5d0: 2853 7472 696e 6729 3a20 6f70 656e 4c69 (String): openLi │ │ │ │ -0026e5e0: 7374 656e 6572 5f6c 7053 7472 696e 675f stener_lpString_ │ │ │ │ -0026e5f0: 7270 2c20 2d2d 206f 7065 6e20 6120 706f rp, -- open a po │ │ │ │ -0026e600: 7274 2066 6f72 0a20 2020 206c 6973 7465 rt for. liste │ │ │ │ -0026e610: 6e69 6e67 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d ning.----------- │ │ │ │ -0026e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026e040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026e050: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +0026e060: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +0026e070: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +0026e080: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +0026e090: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +0026e0a0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +0026e0b0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +0026e0c0: 4d61 6361 756c 6179 3244 6f63 2f6f 765f Macaulay2Doc/ov_ │ │ │ │ +0026e0d0: 6669 6c65 732e 6d32 3a35 3039 3a30 2e0a files.m2:509:0.. │ │ │ │ +0026e0e0: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +0026e0f0: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +0026e100: 206f 7065 6e4c 6973 7465 6e65 725f 6c70 openListener_lp │ │ │ │ +0026e110: 5374 7269 6e67 5f72 702c 204e 6578 743a String_rp, Next: │ │ │ │ +0026e120: 206f 7065 6e49 6e5f 6c70 5374 7269 6e67 openIn_lpString │ │ │ │ +0026e130: 5f72 702c 2055 703a 2075 7369 6e67 2073 _rp, Up: using s │ │ │ │ +0026e140: 6f63 6b65 7473 0a0a 6f70 656e 4c69 7374 ockets..openList │ │ │ │ +0026e150: 656e 6572 2853 7472 696e 6729 202d 2d20 ener(String) -- │ │ │ │ +0026e160: 6f70 656e 2061 2070 6f72 7420 666f 7220 open a port for │ │ │ │ +0026e170: 6c69 7374 656e 696e 670a 2a2a 2a2a 2a2a listening.****** │ │ │ │ +0026e180: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026e190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026e1a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0026e1b0: 2046 756e 6374 696f 6e3a 202a 6e6f 7465 Function: *note │ │ │ │ +0026e1c0: 206f 7065 6e4c 6973 7465 6e65 723a 206f openListener: o │ │ │ │ +0026e1d0: 7065 6e4c 6973 7465 6e65 725f 6c70 5374 penListener_lpSt │ │ │ │ +0026e1e0: 7269 6e67 5f72 702c 0a20 202a 2055 7361 ring_rp,. * Usa │ │ │ │ +0026e1f0: 6765 3a20 0a20 2020 2020 2020 2066 203d ge: . f = │ │ │ │ +0026e200: 206f 7065 6e4c 6973 7465 6e65 7220 730a openListener s. │ │ │ │ +0026e210: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0026e220: 2020 2a20 732c 2061 202a 6e6f 7465 2073 * s, a *note s │ │ │ │ +0026e230: 7472 696e 673a 2053 7472 696e 672c 2c20 tring: String,, │ │ │ │ +0026e240: 6f66 2074 6865 2066 6f72 6d20 2224 696e of the form "$in │ │ │ │ +0026e250: 7465 7266 6163 653a 706f 7274 222e 2020 terface:port". │ │ │ │ +0026e260: 426f 7468 2070 6172 7473 0a20 2020 2020 Both parts. │ │ │ │ +0026e270: 2020 2061 7265 206f 7074 696f 6e61 6c2e are optional. │ │ │ │ +0026e280: 2020 4966 2074 6865 2070 6f72 7420 6973 If the port is │ │ │ │ +0026e290: 206f 6d69 7474 6564 2c20 7468 6520 636f omitted, the co │ │ │ │ +0026e2a0: 6c6f 6e20 6973 206f 7074 696f 6e61 6c2e lon is optional. │ │ │ │ +0026e2b0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +0026e2c0: 2020 2020 2a20 662c 2061 202a 6e6f 7465 * f, a *note │ │ │ │ +0026e2d0: 2066 696c 653a 2046 696c 652c 2c20 616e file: File,, an │ │ │ │ +0026e2e0: 206f 7065 6e20 6c69 7374 656e 6572 206f open listener o │ │ │ │ +0026e2f0: 6e20 7468 6520 7370 6563 6966 6965 6420 n the specified │ │ │ │ +0026e300: 696e 7465 7266 6163 6520 6f66 0a20 2020 interface of. │ │ │ │ +0026e310: 2020 2020 2074 6865 206c 6f63 616c 2068 the local h │ │ │ │ +0026e320: 6f73 7420 6174 2074 6865 2073 7065 6369 ost at the speci │ │ │ │ +0026e330: 6669 6564 2073 6572 7669 6365 2070 6f72 fied service por │ │ │ │ +0026e340: 742e 2049 6620 7468 6520 706f 7274 2069 t. If the port i │ │ │ │ +0026e350: 7320 6f6d 6974 7465 642c 0a20 2020 2020 s omitted,. │ │ │ │ +0026e360: 2020 2069 7420 6973 2074 616b 656e 2074 it is taken t │ │ │ │ +0026e370: 6f20 6265 2070 6f72 7420 3235 3030 2e20 o be port 2500. │ │ │ │ +0026e380: 2049 6620 7468 6520 696e 7465 7266 6163 If the interfac │ │ │ │ +0026e390: 6520 6973 206f 6d69 7474 6564 2c20 7468 e is omitted, th │ │ │ │ +0026e3a0: 6520 6c69 7374 656e 6572 0a20 2020 2020 e listener. │ │ │ │ +0026e3b0: 2020 2061 6363 6570 7473 2063 6f6e 6e65 accepts conne │ │ │ │ +0026e3c0: 6374 696f 6e73 206f 6e20 616c 6c20 696e ctions on all in │ │ │ │ +0026e3d0: 7465 7266 6163 6573 2e0a 0a44 6573 6372 terfaces...Descr │ │ │ │ +0026e3e0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +0026e3f0: 3d3d 0a0a 5573 6520 6f70 656e 496e 4f75 ==..Use openInOu │ │ │ │ +0026e400: 7420 6620 746f 2061 6363 6570 7420 616e t f to accept an │ │ │ │ +0026e410: 2069 6e63 6f6d 696e 6720 636f 6e6e 6563 incoming connec │ │ │ │ +0026e420: 7469 6f6e 206f 6e20 7468 6520 6c69 7374 tion on the list │ │ │ │ +0026e430: 656e 6572 2c20 7265 7475 726e 696e 6720 ener, returning │ │ │ │ +0026e440: 610a 6e65 7720 696e 7075 7420 6f75 7470 a.new input outp │ │ │ │ +0026e450: 7574 2066 696c 6520 7468 6174 2073 6572 ut file that ser │ │ │ │ +0026e460: 7665 7320 6173 2074 6865 2063 6f6e 6e65 ves as the conne │ │ │ │ +0026e470: 6374 696f 6e2e 2020 5468 6520 6675 6e63 ction. The func │ │ │ │ +0026e480: 7469 6f6e 202a 6e6f 7465 0a69 7352 6561 tion *note.isRea │ │ │ │ +0026e490: 6479 3a20 6973 5265 6164 795f 6c70 4669 dy: isReady_lpFi │ │ │ │ +0026e4a0: 6c65 5f72 702c 2063 616e 2062 6520 7573 le_rp, can be us │ │ │ │ +0026e4b0: 6564 2074 6f20 6465 7465 726d 696e 6520 ed to determine │ │ │ │ +0026e4c0: 7768 6574 6865 7220 616e 2069 6e63 6f6d whether an incom │ │ │ │ +0026e4d0: 696e 670a 636f 6e6e 6563 7469 6f6e 2068 ing.connection h │ │ │ │ +0026e4e0: 6173 2061 7272 6976 6564 2c20 7769 7468 as arrived, with │ │ │ │ +0026e4f0: 6f75 7420 626c 6f63 6b69 6e67 2e0a 0a53 out blocking...S │ │ │ │ +0026e500: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +0026e510: 0a0a 2020 2a20 2a6e 6f74 6520 6f70 656e .. * *note open │ │ │ │ +0026e520: 496e 4f75 743a 206f 7065 6e49 6e4f 7574 InOut: openInOut │ │ │ │ +0026e530: 2c20 2d2d 206f 7065 6e20 616e 2069 6e70 , -- open an inp │ │ │ │ +0026e540: 7574 206f 7574 7075 7420 6669 6c65 0a20 ut output file. │ │ │ │ +0026e550: 202a 202a 6e6f 7465 2069 7352 6561 6479 * *note isReady │ │ │ │ +0026e560: 3a20 6973 5265 6164 795f 6c70 4669 6c65 : isReady_lpFile │ │ │ │ +0026e570: 5f72 702c 202d 2d20 7768 6574 6865 7220 _rp, -- whether │ │ │ │ +0026e580: 6120 6669 6c65 2068 6173 2064 6174 6120 a file has data │ │ │ │ +0026e590: 6176 6169 6c61 626c 6520 666f 720a 2020 available for. │ │ │ │ +0026e5a0: 2020 7265 6164 696e 670a 0a57 6179 7320 reading..Ways │ │ │ │ +0026e5b0: 746f 2075 7365 2074 6869 7320 6d65 7468 to use this meth │ │ │ │ +0026e5c0: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d od:.============ │ │ │ │ +0026e5d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +0026e5e0: 2a20 2a6e 6f74 6520 6f70 656e 4c69 7374 * *note openList │ │ │ │ +0026e5f0: 656e 6572 2853 7472 696e 6729 3a20 6f70 ener(String): op │ │ │ │ +0026e600: 656e 4c69 7374 656e 6572 5f6c 7053 7472 enListener_lpStr │ │ │ │ +0026e610: 696e 675f 7270 2c20 2d2d 206f 7065 6e20 ing_rp, -- open │ │ │ │ +0026e620: 6120 706f 7274 2066 6f72 0a20 2020 206c a port for. l │ │ │ │ +0026e630: 6973 7465 6e69 6e67 0a2d 2d2d 2d2d 2d2d istening.------- │ │ │ │ 0026e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0026e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026e660: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -0026e670: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -0026e680: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -0026e690: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -0026e6a0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -0026e6b0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -0026e6c0: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -0026e6d0: 6175 6c61 7932 446f 632f 6f76 5f73 7973 aulay2Doc/ov_sys │ │ │ │ -0026e6e0: 7465 6d2e 6d32 3a34 3232 3a30 2e0a 1f0a tem.m2:422:0.... │ │ │ │ -0026e6f0: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -0026e700: 6f63 2e69 6e66 6f2c 204e 6f64 653a 206f oc.info, Node: o │ │ │ │ -0026e710: 7065 6e49 6e5f 6c70 5374 7269 6e67 5f72 penIn_lpString_r │ │ │ │ -0026e720: 702c 204e 6578 743a 206f 7065 6e49 6e4f p, Next: openInO │ │ │ │ -0026e730: 7574 2c20 5072 6576 3a20 6f70 656e 4c69 ut, Prev: openLi │ │ │ │ -0026e740: 7374 656e 6572 5f6c 7053 7472 696e 675f stener_lpString_ │ │ │ │ -0026e750: 7270 2c20 5570 3a20 7573 696e 6720 736f rp, Up: using so │ │ │ │ -0026e760: 636b 6574 730a 0a6f 7065 6e49 6e28 5374 ckets..openIn(St │ │ │ │ -0026e770: 7269 6e67 2920 2d2d 206f 7065 6e20 616e ring) -- open an │ │ │ │ -0026e780: 2069 6e70 7574 2066 696c 650a 2a2a 2a2a input file.**** │ │ │ │ -0026e790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026e7a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026e7b0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20 .. * Function: │ │ │ │ -0026e7c0: 2a6e 6f74 6520 6f70 656e 496e 3a20 6f70 *note openIn: op │ │ │ │ -0026e7d0: 656e 496e 5f6c 7053 7472 696e 675f 7270 enIn_lpString_rp │ │ │ │ -0026e7e0: 2c0a 2020 2a20 5573 6167 653a 200a 2020 ,. * Usage: . │ │ │ │ -0026e7f0: 2020 2020 2020 6f70 656e 496e 2066 6e0a openIn fn. │ │ │ │ -0026e800: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -0026e810: 2020 2a20 666e 2c20 6120 2a6e 6f74 6520 * fn, a *note │ │ │ │ -0026e820: 7374 7269 6e67 3a20 5374 7269 6e67 2c0a string: String,. │ │ │ │ -0026e830: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -0026e840: 2020 202a 2061 202a 6e6f 7465 2066 696c * a *note fil │ │ │ │ -0026e850: 653a 2046 696c 652c 2c20 616e 206f 7065 e: File,, an ope │ │ │ │ -0026e860: 6e20 696e 7075 7420 6669 6c65 2077 686f n input file who │ │ │ │ -0026e870: 7365 2066 696c 656e 616d 6520 6973 2066 se filename is f │ │ │ │ -0026e880: 6e2e 2046 696c 656e 616d 6573 0a20 2020 n. Filenames. │ │ │ │ -0026e890: 2020 2020 2073 7461 7274 696e 6720 7769 starting wi │ │ │ │ -0026e8a0: 7468 2021 206f 7220 7769 7468 2024 2061 th ! or with $ a │ │ │ │ -0026e8b0: 7265 2074 7265 6174 6564 2073 7065 6369 re treated speci │ │ │ │ -0026e8c0: 616c 6c79 2c20 7365 6520 2a6e 6f74 6520 ally, see *note │ │ │ │ -0026e8d0: 6f70 656e 496e 4f75 743a 0a20 2020 2020 openInOut:. │ │ │ │ -0026e8e0: 2020 206f 7065 6e49 6e4f 7574 2c2e 0a0a openInOut,... │ │ │ │ -0026e8f0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -0026e900: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d =======..+------ │ │ │ │ -0026e910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026e920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026e930: 2d2d 2b0a 7c69 3120 3a20 2274 6573 742d --+.|i1 : "test- │ │ │ │ -0026e940: 6669 6c65 2220 3c3c 2022 6869 2074 6865 file" << "hi the │ │ │ │ -0026e950: 7265 2220 3c3c 2063 6c6f 7365 3b7c 0a2b re" << close;|.+ │ │ │ │ -0026e960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026e970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026e980: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -0026e990: 6720 3d20 6f70 656e 496e 2022 7465 7374 g = openIn "test │ │ │ │ -0026e9a0: 2d66 696c 6522 2020 2020 2020 2020 2020 -file" │ │ │ │ -0026e9b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0026e9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026e9d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0026e9e0: 7c6f 3220 3d20 7465 7374 2d66 696c 6520 |o2 = test-file │ │ │ │ +0026e660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026e680: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0026e690: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0026e6a0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0026e6b0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0026e6c0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0026e6d0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +0026e6e0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0026e6f0: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ +0026e700: 5f73 7973 7465 6d2e 6d32 3a34 3232 3a30 _system.m2:422:0 │ │ │ │ +0026e710: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ +0026e720: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ +0026e730: 653a 206f 7065 6e49 6e5f 6c70 5374 7269 e: openIn_lpStri │ │ │ │ +0026e740: 6e67 5f72 702c 204e 6578 743a 206f 7065 ng_rp, Next: ope │ │ │ │ +0026e750: 6e49 6e4f 7574 2c20 5072 6576 3a20 6f70 nInOut, Prev: op │ │ │ │ +0026e760: 656e 4c69 7374 656e 6572 5f6c 7053 7472 enListener_lpStr │ │ │ │ +0026e770: 696e 675f 7270 2c20 5570 3a20 7573 696e ing_rp, Up: usin │ │ │ │ +0026e780: 6720 736f 636b 6574 730a 0a6f 7065 6e49 g sockets..openI │ │ │ │ +0026e790: 6e28 5374 7269 6e67 2920 2d2d 206f 7065 n(String) -- ope │ │ │ │ +0026e7a0: 6e20 616e 2069 6e70 7574 2066 696c 650a n an input file. │ │ │ │ +0026e7b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026e7c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026e7d0: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ +0026e7e0: 6f6e 3a20 2a6e 6f74 6520 6f70 656e 496e on: *note openIn │ │ │ │ +0026e7f0: 3a20 6f70 656e 496e 5f6c 7053 7472 696e : openIn_lpStrin │ │ │ │ +0026e800: 675f 7270 2c0a 2020 2a20 5573 6167 653a g_rp,. * Usage: │ │ │ │ +0026e810: 200a 2020 2020 2020 2020 6f70 656e 496e . openIn │ │ │ │ +0026e820: 2066 6e0a 2020 2a20 496e 7075 7473 3a0a fn. * Inputs:. │ │ │ │ +0026e830: 2020 2020 2020 2a20 666e 2c20 6120 2a6e * fn, a *n │ │ │ │ +0026e840: 6f74 6520 7374 7269 6e67 3a20 5374 7269 ote string: Stri │ │ │ │ +0026e850: 6e67 2c0a 2020 2a20 4f75 7470 7574 733a ng,. * Outputs: │ │ │ │ +0026e860: 0a20 2020 2020 202a 2061 202a 6e6f 7465 . * a *note │ │ │ │ +0026e870: 2066 696c 653a 2046 696c 652c 2c20 616e file: File,, an │ │ │ │ +0026e880: 206f 7065 6e20 696e 7075 7420 6669 6c65 open input file │ │ │ │ +0026e890: 2077 686f 7365 2066 696c 656e 616d 6520 whose filename │ │ │ │ +0026e8a0: 6973 2066 6e2e 2046 696c 656e 616d 6573 is fn. Filenames │ │ │ │ +0026e8b0: 0a20 2020 2020 2020 2073 7461 7274 696e . startin │ │ │ │ +0026e8c0: 6720 7769 7468 2021 206f 7220 7769 7468 g with ! or with │ │ │ │ +0026e8d0: 2024 2061 7265 2074 7265 6174 6564 2073 $ are treated s │ │ │ │ +0026e8e0: 7065 6369 616c 6c79 2c20 7365 6520 2a6e pecially, see *n │ │ │ │ +0026e8f0: 6f74 6520 6f70 656e 496e 4f75 743a 0a20 ote openInOut:. │ │ │ │ +0026e900: 2020 2020 2020 206f 7065 6e49 6e4f 7574 openInOut │ │ │ │ +0026e910: 2c2e 0a0a 4465 7363 7269 7074 696f 6e0a ,...Description. │ │ │ │ +0026e920: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d ===========..+-- │ │ │ │ +0026e930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026e940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026e950: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 2274 ------+.|i1 : "t │ │ │ │ +0026e960: 6573 742d 6669 6c65 2220 3c3c 2022 6869 est-file" << "hi │ │ │ │ +0026e970: 2074 6865 7265 2220 3c3c 2063 6c6f 7365 there" << close │ │ │ │ +0026e980: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ;|.+------------ │ │ │ │ +0026e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0026e9b0: 3220 3a20 6720 3d20 6f70 656e 496e 2022 2 : g = openIn " │ │ │ │ +0026e9c0: 7465 7374 2d66 696c 6522 2020 2020 2020 test-file" │ │ │ │ +0026e9d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0026e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026e9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ea00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0026ea10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ea20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ea30: 2020 2020 7c0a 7c6f 3220 3a20 4669 6c65 |.|o2 : File │ │ │ │ +0026ea00: 2020 7c0a 7c6f 3220 3d20 7465 7374 2d66 |.|o2 = test-f │ │ │ │ +0026ea10: 696c 6520 2020 2020 2020 2020 2020 2020 ile │ │ │ │ +0026ea20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 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│ │ │ +0026eb00: 2020 2020 7c0a 7c6f 3320 3d20 3820 2020 |.|o3 = 8 │ │ │ │ +0026eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026eb20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026eb30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0026eb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026eb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +0026eb60: 3a20 6174 456e 644f 6646 696c 6520 6720 : atEndOfFile g │ │ │ │ 0026eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026eb80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0026eb90: 3420 3d20 6661 6c73 6520 2020 2020 2020 4 = false │ │ │ │ +0026eb80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0026eb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ebb0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0026ebc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ebd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ebe0: 2d2d 2b0a 7c69 3520 3a20 7265 6164 2067 --+.|i5 : read g │ │ │ │ -0026ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ec00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0026ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ebb0: 7c0a 7c6f 3420 3d20 6661 6c73 6520 2020 |.|o4 = false │ │ │ │ +0026ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ebd0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0026ebe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ec00: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7265 ------+.|i5 : re │ │ │ │ +0026ec10: 6164 2067 2020 2020 2020 2020 2020 2020 ad g │ │ │ │ 0026ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ec30: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -0026ec40: 6869 2074 6865 7265 2020 2020 2020 2020 hi there │ │ │ │ -0026ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ec60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -0026ec70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0026ec90: 7c69 3620 3a20 6174 456e 644f 6646 696c |i6 : atEndOfFil │ │ │ │ -0026eca0: 6520 6720 2020 2020 2020 2020 2020 2020 e g │ │ │ │ -0026ecb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0026ecc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ece0: 2020 2020 7c0a 7c6f 3620 3d20 7472 7565 |.|o6 = true │ │ │ │ +0026ec30: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0026ec40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ec50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0026ec60: 3520 3d20 6869 2074 6865 7265 2020 2020 5 = hi there │ │ │ │ +0026ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ec80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0026ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ecb0: 2d2d 2b0a 7c69 3620 3a20 6174 456e 644f --+.|i6 : atEndO │ │ │ │ +0026ecc0: 6646 696c 6520 6720 2020 2020 2020 2020 fFile g │ │ │ │ +0026ecd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0026ece0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0026ecf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ed00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0026ed10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0026ed20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ed30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ -0026ed40: 3a20 636c 6f73 6520 6720 2020 2020 2020 : close g │ │ │ │ -0026ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ed60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0026ed00: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ +0026ed10: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ +0026ed20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ed30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0026ed40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ed50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0026ed60: 7c69 3720 3a20 636c 6f73 6520 6720 2020 |i7 : close g │ │ │ │ 0026ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ed90: 7c0a 7c6f 3720 3d20 7465 7374 2d66 696c |.|o7 = test-fil │ │ │ │ -0026eda0: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -0026edb0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0026edc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ede0: 2020 2020 2020 7c0a 7c6f 3720 3a20 4669 |.|o7 : Fi │ │ │ │ -0026edf0: 6c65 2020 2020 2020 2020 2020 2020 2020 le │ │ │ │ -0026ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0026ee10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0026ee20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ee30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0026ee40: 3820 3a20 7265 6d6f 7665 4669 6c65 2022 8 : removeFile " │ │ │ │ -0026ee50: 7465 7374 2d66 696c 6522 2020 2020 2020 test-file" │ │ │ │ -0026ee60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0026ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026ee90: 2d2d 2b0a 0a41 2066 696c 656e 616d 6520 --+..A filename │ │ │ │ -0026eea0: 7374 6172 7469 6e67 2077 6974 6820 7e2f starting with ~/ │ │ │ │ -0026eeb0: 2077 696c 6c20 6861 7665 2074 6865 2074 will have the t │ │ │ │ -0026eec0: 696c 6465 2072 6570 6c61 6365 6420 6279 ilde replaced by │ │ │ │ -0026eed0: 2074 6865 2075 7365 7227 7320 686f 6d65 the user's home │ │ │ │ -0026eee0: 0a64 6972 6563 746f 7279 2e0a 0a53 6565 .directory...See │ │ │ │ -0026eef0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -0026ef00: 2020 2a20 2a6e 6f74 6520 6f70 656e 4f75 * *note openOu │ │ │ │ -0026ef10: 743a 206f 7065 6e4f 7574 5f6c 7053 7472 t: openOut_lpStr │ │ │ │ -0026ef20: 696e 675f 7270 2c20 2d2d 206f 7065 6e20 ing_rp, -- open │ │ │ │ -0026ef30: 616e 206f 7574 7075 7420 6669 6c65 0a20 an output file. │ │ │ │ -0026ef40: 202a 202a 6e6f 7465 206f 7065 6e4f 7574 * *note openOut │ │ │ │ -0026ef50: 4170 7065 6e64 3a20 6f70 656e 4f75 7441 Append: openOutA │ │ │ │ -0026ef60: 7070 656e 645f 6c70 5374 7269 6e67 5f72 ppend_lpString_r │ │ │ │ -0026ef70: 702c 202d 2d20 6f70 656e 2061 6e20 6f75 p, -- open an ou │ │ │ │ -0026ef80: 7470 7574 2066 696c 6520 666f 720a 2020 tput file for. │ │ │ │ -0026ef90: 2020 6170 7065 6e64 696e 670a 2020 2a20 appending. * │ │ │ │ -0026efa0: 2a6e 6f74 6520 6f70 656e 496e 4f75 743a *note openInOut: │ │ │ │ -0026efb0: 206f 7065 6e49 6e4f 7574 2c20 2d2d 206f openInOut, -- o │ │ │ │ -0026efc0: 7065 6e20 616e 2069 6e70 7574 206f 7574 pen an input out │ │ │ │ -0026efd0: 7075 7420 6669 6c65 0a20 202a 202a 6e6f put file. * *no │ │ │ │ -0026efe0: 7465 2066 696c 654c 656e 6774 683a 2066 te fileLength: f │ │ │ │ -0026eff0: 696c 654c 656e 6774 682c 202d 2d20 7468 ileLength, -- th │ │ │ │ -0026f000: 6520 6c65 6e67 7468 206f 6620 6120 6669 e length of a fi │ │ │ │ -0026f010: 6c65 0a20 202a 202a 6e6f 7465 2072 6561 le. * *note rea │ │ │ │ -0026f020: 643a 2072 6561 642c 202d 2d20 7265 6164 d: read, -- read │ │ │ │ -0026f030: 2066 726f 6d20 6120 6669 6c65 0a20 202a from a file. * │ │ │ │ -0026f040: 202a 6e6f 7465 2063 6c6f 7365 3a20 636c *note close: cl │ │ │ │ -0026f050: 6f73 652c 202d 2d20 636c 6f73 6520 6120 ose, -- close a │ │ │ │ -0026f060: 6669 6c65 0a20 202a 202a 6e6f 7465 2061 file. * *note a │ │ │ │ -0026f070: 7445 6e64 4f66 4669 6c65 3a20 6174 456e tEndOfFile: atEn │ │ │ │ -0026f080: 644f 6646 696c 655f 6c70 4669 6c65 5f72 dOfFile_lpFile_r │ │ │ │ -0026f090: 702c 202d 2d20 7465 7374 2066 6f72 2065 p, -- test for e │ │ │ │ -0026f0a0: 6e64 206f 6620 6669 6c65 0a20 202a 202a nd of file. * * │ │ │ │ -0026f0b0: 6e6f 7465 2046 696c 6520 3c3c 2054 6869 note File << Thi │ │ │ │ -0026f0c0: 6e67 3a20 7072 696e 7469 6e67 2074 6f20 ng: printing to │ │ │ │ -0026f0d0: 6120 6669 6c65 2c20 2d2d 2070 7269 6e74 a file, -- print │ │ │ │ -0026f0e0: 2074 6f20 6120 6669 6c65 0a0a 5761 7973 to a file..Ways │ │ │ │ -0026f0f0: 2074 6f20 7573 6520 7468 6973 206d 6574 to use this met │ │ │ │ -0026f100: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d hod:.=========== │ │ │ │ -0026f110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0026f120: 202a 202a 6e6f 7465 206f 7065 6e49 6e28 * *note openIn( │ │ │ │ -0026f130: 5374 7269 6e67 293a 206f 7065 6e49 6e5f String): openIn_ │ │ │ │ -0026f140: 6c70 5374 7269 6e67 5f72 702c 202d 2d20 lpString_rp, -- │ │ │ │ -0026f150: 6f70 656e 2061 6e20 696e 7075 7420 6669 open an input fi │ │ │ │ -0026f160: 6c65 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d le.------------- │ │ │ │ -0026f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ed80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0026ed90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026eda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026edb0: 2020 2020 7c0a 7c6f 3720 3d20 7465 7374 |.|o7 = test │ │ │ │ +0026edc0: 2d66 696c 6520 2020 2020 2020 2020 2020 -file │ │ │ │ +0026edd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0026ede0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0026edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ee00: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ +0026ee10: 3a20 4669 6c65 2020 2020 2020 2020 2020 : File │ │ │ │ +0026ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0026ee30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0026ee40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026ee60: 2b0a 7c69 3820 3a20 7265 6d6f 7665 4669 +.|i8 : removeFi │ │ │ │ +0026ee70: 6c65 2022 7465 7374 2d66 696c 6522 2020 le "test-file" │ │ │ │ +0026ee80: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0026ee90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026eeb0: 2d2d 2d2d 2d2d 2b0a 0a41 2066 696c 656e ------+..A filen │ │ │ │ +0026eec0: 616d 6520 7374 6172 7469 6e67 2077 6974 ame starting wit │ │ │ │ +0026eed0: 6820 7e2f 2077 696c 6c20 6861 7665 2074 h ~/ will have t │ │ │ │ +0026eee0: 6865 2074 696c 6465 2072 6570 6c61 6365 he tilde replace │ │ │ │ +0026eef0: 6420 6279 2074 6865 2075 7365 7227 7320 d by the user's │ │ │ │ +0026ef00: 686f 6d65 0a64 6972 6563 746f 7279 2e0a home.directory.. │ │ │ │ +0026ef10: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +0026ef20: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6f70 ==.. * *note op │ │ │ │ +0026ef30: 656e 4f75 743a 206f 7065 6e4f 7574 5f6c enOut: openOut_l │ │ │ │ +0026ef40: 7053 7472 696e 675f 7270 2c20 2d2d 206f pString_rp, -- o │ │ │ │ +0026ef50: 7065 6e20 616e 206f 7574 7075 7420 6669 pen an output fi │ │ │ │ +0026ef60: 6c65 0a20 202a 202a 6e6f 7465 206f 7065 le. * *note ope │ │ │ │ +0026ef70: 6e4f 7574 4170 7065 6e64 3a20 6f70 656e nOutAppend: open │ │ │ │ +0026ef80: 4f75 7441 7070 656e 645f 6c70 5374 7269 OutAppend_lpStri │ │ │ │ +0026ef90: 6e67 5f72 702c 202d 2d20 6f70 656e 2061 ng_rp, -- open a │ │ │ │ +0026efa0: 6e20 6f75 7470 7574 2066 696c 6520 666f n output file fo │ │ │ │ +0026efb0: 720a 2020 2020 6170 7065 6e64 696e 670a r. appending. │ │ │ │ +0026efc0: 2020 2a20 2a6e 6f74 6520 6f70 656e 496e * *note openIn │ │ │ │ +0026efd0: 4f75 743a 206f 7065 6e49 6e4f 7574 2c20 Out: openInOut, │ │ │ │ +0026efe0: 2d2d 206f 7065 6e20 616e 2069 6e70 7574 -- open an input │ │ │ │ +0026eff0: 206f 7574 7075 7420 6669 6c65 0a20 202a output file. * │ │ │ │ +0026f000: 202a 6e6f 7465 2066 696c 654c 656e 6774 *note fileLengt │ │ │ │ +0026f010: 683a 2066 696c 654c 656e 6774 682c 202d h: fileLength, - │ │ │ │ +0026f020: 2d20 7468 6520 6c65 6e67 7468 206f 6620 - the length of │ │ │ │ +0026f030: 6120 6669 6c65 0a20 202a 202a 6e6f 7465 a file. * *note │ │ │ │ +0026f040: 2072 6561 643a 2072 6561 642c 202d 2d20 read: read, -- │ │ │ │ +0026f050: 7265 6164 2066 726f 6d20 6120 6669 6c65 read from a file │ │ │ │ +0026f060: 0a20 202a 202a 6e6f 7465 2063 6c6f 7365 . * *note close │ │ │ │ +0026f070: 3a20 636c 6f73 652c 202d 2d20 636c 6f73 : close, -- clos │ │ │ │ +0026f080: 6520 6120 6669 6c65 0a20 202a 202a 6e6f e a file. * *no │ │ │ │ +0026f090: 7465 2061 7445 6e64 4f66 4669 6c65 3a20 te atEndOfFile: │ │ │ │ +0026f0a0: 6174 456e 644f 6646 696c 655f 6c70 4669 atEndOfFile_lpFi │ │ │ │ +0026f0b0: 6c65 5f72 702c 202d 2d20 7465 7374 2066 le_rp, -- test f │ │ │ │ +0026f0c0: 6f72 2065 6e64 206f 6620 6669 6c65 0a20 or end of file. │ │ │ │ +0026f0d0: 202a 202a 6e6f 7465 2046 696c 6520 3c3c * *note File << │ │ │ │ +0026f0e0: 2054 6869 6e67 3a20 7072 696e 7469 6e67 Thing: printing │ │ │ │ +0026f0f0: 2074 6f20 6120 6669 6c65 2c20 2d2d 2070 to a file, -- p │ │ │ │ +0026f100: 7269 6e74 2074 6f20 6120 6669 6c65 0a0a rint to a file.. │ │ │ │ +0026f110: 5761 7973 2074 6f20 7573 6520 7468 6973 Ways to use this │ │ │ │ +0026f120: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d method:.======= │ │ │ │ +0026f130: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026f140: 3d0a 0a20 202a 202a 6e6f 7465 206f 7065 =.. * *note ope │ │ │ │ +0026f150: 6e49 6e28 5374 7269 6e67 293a 206f 7065 nIn(String): ope │ │ │ │ +0026f160: 6e49 6e5f 6c70 5374 7269 6e67 5f72 702c nIn_lpString_rp, │ │ │ │ +0026f170: 202d 2d20 6f70 656e 2061 6e20 696e 7075 -- open an inpu │ │ │ │ +0026f180: 7420 6669 6c65 0a2d 2d2d 2d2d 2d2d 2d2d t file.--------- │ │ │ │ 0026f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0026f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026f1b0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0026f1c0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0026f1d0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0026f1e0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0026f1f0: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -0026f200: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0026f210: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ -0026f220: 6c61 7932 446f 632f 6f76 5f73 7973 7465 lay2Doc/ov_syste │ │ │ │ -0026f230: 6d2e 6d32 3a34 3434 3a30 2e0a 1f0a 4669 m.m2:444:0....Fi │ │ │ │ -0026f240: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -0026f250: 2e69 6e66 6f2c 204e 6f64 653a 206f 7065 .info, Node: ope │ │ │ │ -0026f260: 6e49 6e4f 7574 2c20 4e65 7874 3a20 6f70 nInOut, Next: op │ │ │ │ -0026f270: 656e 4f75 745f 6c70 5374 7269 6e67 5f72 enOut_lpString_r │ │ │ │ -0026f280: 702c 2050 7265 763a 206f 7065 6e49 6e5f p, Prev: openIn_ │ │ │ │ -0026f290: 6c70 5374 7269 6e67 5f72 702c 2055 703a lpString_rp, Up: │ │ │ │ -0026f2a0: 2075 7369 6e67 2073 6f63 6b65 7473 0a0a using sockets.. │ │ │ │ -0026f2b0: 6f70 656e 496e 4f75 7420 2d2d 206f 7065 openInOut -- ope │ │ │ │ -0026f2c0: 6e20 616e 2069 6e70 7574 206f 7574 7075 n an input outpu │ │ │ │ -0026f2d0: 7420 6669 6c65 0a2a 2a2a 2a2a 2a2a 2a2a t file.********* │ │ │ │ -0026f2e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026f2f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -0026f300: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -0026f310: 2020 206f 7065 6e49 6e4f 7574 2066 0a20 openInOut f. │ │ │ │ -0026f320: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -0026f330: 202a 2066 2c20 6120 2a6e 6f74 6520 7374 * f, a *note st │ │ │ │ -0026f340: 7269 6e67 3a20 5374 7269 6e67 2c20 6f72 ring: String, or │ │ │ │ -0026f350: 2061 202a 6e6f 7465 2066 696c 653a 2046 a *note file: F │ │ │ │ -0026f360: 696c 652c 0a20 202a 204f 7574 7075 7473 ile,. * Outputs │ │ │ │ -0026f370: 3a0a 2020 2020 2020 2a20 616e 206f 7065 :. * an ope │ │ │ │ -0026f380: 6e20 696e 7075 7420 6f75 7470 7574 2066 n input output f │ │ │ │ -0026f390: 696c 650a 0a44 6573 6372 6970 7469 6f6e ile..Description │ │ │ │ -0026f3a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -0026f3b0: 6572 6520 6172 6520 7661 7269 6f75 7320 ere are various │ │ │ │ -0026f3c0: 6f70 7469 6f6e 7320 666f 7220 7468 6520 options for the │ │ │ │ -0026f3d0: 6172 6775 6d65 6e74 2066 2e0a 0a20 202a argument f... * │ │ │ │ -0026f3e0: 2061 2073 7472 696e 6720 6e6f 7420 7374 a string not st │ │ │ │ -0026f3f0: 6172 7469 6e67 2077 6974 6820 2120 6f72 arting with ! or │ │ │ │ -0026f400: 2024 3a20 7468 6520 7374 7269 6e67 2069 $: the string i │ │ │ │ -0026f410: 7320 7461 6b65 6e20 6173 2074 6865 206e s taken as the n │ │ │ │ -0026f420: 616d 6520 6f66 2061 6e0a 2020 2020 696e ame of an. in │ │ │ │ -0026f430: 7075 7420 6f75 7470 7574 2066 696c 6520 put output file │ │ │ │ -0026f440: 746f 206f 7065 6e2e 2020 466f 7220 6578 to open. For ex │ │ │ │ -0026f450: 616d 706c 652c 2069 6e20 556e 6978 2c20 ample, in Unix, │ │ │ │ -0026f460: 6974 206d 6967 6874 2062 6520 6120 6e61 it might be a na │ │ │ │ -0026f470: 6d65 6420 7069 7065 2e0a 2020 2020 4120 med pipe.. A │ │ │ │ -0026f480: 6669 6c65 6e61 6d65 2073 7461 7274 696e filename startin │ │ │ │ -0026f490: 6720 7769 7468 207e 2f20 7769 6c6c 2068 g with ~/ will h │ │ │ │ -0026f4a0: 6176 6520 7468 6520 7469 6c64 6520 7265 ave the tilde re │ │ │ │ -0026f4b0: 706c 6163 6564 2062 7920 7468 6520 7573 placed by the us │ │ │ │ -0026f4c0: 6572 2773 2068 6f6d 650a 2020 2020 6469 er's home. di │ │ │ │ -0026f4d0: 7265 6374 6f72 792e 0a20 202a 2061 2073 rectory.. * a s │ │ │ │ -0026f4e0: 7472 696e 6720 6f66 2074 6865 2066 6f72 tring of the for │ │ │ │ -0026f4f0: 6d20 2221 636d 6422 3a20 7468 6520 636f m "!cmd": the co │ │ │ │ -0026f500: 6d6d 616e 6420 636d 6420 7769 6c6c 2062 mmand cmd will b │ │ │ │ -0026f510: 6520 7374 6172 7465 642c 2061 6e64 2074 e started, and t │ │ │ │ -0026f520: 776f 2070 6970 6573 0a20 2020 2077 696c wo pipes. wil │ │ │ │ -0026f530: 6c20 6265 206f 7065 6e65 642c 2063 6f6e l be opened, con │ │ │ │ -0026f540: 6e65 6374 6564 2074 6f20 6974 7320 7374 nected to its st │ │ │ │ -0026f550: 616e 6461 7264 2069 6e70 7574 2061 6e64 andard input and │ │ │ │ -0026f560: 2073 7461 6e64 6172 6420 6f75 7470 7574 standard output │ │ │ │ -0026f570: 2066 696c 650a 2020 2020 6465 7363 7269 file. descri │ │ │ │ -0026f580: 7074 6f72 732e 2057 6172 6e69 6e67 3a20 ptors. Warning: │ │ │ │ -0026f590: 7069 7065 7320 686f 6c64 206f 6e6c 7920 pipes hold only │ │ │ │ -0026f5a0: 3430 3936 2062 6974 6573 2c20 736f 2069 4096 bites, so i │ │ │ │ -0026f5b0: 6620 796f 7520 7772 6974 6520 6d6f 7265 f you write more │ │ │ │ -0026f5c0: 2074 6861 6e0a 2020 2020 7468 6174 2074 than. that t │ │ │ │ -0026f5d0: 6f20 7468 6520 7265 7375 6c74 696e 6720 o the resulting │ │ │ │ -0026f5e0: 696e 7075 7420 6f75 7470 7574 2066 696c input output fil │ │ │ │ -0026f5f0: 6520 2861 7320 696e 7075 7420 666f 7220 e (as input for │ │ │ │ -0026f600: 7468 6520 636f 6d6d 616e 6429 2077 6974 the command) wit │ │ │ │ -0026f610: 686f 7574 0a20 2020 2072 6561 6469 6e67 hout. reading │ │ │ │ -0026f620: 2061 6e79 2064 6174 612c 2079 6f75 206d any data, you m │ │ │ │ -0026f630: 6179 2062 6c6f 636b 2077 6869 6c65 2074 ay block while t │ │ │ │ -0026f640: 6865 2063 6f6d 6d61 6e64 2069 7320 626c he command is bl │ │ │ │ -0026f650: 6f63 6b65 6420 7761 6974 696e 6720 746f ocked waiting to │ │ │ │ -0026f660: 0a20 2020 2077 7269 7465 206d 6f72 6520 . write more │ │ │ │ -0026f670: 6f75 7470 7574 3b20 696e 2074 6869 7320 output; in this │ │ │ │ -0026f680: 6361 7365 2c20 4d61 6361 756c 6179 3220 case, Macaulay2 │ │ │ │ -0026f690: 7769 6c6c 2061 7070 6561 7220 746f 2068 will appear to h │ │ │ │ -0026f6a0: 616e 672e 0a20 202a 2061 2073 7472 696e ang.. * a strin │ │ │ │ -0026f6b0: 6720 6f66 2074 6865 2066 6f72 6d20 2224 g of the form "$ │ │ │ │ -0026f6c0: 686f 7374 6e61 6d65 3a73 6572 7669 6365 hostname:service │ │ │ │ -0026f6d0: 223a 2061 2063 6f6e 6e65 6374 696f 6e20 ": a connection │ │ │ │ -0026f6e0: 7769 6c6c 2062 6520 6d61 6465 2074 6f20 will be made to │ │ │ │ -0026f6f0: 7468 650a 2020 2020 7370 6563 6966 6965 the. specifie │ │ │ │ -0026f700: 6420 7365 7276 6963 6520 6174 2074 6865 d service at the │ │ │ │ -0026f710: 2073 7065 6369 6669 6564 2068 6f73 742e specified host. │ │ │ │ -0026f720: 2020 4966 2074 6865 2073 6572 7669 6365 If the service │ │ │ │ -0026f730: 2070 6f72 7420 6973 206f 6d69 7474 6564 port is omitted │ │ │ │ -0026f740: 2c0a 2020 2020 616c 6f6e 6720 7769 7468 ,. along with │ │ │ │ -0026f750: 2074 6865 2063 6f6c 6f6e 2c20 7468 656e the colon, then │ │ │ │ -0026f760: 2070 6f72 7420 3235 3030 2069 7320 7573 port 2500 is us │ │ │ │ -0026f770: 6564 2e20 2049 6620 7468 6520 686f 7374 ed. If the host │ │ │ │ -0026f780: 6e61 6d65 2069 7320 6f6d 6974 7465 642c name is omitted, │ │ │ │ -0026f790: 0a20 2020 2061 6e20 696e 636f 6d69 6e67 . an incoming │ │ │ │ -0026f7a0: 2063 6f6e 6e65 6374 696f 6e20 7769 6c6c connection will │ │ │ │ -0026f7b0: 2062 6520 7761 6974 6564 2066 6f72 2e0a be waited for.. │ │ │ │ -0026f7c0: 2020 2a20 6120 6c69 7374 656e 6572 2063 * a listener c │ │ │ │ -0026f7d0: 7265 6174 6564 2070 7265 7669 6f75 736c reated previousl │ │ │ │ -0026f7e0: 7920 6279 202a 6e6f 7465 206f 7065 6e4c y by *note openL │ │ │ │ -0026f7f0: 6973 7465 6e65 723a 0a20 2020 206f 7065 istener:. ope │ │ │ │ -0026f800: 6e4c 6973 7465 6e65 725f 6c70 5374 7269 nListener_lpStri │ │ │ │ -0026f810: 6e67 5f72 702c 3a20 6120 6e65 7720 636f ng_rp,: a new co │ │ │ │ -0026f820: 6e6e 6563 7469 6f6e 2077 696c 6c20 6265 nnection will be │ │ │ │ -0026f830: 2063 7265 6174 6564 2e20 2054 6f20 6176 created. To av │ │ │ │ -0026f840: 6f69 640a 2020 2020 626c 6f63 6b69 6e67 oid. blocking │ │ │ │ -0026f850: 2074 6865 204d 6163 6175 6c61 7932 2070 the Macaulay2 p │ │ │ │ -0026f860: 726f 6365 7373 2077 6869 6c65 2077 6169 rocess while wai │ │ │ │ -0026f870: 7469 6e67 2066 6f72 2074 6865 2069 6e63 ting for the inc │ │ │ │ -0026f880: 6f6d 696e 6720 636f 6e6e 6563 7469 6f6e oming connection │ │ │ │ -0026f890: 2c0a 2020 2020 7573 6520 2a6e 6f74 6520 ,. use *note │ │ │ │ -0026f8a0: 6973 5265 6164 793a 2069 7352 6561 6479 isReady: isReady │ │ │ │ -0026f8b0: 5f6c 7046 696c 655f 7270 2c2e 0a0a 496e _lpFile_rp,...In │ │ │ │ -0026f8c0: 206f 7264 6572 2074 6f20 6f70 656e 2061 order to open a │ │ │ │ -0026f8d0: 2073 6f63 6b65 7420 7375 6363 6573 7366 socket successf │ │ │ │ -0026f8e0: 756c 6c79 2c20 7468 6572 6520 6d75 7374 ully, there must │ │ │ │ -0026f8f0: 2062 6520 6120 7072 6f63 6573 7320 6163 be a process ac │ │ │ │ -0026f900: 6365 7074 696e 670a 636f 6e6e 6563 7469 cepting.connecti │ │ │ │ -0026f910: 6f6e 7320 666f 7220 7468 6520 6465 7369 ons for the desi │ │ │ │ -0026f920: 7265 6420 7365 7276 6963 6520 6f6e 2074 red service on t │ │ │ │ -0026f930: 6865 2073 7065 6369 6669 6564 2068 6f73 he specified hos │ │ │ │ -0026f940: 742e 0a0a 5468 6520 7661 7269 6f75 7320 t...The various │ │ │ │ -0026f950: 666f 726d 7320 6c69 7374 6564 2061 626f forms listed abo │ │ │ │ -0026f960: 7665 2063 616e 2062 6520 7573 6564 2061 ve can be used a │ │ │ │ -0026f970: 6c73 6f20 7769 7468 2061 6c6c 206f 7468 lso with all oth │ │ │ │ -0026f980: 6572 2069 6e70 7574 206f 7574 7075 740a er input output. │ │ │ │ -0026f990: 6f70 6572 6174 696f 6e73 2074 6861 7420 operations that │ │ │ │ -0026f9a0: 6f70 656e 2066 696c 6573 2c20 7375 6368 open files, such │ │ │ │ -0026f9b0: 2061 7320 2a6e 6f74 6520 6f70 656e 496e as *note openIn │ │ │ │ -0026f9c0: 3a20 6f70 656e 496e 5f6c 7053 7472 696e : openIn_lpStrin │ │ │ │ -0026f9d0: 675f 7270 2c2c 202a 6e6f 7465 0a6f 7065 g_rp,, *note.ope │ │ │ │ -0026f9e0: 6e4f 7574 3a20 6f70 656e 4f75 745f 6c70 nOut: openOut_lp │ │ │ │ -0026f9f0: 5374 7269 6e67 5f72 702c 2c20 2a6e 6f74 String_rp,, *not │ │ │ │ -0026fa00: 6520 6765 743a 2067 6574 2c2c 2061 6e64 e get: get,, and │ │ │ │ -0026fa10: 202a 6e6f 7465 203c 3c3a 203c 3c2c 2c20 *note <<: <<,, │ │ │ │ -0026fa20: 7769 7468 2064 6174 610a 7472 616e 7366 with data.transf │ │ │ │ -0026fa30: 6572 2070 6f73 7369 626c 6520 6f6e 6c79 er possible only │ │ │ │ -0026fa40: 2069 6e20 7468 6520 6469 7265 6374 696f in the directio │ │ │ │ -0026fa50: 6e20 7370 6563 6966 6965 642e 2020 4120 n specified. A │ │ │ │ -0026fa60: 706f 7373 6962 6c79 2063 6f6e 6675 7369 possibly confusi │ │ │ │ -0026fa70: 6e67 0a61 7379 6d6d 6574 7279 2069 7320 ng.asymmetry is │ │ │ │ -0026fa80: 7468 6174 2077 6974 6820 6f70 656e 496e that with openIn │ │ │ │ -0026fa90: 2022 2166 6f6f 2220 6f72 2077 6974 6820 "!foo" or with │ │ │ │ -0026faa0: 6765 7420 2221 666f 6f22 2074 6865 2073 get "!foo" the s │ │ │ │ -0026fab0: 7461 6e64 6172 6420 696e 7075 7420 6f66 tandard input of │ │ │ │ -0026fac0: 0a74 6865 2063 6f6d 6d61 6e64 2066 6f6f .the command foo │ │ │ │ -0026fad0: 2069 7320 636c 6f73 6564 2c20 6275 7420 is closed, but │ │ │ │ -0026fae0: 7769 7468 206f 7065 6e4f 7574 2022 2166 with openOut "!f │ │ │ │ -0026faf0: 6f6f 2220 7468 6520 7374 616e 6461 7264 oo" the standard │ │ │ │ -0026fb00: 206f 7574 7075 7420 6f66 2074 6865 0a63 output of the.c │ │ │ │ -0026fb10: 6f6d 6d61 6e64 2066 6f6f 2069 7320 636f ommand foo is co │ │ │ │ -0026fb20: 6e6e 6563 7465 6420 746f 2074 6865 2073 nnected to the s │ │ │ │ -0026fb30: 7461 6e64 6172 6420 6f75 7470 7574 206f tandard output o │ │ │ │ -0026fb40: 6620 7468 6520 7061 7265 6e74 204d 6163 f the parent Mac │ │ │ │ -0026fb50: 6175 6c61 7932 0a70 726f 6365 7373 2e0a aulay2.process.. │ │ │ │ -0026fb60: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -0026fb70: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6f70 ==.. * *note op │ │ │ │ -0026fb80: 656e 496e 3a20 6f70 656e 496e 5f6c 7053 enIn: openIn_lpS │ │ │ │ -0026fb90: 7472 696e 675f 7270 2c20 2d2d 206f 7065 tring_rp, -- ope │ │ │ │ -0026fba0: 6e20 616e 2069 6e70 7574 2066 696c 650a n an input file. │ │ │ │ -0026fbb0: 2020 2a20 2a6e 6f74 6520 6f70 656e 4f75 * *note openOu │ │ │ │ -0026fbc0: 743a 206f 7065 6e4f 7574 5f6c 7053 7472 t: openOut_lpStr │ │ │ │ -0026fbd0: 696e 675f 7270 2c20 2d2d 206f 7065 6e20 ing_rp, -- open │ │ │ │ -0026fbe0: 616e 206f 7574 7075 7420 6669 6c65 0a20 an output file. │ │ │ │ -0026fbf0: 202a 202a 6e6f 7465 206f 7065 6e4c 6973 * *note openLis │ │ │ │ -0026fc00: 7465 6e65 723a 206f 7065 6e4c 6973 7465 tener: openListe │ │ │ │ -0026fc10: 6e65 725f 6c70 5374 7269 6e67 5f72 702c ner_lpString_rp, │ │ │ │ -0026fc20: 202d 2d20 6f70 656e 2061 2070 6f72 7420 -- open a port │ │ │ │ -0026fc30: 666f 7220 6c69 7374 656e 696e 670a 0a57 for listening..W │ │ │ │ -0026fc40: 6179 7320 746f 2075 7365 206f 7065 6e49 ays to use openI │ │ │ │ -0026fc50: 6e4f 7574 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d nOut:.========== │ │ │ │ -0026fc60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0026fc70: 2a20 226f 7065 6e49 6e4f 7574 2846 696c * "openInOut(Fil │ │ │ │ -0026fc80: 6529 220a 2020 2a20 226f 7065 6e49 6e4f e)". * "openInO │ │ │ │ -0026fc90: 7574 2853 7472 696e 6729 220a 0a46 6f72 ut(String)"..For │ │ │ │ -0026fca0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -0026fcb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0026fcc0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -0026fcd0: 6e6f 7465 206f 7065 6e49 6e4f 7574 3a20 note openInOut: │ │ │ │ -0026fce0: 6f70 656e 496e 4f75 742c 2069 7320 6120 openInOut, is a │ │ │ │ -0026fcf0: 2a6e 6f74 6520 636f 6d70 696c 6564 2066 *note compiled f │ │ │ │ -0026fd00: 756e 6374 696f 6e3a 0a43 6f6d 7069 6c65 unction:.Compile │ │ │ │ -0026fd10: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ -0026fd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026fd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026fd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026f1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026f1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026f1d0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0026f1e0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0026f1f0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0026f200: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0026f210: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0026f220: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +0026f230: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ +0026f240: 6163 6175 6c61 7932 446f 632f 6f76 5f73 acaulay2Doc/ov_s │ │ │ │ +0026f250: 7973 7465 6d2e 6d32 3a34 3434 3a30 2e0a ystem.m2:444:0.. │ │ │ │ +0026f260: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +0026f270: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +0026f280: 206f 7065 6e49 6e4f 7574 2c20 4e65 7874 openInOut, Next │ │ │ │ +0026f290: 3a20 6f70 656e 4f75 745f 6c70 5374 7269 : openOut_lpStri │ │ │ │ +0026f2a0: 6e67 5f72 702c 2050 7265 763a 206f 7065 ng_rp, Prev: ope │ │ │ │ +0026f2b0: 6e49 6e5f 6c70 5374 7269 6e67 5f72 702c nIn_lpString_rp, │ │ │ │ +0026f2c0: 2055 703a 2075 7369 6e67 2073 6f63 6b65 Up: using socke │ │ │ │ +0026f2d0: 7473 0a0a 6f70 656e 496e 4f75 7420 2d2d ts..openInOut -- │ │ │ │ +0026f2e0: 206f 7065 6e20 616e 2069 6e70 7574 206f open an input o │ │ │ │ +0026f2f0: 7574 7075 7420 6669 6c65 0a2a 2a2a 2a2a utput file.***** │ │ │ │ +0026f300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026f310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026f320: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0026f330: 2020 2020 2020 206f 7065 6e49 6e4f 7574 openInOut │ │ │ │ +0026f340: 2066 0a20 202a 2049 6e70 7574 733a 0a20 f. * Inputs:. │ │ │ │ +0026f350: 2020 2020 202a 2066 2c20 6120 2a6e 6f74 * f, a *not │ │ │ │ +0026f360: 6520 7374 7269 6e67 3a20 5374 7269 6e67 e string: String │ │ │ │ +0026f370: 2c20 6f72 2061 202a 6e6f 7465 2066 696c , or a *note fil │ │ │ │ +0026f380: 653a 2046 696c 652c 0a20 202a 204f 7574 e: File,. * Out │ │ │ │ +0026f390: 7075 7473 3a0a 2020 2020 2020 2a20 616e puts:. * an │ │ │ │ +0026f3a0: 206f 7065 6e20 696e 7075 7420 6f75 7470 open input outp │ │ │ │ +0026f3b0: 7574 2066 696c 650a 0a44 6573 6372 6970 ut file..Descrip │ │ │ │ +0026f3c0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +0026f3d0: 0a0a 5468 6572 6520 6172 6520 7661 7269 ..There are vari │ │ │ │ +0026f3e0: 6f75 7320 6f70 7469 6f6e 7320 666f 7220 ous options for │ │ │ │ +0026f3f0: 7468 6520 6172 6775 6d65 6e74 2066 2e0a the argument f.. │ │ │ │ +0026f400: 0a20 202a 2061 2073 7472 696e 6720 6e6f . * a string no │ │ │ │ +0026f410: 7420 7374 6172 7469 6e67 2077 6974 6820 t starting with │ │ │ │ +0026f420: 2120 6f72 2024 3a20 7468 6520 7374 7269 ! or $: the stri │ │ │ │ +0026f430: 6e67 2069 7320 7461 6b65 6e20 6173 2074 ng is taken as t │ │ │ │ +0026f440: 6865 206e 616d 6520 6f66 2061 6e0a 2020 he name of an. │ │ │ │ +0026f450: 2020 696e 7075 7420 6f75 7470 7574 2066 input output f │ │ │ │ +0026f460: 696c 6520 746f 206f 7065 6e2e 2020 466f ile to open. Fo │ │ │ │ +0026f470: 7220 6578 616d 706c 652c 2069 6e20 556e r example, in Un │ │ │ │ +0026f480: 6978 2c20 6974 206d 6967 6874 2062 6520 ix, it might be │ │ │ │ +0026f490: 6120 6e61 6d65 6420 7069 7065 2e0a 2020 a named pipe.. │ │ │ │ +0026f4a0: 2020 4120 6669 6c65 6e61 6d65 2073 7461 A filename sta │ │ │ │ +0026f4b0: 7274 696e 6720 7769 7468 207e 2f20 7769 rting with ~/ wi │ │ │ │ +0026f4c0: 6c6c 2068 6176 6520 7468 6520 7469 6c64 ll have the tild │ │ │ │ +0026f4d0: 6520 7265 706c 6163 6564 2062 7920 7468 e replaced by th │ │ │ │ +0026f4e0: 6520 7573 6572 2773 2068 6f6d 650a 2020 e user's home. │ │ │ │ +0026f4f0: 2020 6469 7265 6374 6f72 792e 0a20 202a directory.. * │ │ │ │ +0026f500: 2061 2073 7472 696e 6720 6f66 2074 6865 a string of the │ │ │ │ +0026f510: 2066 6f72 6d20 2221 636d 6422 3a20 7468 form "!cmd": th │ │ │ │ +0026f520: 6520 636f 6d6d 616e 6420 636d 6420 7769 e command cmd wi │ │ │ │ +0026f530: 6c6c 2062 6520 7374 6172 7465 642c 2061 ll be started, a │ │ │ │ +0026f540: 6e64 2074 776f 2070 6970 6573 0a20 2020 nd two pipes. │ │ │ │ +0026f550: 2077 696c 6c20 6265 206f 7065 6e65 642c will be opened, │ │ │ │ +0026f560: 2063 6f6e 6e65 6374 6564 2074 6f20 6974 connected to it │ │ │ │ +0026f570: 7320 7374 616e 6461 7264 2069 6e70 7574 s standard input │ │ │ │ +0026f580: 2061 6e64 2073 7461 6e64 6172 6420 6f75 and standard ou │ │ │ │ +0026f590: 7470 7574 2066 696c 650a 2020 2020 6465 tput file. de │ │ │ │ +0026f5a0: 7363 7269 7074 6f72 732e 2057 6172 6e69 scriptors. Warni │ │ │ │ +0026f5b0: 6e67 3a20 7069 7065 7320 686f 6c64 206f ng: pipes hold o │ │ │ │ +0026f5c0: 6e6c 7920 3430 3936 2062 6974 6573 2c20 nly 4096 bites, │ │ │ │ +0026f5d0: 736f 2069 6620 796f 7520 7772 6974 6520 so if you write │ │ │ │ +0026f5e0: 6d6f 7265 2074 6861 6e0a 2020 2020 7468 more than. th │ │ │ │ +0026f5f0: 6174 2074 6f20 7468 6520 7265 7375 6c74 at to the result │ │ │ │ +0026f600: 696e 6720 696e 7075 7420 6f75 7470 7574 ing input output │ │ │ │ +0026f610: 2066 696c 6520 2861 7320 696e 7075 7420 file (as input │ │ │ │ +0026f620: 666f 7220 7468 6520 636f 6d6d 616e 6429 for the command) │ │ │ │ +0026f630: 2077 6974 686f 7574 0a20 2020 2072 6561 without. rea │ │ │ │ +0026f640: 6469 6e67 2061 6e79 2064 6174 612c 2079 ding any data, y │ │ │ │ +0026f650: 6f75 206d 6179 2062 6c6f 636b 2077 6869 ou may block whi │ │ │ │ +0026f660: 6c65 2074 6865 2063 6f6d 6d61 6e64 2069 le the command i │ │ │ │ +0026f670: 7320 626c 6f63 6b65 6420 7761 6974 696e s blocked waitin │ │ │ │ +0026f680: 6720 746f 0a20 2020 2077 7269 7465 206d g to. write m │ │ │ │ +0026f690: 6f72 6520 6f75 7470 7574 3b20 696e 2074 ore output; in t │ │ │ │ +0026f6a0: 6869 7320 6361 7365 2c20 4d61 6361 756c his case, Macaul │ │ │ │ +0026f6b0: 6179 3220 7769 6c6c 2061 7070 6561 7220 ay2 will appear │ │ │ │ +0026f6c0: 746f 2068 616e 672e 0a20 202a 2061 2073 to hang.. * a s │ │ │ │ +0026f6d0: 7472 696e 6720 6f66 2074 6865 2066 6f72 tring of the for │ │ │ │ +0026f6e0: 6d20 2224 686f 7374 6e61 6d65 3a73 6572 m "$hostname:ser │ │ │ │ +0026f6f0: 7669 6365 223a 2061 2063 6f6e 6e65 6374 vice": a connect │ │ │ │ +0026f700: 696f 6e20 7769 6c6c 2062 6520 6d61 6465 ion will be made │ │ │ │ +0026f710: 2074 6f20 7468 650a 2020 2020 7370 6563 to the. spec │ │ │ │ +0026f720: 6966 6965 6420 7365 7276 6963 6520 6174 ified service at │ │ │ │ +0026f730: 2074 6865 2073 7065 6369 6669 6564 2068 the specified h │ │ │ │ +0026f740: 6f73 742e 2020 4966 2074 6865 2073 6572 ost. If the ser │ │ │ │ +0026f750: 7669 6365 2070 6f72 7420 6973 206f 6d69 vice port is omi │ │ │ │ +0026f760: 7474 6564 2c0a 2020 2020 616c 6f6e 6720 tted,. along │ │ │ │ +0026f770: 7769 7468 2074 6865 2063 6f6c 6f6e 2c20 with the colon, │ │ │ │ +0026f780: 7468 656e 2070 6f72 7420 3235 3030 2069 then port 2500 i │ │ │ │ +0026f790: 7320 7573 6564 2e20 2049 6620 7468 6520 s used. If the │ │ │ │ +0026f7a0: 686f 7374 6e61 6d65 2069 7320 6f6d 6974 hostname is omit │ │ │ │ +0026f7b0: 7465 642c 0a20 2020 2061 6e20 696e 636f ted,. an inco │ │ │ │ +0026f7c0: 6d69 6e67 2063 6f6e 6e65 6374 696f 6e20 ming connection │ │ │ │ +0026f7d0: 7769 6c6c 2062 6520 7761 6974 6564 2066 will be waited f │ │ │ │ +0026f7e0: 6f72 2e0a 2020 2a20 6120 6c69 7374 656e or.. * a listen │ │ │ │ +0026f7f0: 6572 2063 7265 6174 6564 2070 7265 7669 er created previ │ │ │ │ +0026f800: 6f75 736c 7920 6279 202a 6e6f 7465 206f ously by *note o │ │ │ │ +0026f810: 7065 6e4c 6973 7465 6e65 723a 0a20 2020 penListener:. │ │ │ │ +0026f820: 206f 7065 6e4c 6973 7465 6e65 725f 6c70 openListener_lp │ │ │ │ +0026f830: 5374 7269 6e67 5f72 702c 3a20 6120 6e65 String_rp,: a ne │ │ │ │ +0026f840: 7720 636f 6e6e 6563 7469 6f6e 2077 696c w connection wil │ │ │ │ +0026f850: 6c20 6265 2063 7265 6174 6564 2e20 2054 l be created. T │ │ │ │ +0026f860: 6f20 6176 6f69 640a 2020 2020 626c 6f63 o avoid. bloc │ │ │ │ +0026f870: 6b69 6e67 2074 6865 204d 6163 6175 6c61 king the Macaula │ │ │ │ +0026f880: 7932 2070 726f 6365 7373 2077 6869 6c65 y2 process while │ │ │ │ +0026f890: 2077 6169 7469 6e67 2066 6f72 2074 6865 waiting for the │ │ │ │ +0026f8a0: 2069 6e63 6f6d 696e 6720 636f 6e6e 6563 incoming connec │ │ │ │ +0026f8b0: 7469 6f6e 2c0a 2020 2020 7573 6520 2a6e tion,. use *n │ │ │ │ +0026f8c0: 6f74 6520 6973 5265 6164 793a 2069 7352 ote isReady: isR │ │ │ │ +0026f8d0: 6561 6479 5f6c 7046 696c 655f 7270 2c2e eady_lpFile_rp,. │ │ │ │ +0026f8e0: 0a0a 496e 206f 7264 6572 2074 6f20 6f70 ..In order to op │ │ │ │ +0026f8f0: 656e 2061 2073 6f63 6b65 7420 7375 6363 en a socket succ │ │ │ │ +0026f900: 6573 7366 756c 6c79 2c20 7468 6572 6520 essfully, there │ │ │ │ +0026f910: 6d75 7374 2062 6520 6120 7072 6f63 6573 must be a proces │ │ │ │ +0026f920: 7320 6163 6365 7074 696e 670a 636f 6e6e s accepting.conn │ │ │ │ +0026f930: 6563 7469 6f6e 7320 666f 7220 7468 6520 ections for the │ │ │ │ +0026f940: 6465 7369 7265 6420 7365 7276 6963 6520 desired service │ │ │ │ +0026f950: 6f6e 2074 6865 2073 7065 6369 6669 6564 on the specified │ │ │ │ +0026f960: 2068 6f73 742e 0a0a 5468 6520 7661 7269 host...The vari │ │ │ │ +0026f970: 6f75 7320 666f 726d 7320 6c69 7374 6564 ous forms listed │ │ │ │ +0026f980: 2061 626f 7665 2063 616e 2062 6520 7573 above can be us │ │ │ │ +0026f990: 6564 2061 6c73 6f20 7769 7468 2061 6c6c ed also with all │ │ │ │ +0026f9a0: 206f 7468 6572 2069 6e70 7574 206f 7574 other input out │ │ │ │ +0026f9b0: 7075 740a 6f70 6572 6174 696f 6e73 2074 put.operations t │ │ │ │ +0026f9c0: 6861 7420 6f70 656e 2066 696c 6573 2c20 hat open files, │ │ │ │ +0026f9d0: 7375 6368 2061 7320 2a6e 6f74 6520 6f70 such as *note op │ │ │ │ +0026f9e0: 656e 496e 3a20 6f70 656e 496e 5f6c 7053 enIn: openIn_lpS │ │ │ │ +0026f9f0: 7472 696e 675f 7270 2c2c 202a 6e6f 7465 tring_rp,, *note │ │ │ │ +0026fa00: 0a6f 7065 6e4f 7574 3a20 6f70 656e 4f75 .openOut: openOu │ │ │ │ +0026fa10: 745f 6c70 5374 7269 6e67 5f72 702c 2c20 t_lpString_rp,, │ │ │ │ +0026fa20: 2a6e 6f74 6520 6765 743a 2067 6574 2c2c *note get: get,, │ │ │ │ +0026fa30: 2061 6e64 202a 6e6f 7465 203c 3c3a 203c and *note <<: < │ │ │ │ +0026fa40: 3c2c 2c20 7769 7468 2064 6174 610a 7472 <,, with data.tr │ │ │ │ +0026fa50: 616e 7366 6572 2070 6f73 7369 626c 6520 ansfer possible │ │ │ │ +0026fa60: 6f6e 6c79 2069 6e20 7468 6520 6469 7265 only in the dire │ │ │ │ +0026fa70: 6374 696f 6e20 7370 6563 6966 6965 642e ction specified. │ │ │ │ +0026fa80: 2020 4120 706f 7373 6962 6c79 2063 6f6e A possibly con │ │ │ │ +0026fa90: 6675 7369 6e67 0a61 7379 6d6d 6574 7279 fusing.asymmetry │ │ │ │ +0026faa0: 2069 7320 7468 6174 2077 6974 6820 6f70 is that with op │ │ │ │ +0026fab0: 656e 496e 2022 2166 6f6f 2220 6f72 2077 enIn "!foo" or w │ │ │ │ +0026fac0: 6974 6820 6765 7420 2221 666f 6f22 2074 ith get "!foo" t │ │ │ │ +0026fad0: 6865 2073 7461 6e64 6172 6420 696e 7075 he standard inpu │ │ │ │ +0026fae0: 7420 6f66 0a74 6865 2063 6f6d 6d61 6e64 t of.the command │ │ │ │ +0026faf0: 2066 6f6f 2069 7320 636c 6f73 6564 2c20 foo is closed, │ │ │ │ +0026fb00: 6275 7420 7769 7468 206f 7065 6e4f 7574 but with openOut │ │ │ │ +0026fb10: 2022 2166 6f6f 2220 7468 6520 7374 616e "!foo" the stan │ │ │ │ +0026fb20: 6461 7264 206f 7574 7075 7420 6f66 2074 dard output of t │ │ │ │ +0026fb30: 6865 0a63 6f6d 6d61 6e64 2066 6f6f 2069 he.command foo i │ │ │ │ +0026fb40: 7320 636f 6e6e 6563 7465 6420 746f 2074 s connected to t │ │ │ │ +0026fb50: 6865 2073 7461 6e64 6172 6420 6f75 7470 he standard outp │ │ │ │ +0026fb60: 7574 206f 6620 7468 6520 7061 7265 6e74 ut of the parent │ │ │ │ +0026fb70: 204d 6163 6175 6c61 7932 0a70 726f 6365 Macaulay2.proce │ │ │ │ +0026fb80: 7373 2e0a 0a53 6565 2061 6c73 6f0a 3d3d ss...See also.== │ │ │ │ +0026fb90: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +0026fba0: 6520 6f70 656e 496e 3a20 6f70 656e 496e e openIn: openIn │ │ │ │ +0026fbb0: 5f6c 7053 7472 696e 675f 7270 2c20 2d2d _lpString_rp, -- │ │ │ │ +0026fbc0: 206f 7065 6e20 616e 2069 6e70 7574 2066 open an input f │ │ │ │ +0026fbd0: 696c 650a 2020 2a20 2a6e 6f74 6520 6f70 ile. * *note op │ │ │ │ +0026fbe0: 656e 4f75 743a 206f 7065 6e4f 7574 5f6c enOut: openOut_l │ │ │ │ +0026fbf0: 7053 7472 696e 675f 7270 2c20 2d2d 206f pString_rp, -- o │ │ │ │ +0026fc00: 7065 6e20 616e 206f 7574 7075 7420 6669 pen an output fi │ │ │ │ +0026fc10: 6c65 0a20 202a 202a 6e6f 7465 206f 7065 le. * *note ope │ │ │ │ +0026fc20: 6e4c 6973 7465 6e65 723a 206f 7065 6e4c nListener: openL │ │ │ │ +0026fc30: 6973 7465 6e65 725f 6c70 5374 7269 6e67 istener_lpString │ │ │ │ +0026fc40: 5f72 702c 202d 2d20 6f70 656e 2061 2070 _rp, -- open a p │ │ │ │ +0026fc50: 6f72 7420 666f 7220 6c69 7374 656e 696e ort for listenin │ │ │ │ +0026fc60: 670a 0a57 6179 7320 746f 2075 7365 206f g..Ways to use o │ │ │ │ +0026fc70: 7065 6e49 6e4f 7574 3a0a 3d3d 3d3d 3d3d penInOut:.====== │ │ │ │ +0026fc80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0026fc90: 0a0a 2020 2a20 226f 7065 6e49 6e4f 7574 .. * "openInOut │ │ │ │ +0026fca0: 2846 696c 6529 220a 2020 2a20 226f 7065 (File)". * "ope │ │ │ │ +0026fcb0: 6e49 6e4f 7574 2853 7472 696e 6729 220a nInOut(String)". │ │ │ │ +0026fcc0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0026fcd0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0026fce0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0026fcf0: 6374 202a 6e6f 7465 206f 7065 6e49 6e4f ct *note openInO │ │ │ │ +0026fd00: 7574 3a20 6f70 656e 496e 4f75 742c 2069 ut: openInOut, i │ │ │ │ +0026fd10: 7320 6120 2a6e 6f74 6520 636f 6d70 696c s a *note compil │ │ │ │ +0026fd20: 6564 2066 756e 6374 696f 6e3a 0a43 6f6d ed function:.Com │ │ │ │ +0026fd30: 7069 6c65 6446 756e 6374 696f 6e2c 2e0a piledFunction,.. │ │ │ │ +0026fd40: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 0026fd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0026fd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -0026fd70: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -0026fd80: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -0026fd90: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -0026fda0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -0026fdb0: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -0026fdc0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -0026fdd0: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ -0026fde0: 632f 6f76 5f73 7973 7465 6d2e 6d32 3a35 c/ov_system.m2:5 │ │ │ │ -0026fdf0: 3337 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 37:0....File: Ma │ │ │ │ -0026fe00: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ -0026fe10: 204e 6f64 653a 206f 7065 6e4f 7574 5f6c Node: openOut_l │ │ │ │ -0026fe20: 7053 7472 696e 675f 7270 2c20 4e65 7874 pString_rp, Next │ │ │ │ -0026fe30: 3a20 6f70 656e 4f75 7441 7070 656e 645f : openOutAppend_ │ │ │ │ -0026fe40: 6c70 5374 7269 6e67 5f72 702c 2050 7265 lpString_rp, Pre │ │ │ │ -0026fe50: 763a 206f 7065 6e49 6e4f 7574 2c20 5570 v: openInOut, Up │ │ │ │ -0026fe60: 3a20 7573 696e 6720 736f 636b 6574 730a : using sockets. │ │ │ │ -0026fe70: 0a6f 7065 6e4f 7574 2853 7472 696e 6729 .openOut(String) │ │ │ │ -0026fe80: 202d 2d20 6f70 656e 2061 6e20 6f75 7470 -- open an outp │ │ │ │ -0026fe90: 7574 2066 696c 650a 2a2a 2a2a 2a2a 2a2a ut file.******** │ │ │ │ -0026fea0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0026feb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -0026fec0: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e * Function: *n │ │ │ │ -0026fed0: 6f74 6520 6f70 656e 4f75 743a 206f 7065 ote openOut: ope │ │ │ │ -0026fee0: 6e4f 7574 5f6c 7053 7472 696e 675f 7270 nOut_lpString_rp │ │ │ │ -0026fef0: 2c0a 2020 2a20 5573 6167 653a 200a 2020 ,. * Usage: . │ │ │ │ -0026ff00: 2020 2020 2020 6f70 656e 4f75 7420 666e openOut fn │ │ │ │ -0026ff10: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -0026ff20: 2020 202a 2066 6e2c 2061 202a 6e6f 7465 * fn, a *note │ │ │ │ -0026ff30: 2073 7472 696e 673a 2053 7472 696e 672c string: String, │ │ │ │ -0026ff40: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -0026ff50: 2020 2020 2a20 6120 2a6e 6f74 6520 6669 * a *note fi │ │ │ │ -0026ff60: 6c65 3a20 4669 6c65 2c2c 2061 6e20 6f70 le: File,, an op │ │ │ │ -0026ff70: 656e 206f 7574 7075 7420 6669 6c65 2077 en output file w │ │ │ │ -0026ff80: 686f 7365 2066 696c 656e 616d 6520 6973 hose filename is │ │ │ │ -0026ff90: 2066 6e2e 0a20 2020 2020 2020 2046 696c fn.. Fil │ │ │ │ -0026ffa0: 656e 616d 6573 2073 7461 7274 696e 6720 enames starting │ │ │ │ -0026ffb0: 7769 7468 2021 206f 7220 7769 7468 2024 with ! or with $ │ │ │ │ -0026ffc0: 2061 7265 2074 7265 6174 6564 2073 7065 are treated spe │ │ │ │ -0026ffd0: 6369 616c 6c79 2c20 7365 6520 2a6e 6f74 cially, see *not │ │ │ │ -0026ffe0: 650a 2020 2020 2020 2020 6f70 656e 496e e. openIn │ │ │ │ -0026fff0: 4f75 743a 206f 7065 6e49 6e4f 7574 2c2e Out: openInOut,. │ │ │ │ -00270000: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -00270010: 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d =========..+---- │ │ │ │ -00270020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270030: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -00270040: 6720 3d20 6f70 656e 4f75 7420 2274 6573 g = openOut "tes │ │ │ │ -00270050: 742d 6669 6c65 227c 0a7c 2020 2020 2020 t-file"|.| │ │ │ │ -00270060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270070: 2020 2020 2020 7c0a 7c6f 3120 3d20 7465 |.|o1 = te │ │ │ │ -00270080: 7374 2d66 696c 6520 2020 2020 2020 2020 st-file │ │ │ │ -00270090: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -002700a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002700b0: 2020 2020 7c0a 7c6f 3120 3a20 4669 6c65 |.|o1 : File │ │ │ │ +0026fd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026fd70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026fd80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0026fd90: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0026fda0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0026fdb0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0026fdc0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0026fdd0: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +0026fde0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0026fdf0: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ +0026fe00: 7932 446f 632f 6f76 5f73 7973 7465 6d2e y2Doc/ov_system. │ │ │ │ +0026fe10: 6d32 3a35 3337 3a30 2e0a 1f0a 4669 6c65 m2:537:0....File │ │ │ │ +0026fe20: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ +0026fe30: 6e66 6f2c 204e 6f64 653a 206f 7065 6e4f nfo, Node: openO │ │ │ │ +0026fe40: 7574 5f6c 7053 7472 696e 675f 7270 2c20 ut_lpString_rp, │ │ │ │ +0026fe50: 4e65 7874 3a20 6f70 656e 4f75 7441 7070 Next: openOutApp │ │ │ │ +0026fe60: 656e 645f 6c70 5374 7269 6e67 5f72 702c end_lpString_rp, │ │ │ │ +0026fe70: 2050 7265 763a 206f 7065 6e49 6e4f 7574 Prev: openInOut │ │ │ │ +0026fe80: 2c20 5570 3a20 7573 696e 6720 736f 636b , Up: using sock │ │ │ │ +0026fe90: 6574 730a 0a6f 7065 6e4f 7574 2853 7472 ets..openOut(Str │ │ │ │ +0026fea0: 696e 6729 202d 2d20 6f70 656e 2061 6e20 ing) -- open an │ │ │ │ +0026feb0: 6f75 7470 7574 2066 696c 650a 2a2a 2a2a output file.**** │ │ │ │ +0026fec0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026fed0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0026fee0: 2a2a 0a0a 2020 2a20 4675 6e63 7469 6f6e **.. * Function │ │ │ │ +0026fef0: 3a20 2a6e 6f74 6520 6f70 656e 4f75 743a : *note openOut: │ │ │ │ +0026ff00: 206f 7065 6e4f 7574 5f6c 7053 7472 696e openOut_lpStrin │ │ │ │ +0026ff10: 675f 7270 2c0a 2020 2a20 5573 6167 653a g_rp,. * Usage: │ │ │ │ +0026ff20: 200a 2020 2020 2020 2020 6f70 656e 4f75 . openOu │ │ │ │ +0026ff30: 7420 666e 0a20 202a 2049 6e70 7574 733a t fn. * Inputs: │ │ │ │ +0026ff40: 0a20 2020 2020 202a 2066 6e2c 2061 202a . * fn, a * │ │ │ │ +0026ff50: 6e6f 7465 2073 7472 696e 673a 2053 7472 note string: Str │ │ │ │ +0026ff60: 696e 672c 0a20 202a 204f 7574 7075 7473 ing,. * Outputs │ │ │ │ +0026ff70: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ +0026ff80: 6520 6669 6c65 3a20 4669 6c65 2c2c 2061 e file: File,, a │ │ │ │ +0026ff90: 6e20 6f70 656e 206f 7574 7075 7420 6669 n open output fi │ │ │ │ +0026ffa0: 6c65 2077 686f 7365 2066 696c 656e 616d le whose filenam │ │ │ │ +0026ffb0: 6520 6973 2066 6e2e 0a20 2020 2020 2020 e is fn.. │ │ │ │ +0026ffc0: 2046 696c 656e 616d 6573 2073 7461 7274 Filenames start │ │ │ │ +0026ffd0: 696e 6720 7769 7468 2021 206f 7220 7769 ing with ! or wi │ │ │ │ +0026ffe0: 7468 2024 2061 7265 2074 7265 6174 6564 th $ are treated │ │ │ │ +0026fff0: 2073 7065 6369 616c 6c79 2c20 7365 6520 specially, see │ │ │ │ +00270000: 2a6e 6f74 650a 2020 2020 2020 2020 6f70 *note. op │ │ │ │ +00270010: 656e 496e 4f75 743a 206f 7065 6e49 6e4f enInOut: openInO │ │ │ │ +00270020: 7574 2c2e 0a0a 4465 7363 7269 7074 696f ut,...Descriptio │ │ │ │ +00270030: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b n.===========..+ │ │ │ │ +00270040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00270060: 3120 3a20 6720 3d20 6f70 656e 4f75 7420 1 : g = openOut │ │ │ │ +00270070: 2274 6573 742d 6669 6c65 227c 0a7c 2020 "test-file"|.| │ │ │ │ +00270080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270090: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +002700a0: 3d20 7465 7374 2d66 696c 6520 2020 2020 = test-file │ │ │ │ +002700b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 002700c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002700d0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -002700e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002700f0: 2d2d 2b0a 7c69 3220 3a20 6720 3c3c 2022 --+.|i2 : g << " │ │ │ │ -00270100: 6869 2074 6865 7265 2220 2020 2020 2020 hi there" │ │ │ │ -00270110: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00270120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270130: 7c0a 7c6f 3220 3d20 7465 7374 2d66 696c |.|o2 = test-fil │ │ │ │ -00270140: 6520 2020 2020 2020 2020 2020 2020 207c e | │ │ │ │ -00270150: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00270160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00270170: 7c6f 3220 3a20 4669 6c65 2020 2020 2020 |o2 : File │ │ │ │ -00270180: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00270190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002701a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -002701b0: 3320 3a20 6720 3c3c 2063 6c6f 7365 2020 3 : g << close │ │ │ │ -002701c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -002701d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002701e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -002701f0: 3d20 7465 7374 2d66 696c 6520 2020 2020 = test-file │ │ │ │ -00270200: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00270210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270220: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -00270230: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ -00270240: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00270250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270260: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6765 ------+.|i4 : ge │ │ │ │ -00270270: 7420 2274 6573 742d 6669 6c65 2220 2020 t "test-file" │ │ │ │ -00270280: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00270290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002702a0: 2020 2020 7c0a 7c6f 3420 3d20 6869 2074 |.|o4 = hi t │ │ │ │ -002702b0: 6865 7265 2020 2020 2020 2020 2020 2020 here │ │ │ │ -002702c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -002702d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002702e0: 2d2d 2b0a 7c69 3520 3a20 7265 6d6f 7665 --+.|i5 : remove │ │ │ │ -002702f0: 4669 6c65 2022 7465 7374 2d66 696c 6522 File "test-file" │ │ │ │ -00270300: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00270310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270320: 2b0a 0a41 2066 696c 656e 616d 6520 7374 +..A filename st │ │ │ │ -00270330: 6172 7469 6e67 2077 6974 6820 7e2f 2077 arting with ~/ w │ │ │ │ -00270340: 696c 6c20 6861 7665 2074 6865 2074 696c ill have the til │ │ │ │ -00270350: 6465 2072 6570 6c61 6365 6420 6279 2074 de replaced by t │ │ │ │ -00270360: 6865 2075 7365 7227 7320 686f 6d65 0a64 he user's home.d │ │ │ │ -00270370: 6972 6563 746f 7279 2e0a 0a53 6565 2061 irectory...See a │ │ │ │ -00270380: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00270390: 2a20 2a6e 6f74 6520 6f70 656e 496e 3a20 * *note openIn: │ │ │ │ -002703a0: 6f70 656e 496e 5f6c 7053 7472 696e 675f openIn_lpString_ │ │ │ │ -002703b0: 7270 2c20 2d2d 206f 7065 6e20 616e 2069 rp, -- open an i │ │ │ │ -002703c0: 6e70 7574 2066 696c 650a 2020 2a20 2a6e nput file. * *n │ │ │ │ -002703d0: 6f74 6520 6f70 656e 496e 4f75 743a 206f ote openInOut: o │ │ │ │ -002703e0: 7065 6e49 6e4f 7574 2c20 2d2d 206f 7065 penInOut, -- ope │ │ │ │ -002703f0: 6e20 616e 2069 6e70 7574 206f 7574 7075 n an input outpu │ │ │ │ -00270400: 7420 6669 6c65 0a20 202a 202a 6e6f 7465 t file. * *note │ │ │ │ -00270410: 206f 7065 6e4f 7574 4170 7065 6e64 3a20 openOutAppend: │ │ │ │ -00270420: 6f70 656e 4f75 7441 7070 656e 645f 6c70 openOutAppend_lp │ │ │ │ -00270430: 5374 7269 6e67 5f72 702c 202d 2d20 6f70 String_rp, -- op │ │ │ │ -00270440: 656e 2061 6e20 6f75 7470 7574 2066 696c en an output fil │ │ │ │ -00270450: 6520 666f 720a 2020 2020 6170 7065 6e64 e for. append │ │ │ │ -00270460: 696e 670a 2020 2a20 2a6e 6f74 6520 6765 ing. * *note ge │ │ │ │ -00270470: 743a 2067 6574 2c20 2d2d 2067 6574 2074 t: get, -- get t │ │ │ │ -00270480: 6865 2063 6f6e 7465 6e74 7320 6f66 2061 he contents of a │ │ │ │ -00270490: 2066 696c 650a 2020 2a20 2a6e 6f74 6520 file. * *note │ │ │ │ -002704a0: 7265 6d6f 7665 4669 6c65 3a20 7265 6d6f removeFile: remo │ │ │ │ -002704b0: 7665 4669 6c65 2c20 2d2d 2072 656d 6f76 veFile, -- remov │ │ │ │ -002704c0: 6520 6120 6669 6c65 0a20 202a 202a 6e6f e a file. * *no │ │ │ │ -002704d0: 7465 2063 6c6f 7365 3a20 636c 6f73 652c te close: close, │ │ │ │ -002704e0: 202d 2d20 636c 6f73 6520 6120 6669 6c65 -- close a file │ │ │ │ -002704f0: 0a20 202a 202a 6e6f 7465 2046 696c 6520 . * *note File │ │ │ │ -00270500: 3c3c 2054 6869 6e67 3a20 7072 696e 7469 << Thing: printi │ │ │ │ -00270510: 6e67 2074 6f20 6120 6669 6c65 2c20 2d2d ng to a file, -- │ │ │ │ -00270520: 2070 7269 6e74 2074 6f20 6120 6669 6c65 print to a file │ │ │ │ -00270530: 0a0a 5761 7973 2074 6f20 7573 6520 7468 ..Ways to use th │ │ │ │ -00270540: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d is method:.===== │ │ │ │ -00270550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00270560: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206f ===.. * *note o │ │ │ │ -00270570: 7065 6e4f 7574 2853 7472 696e 6729 3a20 penOut(String): │ │ │ │ -00270580: 6f70 656e 4f75 745f 6c70 5374 7269 6e67 openOut_lpString │ │ │ │ -00270590: 5f72 702c 202d 2d20 6f70 656e 2061 6e20 _rp, -- open an │ │ │ │ -002705a0: 6f75 7470 7574 2066 696c 650a 2d2d 2d2d output file.---- │ │ │ │ -002705b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002705c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002700d0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +002700e0: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ +002700f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00270100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270110: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 6720 ------+.|i2 : g │ │ │ │ +00270120: 3c3c 2022 6869 2074 6865 7265 2220 2020 << "hi there" │ │ │ │ +00270130: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00270140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270150: 2020 2020 7c0a 7c6f 3220 3d20 7465 7374 |.|o2 = test │ │ │ │ +00270160: 2d66 696c 6520 2020 2020 2020 2020 2020 -file │ │ │ │ +00270170: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00270180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270190: 2020 7c0a 7c6f 3220 3a20 4669 6c65 2020 |.|o2 : File │ │ │ │ +002701a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002701b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +002701c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002701d0: 2b0a 7c69 3320 3a20 6720 3c3c 2063 6c6f +.|i3 : g << clo │ │ │ │ +002701e0: 7365 2020 2020 2020 2020 2020 2020 207c se | │ │ │ │ +002701f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00270200: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00270210: 7c6f 3320 3d20 7465 7374 2d66 696c 6520 |o3 = test-file │ │ │ │ +00270220: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00270230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270250: 3320 3a20 4669 6c65 2020 2020 2020 2020 3 : File │ │ │ │ +00270260: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00270270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270280: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +00270290: 3a20 6765 7420 2274 6573 742d 6669 6c65 : get "test-file │ │ │ │ +002702a0: 2220 2020 2020 2020 207c 0a7c 2020 2020 " |.| │ │ │ │ +002702b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002702c0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ +002702d0: 6869 2074 6865 7265 2020 2020 2020 2020 hi there │ │ │ │ +002702e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +002702f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270300: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7265 ------+.|i5 : re │ │ │ │ +00270310: 6d6f 7665 4669 6c65 2022 7465 7374 2d66 moveFile "test-f │ │ │ │ +00270320: 696c 6522 207c 0a2b 2d2d 2d2d 2d2d 2d2d ile" |.+-------- │ │ │ │ +00270330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270340: 2d2d 2d2d 2b0a 0a41 2066 696c 656e 616d ----+..A filenam │ │ │ │ +00270350: 6520 7374 6172 7469 6e67 2077 6974 6820 e starting with │ │ │ │ +00270360: 7e2f 2077 696c 6c20 6861 7665 2074 6865 ~/ will have the │ │ │ │ +00270370: 2074 696c 6465 2072 6570 6c61 6365 6420 tilde replaced │ │ │ │ +00270380: 6279 2074 6865 2075 7365 7227 7320 686f by the user's ho │ │ │ │ +00270390: 6d65 0a64 6972 6563 746f 7279 2e0a 0a53 me.directory...S │ │ │ │ +002703a0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +002703b0: 0a0a 2020 2a20 2a6e 6f74 6520 6f70 656e .. * *note open │ │ │ │ +002703c0: 496e 3a20 6f70 656e 496e 5f6c 7053 7472 In: openIn_lpStr │ │ │ │ +002703d0: 696e 675f 7270 2c20 2d2d 206f 7065 6e20 ing_rp, -- open │ │ │ │ +002703e0: 616e 2069 6e70 7574 2066 696c 650a 2020 an input file. │ │ │ │ +002703f0: 2a20 2a6e 6f74 6520 6f70 656e 496e 4f75 * *note openInOu │ │ │ │ +00270400: 743a 206f 7065 6e49 6e4f 7574 2c20 2d2d t: openInOut, -- │ │ │ │ +00270410: 206f 7065 6e20 616e 2069 6e70 7574 206f open an input o │ │ │ │ +00270420: 7574 7075 7420 6669 6c65 0a20 202a 202a utput file. * * │ │ │ │ +00270430: 6e6f 7465 206f 7065 6e4f 7574 4170 7065 note openOutAppe │ │ │ │ +00270440: 6e64 3a20 6f70 656e 4f75 7441 7070 656e nd: openOutAppen │ │ │ │ +00270450: 645f 6c70 5374 7269 6e67 5f72 702c 202d d_lpString_rp, - │ │ │ │ +00270460: 2d20 6f70 656e 2061 6e20 6f75 7470 7574 - open an output │ │ │ │ +00270470: 2066 696c 6520 666f 720a 2020 2020 6170 file for. ap │ │ │ │ +00270480: 7065 6e64 696e 670a 2020 2a20 2a6e 6f74 pending. * *not │ │ │ │ +00270490: 6520 6765 743a 2067 6574 2c20 2d2d 2067 e get: get, -- g │ │ │ │ +002704a0: 6574 2074 6865 2063 6f6e 7465 6e74 7320 et the contents │ │ │ │ +002704b0: 6f66 2061 2066 696c 650a 2020 2a20 2a6e of a file. * *n │ │ │ │ +002704c0: 6f74 6520 7265 6d6f 7665 4669 6c65 3a20 ote removeFile: │ │ │ │ +002704d0: 7265 6d6f 7665 4669 6c65 2c20 2d2d 2072 removeFile, -- r │ │ │ │ +002704e0: 656d 6f76 6520 6120 6669 6c65 0a20 202a emove a file. * │ │ │ │ +002704f0: 202a 6e6f 7465 2063 6c6f 7365 3a20 636c *note close: cl │ │ │ │ +00270500: 6f73 652c 202d 2d20 636c 6f73 6520 6120 ose, -- close a │ │ │ │ +00270510: 6669 6c65 0a20 202a 202a 6e6f 7465 2046 file. * *note F │ │ │ │ +00270520: 696c 6520 3c3c 2054 6869 6e67 3a20 7072 ile << Thing: pr │ │ │ │ +00270530: 696e 7469 6e67 2074 6f20 6120 6669 6c65 inting to a file │ │ │ │ +00270540: 2c20 2d2d 2070 7269 6e74 2074 6f20 6120 , -- print to a │ │ │ │ +00270550: 6669 6c65 0a0a 5761 7973 2074 6f20 7573 file..Ways to us │ │ │ │ +00270560: 6520 7468 6973 206d 6574 686f 643a 0a3d e this method:.= │ │ │ │ +00270570: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00270580: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +00270590: 7465 206f 7065 6e4f 7574 2853 7472 696e te openOut(Strin │ │ │ │ +002705a0: 6729 3a20 6f70 656e 4f75 745f 6c70 5374 g): openOut_lpSt │ │ │ │ +002705b0: 7269 6e67 5f72 702c 202d 2d20 6f70 656e ring_rp, -- open │ │ │ │ +002705c0: 2061 6e20 6f75 7470 7574 2066 696c 650a an output file. │ │ │ │ 002705d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002705e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002705f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -00270600: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -00270610: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -00270620: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -00270630: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -00270640: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ -00270650: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -00270660: 6573 2f0a 4d61 6361 756c 6179 3244 6f63 es/.Macaulay2Doc │ │ │ │ -00270670: 2f6f 765f 7379 7374 656d 2e6d 323a 3436 /ov_system.m2:46 │ │ │ │ -00270680: 333a 302e 0a1f 0a46 696c 653a 204d 6163 3:0....File: Mac │ │ │ │ -00270690: 6175 6c61 7932 446f 632e 696e 666f 2c20 aulay2Doc.info, │ │ │ │ -002706a0: 4e6f 6465 3a20 6f70 656e 4f75 7441 7070 Node: openOutApp │ │ │ │ -002706b0: 656e 645f 6c70 5374 7269 6e67 5f72 702c end_lpString_rp, │ │ │ │ -002706c0: 204e 6578 743a 2065 6e64 6c2c 2050 7265 Next: endl, Pre │ │ │ │ -002706d0: 763a 206f 7065 6e4f 7574 5f6c 7053 7472 v: openOut_lpStr │ │ │ │ -002706e0: 696e 675f 7270 2c20 5570 3a20 7573 696e ing_rp, Up: usin │ │ │ │ -002706f0: 6720 736f 636b 6574 730a 0a6f 7065 6e4f g sockets..openO │ │ │ │ -00270700: 7574 4170 7065 6e64 2853 7472 696e 6729 utAppend(String) │ │ │ │ -00270710: 202d 2d20 6f70 656e 2061 6e20 6f75 7470 -- open an outp │ │ │ │ -00270720: 7574 2066 696c 6520 666f 7220 6170 7065 ut file for appe │ │ │ │ -00270730: 6e64 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a nding.********** │ │ │ │ -00270740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00270750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002705f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00270620: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00270630: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00270640: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00270650: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00270660: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +00270670: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00270680: 636b 6167 6573 2f0a 4d61 6361 756c 6179 ckages/.Macaulay │ │ │ │ +00270690: 3244 6f63 2f6f 765f 7379 7374 656d 2e6d 2Doc/ov_system.m │ │ │ │ +002706a0: 323a 3436 333a 302e 0a1f 0a46 696c 653a 2:463:0....File: │ │ │ │ +002706b0: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ +002706c0: 666f 2c20 4e6f 6465 3a20 6f70 656e 4f75 fo, Node: openOu │ │ │ │ +002706d0: 7441 7070 656e 645f 6c70 5374 7269 6e67 tAppend_lpString │ │ │ │ +002706e0: 5f72 702c 204e 6578 743a 2065 6e64 6c2c _rp, Next: endl, │ │ │ │ +002706f0: 2050 7265 763a 206f 7065 6e4f 7574 5f6c Prev: openOut_l │ │ │ │ +00270700: 7053 7472 696e 675f 7270 2c20 5570 3a20 pString_rp, Up: │ │ │ │ +00270710: 7573 696e 6720 736f 636b 6574 730a 0a6f using sockets..o │ │ │ │ +00270720: 7065 6e4f 7574 4170 7065 6e64 2853 7472 penOutAppend(Str │ │ │ │ +00270730: 696e 6729 202d 2d20 6f70 656e 2061 6e20 ing) -- open an │ │ │ │ +00270740: 6f75 7470 7574 2066 696c 6520 666f 7220 output file for │ │ │ │ +00270750: 6170 7065 6e64 696e 670a 2a2a 2a2a 2a2a appending.****** │ │ │ │ 00270760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00270770: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20 .. * Function: │ │ │ │ -00270780: 2a6e 6f74 6520 6f70 656e 4f75 7441 7070 *note openOutApp │ │ │ │ -00270790: 656e 643a 206f 7065 6e4f 7574 4170 7065 end: openOutAppe │ │ │ │ -002707a0: 6e64 5f6c 7053 7472 696e 675f 7270 2c0a nd_lpString_rp,. │ │ │ │ -002707b0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -002707c0: 2020 2020 6f70 656e 4f75 7441 7070 656e openOutAppen │ │ │ │ -002707d0: 6420 666e 0a20 202a 2049 6e70 7574 733a d fn. * Inputs: │ │ │ │ -002707e0: 0a20 2020 2020 202a 2066 6e2c 2061 202a . * fn, a * │ │ │ │ -002707f0: 6e6f 7465 2073 7472 696e 673a 2053 7472 note string: Str │ │ │ │ -00270800: 696e 672c 0a20 202a 204f 7574 7075 7473 ing,. * Outputs │ │ │ │ -00270810: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ -00270820: 6520 6669 6c65 3a20 4669 6c65 2c2c 2061 e file: File,, a │ │ │ │ -00270830: 6e20 6f70 656e 206f 7574 7075 7420 6669 n open output fi │ │ │ │ -00270840: 6c65 2077 686f 7365 2066 696c 656e 616d le whose filenam │ │ │ │ -00270850: 6520 6973 2066 6e0a 0a44 6573 6372 6970 e is fn..Descrip │ │ │ │ -00270860: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00270870: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ -00270880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270890: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -002708a0: 6720 3d20 6f70 656e 4f75 7420 2274 6573 g = openOut "tes │ │ │ │ -002708b0: 742d 6669 6c65 2220 2020 2020 2020 2020 t-file" │ │ │ │ -002708c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -002708d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002708e0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -002708f0: 7465 7374 2d66 696c 6520 2020 2020 2020 test-file │ │ │ │ -00270900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270910: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00270770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00270780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00270790: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ +002707a0: 6f6e 3a20 2a6e 6f74 6520 6f70 656e 4f75 on: *note openOu │ │ │ │ +002707b0: 7441 7070 656e 643a 206f 7065 6e4f 7574 tAppend: openOut │ │ │ │ +002707c0: 4170 7065 6e64 5f6c 7053 7472 696e 675f Append_lpString_ │ │ │ │ +002707d0: 7270 2c0a 2020 2a20 5573 6167 653a 200a rp,. * Usage: . │ │ │ │ +002707e0: 2020 2020 2020 2020 6f70 656e 4f75 7441 openOutA │ │ │ │ +002707f0: 7070 656e 6420 666e 0a20 202a 2049 6e70 ppend fn. * Inp │ │ │ │ +00270800: 7574 733a 0a20 2020 2020 202a 2066 6e2c uts:. * fn, │ │ │ │ +00270810: 2061 202a 6e6f 7465 2073 7472 696e 673a a *note string: │ │ │ │ +00270820: 2053 7472 696e 672c 0a20 202a 204f 7574 String,. * Out │ │ │ │ +00270830: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ +00270840: 2a6e 6f74 6520 6669 6c65 3a20 4669 6c65 *note file: File │ │ │ │ +00270850: 2c2c 2061 6e20 6f70 656e 206f 7574 7075 ,, an open outpu │ │ │ │ +00270860: 7420 6669 6c65 2077 686f 7365 2066 696c t file whose fil │ │ │ │ +00270870: 656e 616d 6520 6973 2066 6e0a 0a44 6573 ename is fn..Des │ │ │ │ +00270880: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +00270890: 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ====..+--------- │ │ │ │ +002708a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002708b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +002708c0: 3120 3a20 6720 3d20 6f70 656e 4f75 7420 1 : g = openOut │ │ │ │ +002708d0: 2274 6573 742d 6669 6c65 2220 2020 2020 "test-file" │ │ │ │ +002708e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +002708f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270900: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270910: 3120 3d20 7465 7374 2d66 696c 6520 2020 1 = test-file │ │ │ │ 00270920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270930: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -00270940: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ -00270950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270960: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00270970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270980: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -00270990: 6720 3c3c 2022 6869 2074 6865 7265 2220 g << "hi there" │ │ │ │ -002709a0: 3c3c 2065 6e64 6c20 3c3c 2063 6c6f 7365 << endl << close │ │ │ │ -002709b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -002709c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002709d0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -002709e0: 7465 7374 2d66 696c 6520 2020 2020 2020 test-file │ │ │ │ -002709f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270a00: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00270930: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00270940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270950: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270960: 3120 3a20 4669 6c65 2020 2020 2020 2020 1 : File │ │ │ │ +00270970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270980: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00270990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002709a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +002709b0: 3220 3a20 6720 3c3c 2022 6869 2074 6865 2 : g << "hi the │ │ │ │ +002709c0: 7265 2220 3c3c 2065 6e64 6c20 3c3c 2063 re" << endl << c │ │ │ │ +002709d0: 6c6f 7365 7c0a 7c20 2020 2020 2020 2020 lose|.| │ │ │ │ +002709e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002709f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270a00: 3220 3d20 7465 7374 2d66 696c 6520 2020 2 = test-file │ │ │ │ 00270a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270a20: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -00270a30: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ -00270a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270a50: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00270a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270a70: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -00270a80: 6820 3d20 6f70 656e 4f75 7441 7070 656e h = openOutAppen │ │ │ │ -00270a90: 6420 2274 6573 742d 6669 6c65 2220 2020 d "test-file" │ │ │ │ -00270aa0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00270ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270ac0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00270ad0: 7465 7374 2d66 696c 6520 2020 2020 2020 test-file │ │ │ │ -00270ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270af0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00270a20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00270a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270a40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270a50: 3220 3a20 4669 6c65 2020 2020 2020 2020 2 : File │ │ │ │ +00270a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270a70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00270a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00270aa0: 3320 3a20 6820 3d20 6f70 656e 4f75 7441 3 : h = openOutA │ │ │ │ +00270ab0: 7070 656e 6420 2274 6573 742d 6669 6c65 ppend "test-file │ │ │ │ +00270ac0: 2220 2020 7c0a 7c20 2020 2020 2020 2020 " |.| │ │ │ │ +00270ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270ae0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270af0: 3320 3d20 7465 7374 2d66 696c 6520 2020 3 = test-file │ │ │ │ 00270b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270b10: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -00270b20: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ -00270b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270b40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00270b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270b60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00270b70: 6820 3c3c 2022 686f 2074 6865 7265 2220 h << "ho there" │ │ │ │ -00270b80: 3c3c 2065 6e64 6c20 3c3c 2063 6c6f 7365 << endl << close │ │ │ │ -00270b90: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00270ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270bb0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -00270bc0: 7465 7374 2d66 696c 6520 2020 2020 2020 test-file │ │ │ │ -00270bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270be0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00270b10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00270b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270b30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270b40: 3320 3a20 4669 6c65 2020 2020 2020 2020 3 : File │ │ │ │ +00270b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270b60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00270b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00270b90: 3420 3a20 6820 3c3c 2022 686f 2074 6865 4 : h << "ho the │ │ │ │ +00270ba0: 7265 2220 3c3c 2065 6e64 6c20 3c3c 2063 re" << endl << c │ │ │ │ +00270bb0: 6c6f 7365 7c0a 7c20 2020 2020 2020 2020 lose|.| │ │ │ │ +00270bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270bd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270be0: 3420 3d20 7465 7374 2d66 696c 6520 2020 4 = test-file │ │ │ │ 00270bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270c00: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ -00270c10: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ -00270c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270c30: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00270c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270c50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ -00270c60: 6765 7420 2274 6573 742d 6669 6c65 2220 get "test-file" │ │ │ │ -00270c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270c80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00270c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270ca0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -00270cb0: 6869 2074 6865 7265 2020 2020 2020 2020 hi there │ │ │ │ -00270cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270cd0: 7c0a 7c20 2020 2020 686f 2074 6865 7265 |.| ho there │ │ │ │ +00270c00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00270c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270c20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270c30: 3420 3a20 4669 6c65 2020 2020 2020 2020 4 : File │ │ │ │ +00270c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270c50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00270c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00270c80: 3520 3a20 6765 7420 2274 6573 742d 6669 5 : get "test-fi │ │ │ │ +00270c90: 6c65 2220 2020 2020 2020 2020 2020 2020 le" │ │ │ │ +00270ca0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00270cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00270cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00270cd0: 3520 3d20 6869 2074 6865 7265 2020 2020 5 = hi there │ │ │ │ 00270ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00270cf0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00270d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270d20: 2b0a 7c69 3620 3a20 7265 6d6f 7665 4669 +.|i6 : removeFi │ │ │ │ -00270d30: 6c65 2022 7465 7374 2d66 696c 6522 2020 le "test-file" │ │ │ │ -00270d40: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00270d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00270d70: 2b0a 0a41 2066 696c 656e 616d 6520 7374 +..A filename st │ │ │ │ -00270d80: 6172 7469 6e67 2077 6974 6820 7e2f 2077 arting with ~/ w │ │ │ │ -00270d90: 696c 6c20 6861 7665 2074 6865 2074 696c ill have the til │ │ │ │ -00270da0: 6465 2072 6570 6c61 6365 6420 6279 2074 de replaced by t │ │ │ │ -00270db0: 6865 2075 7365 7227 7320 686f 6d65 0a64 he user's home.d │ │ │ │ -00270dc0: 6972 6563 746f 7279 2e0a 0a53 6565 2061 irectory...See a │ │ │ │ -00270dd0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00270de0: 2a20 2a6e 6f74 6520 6f70 656e 496e 3a20 * *note openIn: │ │ │ │ -00270df0: 6f70 656e 496e 5f6c 7053 7472 696e 675f openIn_lpString_ │ │ │ │ -00270e00: 7270 2c20 2d2d 206f 7065 6e20 616e 2069 rp, -- open an i │ │ │ │ -00270e10: 6e70 7574 2066 696c 650a 2020 2a20 2a6e nput file. * *n │ │ │ │ -00270e20: 6f74 6520 6f70 656e 496e 4f75 743a 206f ote openInOut: o │ │ │ │ -00270e30: 7065 6e49 6e4f 7574 2c20 2d2d 206f 7065 penInOut, -- ope │ │ │ │ -00270e40: 6e20 616e 2069 6e70 7574 206f 7574 7075 n an input outpu │ │ │ │ -00270e50: 7420 6669 6c65 0a20 202a 202a 6e6f 7465 t file. * *note │ │ │ │ -00270e60: 206f 7065 6e4f 7574 3a20 6f70 656e 4f75 openOut: openOu │ │ │ │ -00270e70: 745f 6c70 5374 7269 6e67 5f72 702c 202d t_lpString_rp, - │ │ │ │ -00270e80: 2d20 6f70 656e 2061 6e20 6f75 7470 7574 - open an output │ │ │ │ -00270e90: 2066 696c 650a 2020 2a20 2a6e 6f74 6520 file. * *note │ │ │ │ -00270ea0: 4669 6c65 3a20 4669 6c65 2c20 2d2d 2074 File: File, -- t │ │ │ │ -00270eb0: 6865 2063 6c61 7373 206f 6620 616c 6c20 he class of all │ │ │ │ -00270ec0: 6669 6c65 730a 2020 2a20 2a6e 6f74 6520 files. * *note │ │ │ │ -00270ed0: 6765 743a 2067 6574 2c20 2d2d 2067 6574 get: get, -- get │ │ │ │ -00270ee0: 2074 6865 2063 6f6e 7465 6e74 7320 6f66 the contents of │ │ │ │ -00270ef0: 2061 2066 696c 650a 2020 2a20 2a6e 6f74 a file. * *not │ │ │ │ -00270f00: 6520 7265 6d6f 7665 4669 6c65 3a20 7265 e removeFile: re │ │ │ │ -00270f10: 6d6f 7665 4669 6c65 2c20 2d2d 2072 656d moveFile, -- rem │ │ │ │ -00270f20: 6f76 6520 6120 6669 6c65 0a20 202a 202a ove a file. * * │ │ │ │ -00270f30: 6e6f 7465 2046 696c 6520 3c3c 2054 6869 note File << Thi │ │ │ │ -00270f40: 6e67 3a20 7072 696e 7469 6e67 2074 6f20 ng: printing to │ │ │ │ -00270f50: 6120 6669 6c65 2c20 2d2d 2070 7269 6e74 a file, -- print │ │ │ │ -00270f60: 2074 6f20 6120 6669 6c65 0a0a 5761 7973 to a file..Ways │ │ │ │ -00270f70: 2074 6f20 7573 6520 7468 6973 206d 6574 to use this met │ │ │ │ -00270f80: 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d hod:.=========== │ │ │ │ -00270f90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -00270fa0: 202a 202a 6e6f 7465 206f 7065 6e4f 7574 * *note openOut │ │ │ │ -00270fb0: 4170 7065 6e64 2853 7472 696e 6729 3a20 Append(String): │ │ │ │ -00270fc0: 6f70 656e 4f75 7441 7070 656e 645f 6c70 openOutAppend_lp │ │ │ │ -00270fd0: 5374 7269 6e67 5f72 702c 202d 2d20 6f70 String_rp, -- op │ │ │ │ -00270fe0: 656e 2061 6e20 6f75 7470 7574 0a20 2020 en an output. │ │ │ │ -00270ff0: 2066 696c 6520 666f 7220 6170 7065 6e64 file for append │ │ │ │ -00271000: 696e 670a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ing.------------ │ │ │ │ -00271010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00271020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270cf0: 2020 2020 7c0a 7c20 2020 2020 686f 2074 |.| ho t │ │ │ │ +00270d00: 6865 7265 2020 2020 2020 2020 2020 2020 here │ │ │ │ +00270d10: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00270d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270d40: 2d2d 2d2d 2b0a 7c69 3620 3a20 7265 6d6f ----+.|i6 : remo │ │ │ │ +00270d50: 7665 4669 6c65 2022 7465 7374 2d66 696c veFile "test-fil │ │ │ │ +00270d60: 6522 2020 2020 2020 2020 2020 7c0a 2b2d e" |.+- │ │ │ │ +00270d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00270d90: 2d2d 2d2d 2b0a 0a41 2066 696c 656e 616d ----+..A filenam │ │ │ │ +00270da0: 6520 7374 6172 7469 6e67 2077 6974 6820 e starting with │ │ │ │ +00270db0: 7e2f 2077 696c 6c20 6861 7665 2074 6865 ~/ will have the │ │ │ │ +00270dc0: 2074 696c 6465 2072 6570 6c61 6365 6420 tilde replaced │ │ │ │ +00270dd0: 6279 2074 6865 2075 7365 7227 7320 686f by the user's ho │ │ │ │ +00270de0: 6d65 0a64 6972 6563 746f 7279 2e0a 0a53 me.directory...S │ │ │ │ +00270df0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00270e00: 0a0a 2020 2a20 2a6e 6f74 6520 6f70 656e .. * *note open │ │ │ │ +00270e10: 496e 3a20 6f70 656e 496e 5f6c 7053 7472 In: openIn_lpStr │ │ │ │ +00270e20: 696e 675f 7270 2c20 2d2d 206f 7065 6e20 ing_rp, -- open │ │ │ │ +00270e30: 616e 2069 6e70 7574 2066 696c 650a 2020 an input file. │ │ │ │ +00270e40: 2a20 2a6e 6f74 6520 6f70 656e 496e 4f75 * *note openInOu │ │ │ │ +00270e50: 743a 206f 7065 6e49 6e4f 7574 2c20 2d2d t: openInOut, -- │ │ │ │ +00270e60: 206f 7065 6e20 616e 2069 6e70 7574 206f open an input o │ │ │ │ +00270e70: 7574 7075 7420 6669 6c65 0a20 202a 202a utput file. * * │ │ │ │ +00270e80: 6e6f 7465 206f 7065 6e4f 7574 3a20 6f70 note openOut: op │ │ │ │ +00270e90: 656e 4f75 745f 6c70 5374 7269 6e67 5f72 enOut_lpString_r │ │ │ │ +00270ea0: 702c 202d 2d20 6f70 656e 2061 6e20 6f75 p, -- open an ou │ │ │ │ +00270eb0: 7470 7574 2066 696c 650a 2020 2a20 2a6e tput file. * *n │ │ │ │ +00270ec0: 6f74 6520 4669 6c65 3a20 4669 6c65 2c20 ote File: File, │ │ │ │ +00270ed0: 2d2d 2074 6865 2063 6c61 7373 206f 6620 -- the class of │ │ │ │ +00270ee0: 616c 6c20 6669 6c65 730a 2020 2a20 2a6e all files. * *n │ │ │ │ +00270ef0: 6f74 6520 6765 743a 2067 6574 2c20 2d2d ote get: get, -- │ │ │ │ +00270f00: 2067 6574 2074 6865 2063 6f6e 7465 6e74 get the content │ │ │ │ +00270f10: 7320 6f66 2061 2066 696c 650a 2020 2a20 s of a file. * │ │ │ │ +00270f20: 2a6e 6f74 6520 7265 6d6f 7665 4669 6c65 *note removeFile │ │ │ │ +00270f30: 3a20 7265 6d6f 7665 4669 6c65 2c20 2d2d : removeFile, -- │ │ │ │ +00270f40: 2072 656d 6f76 6520 6120 6669 6c65 0a20 remove a file. │ │ │ │ +00270f50: 202a 202a 6e6f 7465 2046 696c 6520 3c3c * *note File << │ │ │ │ +00270f60: 2054 6869 6e67 3a20 7072 696e 7469 6e67 Thing: printing │ │ │ │ +00270f70: 2074 6f20 6120 6669 6c65 2c20 2d2d 2070 to a file, -- p │ │ │ │ +00270f80: 7269 6e74 2074 6f20 6120 6669 6c65 0a0a rint to a file.. │ │ │ │ +00270f90: 5761 7973 2074 6f20 7573 6520 7468 6973 Ways to use this │ │ │ │ +00270fa0: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d method:.======= │ │ │ │ +00270fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00270fc0: 3d0a 0a20 202a 202a 6e6f 7465 206f 7065 =.. * *note ope │ │ │ │ +00270fd0: 6e4f 7574 4170 7065 6e64 2853 7472 696e nOutAppend(Strin │ │ │ │ +00270fe0: 6729 3a20 6f70 656e 4f75 7441 7070 656e g): openOutAppen │ │ │ │ +00270ff0: 645f 6c70 5374 7269 6e67 5f72 702c 202d d_lpString_rp, - │ │ │ │ +00271000: 2d20 6f70 656e 2061 6e20 6f75 7470 7574 - open an output │ │ │ │ +00271010: 0a20 2020 2066 696c 6520 666f 7220 6170 . file for ap │ │ │ │ +00271020: 7065 6e64 696e 670a 2d2d 2d2d 2d2d 2d2d pending.-------- │ │ │ │ 00271030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00271040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00271050: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -00271060: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -00271070: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -00271080: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -00271090: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -002710a0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -002710b0: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -002710c0: 756c 6179 3244 6f63 2f6f 765f 7379 7374 ulay2Doc/ov_syst │ │ │ │ -002710d0: 656d 2e6d 323a 3438 323a 302e 0a1f 0a46 em.m2:482:0....F │ │ │ │ -002710e0: 696c 653a 204d 6163 6175 6c61 7932 446f ile: Macaulay2Do │ │ │ │ -002710f0: 632e 696e 666f 2c20 4e6f 6465 3a20 656e c.info, Node: en │ │ │ │ -00271100: 646c 2c20 4e65 7874 3a20 666c 7573 682c dl, Next: flush, │ │ │ │ -00271110: 2050 7265 763a 206f 7065 6e4f 7574 4170 Prev: openOutAp │ │ │ │ -00271120: 7065 6e64 5f6c 7053 7472 696e 675f 7270 pend_lpString_rp │ │ │ │ -00271130: 2c20 5570 3a20 7573 696e 6720 736f 636b , Up: using sock │ │ │ │ -00271140: 6574 730a 0a65 6e64 6c20 2d2d 2065 6e64 ets..endl -- end │ │ │ │ -00271150: 2061 6e20 6f75 7470 7574 206c 696e 650a an output line. │ │ │ │ -00271160: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00271170: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 **********..Desc │ │ │ │ -00271180: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -00271190: 3d3d 3d0a 0a66 203c 3c20 656e 646c 202d ===..f << endl - │ │ │ │ -002711a0: 2d20 656e 6473 2074 6865 206c 696e 6520 - ends the line │ │ │ │ -002711b0: 6375 7272 656e 746c 7920 6265 696e 6720 currently being │ │ │ │ -002711c0: 7075 7420 6f75 7420 746f 2074 6865 2066 put out to the f │ │ │ │ -002711d0: 696c 6520 662e 0a0a 0a49 7420 6973 2061 ile f....It is a │ │ │ │ -002711e0: 6e20 6573 7365 6e74 6961 6c20 706f 7274 n essential port │ │ │ │ -002711f0: 6162 6c65 2070 726f 6772 616d 6d69 6e67 able programming │ │ │ │ -00271200: 2070 7261 6374 6963 6520 746f 2075 7365 practice to use │ │ │ │ -00271210: 2065 6e64 6c20 616c 7761 7973 2c20 666f endl always, fo │ │ │ │ -00271220: 720a 7772 6974 696e 6720 6e65 776c 696e r.writing newlin │ │ │ │ -00271230: 6520 6368 6172 6163 7465 7273 2028 7365 e characters (se │ │ │ │ -00271240: 6520 2a6e 6f74 6520 6e65 776c 696e 653a e *note newline: │ │ │ │ -00271250: 206e 6577 6c69 6e65 2c29 2074 6f20 6120 newline,) to a │ │ │ │ -00271260: 6669 6c65 2077 696c 6c20 6e6f 740a 7465 file will not.te │ │ │ │ -00271270: 726d 696e 6174 6520 6120 6c69 6e65 2063 rminate a line c │ │ │ │ -00271280: 6f6e 7461 696e 696e 6720 6e65 7473 2070 ontaining nets p │ │ │ │ -00271290: 726f 7065 726c 792c 2061 6e64 2069 7420 roperly, and it │ │ │ │ -002712a0: 7769 6c6c 206e 6f74 2066 6c75 7368 2074 will not flush t │ │ │ │ -002712b0: 6865 206f 7574 7075 740a 6275 6666 6572 he output.buffer │ │ │ │ -002712c0: 2e0a 0a57 6179 7320 746f 2075 7365 2065 ...Ways to use e │ │ │ │ -002712d0: 6e64 6c3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ndl:.=========== │ │ │ │ -002712e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 224d 616e ======.. * "Man │ │ │ │ -002712f0: 6970 756c 6174 6f72 2044 6174 6162 6173 ipulator Databas │ │ │ │ -00271300: 6522 202d 2d20 7365 6520 2a6e 6f74 6520 e" -- see *note │ │ │ │ -00271310: 4d61 6e69 7075 6c61 746f 723a 204d 616e Manipulator: Man │ │ │ │ -00271320: 6970 756c 6174 6f72 2c20 2d2d 2074 6865 ipulator, -- the │ │ │ │ -00271330: 2063 6c61 7373 0a20 2020 206f 6620 616c class. of al │ │ │ │ -00271340: 6c20 6669 6c65 206d 616e 6970 756c 6174 l file manipulat │ │ │ │ -00271350: 6f72 730a 2020 2a20 224d 616e 6970 756c ors. * "Manipul │ │ │ │ -00271360: 6174 6f72 2046 696c 6522 202d 2d20 7365 ator File" -- se │ │ │ │ -00271370: 6520 2a6e 6f74 6520 4d61 6e69 7075 6c61 e *note Manipula │ │ │ │ -00271380: 746f 723a 204d 616e 6970 756c 6174 6f72 tor: Manipulator │ │ │ │ -00271390: 2c20 2d2d 2074 6865 2063 6c61 7373 206f , -- the class o │ │ │ │ -002713a0: 660a 2020 2020 616c 6c20 6669 6c65 206d f. all file m │ │ │ │ -002713b0: 616e 6970 756c 6174 6f72 730a 2020 2a20 anipulators. * │ │ │ │ -002713c0: 224d 616e 6970 756c 6174 6f72 204e 6f74 "Manipulator Not │ │ │ │ -002713d0: 6869 6e67 2220 2d2d 2073 6565 202a 6e6f hing" -- see *no │ │ │ │ -002713e0: 7465 204d 616e 6970 756c 6174 6f72 3a20 te Manipulator: │ │ │ │ -002713f0: 4d61 6e69 7075 6c61 746f 722c 202d 2d20 Manipulator, -- │ │ │ │ -00271400: 7468 6520 636c 6173 730a 2020 2020 6f66 the class. of │ │ │ │ -00271410: 2061 6c6c 2066 696c 6520 6d61 6e69 7075 all file manipu │ │ │ │ -00271420: 6c61 746f 7273 0a20 202a 2022 6e65 7720 lators. * "new │ │ │ │ -00271430: 4d61 6e69 7075 6c61 746f 7220 6672 6f6d Manipulator from │ │ │ │ -00271440: 2046 756e 6374 696f 6e22 202d 2d20 7365 Function" -- se │ │ │ │ -00271450: 6520 2a6e 6f74 6520 4d61 6e69 7075 6c61 e *note Manipula │ │ │ │ -00271460: 746f 723a 204d 616e 6970 756c 6174 6f72 tor: Manipulator │ │ │ │ -00271470: 2c20 2d2d 0a20 2020 2074 6865 2063 6c61 , --. the cla │ │ │ │ -00271480: 7373 206f 6620 616c 6c20 6669 6c65 206d ss of all file m │ │ │ │ -00271490: 616e 6970 756c 6174 6f72 730a 2020 2a20 anipulators. * │ │ │ │ -002714a0: 226d 6574 686f 6473 284d 616e 6970 756c "methods(Manipul │ │ │ │ -002714b0: 6174 6f72 2922 202d 2d20 7365 6520 2a6e ator)" -- see *n │ │ │ │ -002714c0: 6f74 6520 6d65 7468 6f64 733a 206d 6574 ote methods: met │ │ │ │ -002714d0: 686f 6473 2c20 2d2d 206c 6973 7420 6d65 hods, -- list me │ │ │ │ -002714e0: 7468 6f64 730a 2020 2a20 224d 616e 6970 thods. * "Manip │ │ │ │ -002714f0: 756c 6174 6f72 204e 6574 4669 6c65 2220 ulator NetFile" │ │ │ │ -00271500: 2d2d 2073 6565 202a 6e6f 7465 204e 6574 -- see *note Net │ │ │ │ -00271510: 4669 6c65 3a20 4e65 7446 696c 652c 202d File: NetFile, - │ │ │ │ -00271520: 2d20 7468 6520 636c 6173 7320 6f66 2061 - the class of a │ │ │ │ -00271530: 6c6c 0a20 2020 206e 6574 2066 696c 6573 ll. net files │ │ │ │ -00271540: 0a20 202a 2022 4e65 7446 696c 6520 3c3c . * "NetFile << │ │ │ │ -00271550: 204d 616e 6970 756c 6174 6f72 2220 2d2d Manipulator" -- │ │ │ │ -00271560: 2073 6565 202a 6e6f 7465 204e 6574 4669 see *note NetFi │ │ │ │ -00271570: 6c65 3a20 4e65 7446 696c 652c 202d 2d20 le: NetFile, -- │ │ │ │ -00271580: 7468 6520 636c 6173 7320 6f66 2061 6c6c the class of all │ │ │ │ -00271590: 0a20 2020 206e 6574 2066 696c 6573 0a20 . net files. │ │ │ │ -002715a0: 202a 2022 4669 6c65 203c 3c20 4d61 6e69 * "File << Mani │ │ │ │ -002715b0: 7075 6c61 746f 7222 202d 2d20 7365 6520 pulator" -- see │ │ │ │ -002715c0: 2a6e 6f74 6520 7072 696e 7469 6e67 2074 *note printing t │ │ │ │ -002715d0: 6f20 6120 6669 6c65 3a20 7072 696e 7469 o a file: printi │ │ │ │ -002715e0: 6e67 2074 6f20 6120 6669 6c65 2c0a 2020 ng to a file,. │ │ │ │ -002715f0: 2020 2d2d 2070 7269 6e74 2074 6f20 6120 -- print to a │ │ │ │ -00271600: 6669 6c65 0a20 202a 2022 4e6f 7468 696e file. * "Nothin │ │ │ │ -00271610: 6720 3c3c 204d 616e 6970 756c 6174 6f72 g << Manipulator │ │ │ │ -00271620: 2220 2d2d 2073 6565 202a 6e6f 7465 2070 " -- see *note p │ │ │ │ -00271630: 7269 6e74 696e 6720 746f 2061 2066 696c rinting to a fil │ │ │ │ -00271640: 653a 2070 7269 6e74 696e 6720 746f 2061 e: printing to a │ │ │ │ -00271650: 0a20 2020 2066 696c 652c 202d 2d20 7072 . file, -- pr │ │ │ │ -00271660: 696e 7420 746f 2061 2066 696c 650a 0a46 int to a file..F │ │ │ │ -00271670: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00271680: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00271690: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -002716a0: 202a 6e6f 7465 2065 6e64 6c3a 2065 6e64 *note endl: end │ │ │ │ -002716b0: 6c2c 2069 7320 6120 2a6e 6f74 6520 6d61 l, is a *note ma │ │ │ │ -002716c0: 6e69 7075 6c61 746f 723a 204d 616e 6970 nipulator: Manip │ │ │ │ -002716d0: 756c 6174 6f72 2c2e 0a0a 2d2d 2d2d 2d2d ulator,...------ │ │ │ │ -002716e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002716f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271070: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00271080: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00271090: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +002710a0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +002710b0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +002710c0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +002710d0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +002710e0: 4d61 6361 756c 6179 3244 6f63 2f6f 765f Macaulay2Doc/ov_ │ │ │ │ +002710f0: 7379 7374 656d 2e6d 323a 3438 323a 302e system.m2:482:0. │ │ │ │ +00271100: 0a1f 0a46 696c 653a 204d 6163 6175 6c61 ...File: Macaula │ │ │ │ +00271110: 7932 446f 632e 696e 666f 2c20 4e6f 6465 y2Doc.info, Node │ │ │ │ +00271120: 3a20 656e 646c 2c20 4e65 7874 3a20 666c : endl, Next: fl │ │ │ │ +00271130: 7573 682c 2050 7265 763a 206f 7065 6e4f ush, Prev: openO │ │ │ │ +00271140: 7574 4170 7065 6e64 5f6c 7053 7472 696e utAppend_lpStrin │ │ │ │ +00271150: 675f 7270 2c20 5570 3a20 7573 696e 6720 g_rp, Up: using │ │ │ │ +00271160: 736f 636b 6574 730a 0a65 6e64 6c20 2d2d sockets..endl -- │ │ │ │ +00271170: 2065 6e64 2061 6e20 6f75 7470 7574 206c end an output l │ │ │ │ +00271180: 696e 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ine.************ │ │ │ │ +00271190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +002711a0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +002711b0: 3d3d 3d3d 3d3d 3d0a 0a66 203c 3c20 656e =======..f << en │ │ │ │ +002711c0: 646c 202d 2d20 656e 6473 2074 6865 206c dl -- ends the l │ │ │ │ +002711d0: 696e 6520 6375 7272 656e 746c 7920 6265 ine currently be │ │ │ │ +002711e0: 696e 6720 7075 7420 6f75 7420 746f 2074 ing put out to t │ │ │ │ +002711f0: 6865 2066 696c 6520 662e 0a0a 0a49 7420 he file f....It │ │ │ │ +00271200: 6973 2061 6e20 6573 7365 6e74 6961 6c20 is an essential │ │ │ │ +00271210: 706f 7274 6162 6c65 2070 726f 6772 616d portable program │ │ │ │ +00271220: 6d69 6e67 2070 7261 6374 6963 6520 746f ming practice to │ │ │ │ +00271230: 2075 7365 2065 6e64 6c20 616c 7761 7973 use endl always │ │ │ │ +00271240: 2c20 666f 720a 7772 6974 696e 6720 6e65 , for.writing ne │ │ │ │ +00271250: 776c 696e 6520 6368 6172 6163 7465 7273 wline characters │ │ │ │ +00271260: 2028 7365 6520 2a6e 6f74 6520 6e65 776c (see *note newl │ │ │ │ +00271270: 696e 653a 206e 6577 6c69 6e65 2c29 2074 ine: newline,) t │ │ │ │ +00271280: 6f20 6120 6669 6c65 2077 696c 6c20 6e6f o a file will no │ │ │ │ +00271290: 740a 7465 726d 696e 6174 6520 6120 6c69 t.terminate a li │ │ │ │ +002712a0: 6e65 2063 6f6e 7461 696e 696e 6720 6e65 ne containing ne │ │ │ │ +002712b0: 7473 2070 726f 7065 726c 792c 2061 6e64 ts properly, and │ │ │ │ +002712c0: 2069 7420 7769 6c6c 206e 6f74 2066 6c75 it will not flu │ │ │ │ +002712d0: 7368 2074 6865 206f 7574 7075 740a 6275 sh the output.bu │ │ │ │ +002712e0: 6666 6572 2e0a 0a57 6179 7320 746f 2075 ffer...Ways to u │ │ │ │ +002712f0: 7365 2065 6e64 6c3a 0a3d 3d3d 3d3d 3d3d se endl:.======= │ │ │ │ +00271300: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00271310: 224d 616e 6970 756c 6174 6f72 2044 6174 "Manipulator Dat │ │ │ │ +00271320: 6162 6173 6522 202d 2d20 7365 6520 2a6e abase" -- see *n │ │ │ │ +00271330: 6f74 6520 4d61 6e69 7075 6c61 746f 723a ote Manipulator: │ │ │ │ +00271340: 204d 616e 6970 756c 6174 6f72 2c20 2d2d Manipulator, -- │ │ │ │ +00271350: 2074 6865 2063 6c61 7373 0a20 2020 206f the class. o │ │ │ │ +00271360: 6620 616c 6c20 6669 6c65 206d 616e 6970 f all file manip │ │ │ │ +00271370: 756c 6174 6f72 730a 2020 2a20 224d 616e ulators. * "Man │ │ │ │ +00271380: 6970 756c 6174 6f72 2046 696c 6522 202d ipulator File" - │ │ │ │ +00271390: 2d20 7365 6520 2a6e 6f74 6520 4d61 6e69 - see *note Mani │ │ │ │ +002713a0: 7075 6c61 746f 723a 204d 616e 6970 756c pulator: Manipul │ │ │ │ +002713b0: 6174 6f72 2c20 2d2d 2074 6865 2063 6c61 ator, -- the cla │ │ │ │ +002713c0: 7373 206f 660a 2020 2020 616c 6c20 6669 ss of. all fi │ │ │ │ +002713d0: 6c65 206d 616e 6970 756c 6174 6f72 730a le manipulators. │ │ │ │ +002713e0: 2020 2a20 224d 616e 6970 756c 6174 6f72 * "Manipulator │ │ │ │ +002713f0: 204e 6f74 6869 6e67 2220 2d2d 2073 6565 Nothing" -- see │ │ │ │ +00271400: 202a 6e6f 7465 204d 616e 6970 756c 6174 *note Manipulat │ │ │ │ +00271410: 6f72 3a20 4d61 6e69 7075 6c61 746f 722c or: Manipulator, │ │ │ │ +00271420: 202d 2d20 7468 6520 636c 6173 730a 2020 -- the class. │ │ │ │ +00271430: 2020 6f66 2061 6c6c 2066 696c 6520 6d61 of all file ma │ │ │ │ +00271440: 6e69 7075 6c61 746f 7273 0a20 202a 2022 nipulators. * " │ │ │ │ +00271450: 6e65 7720 4d61 6e69 7075 6c61 746f 7220 new Manipulator │ │ │ │ +00271460: 6672 6f6d 2046 756e 6374 696f 6e22 202d from Function" - │ │ │ │ +00271470: 2d20 7365 6520 2a6e 6f74 6520 4d61 6e69 - see *note Mani │ │ │ │ +00271480: 7075 6c61 746f 723a 204d 616e 6970 756c pulator: Manipul │ │ │ │ +00271490: 6174 6f72 2c20 2d2d 0a20 2020 2074 6865 ator, --. the │ │ │ │ +002714a0: 2063 6c61 7373 206f 6620 616c 6c20 6669 class of all fi │ │ │ │ +002714b0: 6c65 206d 616e 6970 756c 6174 6f72 730a le manipulators. │ │ │ │ +002714c0: 2020 2a20 226d 6574 686f 6473 284d 616e * "methods(Man │ │ │ │ +002714d0: 6970 756c 6174 6f72 2922 202d 2d20 7365 ipulator)" -- se │ │ │ │ +002714e0: 6520 2a6e 6f74 6520 6d65 7468 6f64 733a e *note methods: │ │ │ │ +002714f0: 206d 6574 686f 6473 2c20 2d2d 206c 6973 methods, -- lis │ │ │ │ +00271500: 7420 6d65 7468 6f64 730a 2020 2a20 224d t methods. * "M │ │ │ │ +00271510: 616e 6970 756c 6174 6f72 204e 6574 4669 anipulator NetFi │ │ │ │ +00271520: 6c65 2220 2d2d 2073 6565 202a 6e6f 7465 le" -- see *note │ │ │ │ +00271530: 204e 6574 4669 6c65 3a20 4e65 7446 696c NetFile: NetFil │ │ │ │ +00271540: 652c 202d 2d20 7468 6520 636c 6173 7320 e, -- the class │ │ │ │ +00271550: 6f66 2061 6c6c 0a20 2020 206e 6574 2066 of all. net f │ │ │ │ +00271560: 696c 6573 0a20 202a 2022 4e65 7446 696c iles. * "NetFil │ │ │ │ +00271570: 6520 3c3c 204d 616e 6970 756c 6174 6f72 e << Manipulator │ │ │ │ +00271580: 2220 2d2d 2073 6565 202a 6e6f 7465 204e " -- see *note N │ │ │ │ +00271590: 6574 4669 6c65 3a20 4e65 7446 696c 652c etFile: NetFile, │ │ │ │ +002715a0: 202d 2d20 7468 6520 636c 6173 7320 6f66 -- the class of │ │ │ │ +002715b0: 2061 6c6c 0a20 2020 206e 6574 2066 696c all. net fil │ │ │ │ +002715c0: 6573 0a20 202a 2022 4669 6c65 203c 3c20 es. * "File << │ │ │ │ +002715d0: 4d61 6e69 7075 6c61 746f 7222 202d 2d20 Manipulator" -- │ │ │ │ +002715e0: 7365 6520 2a6e 6f74 6520 7072 696e 7469 see *note printi │ │ │ │ +002715f0: 6e67 2074 6f20 6120 6669 6c65 3a20 7072 ng to a file: pr │ │ │ │ +00271600: 696e 7469 6e67 2074 6f20 6120 6669 6c65 inting to a file │ │ │ │ +00271610: 2c0a 2020 2020 2d2d 2070 7269 6e74 2074 ,. -- print t │ │ │ │ +00271620: 6f20 6120 6669 6c65 0a20 202a 2022 4e6f o a file. * "No │ │ │ │ +00271630: 7468 696e 6720 3c3c 204d 616e 6970 756c thing << Manipul │ │ │ │ +00271640: 6174 6f72 2220 2d2d 2073 6565 202a 6e6f ator" -- see *no │ │ │ │ +00271650: 7465 2070 7269 6e74 696e 6720 746f 2061 te printing to a │ │ │ │ +00271660: 2066 696c 653a 2070 7269 6e74 696e 6720 file: printing │ │ │ │ +00271670: 746f 2061 0a20 2020 2066 696c 652c 202d to a. file, - │ │ │ │ +00271680: 2d20 7072 696e 7420 746f 2061 2066 696c - print to a fil │ │ │ │ +00271690: 650a 0a46 6f72 2074 6865 2070 726f 6772 e..For the progr │ │ │ │ +002716a0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +002716b0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +002716c0: 6a65 6374 202a 6e6f 7465 2065 6e64 6c3a ject *note endl: │ │ │ │ +002716d0: 2065 6e64 6c2c 2069 7320 6120 2a6e 6f74 endl, is a *not │ │ │ │ +002716e0: 6520 6d61 6e69 7075 6c61 746f 723a 204d e manipulator: M │ │ │ │ +002716f0: 616e 6970 756c 6174 6f72 2c2e 0a0a 2d2d anipulator,...-- │ │ │ │ 00271700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00271710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00271720: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00271730: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00271740: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -00271750: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -00271760: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -00271770: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -00271780: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -00271790: 2f0a 4d61 6361 756c 6179 3244 6f63 2f6f /.Macaulay2Doc/o │ │ │ │ -002717a0: 765f 7379 7374 656d 2e6d 323a 3135 3533 v_system.m2:1553 │ │ │ │ -002717b0: 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 6361 :0....File: Maca │ │ │ │ -002717c0: 756c 6179 3244 6f63 2e69 6e66 6f2c 204e ulay2Doc.info, N │ │ │ │ -002717d0: 6f64 653a 2066 6c75 7368 2c20 4e65 7874 ode: flush, Next │ │ │ │ -002717e0: 3a20 636c 6f73 652c 2050 7265 763a 2065 : close, Prev: e │ │ │ │ -002717f0: 6e64 6c2c 2055 703a 2075 7369 6e67 2073 ndl, Up: using s │ │ │ │ -00271800: 6f63 6b65 7473 0a0a 666c 7573 6820 2d2d ockets..flush -- │ │ │ │ -00271810: 2066 6c75 7368 206f 7574 7075 7420 746f flush output to │ │ │ │ -00271820: 2066 696c 650a 2a2a 2a2a 2a2a 2a2a 2a2a file.********** │ │ │ │ -00271830: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00271840: 2a2a 2a0a 0a44 6573 6372 6970 7469 6f6e ***..Description │ │ │ │ -00271850: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 6620 .===========..f │ │ │ │ -00271860: 3c3c 2066 6c75 7368 202d 2d20 7772 6974 << flush -- writ │ │ │ │ -00271870: 6573 206f 7574 2061 6e79 2062 7566 6665 es out any buffe │ │ │ │ -00271880: 7265 6420 6f75 7470 7574 2066 6f72 2074 red output for t │ │ │ │ -00271890: 6865 206f 7574 7075 7420 6669 6c65 2066 he output file f │ │ │ │ -002718a0: 2e0a 0a57 6179 7320 746f 2075 7365 2066 ...Ways to use f │ │ │ │ -002718b0: 6c75 7368 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d lush:.========== │ │ │ │ -002718c0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 224d ========.. * "M │ │ │ │ -002718d0: 616e 6970 756c 6174 6f72 2044 6174 6162 anipulator Datab │ │ │ │ -002718e0: 6173 6522 202d 2d20 7365 6520 2a6e 6f74 ase" -- see *not │ │ │ │ -002718f0: 6520 4d61 6e69 7075 6c61 746f 723a 204d e Manipulator: M │ │ │ │ -00271900: 616e 6970 756c 6174 6f72 2c20 2d2d 2074 anipulator, -- t │ │ │ │ -00271910: 6865 2063 6c61 7373 0a20 2020 206f 6620 he class. of │ │ │ │ -00271920: 616c 6c20 6669 6c65 206d 616e 6970 756c all file manipul │ │ │ │ -00271930: 6174 6f72 730a 2020 2a20 224d 616e 6970 ators. * "Manip │ │ │ │ -00271940: 756c 6174 6f72 2046 696c 6522 202d 2d20 ulator File" -- │ │ │ │ -00271950: 7365 6520 2a6e 6f74 6520 4d61 6e69 7075 see *note Manipu │ │ │ │ -00271960: 6c61 746f 723a 204d 616e 6970 756c 6174 lator: Manipulat │ │ │ │ -00271970: 6f72 2c20 2d2d 2074 6865 2063 6c61 7373 or, -- the class │ │ │ │ -00271980: 206f 660a 2020 2020 616c 6c20 6669 6c65 of. all file │ │ │ │ -00271990: 206d 616e 6970 756c 6174 6f72 730a 2020 manipulators. │ │ │ │ -002719a0: 2a20 224d 616e 6970 756c 6174 6f72 204e * "Manipulator N │ │ │ │ -002719b0: 6f74 6869 6e67 2220 2d2d 2073 6565 202a othing" -- see * │ │ │ │ -002719c0: 6e6f 7465 204d 616e 6970 756c 6174 6f72 note Manipulator │ │ │ │ -002719d0: 3a20 4d61 6e69 7075 6c61 746f 722c 202d : Manipulator, - │ │ │ │ -002719e0: 2d20 7468 6520 636c 6173 730a 2020 2020 - the class. │ │ │ │ -002719f0: 6f66 2061 6c6c 2066 696c 6520 6d61 6e69 of all file mani │ │ │ │ -00271a00: 7075 6c61 746f 7273 0a20 202a 2022 6e65 pulators. * "ne │ │ │ │ -00271a10: 7720 4d61 6e69 7075 6c61 746f 7220 6672 w Manipulator fr │ │ │ │ -00271a20: 6f6d 2046 756e 6374 696f 6e22 202d 2d20 om Function" -- │ │ │ │ -00271a30: 7365 6520 2a6e 6f74 6520 4d61 6e69 7075 see *note Manipu │ │ │ │ -00271a40: 6c61 746f 723a 204d 616e 6970 756c 6174 lator: Manipulat │ │ │ │ -00271a50: 6f72 2c20 2d2d 0a20 2020 2074 6865 2063 or, --. the c │ │ │ │ -00271a60: 6c61 7373 206f 6620 616c 6c20 6669 6c65 lass of all file │ │ │ │ -00271a70: 206d 616e 6970 756c 6174 6f72 730a 2020 manipulators. │ │ │ │ -00271a80: 2a20 226d 6574 686f 6473 284d 616e 6970 * "methods(Manip │ │ │ │ -00271a90: 756c 6174 6f72 2922 202d 2d20 7365 6520 ulator)" -- see │ │ │ │ -00271aa0: 2a6e 6f74 6520 6d65 7468 6f64 733a 206d *note methods: m │ │ │ │ -00271ab0: 6574 686f 6473 2c20 2d2d 206c 6973 7420 ethods, -- list │ │ │ │ -00271ac0: 6d65 7468 6f64 730a 2020 2a20 224d 616e methods. * "Man │ │ │ │ -00271ad0: 6970 756c 6174 6f72 204e 6574 4669 6c65 ipulator NetFile │ │ │ │ -00271ae0: 2220 2d2d 2073 6565 202a 6e6f 7465 204e " -- see *note N │ │ │ │ -00271af0: 6574 4669 6c65 3a20 4e65 7446 696c 652c etFile: NetFile, │ │ │ │ -00271b00: 202d 2d20 7468 6520 636c 6173 7320 6f66 -- the class of │ │ │ │ -00271b10: 2061 6c6c 0a20 2020 206e 6574 2066 696c all. net fil │ │ │ │ -00271b20: 6573 0a20 202a 2022 4e65 7446 696c 6520 es. * "NetFile │ │ │ │ -00271b30: 3c3c 204d 616e 6970 756c 6174 6f72 2220 << Manipulator" │ │ │ │ -00271b40: 2d2d 2073 6565 202a 6e6f 7465 204e 6574 -- see *note Net │ │ │ │ -00271b50: 4669 6c65 3a20 4e65 7446 696c 652c 202d File: NetFile, - │ │ │ │ -00271b60: 2d20 7468 6520 636c 6173 7320 6f66 2061 - the class of a │ │ │ │ -00271b70: 6c6c 0a20 2020 206e 6574 2066 696c 6573 ll. net files │ │ │ │ -00271b80: 0a20 202a 2022 4669 6c65 203c 3c20 4d61 . * "File << Ma │ │ │ │ -00271b90: 6e69 7075 6c61 746f 7222 202d 2d20 7365 nipulator" -- se │ │ │ │ -00271ba0: 6520 2a6e 6f74 6520 7072 696e 7469 6e67 e *note printing │ │ │ │ -00271bb0: 2074 6f20 6120 6669 6c65 3a20 7072 696e to a file: prin │ │ │ │ -00271bc0: 7469 6e67 2074 6f20 6120 6669 6c65 2c0a ting to a file,. │ │ │ │ -00271bd0: 2020 2020 2d2d 2070 7269 6e74 2074 6f20 -- print to │ │ │ │ -00271be0: 6120 6669 6c65 0a20 202a 2022 4e6f 7468 a file. * "Noth │ │ │ │ -00271bf0: 696e 6720 3c3c 204d 616e 6970 756c 6174 ing << Manipulat │ │ │ │ -00271c00: 6f72 2220 2d2d 2073 6565 202a 6e6f 7465 or" -- see *note │ │ │ │ -00271c10: 2070 7269 6e74 696e 6720 746f 2061 2066 printing to a f │ │ │ │ -00271c20: 696c 653a 2070 7269 6e74 696e 6720 746f ile: printing to │ │ │ │ -00271c30: 2061 0a20 2020 2066 696c 652c 202d 2d20 a. file, -- │ │ │ │ -00271c40: 7072 696e 7420 746f 2061 2066 696c 650a print to a file. │ │ │ │ -00271c50: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00271c60: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00271c70: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00271c80: 6374 202a 6e6f 7465 2066 6c75 7368 3a20 ct *note flush: │ │ │ │ -00271c90: 666c 7573 682c 2069 7320 6120 2a6e 6f74 flush, is a *not │ │ │ │ -00271ca0: 6520 6d61 6e69 7075 6c61 746f 723a 204d e manipulator: M │ │ │ │ -00271cb0: 616e 6970 756c 6174 6f72 2c2e 0a0a 2d2d anipulator,...-- │ │ │ │ -00271cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00271cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00271ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +00271750: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +00271760: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +00271770: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +00271780: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +00271790: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ +002717a0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +002717b0: 6167 6573 2f0a 4d61 6361 756c 6179 3244 ages/.Macaulay2D │ │ │ │ +002717c0: 6f63 2f6f 765f 7379 7374 656d 2e6d 323a oc/ov_system.m2: │ │ │ │ +002717d0: 3135 3533 3a30 2e0a 1f0a 4669 6c65 3a20 1553:0....File: │ │ │ │ +002717e0: 4d61 6361 756c 6179 3244 6f63 2e69 6e66 Macaulay2Doc.inf │ │ │ │ +002717f0: 6f2c 204e 6f64 653a 2066 6c75 7368 2c20 o, Node: flush, │ │ │ │ +00271800: 4e65 7874 3a20 636c 6f73 652c 2050 7265 Next: close, Pre │ │ │ │ +00271810: 763a 2065 6e64 6c2c 2055 703a 2075 7369 v: endl, Up: usi │ │ │ │ +00271820: 6e67 2073 6f63 6b65 7473 0a0a 666c 7573 ng sockets..flus │ │ │ │ +00271830: 6820 2d2d 2066 6c75 7368 206f 7574 7075 h -- flush outpu │ │ │ │ +00271840: 7420 746f 2066 696c 650a 2a2a 2a2a 2a2a t to file.****** │ │ │ │ +00271850: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00271860: 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 6970 *******..Descrip │ │ │ │ +00271870: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00271880: 0a0a 6620 3c3c 2066 6c75 7368 202d 2d20 ..f << flush -- │ │ │ │ +00271890: 7772 6974 6573 206f 7574 2061 6e79 2062 writes out any b │ │ │ │ +002718a0: 7566 6665 7265 6420 6f75 7470 7574 2066 uffered output f │ │ │ │ +002718b0: 6f72 2074 6865 206f 7574 7075 7420 6669 or the output fi │ │ │ │ +002718c0: 6c65 2066 2e0a 0a57 6179 7320 746f 2075 le f...Ways to u │ │ │ │ +002718d0: 7365 2066 6c75 7368 3a0a 3d3d 3d3d 3d3d se flush:.====== │ │ │ │ +002718e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +002718f0: 2a20 224d 616e 6970 756c 6174 6f72 2044 * "Manipulator D │ │ │ │ +00271900: 6174 6162 6173 6522 202d 2d20 7365 6520 atabase" -- see │ │ │ │ +00271910: 2a6e 6f74 6520 4d61 6e69 7075 6c61 746f *note Manipulato │ │ │ │ +00271920: 723a 204d 616e 6970 756c 6174 6f72 2c20 r: Manipulator, │ │ │ │ +00271930: 2d2d 2074 6865 2063 6c61 7373 0a20 2020 -- the class. │ │ │ │ +00271940: 206f 6620 616c 6c20 6669 6c65 206d 616e of all file man │ │ │ │ +00271950: 6970 756c 6174 6f72 730a 2020 2a20 224d ipulators. * "M │ │ │ │ +00271960: 616e 6970 756c 6174 6f72 2046 696c 6522 anipulator File" │ │ │ │ +00271970: 202d 2d20 7365 6520 2a6e 6f74 6520 4d61 -- see *note Ma │ │ │ │ +00271980: 6e69 7075 6c61 746f 723a 204d 616e 6970 nipulator: Manip │ │ │ │ +00271990: 756c 6174 6f72 2c20 2d2d 2074 6865 2063 ulator, -- the c │ │ │ │ +002719a0: 6c61 7373 206f 660a 2020 2020 616c 6c20 lass of. all │ │ │ │ +002719b0: 6669 6c65 206d 616e 6970 756c 6174 6f72 file manipulator │ │ │ │ +002719c0: 730a 2020 2a20 224d 616e 6970 756c 6174 s. * "Manipulat │ │ │ │ +002719d0: 6f72 204e 6f74 6869 6e67 2220 2d2d 2073 or Nothing" -- s │ │ │ │ +002719e0: 6565 202a 6e6f 7465 204d 616e 6970 756c ee *note Manipul │ │ │ │ +002719f0: 6174 6f72 3a20 4d61 6e69 7075 6c61 746f ator: Manipulato │ │ │ │ +00271a00: 722c 202d 2d20 7468 6520 636c 6173 730a r, -- the class. │ │ │ │ +00271a10: 2020 2020 6f66 2061 6c6c 2066 696c 6520 of all file │ │ │ │ +00271a20: 6d61 6e69 7075 6c61 746f 7273 0a20 202a manipulators. * │ │ │ │ +00271a30: 2022 6e65 7720 4d61 6e69 7075 6c61 746f "new Manipulato │ │ │ │ +00271a40: 7220 6672 6f6d 2046 756e 6374 696f 6e22 r from Function" │ │ │ │ +00271a50: 202d 2d20 7365 6520 2a6e 6f74 6520 4d61 -- see *note Ma │ │ │ │ +00271a60: 6e69 7075 6c61 746f 723a 204d 616e 6970 nipulator: Manip │ │ │ │ +00271a70: 756c 6174 6f72 2c20 2d2d 0a20 2020 2074 ulator, --. t │ │ │ │ +00271a80: 6865 2063 6c61 7373 206f 6620 616c 6c20 he class of all │ │ │ │ +00271a90: 6669 6c65 206d 616e 6970 756c 6174 6f72 file manipulator │ │ │ │ +00271aa0: 730a 2020 2a20 226d 6574 686f 6473 284d s. * "methods(M │ │ │ │ +00271ab0: 616e 6970 756c 6174 6f72 2922 202d 2d20 anipulator)" -- │ │ │ │ +00271ac0: 7365 6520 2a6e 6f74 6520 6d65 7468 6f64 see *note method │ │ │ │ +00271ad0: 733a 206d 6574 686f 6473 2c20 2d2d 206c s: methods, -- l │ │ │ │ +00271ae0: 6973 7420 6d65 7468 6f64 730a 2020 2a20 ist methods. * │ │ │ │ +00271af0: 224d 616e 6970 756c 6174 6f72 204e 6574 "Manipulator Net │ │ │ │ +00271b00: 4669 6c65 2220 2d2d 2073 6565 202a 6e6f File" -- see *no │ │ │ │ +00271b10: 7465 204e 6574 4669 6c65 3a20 4e65 7446 te NetFile: NetF │ │ │ │ +00271b20: 696c 652c 202d 2d20 7468 6520 636c 6173 ile, -- the clas │ │ │ │ +00271b30: 7320 6f66 2061 6c6c 0a20 2020 206e 6574 s of all. net │ │ │ │ +00271b40: 2066 696c 6573 0a20 202a 2022 4e65 7446 files. * "NetF │ │ │ │ +00271b50: 696c 6520 3c3c 204d 616e 6970 756c 6174 ile << Manipulat │ │ │ │ +00271b60: 6f72 2220 2d2d 2073 6565 202a 6e6f 7465 or" -- see *note │ │ │ │ +00271b70: 204e 6574 4669 6c65 3a20 4e65 7446 696c NetFile: NetFil │ │ │ │ +00271b80: 652c 202d 2d20 7468 6520 636c 6173 7320 e, -- the class │ │ │ │ +00271b90: 6f66 2061 6c6c 0a20 2020 206e 6574 2066 of all. net f │ │ │ │ +00271ba0: 696c 6573 0a20 202a 2022 4669 6c65 203c iles. * "File < │ │ │ │ +00271bb0: 3c20 4d61 6e69 7075 6c61 746f 7222 202d < Manipulator" - │ │ │ │ +00271bc0: 2d20 7365 6520 2a6e 6f74 6520 7072 696e - see *note prin │ │ │ │ +00271bd0: 7469 6e67 2074 6f20 6120 6669 6c65 3a20 ting to a file: │ │ │ │ +00271be0: 7072 696e 7469 6e67 2074 6f20 6120 6669 printing to a fi │ │ │ │ +00271bf0: 6c65 2c0a 2020 2020 2d2d 2070 7269 6e74 le,. -- print │ │ │ │ +00271c00: 2074 6f20 6120 6669 6c65 0a20 202a 2022 to a file. * " │ │ │ │ +00271c10: 4e6f 7468 696e 6720 3c3c 204d 616e 6970 Nothing << Manip │ │ │ │ +00271c20: 756c 6174 6f72 2220 2d2d 2073 6565 202a ulator" -- see * │ │ │ │ +00271c30: 6e6f 7465 2070 7269 6e74 696e 6720 746f note printing to │ │ │ │ +00271c40: 2061 2066 696c 653a 2070 7269 6e74 696e a file: printin │ │ │ │ +00271c50: 6720 746f 2061 0a20 2020 2066 696c 652c g to a. file, │ │ │ │ +00271c60: 202d 2d20 7072 696e 7420 746f 2061 2066 -- print to a f │ │ │ │ +00271c70: 696c 650a 0a46 6f72 2074 6865 2070 726f ile..For the pro │ │ │ │ +00271c80: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00271c90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00271ca0: 6f62 6a65 6374 202a 6e6f 7465 2066 6c75 object *note flu │ │ │ │ +00271cb0: 7368 3a20 666c 7573 682c 2069 7320 6120 sh: flush, is a │ │ │ │ +00271cc0: 2a6e 6f74 6520 6d61 6e69 7075 6c61 746f *note manipulato │ │ │ │ +00271cd0: 723a 204d 616e 6970 756c 6174 6f72 2c2e r: Manipulator,. │ │ │ │ +00271ce0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ 00271cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00271d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -00271d10: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -00271d20: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -00271d30: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -00271d40: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -00271d50: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ -00271d60: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -00271d70: 6167 6573 2f0a 4d61 6361 756c 6179 3244 ages/.Macaulay2D │ │ │ │ -00271d80: 6f63 2f6f 765f 7379 7374 656d 2e6d 323a oc/ov_system.m2: │ │ │ │ -00271d90: 3135 3431 3a30 2e0a 1f0a 4669 6c65 3a20 1541:0....File: │ │ │ │ -00271da0: 4d61 6361 756c 6179 3244 6f63 2e69 6e66 Macaulay2Doc.inf │ │ │ │ -00271db0: 6f2c 204e 6f64 653a 2063 6c6f 7365 2c20 o, Node: close, │ │ │ │ -00271dc0: 4e65 7874 3a20 636c 6f73 6549 6e2c 2050 Next: closeIn, P │ │ │ │ -00271dd0: 7265 763a 2066 6c75 7368 2c20 5570 3a20 rev: flush, Up: │ │ │ │ -00271de0: 7573 696e 6720 736f 636b 6574 730a 0a63 using sockets..c │ │ │ │ -00271df0: 6c6f 7365 202d 2d20 636c 6f73 6520 6120 lose -- close a │ │ │ │ -00271e00: 6669 6c65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a file.*********** │ │ │ │ -00271e10: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 **********..Desc │ │ │ │ -00271e20: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -00271e30: 3d3d 3d0a 0a66 203c 3c20 636c 6f73 6520 ===..f << close │ │ │ │ -00271e40: 2d2d 2063 6c6f 7365 7320 7468 6520 6669 -- closes the fi │ │ │ │ -00271e50: 6c65 2066 2e0a 636c 6f73 6520 6620 2d2d le f..close f -- │ │ │ │ -00271e60: 2063 6c6f 7365 7320 7468 6520 6669 6c65 closes the file │ │ │ │ -00271e70: 2066 2e0a 0a0a 496e 2074 6865 2063 6173 f....In the cas │ │ │ │ -00271e80: 6520 6f66 2061 6e20 6f75 7470 7574 2066 e of an output f │ │ │ │ -00271e90: 696c 652c 2061 6e79 2062 7566 6665 7265 ile, any buffere │ │ │ │ -00271ea0: 6420 6f75 7470 7574 2069 7320 6669 7273 d output is firs │ │ │ │ -00271eb0: 7420 7772 6974 7465 6e20 746f 2074 6865 t written to the │ │ │ │ -00271ec0: 0a66 696c 652c 2061 6e64 2074 6865 2072 .file, and the r │ │ │ │ -00271ed0: 6574 7572 6e20 7661 6c75 6520 6973 2061 eturn value is a │ │ │ │ -00271ee0: 6e20 696e 7465 6765 722c 206e 6f72 6d61 n integer, norma │ │ │ │ -00271ef0: 6c6c 7920 302c 206f 7220 2d31 206f 6e20 lly 0, or -1 on │ │ │ │ -00271f00: 6572 726f 722c 206f 7220 7468 650a 7265 error, or the.re │ │ │ │ -00271f10: 7475 726e 2073 7461 7475 7320 6f66 2074 turn status of t │ │ │ │ -00271f20: 6865 2063 6869 6c64 2070 726f 6365 7373 he child process │ │ │ │ -00271f30: 2069 6e20 6361 7365 2074 6865 2066 696c in case the fil │ │ │ │ -00271f40: 6520 7761 7320 6120 7069 7065 2e0a 0a0a e was a pipe.... │ │ │ │ -00271f50: 4966 2074 6865 2066 696c 6520 7761 7320 If the file was │ │ │ │ -00271f60: 6f70 656e 2066 6f72 2062 6f74 6820 696e open for both in │ │ │ │ -00271f70: 7075 7420 616e 6420 6f75 7470 7574 2c20 put and output, │ │ │ │ -00271f80: 626f 7468 2064 6972 6563 7469 6f6e 7320 both directions │ │ │ │ -00271f90: 6172 6520 636c 6f73 6564 2e0a 0a0a 4966 are closed....If │ │ │ │ -00271fa0: 2074 6865 2066 696c 6520 6973 2061 2070 the file is a p │ │ │ │ -00271fb0: 6970 6520 746f 2061 6e6f 7468 6572 2070 ipe to another p │ │ │ │ -00271fc0: 726f 6365 7373 2c20 692e 652e 2c20 7468 rocess, i.e., th │ │ │ │ -00271fd0: 6520 6669 6c65 6e61 6d65 2062 6567 616e e filename began │ │ │ │ -00271fe0: 2077 6974 6820 7468 650a 6368 6172 6163 with the.charac │ │ │ │ -00271ff0: 7465 7220 212c 2077 6520 7769 6c6c 2077 ter !, we will w │ │ │ │ -00272000: 6169 7420 666f 7220 7468 6520 7072 6f63 ait for the proc │ │ │ │ -00272010: 6573 7320 746f 2074 6572 6d69 6e61 7465 ess to terminate │ │ │ │ -00272020: 2e20 2049 6620 796f 7520 646f 6e27 7420 . If you don't │ │ │ │ -00272030: 7761 6e74 2074 6f0a 7761 6974 2066 6f72 want to.wait for │ │ │ │ -00272040: 2074 6865 2070 726f 6365 7373 2074 6f20 the process to │ │ │ │ -00272050: 7465 726d 696e 6174 652c 206f 7065 6e20 terminate, open │ │ │ │ -00272060: 7468 6520 6669 6c65 2077 6974 6820 2a6e the file with *n │ │ │ │ -00272070: 6f74 6520 6f70 656e 496e 4f75 743a 0a6f ote openInOut:.o │ │ │ │ -00272080: 7065 6e49 6e4f 7574 2c2c 2061 6e64 2069 penInOut,, and i │ │ │ │ -00272090: 6620 6e65 6365 7373 6172 792c 2075 7365 f necessary, use │ │ │ │ -002720a0: 202a 6e6f 7465 2063 6c6f 7365 496e 3a20 *note closeIn: │ │ │ │ -002720b0: 636c 6f73 6549 6e2c 2074 6f20 636c 6f73 closeIn, to clos │ │ │ │ -002720c0: 6520 6974 2c20 746f 0a69 6e64 6963 6174 e it, to.indicat │ │ │ │ -002720d0: 6520 7468 6174 2069 7420 6861 7320 7265 e that it has re │ │ │ │ -002720e0: 6365 6976 6564 2061 6c6c 2069 7473 2069 ceived all its i │ │ │ │ -002720f0: 6e70 7574 2e0a 0a0a 4966 2074 6865 2066 nput....If the f │ │ │ │ -00272100: 696c 6520 6973 2073 7464 696f 2074 6865 ile is stdio the │ │ │ │ -00272110: 6e20 6974 2069 7320 6c65 6674 206f 7065 n it is left ope │ │ │ │ -00272120: 6e2c 2061 6e64 206e 6f20 6572 726f 7220 n, and no error │ │ │ │ -00272130: 6973 2073 6967 6e61 6c65 642e 0a0a 5761 is signaled...Wa │ │ │ │ -00272140: 7973 2074 6f20 7573 6520 636c 6f73 653a ys to use close: │ │ │ │ -00272150: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00272160: 3d3d 3d0a 0a20 202a 2022 4d61 6e69 7075 ===.. * "Manipu │ │ │ │ -00272170: 6c61 746f 7220 4461 7461 6261 7365 2220 lator Database" │ │ │ │ -00272180: 2d2d 2073 6565 202a 6e6f 7465 204d 616e -- see *note Man │ │ │ │ -00272190: 6970 756c 6174 6f72 3a20 4d61 6e69 7075 ipulator: Manipu │ │ │ │ -002721a0: 6c61 746f 722c 202d 2d20 7468 6520 636c lator, -- the cl │ │ │ │ -002721b0: 6173 730a 2020 2020 6f66 2061 6c6c 2066 ass. of all f │ │ │ │ -002721c0: 696c 6520 6d61 6e69 7075 6c61 746f 7273 ile manipulators │ │ │ │ -002721d0: 0a20 202a 2022 4d61 6e69 7075 6c61 746f . * "Manipulato │ │ │ │ -002721e0: 7220 4669 6c65 2220 2d2d 2073 6565 202a r File" -- see * │ │ │ │ -002721f0: 6e6f 7465 204d 616e 6970 756c 6174 6f72 note Manipulator │ │ │ │ -00272200: 3a20 4d61 6e69 7075 6c61 746f 722c 202d : Manipulator, - │ │ │ │ -00272210: 2d20 7468 6520 636c 6173 7320 6f66 0a20 - the class of. │ │ │ │ -00272220: 2020 2061 6c6c 2066 696c 6520 6d61 6e69 all file mani │ │ │ │ -00272230: 7075 6c61 746f 7273 0a20 202a 2022 4d61 pulators. * "Ma │ │ │ │ -00272240: 6e69 7075 6c61 746f 7220 4e6f 7468 696e nipulator Nothin │ │ │ │ -00272250: 6722 202d 2d20 7365 6520 2a6e 6f74 6520 g" -- see *note │ │ │ │ -00272260: 4d61 6e69 7075 6c61 746f 723a 204d 616e Manipulator: Man │ │ │ │ -00272270: 6970 756c 6174 6f72 2c20 2d2d 2074 6865 ipulator, -- the │ │ │ │ -00272280: 2063 6c61 7373 0a20 2020 206f 6620 616c class. of al │ │ │ │ -00272290: 6c20 6669 6c65 206d 616e 6970 756c 6174 l file manipulat │ │ │ │ -002722a0: 6f72 730a 2020 2a20 226e 6577 204d 616e ors. * "new Man │ │ │ │ -002722b0: 6970 756c 6174 6f72 2066 726f 6d20 4675 ipulator from Fu │ │ │ │ -002722c0: 6e63 7469 6f6e 2220 2d2d 2073 6565 202a nction" -- see * │ │ │ │ -002722d0: 6e6f 7465 204d 616e 6970 756c 6174 6f72 note Manipulator │ │ │ │ -002722e0: 3a20 4d61 6e69 7075 6c61 746f 722c 202d : Manipulator, - │ │ │ │ -002722f0: 2d0a 2020 2020 7468 6520 636c 6173 7320 -. the class │ │ │ │ -00272300: 6f66 2061 6c6c 2066 696c 6520 6d61 6e69 of all file mani │ │ │ │ -00272310: 7075 6c61 746f 7273 0a20 202a 2022 6d65 pulators. * "me │ │ │ │ -00272320: 7468 6f64 7328 4d61 6e69 7075 6c61 746f thods(Manipulato │ │ │ │ -00272330: 7229 2220 2d2d 2073 6565 202a 6e6f 7465 r)" -- see *note │ │ │ │ -00272340: 206d 6574 686f 6473 3a20 6d65 7468 6f64 methods: method │ │ │ │ -00272350: 732c 202d 2d20 6c69 7374 206d 6574 686f s, -- list metho │ │ │ │ -00272360: 6473 0a20 202a 2022 4d61 6e69 7075 6c61 ds. * "Manipula │ │ │ │ -00272370: 746f 7220 4e65 7446 696c 6522 202d 2d20 tor NetFile" -- │ │ │ │ -00272380: 7365 6520 2a6e 6f74 6520 4e65 7446 696c see *note NetFil │ │ │ │ -00272390: 653a 204e 6574 4669 6c65 2c20 2d2d 2074 e: NetFile, -- t │ │ │ │ -002723a0: 6865 2063 6c61 7373 206f 6620 616c 6c0a he class of all. │ │ │ │ -002723b0: 2020 2020 6e65 7420 6669 6c65 730a 2020 net files. │ │ │ │ -002723c0: 2a20 224e 6574 4669 6c65 203c 3c20 4d61 * "NetFile << Ma │ │ │ │ -002723d0: 6e69 7075 6c61 746f 7222 202d 2d20 7365 nipulator" -- se │ │ │ │ -002723e0: 6520 2a6e 6f74 6520 4e65 7446 696c 653a e *note NetFile: │ │ │ │ -002723f0: 204e 6574 4669 6c65 2c20 2d2d 2074 6865 NetFile, -- the │ │ │ │ -00272400: 2063 6c61 7373 206f 6620 616c 6c0a 2020 class of all. │ │ │ │ -00272410: 2020 6e65 7420 6669 6c65 730a 2020 2a20 net files. * │ │ │ │ -00272420: 2246 696c 6520 3c3c 204d 616e 6970 756c "File << Manipul │ │ │ │ -00272430: 6174 6f72 2220 2d2d 2073 6565 202a 6e6f ator" -- see *no │ │ │ │ -00272440: 7465 2070 7269 6e74 696e 6720 746f 2061 te printing to a │ │ │ │ -00272450: 2066 696c 653a 2070 7269 6e74 696e 6720 file: printing │ │ │ │ -00272460: 746f 2061 2066 696c 652c 0a20 2020 202d to a file,. - │ │ │ │ -00272470: 2d20 7072 696e 7420 746f 2061 2066 696c - print to a fil │ │ │ │ -00272480: 650a 2020 2a20 224e 6f74 6869 6e67 203c e. * "Nothing < │ │ │ │ -00272490: 3c20 4d61 6e69 7075 6c61 746f 7222 202d < Manipulator" - │ │ │ │ -002724a0: 2d20 7365 6520 2a6e 6f74 6520 7072 696e - see *note prin │ │ │ │ -002724b0: 7469 6e67 2074 6f20 6120 6669 6c65 3a20 ting to a file: │ │ │ │ -002724c0: 7072 696e 7469 6e67 2074 6f20 610a 2020 printing to a. │ │ │ │ -002724d0: 2020 6669 6c65 2c20 2d2d 2070 7269 6e74 file, -- print │ │ │ │ -002724e0: 2074 6f20 6120 6669 6c65 0a0a 466f 7220 to a file..For │ │ │ │ -002724f0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00272500: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00272510: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00272520: 6f74 6520 636c 6f73 653a 2063 6c6f 7365 ote close: close │ │ │ │ -00272530: 2c20 6973 2061 202a 6e6f 7465 206d 616e , is a *note man │ │ │ │ -00272540: 6970 756c 6174 6f72 3a20 4d61 6e69 7075 ipulator: Manipu │ │ │ │ -00272550: 6c61 746f 722c 2e0a 0a2d 2d2d 2d2d 2d2d lator,...------- │ │ │ │ -00272560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00272570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00271d30: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +00271d40: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +00271d50: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +00271d60: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +00271d70: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +00271d80: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +00271d90: 7061 636b 6167 6573 2f0a 4d61 6361 756c packages/.Macaul │ │ │ │ +00271da0: 6179 3244 6f63 2f6f 765f 7379 7374 656d ay2Doc/ov_system │ │ │ │ +00271db0: 2e6d 323a 3135 3431 3a30 2e0a 1f0a 4669 .m2:1541:0....Fi │ │ │ │ +00271dc0: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ +00271dd0: 2e69 6e66 6f2c 204e 6f64 653a 2063 6c6f .info, Node: clo │ │ │ │ +00271de0: 7365 2c20 4e65 7874 3a20 636c 6f73 6549 se, Next: closeI │ │ │ │ +00271df0: 6e2c 2050 7265 763a 2066 6c75 7368 2c20 n, Prev: flush, │ │ │ │ +00271e00: 5570 3a20 7573 696e 6720 736f 636b 6574 Up: using socket │ │ │ │ +00271e10: 730a 0a63 6c6f 7365 202d 2d20 636c 6f73 s..close -- clos │ │ │ │ +00271e20: 6520 6120 6669 6c65 0a2a 2a2a 2a2a 2a2a e a file.******* │ │ │ │ +00271e30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00271e40: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00271e50: 3d3d 3d3d 3d3d 3d0a 0a66 203c 3c20 636c =======..f << cl │ │ │ │ +00271e60: 6f73 6520 2d2d 2063 6c6f 7365 7320 7468 ose -- closes th │ │ │ │ +00271e70: 6520 6669 6c65 2066 2e0a 636c 6f73 6520 e file f..close │ │ │ │ +00271e80: 6620 2d2d 2063 6c6f 7365 7320 7468 6520 f -- closes the │ │ │ │ +00271e90: 6669 6c65 2066 2e0a 0a0a 496e 2074 6865 file f....In the │ │ │ │ +00271ea0: 2063 6173 6520 6f66 2061 6e20 6f75 7470 case of an outp │ │ │ │ +00271eb0: 7574 2066 696c 652c 2061 6e79 2062 7566 ut file, any buf │ │ │ │ +00271ec0: 6665 7265 6420 6f75 7470 7574 2069 7320 fered output is │ │ │ │ +00271ed0: 6669 7273 7420 7772 6974 7465 6e20 746f first written to │ │ │ │ +00271ee0: 2074 6865 0a66 696c 652c 2061 6e64 2074 the.file, and t │ │ │ │ +00271ef0: 6865 2072 6574 7572 6e20 7661 6c75 6520 he return value │ │ │ │ +00271f00: 6973 2061 6e20 696e 7465 6765 722c 206e is an integer, n │ │ │ │ +00271f10: 6f72 6d61 6c6c 7920 302c 206f 7220 2d31 ormally 0, or -1 │ │ │ │ +00271f20: 206f 6e20 6572 726f 722c 206f 7220 7468 on error, or th │ │ │ │ +00271f30: 650a 7265 7475 726e 2073 7461 7475 7320 e.return status │ │ │ │ +00271f40: 6f66 2074 6865 2063 6869 6c64 2070 726f of the child pro │ │ │ │ +00271f50: 6365 7373 2069 6e20 6361 7365 2074 6865 cess in case the │ │ │ │ +00271f60: 2066 696c 6520 7761 7320 6120 7069 7065 file was a pipe │ │ │ │ +00271f70: 2e0a 0a0a 4966 2074 6865 2066 696c 6520 ....If the file │ │ │ │ +00271f80: 7761 7320 6f70 656e 2066 6f72 2062 6f74 was open for bot │ │ │ │ +00271f90: 6820 696e 7075 7420 616e 6420 6f75 7470 h input and outp │ │ │ │ +00271fa0: 7574 2c20 626f 7468 2064 6972 6563 7469 ut, both directi │ │ │ │ +00271fb0: 6f6e 7320 6172 6520 636c 6f73 6564 2e0a ons are closed.. │ │ │ │ +00271fc0: 0a0a 4966 2074 6865 2066 696c 6520 6973 ..If the file is │ │ │ │ +00271fd0: 2061 2070 6970 6520 746f 2061 6e6f 7468 a pipe to anoth │ │ │ │ +00271fe0: 6572 2070 726f 6365 7373 2c20 692e 652e er process, i.e. │ │ │ │ +00271ff0: 2c20 7468 6520 6669 6c65 6e61 6d65 2062 , the filename b │ │ │ │ +00272000: 6567 616e 2077 6974 6820 7468 650a 6368 egan with the.ch │ │ │ │ +00272010: 6172 6163 7465 7220 212c 2077 6520 7769 aracter !, we wi │ │ │ │ +00272020: 6c6c 2077 6169 7420 666f 7220 7468 6520 ll wait for the │ │ │ │ +00272030: 7072 6f63 6573 7320 746f 2074 6572 6d69 process to termi │ │ │ │ +00272040: 6e61 7465 2e20 2049 6620 796f 7520 646f nate. If you do │ │ │ │ +00272050: 6e27 7420 7761 6e74 2074 6f0a 7761 6974 n't want to.wait │ │ │ │ +00272060: 2066 6f72 2074 6865 2070 726f 6365 7373 for the process │ │ │ │ +00272070: 2074 6f20 7465 726d 696e 6174 652c 206f to terminate, o │ │ │ │ +00272080: 7065 6e20 7468 6520 6669 6c65 2077 6974 pen the file wit │ │ │ │ +00272090: 6820 2a6e 6f74 6520 6f70 656e 496e 4f75 h *note openInOu │ │ │ │ +002720a0: 743a 0a6f 7065 6e49 6e4f 7574 2c2c 2061 t:.openInOut,, a │ │ │ │ +002720b0: 6e64 2069 6620 6e65 6365 7373 6172 792c nd if necessary, │ │ │ │ +002720c0: 2075 7365 202a 6e6f 7465 2063 6c6f 7365 use *note close │ │ │ │ +002720d0: 496e 3a20 636c 6f73 6549 6e2c 2074 6f20 In: closeIn, to │ │ │ │ +002720e0: 636c 6f73 6520 6974 2c20 746f 0a69 6e64 close it, to.ind │ │ │ │ +002720f0: 6963 6174 6520 7468 6174 2069 7420 6861 icate that it ha │ │ │ │ +00272100: 7320 7265 6365 6976 6564 2061 6c6c 2069 s received all i │ │ │ │ +00272110: 7473 2069 6e70 7574 2e0a 0a0a 4966 2074 ts input....If t │ │ │ │ +00272120: 6865 2066 696c 6520 6973 2073 7464 696f he file is stdio │ │ │ │ +00272130: 2074 6865 6e20 6974 2069 7320 6c65 6674 then it is left │ │ │ │ +00272140: 206f 7065 6e2c 2061 6e64 206e 6f20 6572 open, and no er │ │ │ │ +00272150: 726f 7220 6973 2073 6967 6e61 6c65 642e ror is signaled. │ │ │ │ +00272160: 0a0a 5761 7973 2074 6f20 7573 6520 636c ..Ways to use cl │ │ │ │ +00272170: 6f73 653a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ose:.=========== │ │ │ │ +00272180: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4d61 =======.. * "Ma │ │ │ │ +00272190: 6e69 7075 6c61 746f 7220 4461 7461 6261 nipulator Databa │ │ │ │ +002721a0: 7365 2220 2d2d 2073 6565 202a 6e6f 7465 se" -- see *note │ │ │ │ +002721b0: 204d 616e 6970 756c 6174 6f72 3a20 4d61 Manipulator: Ma │ │ │ │ +002721c0: 6e69 7075 6c61 746f 722c 202d 2d20 7468 nipulator, -- th │ │ │ │ +002721d0: 6520 636c 6173 730a 2020 2020 6f66 2061 e class. of a │ │ │ │ +002721e0: 6c6c 2066 696c 6520 6d61 6e69 7075 6c61 ll file manipula │ │ │ │ +002721f0: 746f 7273 0a20 202a 2022 4d61 6e69 7075 tors. * "Manipu │ │ │ │ +00272200: 6c61 746f 7220 4669 6c65 2220 2d2d 2073 lator File" -- s │ │ │ │ +00272210: 6565 202a 6e6f 7465 204d 616e 6970 756c ee *note Manipul │ │ │ │ +00272220: 6174 6f72 3a20 4d61 6e69 7075 6c61 746f ator: Manipulato │ │ │ │ +00272230: 722c 202d 2d20 7468 6520 636c 6173 7320 r, -- the class │ │ │ │ +00272240: 6f66 0a20 2020 2061 6c6c 2066 696c 6520 of. all file │ │ │ │ +00272250: 6d61 6e69 7075 6c61 746f 7273 0a20 202a manipulators. * │ │ │ │ +00272260: 2022 4d61 6e69 7075 6c61 746f 7220 4e6f "Manipulator No │ │ │ │ +00272270: 7468 696e 6722 202d 2d20 7365 6520 2a6e thing" -- see *n │ │ │ │ +00272280: 6f74 6520 4d61 6e69 7075 6c61 746f 723a ote Manipulator: │ │ │ │ +00272290: 204d 616e 6970 756c 6174 6f72 2c20 2d2d Manipulator, -- │ │ │ │ +002722a0: 2074 6865 2063 6c61 7373 0a20 2020 206f the class. o │ │ │ │ +002722b0: 6620 616c 6c20 6669 6c65 206d 616e 6970 f all file manip │ │ │ │ +002722c0: 756c 6174 6f72 730a 2020 2a20 226e 6577 ulators. * "new │ │ │ │ +002722d0: 204d 616e 6970 756c 6174 6f72 2066 726f Manipulator fro │ │ │ │ +002722e0: 6d20 4675 6e63 7469 6f6e 2220 2d2d 2073 m Function" -- s │ │ │ │ +002722f0: 6565 202a 6e6f 7465 204d 616e 6970 756c ee *note Manipul │ │ │ │ +00272300: 6174 6f72 3a20 4d61 6e69 7075 6c61 746f ator: Manipulato │ │ │ │ +00272310: 722c 202d 2d0a 2020 2020 7468 6520 636c r, --. the cl │ │ │ │ +00272320: 6173 7320 6f66 2061 6c6c 2066 696c 6520 ass of all file │ │ │ │ +00272330: 6d61 6e69 7075 6c61 746f 7273 0a20 202a manipulators. * │ │ │ │ +00272340: 2022 6d65 7468 6f64 7328 4d61 6e69 7075 "methods(Manipu │ │ │ │ +00272350: 6c61 746f 7229 2220 2d2d 2073 6565 202a lator)" -- see * │ │ │ │ +00272360: 6e6f 7465 206d 6574 686f 6473 3a20 6d65 note methods: me │ │ │ │ +00272370: 7468 6f64 732c 202d 2d20 6c69 7374 206d thods, -- list m │ │ │ │ +00272380: 6574 686f 6473 0a20 202a 2022 4d61 6e69 ethods. * "Mani │ │ │ │ +00272390: 7075 6c61 746f 7220 4e65 7446 696c 6522 pulator NetFile" │ │ │ │ +002723a0: 202d 2d20 7365 6520 2a6e 6f74 6520 4e65 -- see *note Ne │ │ │ │ +002723b0: 7446 696c 653a 204e 6574 4669 6c65 2c20 tFile: NetFile, │ │ │ │ +002723c0: 2d2d 2074 6865 2063 6c61 7373 206f 6620 -- the class of │ │ │ │ +002723d0: 616c 6c0a 2020 2020 6e65 7420 6669 6c65 all. net file │ │ │ │ +002723e0: 730a 2020 2a20 224e 6574 4669 6c65 203c s. * "NetFile < │ │ │ │ +002723f0: 3c20 4d61 6e69 7075 6c61 746f 7222 202d < Manipulator" - │ │ │ │ +00272400: 2d20 7365 6520 2a6e 6f74 6520 4e65 7446 - see *note NetF │ │ │ │ +00272410: 696c 653a 204e 6574 4669 6c65 2c20 2d2d ile: NetFile, -- │ │ │ │ +00272420: 2074 6865 2063 6c61 7373 206f 6620 616c the class of al │ │ │ │ +00272430: 6c0a 2020 2020 6e65 7420 6669 6c65 730a l. net files. │ │ │ │ +00272440: 2020 2a20 2246 696c 6520 3c3c 204d 616e * "File << Man │ │ │ │ +00272450: 6970 756c 6174 6f72 2220 2d2d 2073 6565 ipulator" -- see │ │ │ │ +00272460: 202a 6e6f 7465 2070 7269 6e74 696e 6720 *note printing │ │ │ │ +00272470: 746f 2061 2066 696c 653a 2070 7269 6e74 to a file: print │ │ │ │ +00272480: 696e 6720 746f 2061 2066 696c 652c 0a20 ing to a file,. │ │ │ │ +00272490: 2020 202d 2d20 7072 696e 7420 746f 2061 -- print to a │ │ │ │ +002724a0: 2066 696c 650a 2020 2a20 224e 6f74 6869 file. * "Nothi │ │ │ │ +002724b0: 6e67 203c 3c20 4d61 6e69 7075 6c61 746f ng << Manipulato │ │ │ │ +002724c0: 7222 202d 2d20 7365 6520 2a6e 6f74 6520 r" -- see *note │ │ │ │ +002724d0: 7072 696e 7469 6e67 2074 6f20 6120 6669 printing to a fi │ │ │ │ +002724e0: 6c65 3a20 7072 696e 7469 6e67 2074 6f20 le: printing to │ │ │ │ +002724f0: 610a 2020 2020 6669 6c65 2c20 2d2d 2070 a. file, -- p │ │ │ │ +00272500: 7269 6e74 2074 6f20 6120 6669 6c65 0a0a rint to a file.. │ │ │ │ +00272510: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +00272520: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +00272530: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +00272540: 7420 2a6e 6f74 6520 636c 6f73 653a 2063 t *note close: c │ │ │ │ +00272550: 6c6f 7365 2c20 6973 2061 202a 6e6f 7465 lose, is a *note │ │ │ │ +00272560: 206d 616e 6970 756c 6174 6f72 3a20 4d61 manipulator: Ma │ │ │ │ +00272570: 6e69 7075 6c61 746f 722c 2e0a 0a2d 2d2d nipulator,...--- │ │ │ │ 00272580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00272590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002725a0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -002725b0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -002725c0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -002725d0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -002725e0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -002725f0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -00272600: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00272610: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ -00272620: 5f73 7973 7465 6d2e 6d32 3a31 3438 333a _system.m2:1483: │ │ │ │ -00272630: 302e 0a1f 0a46 696c 653a 204d 6163 6175 0....File: Macau │ │ │ │ -00272640: 6c61 7932 446f 632e 696e 666f 2c20 4e6f lay2Doc.info, No │ │ │ │ -00272650: 6465 3a20 636c 6f73 6549 6e2c 204e 6578 de: closeIn, Nex │ │ │ │ -00272660: 743a 2063 6c6f 7365 4f75 742c 2050 7265 t: closeOut, Pre │ │ │ │ -00272670: 763a 2063 6c6f 7365 2c20 5570 3a20 7573 v: close, Up: us │ │ │ │ -00272680: 696e 6720 736f 636b 6574 730a 0a63 6c6f ing sockets..clo │ │ │ │ -00272690: 7365 496e 202d 2d20 636c 6f73 6520 616e seIn -- close an │ │ │ │ -002726a0: 2069 6e70 7574 2066 696c 650a 2a2a 2a2a input file.**** │ │ │ │ -002726b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002726c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 **********..Desc │ │ │ │ -002726d0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -002726e0: 3d3d 3d0a 0a66 203c 3c20 636c 6f73 6549 ===..f << closeI │ │ │ │ -002726f0: 6e20 2d2d 2063 6c6f 7365 7320 7468 6520 n -- closes the │ │ │ │ -00272700: 696e 7075 7420 6669 6c65 2066 2e0a 636c input file f..cl │ │ │ │ -00272710: 6f73 6549 6e20 6620 2d2d 2063 6c6f 7365 oseIn f -- close │ │ │ │ -00272720: 7320 7468 6520 696e 7075 7420 6669 6c65 s the input file │ │ │ │ -00272730: 2066 2e0a 0a0a 4966 2074 6865 2066 696c f....If the fil │ │ │ │ -00272740: 6520 7761 7320 6f70 656e 206f 6e6c 7920 e was open only │ │ │ │ -00272750: 666f 7220 696e 7075 742c 2074 6865 6e20 for input, then │ │ │ │ -00272760: 2a6e 6f74 6520 636c 6f73 653a 2063 6c6f *note close: clo │ │ │ │ -00272770: 7365 2c20 6973 2065 6173 6965 7220 746f se, is easier to │ │ │ │ -00272780: 2075 7365 0a61 6e64 2068 6173 2074 6865 use.and has the │ │ │ │ -00272790: 2073 616d 6520 6566 6665 6374 2e0a 0a0a same effect.... │ │ │ │ -002727a0: 4966 2074 6865 2066 696c 6520 7761 7320 If the file was │ │ │ │ -002727b0: 6f70 656e 2066 6f72 2062 6f74 6820 696e open for both in │ │ │ │ -002727c0: 7075 7420 616e 6420 6f75 7470 7574 2c20 put and output, │ │ │ │ -002727d0: 6974 2072 656d 6169 6e73 206f 7065 6e20 it remains open │ │ │ │ -002727e0: 666f 7220 6f75 7470 7574 2e0a 0a57 6179 for output...Way │ │ │ │ -002727f0: 7320 746f 2075 7365 2063 6c6f 7365 496e s to use closeIn │ │ │ │ -00272800: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00272810: 3d3d 3d3d 3d3d 0a0a 2020 2a20 224d 616e ======.. * "Man │ │ │ │ -00272820: 6970 756c 6174 6f72 2044 6174 6162 6173 ipulator Databas │ │ │ │ -00272830: 6522 202d 2d20 7365 6520 2a6e 6f74 6520 e" -- see *note │ │ │ │ -00272840: 4d61 6e69 7075 6c61 746f 723a 204d 616e Manipulator: Man │ │ │ │ -00272850: 6970 756c 6174 6f72 2c20 2d2d 2074 6865 ipulator, -- the │ │ │ │ -00272860: 2063 6c61 7373 0a20 2020 206f 6620 616c class. of al │ │ │ │ -00272870: 6c20 6669 6c65 206d 616e 6970 756c 6174 l file manipulat │ │ │ │ -00272880: 6f72 730a 2020 2a20 224d 616e 6970 756c ors. * "Manipul │ │ │ │ -00272890: 6174 6f72 2046 696c 6522 202d 2d20 7365 ator File" -- se │ │ │ │ -002728a0: 6520 2a6e 6f74 6520 4d61 6e69 7075 6c61 e *note Manipula │ │ │ │ -002728b0: 746f 723a 204d 616e 6970 756c 6174 6f72 tor: Manipulator │ │ │ │ -002728c0: 2c20 2d2d 2074 6865 2063 6c61 7373 206f , -- the class o │ │ │ │ -002728d0: 660a 2020 2020 616c 6c20 6669 6c65 206d f. all file m │ │ │ │ -002728e0: 616e 6970 756c 6174 6f72 730a 2020 2a20 anipulators. * │ │ │ │ -002728f0: 224d 616e 6970 756c 6174 6f72 204e 6f74 "Manipulator Not │ │ │ │ -00272900: 6869 6e67 2220 2d2d 2073 6565 202a 6e6f hing" -- see *no │ │ │ │ -00272910: 7465 204d 616e 6970 756c 6174 6f72 3a20 te Manipulator: │ │ │ │ -00272920: 4d61 6e69 7075 6c61 746f 722c 202d 2d20 Manipulator, -- │ │ │ │ -00272930: 7468 6520 636c 6173 730a 2020 2020 6f66 the class. of │ │ │ │ -00272940: 2061 6c6c 2066 696c 6520 6d61 6e69 7075 all file manipu │ │ │ │ -00272950: 6c61 746f 7273 0a20 202a 2022 6e65 7720 lators. * "new │ │ │ │ -00272960: 4d61 6e69 7075 6c61 746f 7220 6672 6f6d Manipulator from │ │ │ │ -00272970: 2046 756e 6374 696f 6e22 202d 2d20 7365 Function" -- se │ │ │ │ -00272980: 6520 2a6e 6f74 6520 4d61 6e69 7075 6c61 e *note Manipula │ │ │ │ -00272990: 746f 723a 204d 616e 6970 756c 6174 6f72 tor: Manipulator │ │ │ │ -002729a0: 2c20 2d2d 0a20 2020 2074 6865 2063 6c61 , --. the cla │ │ │ │ -002729b0: 7373 206f 6620 616c 6c20 6669 6c65 206d ss of all file m │ │ │ │ -002729c0: 616e 6970 756c 6174 6f72 730a 2020 2a20 anipulators. * │ │ │ │ -002729d0: 226d 6574 686f 6473 284d 616e 6970 756c "methods(Manipul │ │ │ │ -002729e0: 6174 6f72 2922 202d 2d20 7365 6520 2a6e ator)" -- see *n │ │ │ │ -002729f0: 6f74 6520 6d65 7468 6f64 733a 206d 6574 ote methods: met │ │ │ │ -00272a00: 686f 6473 2c20 2d2d 206c 6973 7420 6d65 hods, -- list me │ │ │ │ -00272a10: 7468 6f64 730a 2020 2a20 224d 616e 6970 thods. * "Manip │ │ │ │ -00272a20: 756c 6174 6f72 204e 6574 4669 6c65 2220 ulator NetFile" │ │ │ │ -00272a30: 2d2d 2073 6565 202a 6e6f 7465 204e 6574 -- see *note Net │ │ │ │ -00272a40: 4669 6c65 3a20 4e65 7446 696c 652c 202d File: NetFile, - │ │ │ │ -00272a50: 2d20 7468 6520 636c 6173 7320 6f66 2061 - the class of a │ │ │ │ -00272a60: 6c6c 0a20 2020 206e 6574 2066 696c 6573 ll. net files │ │ │ │ -00272a70: 0a20 202a 2022 4e65 7446 696c 6520 3c3c . * "NetFile << │ │ │ │ -00272a80: 204d 616e 6970 756c 6174 6f72 2220 2d2d Manipulator" -- │ │ │ │ -00272a90: 2073 6565 202a 6e6f 7465 204e 6574 4669 see *note NetFi │ │ │ │ -00272aa0: 6c65 3a20 4e65 7446 696c 652c 202d 2d20 le: NetFile, -- │ │ │ │ -00272ab0: 7468 6520 636c 6173 7320 6f66 2061 6c6c the class of all │ │ │ │ -00272ac0: 0a20 2020 206e 6574 2066 696c 6573 0a20 . net files. │ │ │ │ -00272ad0: 202a 2022 4669 6c65 203c 3c20 4d61 6e69 * "File << Mani │ │ │ │ -00272ae0: 7075 6c61 746f 7222 202d 2d20 7365 6520 pulator" -- see │ │ │ │ -00272af0: 2a6e 6f74 6520 7072 696e 7469 6e67 2074 *note printing t │ │ │ │ -00272b00: 6f20 6120 6669 6c65 3a20 7072 696e 7469 o a file: printi │ │ │ │ -00272b10: 6e67 2074 6f20 6120 6669 6c65 2c0a 2020 ng to a file,. │ │ │ │ -00272b20: 2020 2d2d 2070 7269 6e74 2074 6f20 6120 -- print to a │ │ │ │ -00272b30: 6669 6c65 0a20 202a 2022 4e6f 7468 696e file. * "Nothin │ │ │ │ -00272b40: 6720 3c3c 204d 616e 6970 756c 6174 6f72 g << Manipulator │ │ │ │ -00272b50: 2220 2d2d 2073 6565 202a 6e6f 7465 2070 " -- see *note p │ │ │ │ -00272b60: 7269 6e74 696e 6720 746f 2061 2066 696c rinting to a fil │ │ │ │ -00272b70: 653a 2070 7269 6e74 696e 6720 746f 2061 e: printing to a │ │ │ │ -00272b80: 0a20 2020 2066 696c 652c 202d 2d20 7072 . file, -- pr │ │ │ │ -00272b90: 696e 7420 746f 2061 2066 696c 650a 0a46 int to a file..F │ │ │ │ -00272ba0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00272bb0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00272bc0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00272bd0: 202a 6e6f 7465 2063 6c6f 7365 496e 3a20 *note closeIn: │ │ │ │ -00272be0: 636c 6f73 6549 6e2c 2069 7320 6120 2a6e closeIn, is a *n │ │ │ │ -00272bf0: 6f74 6520 6d61 6e69 7075 6c61 746f 723a ote manipulator: │ │ │ │ -00272c00: 204d 616e 6970 756c 6174 6f72 2c2e 0a0a Manipulator,... │ │ │ │ -00272c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00272c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00272c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002725a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002725b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002725c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +002725d0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +002725e0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +002725f0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00272600: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00272610: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ +00272620: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00272630: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ +00272640: 632f 6f76 5f73 7973 7465 6d2e 6d32 3a31 c/ov_system.m2:1 │ │ │ │ +00272650: 3438 333a 302e 0a1f 0a46 696c 653a 204d 483:0....File: M │ │ │ │ +00272660: 6163 6175 6c61 7932 446f 632e 696e 666f acaulay2Doc.info │ │ │ │ +00272670: 2c20 4e6f 6465 3a20 636c 6f73 6549 6e2c , Node: closeIn, │ │ │ │ +00272680: 204e 6578 743a 2063 6c6f 7365 4f75 742c Next: closeOut, │ │ │ │ +00272690: 2050 7265 763a 2063 6c6f 7365 2c20 5570 Prev: close, Up │ │ │ │ +002726a0: 3a20 7573 696e 6720 736f 636b 6574 730a : using sockets. │ │ │ │ +002726b0: 0a63 6c6f 7365 496e 202d 2d20 636c 6f73 .closeIn -- clos │ │ │ │ +002726c0: 6520 616e 2069 6e70 7574 2066 696c 650a e an input file. │ │ │ │ +002726d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002726e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +002726f0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00272700: 3d3d 3d3d 3d3d 3d0a 0a66 203c 3c20 636c =======..f << cl │ │ │ │ +00272710: 6f73 6549 6e20 2d2d 2063 6c6f 7365 7320 oseIn -- closes │ │ │ │ +00272720: 7468 6520 696e 7075 7420 6669 6c65 2066 the input file f │ │ │ │ +00272730: 2e0a 636c 6f73 6549 6e20 6620 2d2d 2063 ..closeIn f -- c │ │ │ │ +00272740: 6c6f 7365 7320 7468 6520 696e 7075 7420 loses the input │ │ │ │ +00272750: 6669 6c65 2066 2e0a 0a0a 4966 2074 6865 file f....If the │ │ │ │ +00272760: 2066 696c 6520 7761 7320 6f70 656e 206f file was open o │ │ │ │ +00272770: 6e6c 7920 666f 7220 696e 7075 742c 2074 nly for input, t │ │ │ │ +00272780: 6865 6e20 2a6e 6f74 6520 636c 6f73 653a hen *note close: │ │ │ │ +00272790: 2063 6c6f 7365 2c20 6973 2065 6173 6965 close, is easie │ │ │ │ +002727a0: 7220 746f 2075 7365 0a61 6e64 2068 6173 r to use.and has │ │ │ │ +002727b0: 2074 6865 2073 616d 6520 6566 6665 6374 the same effect │ │ │ │ +002727c0: 2e0a 0a0a 4966 2074 6865 2066 696c 6520 ....If the file │ │ │ │ +002727d0: 7761 7320 6f70 656e 2066 6f72 2062 6f74 was open for bot │ │ │ │ +002727e0: 6820 696e 7075 7420 616e 6420 6f75 7470 h input and outp │ │ │ │ +002727f0: 7574 2c20 6974 2072 656d 6169 6e73 206f ut, it remains o │ │ │ │ +00272800: 7065 6e20 666f 7220 6f75 7470 7574 2e0a pen for output.. │ │ │ │ +00272810: 0a57 6179 7320 746f 2075 7365 2063 6c6f .Ways to use clo │ │ │ │ +00272820: 7365 496e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d seIn:.========== │ │ │ │ +00272830: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00272840: 224d 616e 6970 756c 6174 6f72 2044 6174 "Manipulator Dat │ │ │ │ +00272850: 6162 6173 6522 202d 2d20 7365 6520 2a6e abase" -- see *n │ │ │ │ +00272860: 6f74 6520 4d61 6e69 7075 6c61 746f 723a ote Manipulator: │ │ │ │ +00272870: 204d 616e 6970 756c 6174 6f72 2c20 2d2d Manipulator, -- │ │ │ │ +00272880: 2074 6865 2063 6c61 7373 0a20 2020 206f the class. o │ │ │ │ +00272890: 6620 616c 6c20 6669 6c65 206d 616e 6970 f all file manip │ │ │ │ +002728a0: 756c 6174 6f72 730a 2020 2a20 224d 616e ulators. * "Man │ │ │ │ +002728b0: 6970 756c 6174 6f72 2046 696c 6522 202d ipulator File" - │ │ │ │ +002728c0: 2d20 7365 6520 2a6e 6f74 6520 4d61 6e69 - see *note Mani │ │ │ │ +002728d0: 7075 6c61 746f 723a 204d 616e 6970 756c pulator: Manipul │ │ │ │ +002728e0: 6174 6f72 2c20 2d2d 2074 6865 2063 6c61 ator, -- the cla │ │ │ │ +002728f0: 7373 206f 660a 2020 2020 616c 6c20 6669 ss of. all fi │ │ │ │ +00272900: 6c65 206d 616e 6970 756c 6174 6f72 730a le manipulators. │ │ │ │ +00272910: 2020 2a20 224d 616e 6970 756c 6174 6f72 * "Manipulator │ │ │ │ +00272920: 204e 6f74 6869 6e67 2220 2d2d 2073 6565 Nothing" -- see │ │ │ │ +00272930: 202a 6e6f 7465 204d 616e 6970 756c 6174 *note Manipulat │ │ │ │ +00272940: 6f72 3a20 4d61 6e69 7075 6c61 746f 722c or: Manipulator, │ │ │ │ +00272950: 202d 2d20 7468 6520 636c 6173 730a 2020 -- the class. │ │ │ │ +00272960: 2020 6f66 2061 6c6c 2066 696c 6520 6d61 of all file ma │ │ │ │ +00272970: 6e69 7075 6c61 746f 7273 0a20 202a 2022 nipulators. * " │ │ │ │ +00272980: 6e65 7720 4d61 6e69 7075 6c61 746f 7220 new Manipulator │ │ │ │ +00272990: 6672 6f6d 2046 756e 6374 696f 6e22 202d from Function" - │ │ │ │ +002729a0: 2d20 7365 6520 2a6e 6f74 6520 4d61 6e69 - see *note Mani │ │ │ │ +002729b0: 7075 6c61 746f 723a 204d 616e 6970 756c pulator: Manipul │ │ │ │ +002729c0: 6174 6f72 2c20 2d2d 0a20 2020 2074 6865 ator, --. the │ │ │ │ +002729d0: 2063 6c61 7373 206f 6620 616c 6c20 6669 class of all fi │ │ │ │ +002729e0: 6c65 206d 616e 6970 756c 6174 6f72 730a le manipulators. │ │ │ │ +002729f0: 2020 2a20 226d 6574 686f 6473 284d 616e * "methods(Man │ │ │ │ +00272a00: 6970 756c 6174 6f72 2922 202d 2d20 7365 ipulator)" -- se │ │ │ │ +00272a10: 6520 2a6e 6f74 6520 6d65 7468 6f64 733a e *note methods: │ │ │ │ +00272a20: 206d 6574 686f 6473 2c20 2d2d 206c 6973 methods, -- lis │ │ │ │ +00272a30: 7420 6d65 7468 6f64 730a 2020 2a20 224d t methods. * "M │ │ │ │ +00272a40: 616e 6970 756c 6174 6f72 204e 6574 4669 anipulator NetFi │ │ │ │ +00272a50: 6c65 2220 2d2d 2073 6565 202a 6e6f 7465 le" -- see *note │ │ │ │ +00272a60: 204e 6574 4669 6c65 3a20 4e65 7446 696c NetFile: NetFil │ │ │ │ +00272a70: 652c 202d 2d20 7468 6520 636c 6173 7320 e, -- the class │ │ │ │ +00272a80: 6f66 2061 6c6c 0a20 2020 206e 6574 2066 of all. net f │ │ │ │ +00272a90: 696c 6573 0a20 202a 2022 4e65 7446 696c iles. * "NetFil │ │ │ │ +00272aa0: 6520 3c3c 204d 616e 6970 756c 6174 6f72 e << Manipulator │ │ │ │ +00272ab0: 2220 2d2d 2073 6565 202a 6e6f 7465 204e " -- see *note N │ │ │ │ +00272ac0: 6574 4669 6c65 3a20 4e65 7446 696c 652c etFile: NetFile, │ │ │ │ +00272ad0: 202d 2d20 7468 6520 636c 6173 7320 6f66 -- the class of │ │ │ │ +00272ae0: 2061 6c6c 0a20 2020 206e 6574 2066 696c all. net fil │ │ │ │ +00272af0: 6573 0a20 202a 2022 4669 6c65 203c 3c20 es. * "File << │ │ │ │ +00272b00: 4d61 6e69 7075 6c61 746f 7222 202d 2d20 Manipulator" -- │ │ │ │ +00272b10: 7365 6520 2a6e 6f74 6520 7072 696e 7469 see *note printi │ │ │ │ +00272b20: 6e67 2074 6f20 6120 6669 6c65 3a20 7072 ng to a file: pr │ │ │ │ +00272b30: 696e 7469 6e67 2074 6f20 6120 6669 6c65 inting to a file │ │ │ │ +00272b40: 2c0a 2020 2020 2d2d 2070 7269 6e74 2074 ,. -- print t │ │ │ │ +00272b50: 6f20 6120 6669 6c65 0a20 202a 2022 4e6f o a file. * "No │ │ │ │ +00272b60: 7468 696e 6720 3c3c 204d 616e 6970 756c thing << Manipul │ │ │ │ +00272b70: 6174 6f72 2220 2d2d 2073 6565 202a 6e6f ator" -- see *no │ │ │ │ +00272b80: 7465 2070 7269 6e74 696e 6720 746f 2061 te printing to a │ │ │ │ +00272b90: 2066 696c 653a 2070 7269 6e74 696e 6720 file: printing │ │ │ │ +00272ba0: 746f 2061 0a20 2020 2066 696c 652c 202d to a. file, - │ │ │ │ +00272bb0: 2d20 7072 696e 7420 746f 2061 2066 696c - print to a fil │ │ │ │ +00272bc0: 650a 0a46 6f72 2074 6865 2070 726f 6772 e..For the progr │ │ │ │ +00272bd0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00272be0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00272bf0: 6a65 6374 202a 6e6f 7465 2063 6c6f 7365 ject *note close │ │ │ │ +00272c00: 496e 3a20 636c 6f73 6549 6e2c 2069 7320 In: closeIn, is │ │ │ │ +00272c10: 6120 2a6e 6f74 6520 6d61 6e69 7075 6c61 a *note manipula │ │ │ │ +00272c20: 746f 723a 204d 616e 6970 756c 6174 6f72 tor: Manipulator │ │ │ │ +00272c30: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ 00272c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00272c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00272c60: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00272c70: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00272c80: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00272c90: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -00272ca0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -00272cb0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -00272cc0: 636b 6167 6573 2f0a 4d61 6361 756c 6179 ckages/.Macaulay │ │ │ │ -00272cd0: 3244 6f63 2f6f 765f 7379 7374 656d 2e6d 2Doc/ov_system.m │ │ │ │ -00272ce0: 323a 3135 3136 3a30 2e0a 1f0a 4669 6c65 2:1516:0....File │ │ │ │ -00272cf0: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ -00272d00: 6e66 6f2c 204e 6f64 653a 2063 6c6f 7365 nfo, Node: close │ │ │ │ -00272d10: 4f75 742c 204e 6578 743a 2067 6574 2c20 Out, Next: get, │ │ │ │ -00272d20: 5072 6576 3a20 636c 6f73 6549 6e2c 2055 Prev: closeIn, U │ │ │ │ -00272d30: 703a 2075 7369 6e67 2073 6f63 6b65 7473 p: using sockets │ │ │ │ -00272d40: 0a0a 636c 6f73 654f 7574 202d 2d20 636c ..closeOut -- cl │ │ │ │ -00272d50: 6f73 6520 616e 206f 7574 7075 7420 6669 ose an output fi │ │ │ │ -00272d60: 6c65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a le.************* │ │ │ │ -00272d70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00272d80: 2a2a 2a0a 0a44 6573 6372 6970 7469 6f6e ***..Description │ │ │ │ -00272d90: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 6620 .===========..f │ │ │ │ -00272da0: 3c3c 2063 6c6f 7365 4f75 7420 2d2d 2063 << closeOut -- c │ │ │ │ -00272db0: 6c6f 7365 7320 7468 6520 6f75 7470 7574 loses the output │ │ │ │ -00272dc0: 2066 696c 6520 662e 0a63 6c6f 7365 4f75 file f..closeOu │ │ │ │ -00272dd0: 7420 6620 2d2d 2063 6c6f 7365 7320 7468 t f -- closes th │ │ │ │ -00272de0: 6520 6f75 7470 7574 2066 696c 6520 662e e output file f. │ │ │ │ -00272df0: 0a0a 0a41 6e79 2062 7566 6665 7265 6420 ...Any buffered │ │ │ │ -00272e00: 6f75 7470 7574 2069 7320 6669 7273 7420 output is first │ │ │ │ -00272e10: 7772 6974 7465 6e20 746f 2074 6865 2066 written to the f │ │ │ │ -00272e20: 696c 652c 2061 6e64 2074 6865 2072 6574 ile, and the ret │ │ │ │ -00272e30: 7572 6e20 7661 6c75 6520 6973 2061 6e0a urn value is an. │ │ │ │ -00272e40: 696e 7465 6765 722c 206e 6f72 6d61 6c6c integer, normall │ │ │ │ -00272e50: 7920 302c 206f 7220 2d31 206f 6e20 6572 y 0, or -1 on er │ │ │ │ -00272e60: 726f 722c 206f 7220 7468 6520 7265 7475 ror, or the retu │ │ │ │ -00272e70: 726e 2073 7461 7475 7320 6f66 2074 6865 rn status of the │ │ │ │ -00272e80: 2063 6869 6c64 2070 726f 6365 7373 0a69 child process.i │ │ │ │ -00272e90: 6e20 6361 7365 2074 6865 2066 696c 6520 n case the file │ │ │ │ -00272ea0: 7761 7320 6120 7069 7065 2e0a 0a0a 4966 was a pipe....If │ │ │ │ -00272eb0: 2074 6865 2066 696c 6520 7761 7320 6f70 the file was op │ │ │ │ -00272ec0: 656e 206f 6e6c 7920 666f 7220 6f75 7470 en only for outp │ │ │ │ -00272ed0: 7574 2c20 7468 656e 202a 6e6f 7465 2063 ut, then *note c │ │ │ │ -00272ee0: 6c6f 7365 3a20 636c 6f73 652c 2069 7320 lose: close, is │ │ │ │ -00272ef0: 6561 7369 6572 2074 6f20 7573 650a 616e easier to use.an │ │ │ │ -00272f00: 6420 6861 7320 7468 6520 7361 6d65 2065 d has the same e │ │ │ │ -00272f10: 6666 6563 742e 0a0a 0a49 6620 7468 6520 ffect....If the │ │ │ │ -00272f20: 6669 6c65 2077 6173 206f 7065 6e20 666f file was open fo │ │ │ │ -00272f30: 7220 626f 7468 2069 6e70 7574 2061 6e64 r both input and │ │ │ │ -00272f40: 206f 7574 7075 742c 2069 7420 7265 6d61 output, it rema │ │ │ │ -00272f50: 696e 7320 6f70 656e 2066 6f72 2069 6e70 ins open for inp │ │ │ │ -00272f60: 7574 2e0a 0a57 6179 7320 746f 2075 7365 ut...Ways to use │ │ │ │ -00272f70: 2063 6c6f 7365 4f75 743a 0a3d 3d3d 3d3d closeOut:.===== │ │ │ │ -00272f80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00272f90: 0a0a 2020 2a20 224d 616e 6970 756c 6174 .. * "Manipulat │ │ │ │ -00272fa0: 6f72 2044 6174 6162 6173 6522 202d 2d20 or Database" -- │ │ │ │ -00272fb0: 7365 6520 2a6e 6f74 6520 4d61 6e69 7075 see *note Manipu │ │ │ │ -00272fc0: 6c61 746f 723a 204d 616e 6970 756c 6174 lator: Manipulat │ │ │ │ -00272fd0: 6f72 2c20 2d2d 2074 6865 2063 6c61 7373 or, -- the class │ │ │ │ -00272fe0: 0a20 2020 206f 6620 616c 6c20 6669 6c65 . of all file │ │ │ │ -00272ff0: 206d 616e 6970 756c 6174 6f72 730a 2020 manipulators. │ │ │ │ -00273000: 2a20 224d 616e 6970 756c 6174 6f72 2046 * "Manipulator F │ │ │ │ -00273010: 696c 6522 202d 2d20 7365 6520 2a6e 6f74 ile" -- see *not │ │ │ │ -00273020: 6520 4d61 6e69 7075 6c61 746f 723a 204d e Manipulator: M │ │ │ │ -00273030: 616e 6970 756c 6174 6f72 2c20 2d2d 2074 anipulator, -- t │ │ │ │ -00273040: 6865 2063 6c61 7373 206f 660a 2020 2020 he class of. │ │ │ │ -00273050: 616c 6c20 6669 6c65 206d 616e 6970 756c all file manipul │ │ │ │ -00273060: 6174 6f72 730a 2020 2a20 224d 616e 6970 ators. * "Manip │ │ │ │ -00273070: 756c 6174 6f72 204e 6f74 6869 6e67 2220 ulator Nothing" │ │ │ │ -00273080: 2d2d 2073 6565 202a 6e6f 7465 204d 616e -- see *note Man │ │ │ │ -00273090: 6970 756c 6174 6f72 3a20 4d61 6e69 7075 ipulator: Manipu │ │ │ │ -002730a0: 6c61 746f 722c 202d 2d20 7468 6520 636c lator, -- the cl │ │ │ │ -002730b0: 6173 730a 2020 2020 6f66 2061 6c6c 2066 ass. of all f │ │ │ │ -002730c0: 696c 6520 6d61 6e69 7075 6c61 746f 7273 ile manipulators │ │ │ │ -002730d0: 0a20 202a 2022 6e65 7720 4d61 6e69 7075 . * "new Manipu │ │ │ │ -002730e0: 6c61 746f 7220 6672 6f6d 2046 756e 6374 lator from Funct │ │ │ │ -002730f0: 696f 6e22 202d 2d20 7365 6520 2a6e 6f74 ion" -- see *not │ │ │ │ -00273100: 6520 4d61 6e69 7075 6c61 746f 723a 204d e Manipulator: M │ │ │ │ -00273110: 616e 6970 756c 6174 6f72 2c20 2d2d 0a20 anipulator, --. │ │ │ │ -00273120: 2020 2074 6865 2063 6c61 7373 206f 6620 the class of │ │ │ │ -00273130: 616c 6c20 6669 6c65 206d 616e 6970 756c all file manipul │ │ │ │ -00273140: 6174 6f72 730a 2020 2a20 226d 6574 686f ators. * "metho │ │ │ │ -00273150: 6473 284d 616e 6970 756c 6174 6f72 2922 ds(Manipulator)" │ │ │ │ -00273160: 202d 2d20 7365 6520 2a6e 6f74 6520 6d65 -- see *note me │ │ │ │ -00273170: 7468 6f64 733a 206d 6574 686f 6473 2c20 thods: methods, │ │ │ │ -00273180: 2d2d 206c 6973 7420 6d65 7468 6f64 730a -- list methods. │ │ │ │ -00273190: 2020 2a20 224d 616e 6970 756c 6174 6f72 * "Manipulator │ │ │ │ -002731a0: 204e 6574 4669 6c65 2220 2d2d 2073 6565 NetFile" -- see │ │ │ │ -002731b0: 202a 6e6f 7465 204e 6574 4669 6c65 3a20 *note NetFile: │ │ │ │ -002731c0: 4e65 7446 696c 652c 202d 2d20 7468 6520 NetFile, -- the │ │ │ │ -002731d0: 636c 6173 7320 6f66 2061 6c6c 0a20 2020 class of all. │ │ │ │ -002731e0: 206e 6574 2066 696c 6573 0a20 202a 2022 net files. * " │ │ │ │ -002731f0: 4e65 7446 696c 6520 3c3c 204d 616e 6970 NetFile << Manip │ │ │ │ -00273200: 756c 6174 6f72 2220 2d2d 2073 6565 202a ulator" -- see * │ │ │ │ -00273210: 6e6f 7465 204e 6574 4669 6c65 3a20 4e65 note NetFile: Ne │ │ │ │ -00273220: 7446 696c 652c 202d 2d20 7468 6520 636c tFile, -- the cl │ │ │ │ -00273230: 6173 7320 6f66 2061 6c6c 0a20 2020 206e ass of all. n │ │ │ │ -00273240: 6574 2066 696c 6573 0a20 202a 2022 4669 et files. * "Fi │ │ │ │ -00273250: 6c65 203c 3c20 4d61 6e69 7075 6c61 746f le << Manipulato │ │ │ │ -00273260: 7222 202d 2d20 7365 6520 2a6e 6f74 6520 r" -- see *note │ │ │ │ -00273270: 7072 696e 7469 6e67 2074 6f20 6120 6669 printing to a fi │ │ │ │ -00273280: 6c65 3a20 7072 696e 7469 6e67 2074 6f20 le: printing to │ │ │ │ -00273290: 6120 6669 6c65 2c0a 2020 2020 2d2d 2070 a file,. -- p │ │ │ │ -002732a0: 7269 6e74 2074 6f20 6120 6669 6c65 0a20 rint to a file. │ │ │ │ -002732b0: 202a 2022 4e6f 7468 696e 6720 3c3c 204d * "Nothing << M │ │ │ │ -002732c0: 616e 6970 756c 6174 6f72 2220 2d2d 2073 anipulator" -- s │ │ │ │ -002732d0: 6565 202a 6e6f 7465 2070 7269 6e74 696e ee *note printin │ │ │ │ -002732e0: 6720 746f 2061 2066 696c 653a 2070 7269 g to a file: pri │ │ │ │ -002732f0: 6e74 696e 6720 746f 2061 0a20 2020 2066 nting to a. f │ │ │ │ -00273300: 696c 652c 202d 2d20 7072 696e 7420 746f ile, -- print to │ │ │ │ -00273310: 2061 2066 696c 650a 0a46 6f72 2074 6865 a file..For the │ │ │ │ -00273320: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00273330: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00273340: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00273350: 2063 6c6f 7365 4f75 743a 2063 6c6f 7365 closeOut: close │ │ │ │ -00273360: 4f75 742c 2069 7320 6120 2a6e 6f74 6520 Out, is a *note │ │ │ │ -00273370: 6d61 6e69 7075 6c61 746f 723a 204d 616e manipulator: Man │ │ │ │ -00273380: 6970 756c 6174 6f72 2c2e 0a0a 2d2d 2d2d ipulator,...---- │ │ │ │ -00273390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002733a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00272c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00272c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00272c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00272c80: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +00272c90: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +00272ca0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +00272cb0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +00272cc0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +00272cd0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +00272ce0: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ +00272cf0: 756c 6179 3244 6f63 2f6f 765f 7379 7374 ulay2Doc/ov_syst │ │ │ │ +00272d00: 656d 2e6d 323a 3135 3136 3a30 2e0a 1f0a em.m2:1516:0.... │ │ │ │ +00272d10: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ +00272d20: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2063 oc.info, Node: c │ │ │ │ +00272d30: 6c6f 7365 4f75 742c 204e 6578 743a 2067 loseOut, Next: g │ │ │ │ +00272d40: 6574 2c20 5072 6576 3a20 636c 6f73 6549 et, Prev: closeI │ │ │ │ +00272d50: 6e2c 2055 703a 2075 7369 6e67 2073 6f63 n, Up: using soc │ │ │ │ +00272d60: 6b65 7473 0a0a 636c 6f73 654f 7574 202d kets..closeOut - │ │ │ │ +00272d70: 2d20 636c 6f73 6520 616e 206f 7574 7075 - close an outpu │ │ │ │ +00272d80: 7420 6669 6c65 0a2a 2a2a 2a2a 2a2a 2a2a t file.********* │ │ │ │ +00272d90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00272da0: 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 6970 *******..Descrip │ │ │ │ +00272db0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00272dc0: 0a0a 6620 3c3c 2063 6c6f 7365 4f75 7420 ..f << closeOut │ │ │ │ +00272dd0: 2d2d 2063 6c6f 7365 7320 7468 6520 6f75 -- closes the ou │ │ │ │ +00272de0: 7470 7574 2066 696c 6520 662e 0a63 6c6f tput file f..clo │ │ │ │ +00272df0: 7365 4f75 7420 6620 2d2d 2063 6c6f 7365 seOut f -- close │ │ │ │ +00272e00: 7320 7468 6520 6f75 7470 7574 2066 696c s the output fil │ │ │ │ +00272e10: 6520 662e 0a0a 0a41 6e79 2062 7566 6665 e f....Any buffe │ │ │ │ +00272e20: 7265 6420 6f75 7470 7574 2069 7320 6669 red output is fi │ │ │ │ +00272e30: 7273 7420 7772 6974 7465 6e20 746f 2074 rst written to t │ │ │ │ +00272e40: 6865 2066 696c 652c 2061 6e64 2074 6865 he file, and the │ │ │ │ +00272e50: 2072 6574 7572 6e20 7661 6c75 6520 6973 return value is │ │ │ │ +00272e60: 2061 6e0a 696e 7465 6765 722c 206e 6f72 an.integer, nor │ │ │ │ +00272e70: 6d61 6c6c 7920 302c 206f 7220 2d31 206f mally 0, or -1 o │ │ │ │ +00272e80: 6e20 6572 726f 722c 206f 7220 7468 6520 n error, or the │ │ │ │ +00272e90: 7265 7475 726e 2073 7461 7475 7320 6f66 return status of │ │ │ │ +00272ea0: 2074 6865 2063 6869 6c64 2070 726f 6365 the child proce │ │ │ │ +00272eb0: 7373 0a69 6e20 6361 7365 2074 6865 2066 ss.in case the f │ │ │ │ +00272ec0: 696c 6520 7761 7320 6120 7069 7065 2e0a ile was a pipe.. │ │ │ │ +00272ed0: 0a0a 4966 2074 6865 2066 696c 6520 7761 ..If the file wa │ │ │ │ +00272ee0: 7320 6f70 656e 206f 6e6c 7920 666f 7220 s open only for │ │ │ │ +00272ef0: 6f75 7470 7574 2c20 7468 656e 202a 6e6f output, then *no │ │ │ │ +00272f00: 7465 2063 6c6f 7365 3a20 636c 6f73 652c te close: close, │ │ │ │ +00272f10: 2069 7320 6561 7369 6572 2074 6f20 7573 is easier to us │ │ │ │ +00272f20: 650a 616e 6420 6861 7320 7468 6520 7361 e.and has the sa │ │ │ │ +00272f30: 6d65 2065 6666 6563 742e 0a0a 0a49 6620 me effect....If │ │ │ │ +00272f40: 7468 6520 6669 6c65 2077 6173 206f 7065 the file was ope │ │ │ │ +00272f50: 6e20 666f 7220 626f 7468 2069 6e70 7574 n for both input │ │ │ │ +00272f60: 2061 6e64 206f 7574 7075 742c 2069 7420 and output, it │ │ │ │ +00272f70: 7265 6d61 696e 7320 6f70 656e 2066 6f72 remains open for │ │ │ │ +00272f80: 2069 6e70 7574 2e0a 0a57 6179 7320 746f input...Ways to │ │ │ │ +00272f90: 2075 7365 2063 6c6f 7365 4f75 743a 0a3d use closeOut:.= │ │ │ │ +00272fa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00272fb0: 3d3d 3d3d 0a0a 2020 2a20 224d 616e 6970 ====.. * "Manip │ │ │ │ +00272fc0: 756c 6174 6f72 2044 6174 6162 6173 6522 ulator Database" │ │ │ │ +00272fd0: 202d 2d20 7365 6520 2a6e 6f74 6520 4d61 -- see *note Ma │ │ │ │ +00272fe0: 6e69 7075 6c61 746f 723a 204d 616e 6970 nipulator: Manip │ │ │ │ +00272ff0: 756c 6174 6f72 2c20 2d2d 2074 6865 2063 ulator, -- the c │ │ │ │ +00273000: 6c61 7373 0a20 2020 206f 6620 616c 6c20 lass. of all │ │ │ │ +00273010: 6669 6c65 206d 616e 6970 756c 6174 6f72 file manipulator │ │ │ │ +00273020: 730a 2020 2a20 224d 616e 6970 756c 6174 s. * "Manipulat │ │ │ │ +00273030: 6f72 2046 696c 6522 202d 2d20 7365 6520 or File" -- see │ │ │ │ +00273040: 2a6e 6f74 6520 4d61 6e69 7075 6c61 746f *note Manipulato │ │ │ │ +00273050: 723a 204d 616e 6970 756c 6174 6f72 2c20 r: Manipulator, │ │ │ │ +00273060: 2d2d 2074 6865 2063 6c61 7373 206f 660a -- the class of. │ │ │ │ +00273070: 2020 2020 616c 6c20 6669 6c65 206d 616e all file man │ │ │ │ +00273080: 6970 756c 6174 6f72 730a 2020 2a20 224d ipulators. * "M │ │ │ │ +00273090: 616e 6970 756c 6174 6f72 204e 6f74 6869 anipulator Nothi │ │ │ │ +002730a0: 6e67 2220 2d2d 2073 6565 202a 6e6f 7465 ng" -- see *note │ │ │ │ +002730b0: 204d 616e 6970 756c 6174 6f72 3a20 4d61 Manipulator: Ma │ │ │ │ +002730c0: 6e69 7075 6c61 746f 722c 202d 2d20 7468 nipulator, -- th │ │ │ │ +002730d0: 6520 636c 6173 730a 2020 2020 6f66 2061 e class. of a │ │ │ │ +002730e0: 6c6c 2066 696c 6520 6d61 6e69 7075 6c61 ll file manipula │ │ │ │ +002730f0: 746f 7273 0a20 202a 2022 6e65 7720 4d61 tors. * "new Ma │ │ │ │ +00273100: 6e69 7075 6c61 746f 7220 6672 6f6d 2046 nipulator from F │ │ │ │ +00273110: 756e 6374 696f 6e22 202d 2d20 7365 6520 unction" -- see │ │ │ │ +00273120: 2a6e 6f74 6520 4d61 6e69 7075 6c61 746f *note Manipulato │ │ │ │ +00273130: 723a 204d 616e 6970 756c 6174 6f72 2c20 r: Manipulator, │ │ │ │ +00273140: 2d2d 0a20 2020 2074 6865 2063 6c61 7373 --. the class │ │ │ │ +00273150: 206f 6620 616c 6c20 6669 6c65 206d 616e of all file man │ │ │ │ +00273160: 6970 756c 6174 6f72 730a 2020 2a20 226d ipulators. * "m │ │ │ │ +00273170: 6574 686f 6473 284d 616e 6970 756c 6174 ethods(Manipulat │ │ │ │ +00273180: 6f72 2922 202d 2d20 7365 6520 2a6e 6f74 or)" -- see *not │ │ │ │ +00273190: 6520 6d65 7468 6f64 733a 206d 6574 686f e methods: metho │ │ │ │ +002731a0: 6473 2c20 2d2d 206c 6973 7420 6d65 7468 ds, -- list meth │ │ │ │ +002731b0: 6f64 730a 2020 2a20 224d 616e 6970 756c ods. * "Manipul │ │ │ │ +002731c0: 6174 6f72 204e 6574 4669 6c65 2220 2d2d ator NetFile" -- │ │ │ │ +002731d0: 2073 6565 202a 6e6f 7465 204e 6574 4669 see *note NetFi │ │ │ │ +002731e0: 6c65 3a20 4e65 7446 696c 652c 202d 2d20 le: NetFile, -- │ │ │ │ +002731f0: 7468 6520 636c 6173 7320 6f66 2061 6c6c the class of all │ │ │ │ +00273200: 0a20 2020 206e 6574 2066 696c 6573 0a20 . net files. │ │ │ │ +00273210: 202a 2022 4e65 7446 696c 6520 3c3c 204d * "NetFile << M │ │ │ │ +00273220: 616e 6970 756c 6174 6f72 2220 2d2d 2073 anipulator" -- s │ │ │ │ +00273230: 6565 202a 6e6f 7465 204e 6574 4669 6c65 ee *note NetFile │ │ │ │ +00273240: 3a20 4e65 7446 696c 652c 202d 2d20 7468 : NetFile, -- th │ │ │ │ +00273250: 6520 636c 6173 7320 6f66 2061 6c6c 0a20 e class of all. │ │ │ │ +00273260: 2020 206e 6574 2066 696c 6573 0a20 202a net files. * │ │ │ │ +00273270: 2022 4669 6c65 203c 3c20 4d61 6e69 7075 "File << Manipu │ │ │ │ +00273280: 6c61 746f 7222 202d 2d20 7365 6520 2a6e lator" -- see *n │ │ │ │ +00273290: 6f74 6520 7072 696e 7469 6e67 2074 6f20 ote printing to │ │ │ │ +002732a0: 6120 6669 6c65 3a20 7072 696e 7469 6e67 a file: printing │ │ │ │ +002732b0: 2074 6f20 6120 6669 6c65 2c0a 2020 2020 to a file,. │ │ │ │ +002732c0: 2d2d 2070 7269 6e74 2074 6f20 6120 6669 -- print to a fi │ │ │ │ +002732d0: 6c65 0a20 202a 2022 4e6f 7468 696e 6720 le. * "Nothing │ │ │ │ +002732e0: 3c3c 204d 616e 6970 756c 6174 6f72 2220 << Manipulator" │ │ │ │ +002732f0: 2d2d 2073 6565 202a 6e6f 7465 2070 7269 -- see *note pri │ │ │ │ +00273300: 6e74 696e 6720 746f 2061 2066 696c 653a nting to a file: │ │ │ │ +00273310: 2070 7269 6e74 696e 6720 746f 2061 0a20 printing to a. │ │ │ │ +00273320: 2020 2066 696c 652c 202d 2d20 7072 696e file, -- prin │ │ │ │ +00273330: 7420 746f 2061 2066 696c 650a 0a46 6f72 t to a file..For │ │ │ │ +00273340: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00273350: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00273360: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00273370: 6e6f 7465 2063 6c6f 7365 4f75 743a 2063 note closeOut: c │ │ │ │ +00273380: 6c6f 7365 4f75 742c 2069 7320 6120 2a6e loseOut, is a *n │ │ │ │ +00273390: 6f74 6520 6d61 6e69 7075 6c61 746f 723a ote manipulator: │ │ │ │ +002733a0: 204d 616e 6970 756c 6174 6f72 2c2e 0a0a Manipulator,... │ │ │ │ 002733b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002733c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002733d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -002733e0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -002733f0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -00273400: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -00273410: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -00273420: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ -00273430: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -00273440: 6573 2f0a 4d61 6361 756c 6179 3244 6f63 es/.Macaulay2Doc │ │ │ │ -00273450: 2f6f 765f 7379 7374 656d 2e6d 323a 3135 /ov_system.m2:15 │ │ │ │ -00273460: 3335 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 35:0....File: Ma │ │ │ │ -00273470: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ -00273480: 204e 6f64 653a 2067 6574 2c20 4e65 7874 Node: get, Next │ │ │ │ -00273490: 3a20 6765 7463 2c20 5072 6576 3a20 636c : getc, Prev: cl │ │ │ │ -002734a0: 6f73 654f 7574 2c20 5570 3a20 7573 696e oseOut, Up: usin │ │ │ │ -002734b0: 6720 736f 636b 6574 730a 0a67 6574 202d g sockets..get - │ │ │ │ -002734c0: 2d20 6765 7420 7468 6520 636f 6e74 656e - get the conten │ │ │ │ -002734d0: 7473 206f 6620 6120 6669 6c65 0a2a 2a2a ts of a file.*** │ │ │ │ -002734e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002734f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00273500: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00273510: 2020 2020 6765 7420 660a 2020 2a20 496e get f. * In │ │ │ │ -00273520: 7075 7473 3a0a 2020 2020 2020 2a20 662c puts:. * f, │ │ │ │ -00273530: 2061 202a 6e6f 7465 2066 696c 653a 2046 a *note file: F │ │ │ │ -00273540: 696c 652c 206f 7220 6120 2a6e 6f74 6520 ile, or a *note │ │ │ │ -00273550: 7374 7269 6e67 3a20 5374 7269 6e67 2c2e string: String,. │ │ │ │ -00273560: 2020 4966 2066 2069 7320 6120 7374 7269 If f is a stri │ │ │ │ -00273570: 6e67 2c0a 2020 2020 2020 2020 7468 656e ng,. then │ │ │ │ -00273580: 2069 7420 6973 206f 7065 6e65 642c 2061 it is opened, a │ │ │ │ -00273590: 7320 7769 7468 202a 6e6f 7465 206f 7065 s with *note ope │ │ │ │ -002735a0: 6e49 6e3a 206f 7065 6e49 6e5f 6c70 5374 nIn: openIn_lpSt │ │ │ │ -002735b0: 7269 6e67 5f72 702c 2e0a 2020 2020 2020 ring_rp,.. │ │ │ │ -002735c0: 2020 4669 6c65 6e61 6d65 7320 7374 6172 Filenames star │ │ │ │ -002735d0: 7469 6e67 2077 6974 6820 2120 6f72 2077 ting with ! or w │ │ │ │ -002735e0: 6974 6820 2420 6172 6520 7472 6561 7465 ith $ are treate │ │ │ │ -002735f0: 6420 7370 6563 6961 6c6c 792c 2073 6565 d specially, see │ │ │ │ -00273600: 202a 6e6f 7465 0a20 2020 2020 2020 206f *note. o │ │ │ │ -00273610: 7065 6e49 6e4f 7574 3a20 6f70 656e 496e penInOut: openIn │ │ │ │ -00273620: 4f75 742c 2e0a 2020 2a20 4f75 7470 7574 Out,.. * Output │ │ │ │ -00273630: 733a 0a20 2020 2020 202a 2061 2073 7472 s:. * a str │ │ │ │ -00273640: 696e 6720 636f 6e74 6169 6e69 6e67 2074 ing containing t │ │ │ │ -00273650: 6865 2063 6f6e 7465 6e74 7320 6f66 2074 he contents of t │ │ │ │ -00273660: 6865 2066 696c 652e 2020 4966 2074 6865 he file. If the │ │ │ │ -00273670: 2066 696c 6520 7761 7320 616c 7265 6164 file was alread │ │ │ │ -00273680: 790a 2020 2020 2020 2020 6f70 656e 2061 y. open a │ │ │ │ -00273690: 6e64 2070 6172 7469 616c 6c79 2072 6561 nd partially rea │ │ │ │ -002736a0: 642c 2074 6865 2072 656d 6169 6e64 6572 d, the remainder │ │ │ │ -002736b0: 206f 6620 7468 6520 636f 6e74 656e 7473 of the contents │ │ │ │ -002736c0: 206f 6620 7468 6520 6669 6c65 2061 7265 of the file are │ │ │ │ -002736d0: 0a20 2020 2020 2020 2072 6574 7572 6e65 . returne │ │ │ │ -002736e0: 642e 0a20 202a 2043 6f6e 7365 7175 656e d.. * Consequen │ │ │ │ -002736f0: 6365 733a 0a20 2020 2020 202a 2049 6e20 ces:. * In │ │ │ │ -00273700: 7468 6520 6361 7365 2077 6865 7265 2066 the case where f │ │ │ │ -00273710: 2069 7320 6120 6669 6c65 2c20 6974 7320 is a file, its │ │ │ │ -00273720: 696e 7075 7420 7369 6465 2069 7320 636c input side is cl │ │ │ │ -00273730: 6f73 6564 2e0a 0a44 6573 6372 6970 7469 osed...Descripti │ │ │ │ -00273740: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00273750: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00273760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273770: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -00273780: 2274 6573 742d 6669 6c65 2220 3c3c 2022 "test-file" << " │ │ │ │ -00273790: 6869 2074 6865 7265 2220 3c3c 2063 6c6f hi there" << clo │ │ │ │ -002737a0: 7365 7c0a 7c20 2020 2020 2020 2020 2020 se|.| │ │ │ │ -002737b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002737c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -002737d0: 3120 3d20 7465 7374 2d66 696c 6520 2020 1 = test-file │ │ │ │ +002733d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002733e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002733f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00273400: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00273410: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00273420: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00273430: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00273440: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +00273450: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00273460: 636b 6167 6573 2f0a 4d61 6361 756c 6179 ckages/.Macaulay │ │ │ │ +00273470: 3244 6f63 2f6f 765f 7379 7374 656d 2e6d 2Doc/ov_system.m │ │ │ │ +00273480: 323a 3135 3335 3a30 2e0a 1f0a 4669 6c65 2:1535:0....File │ │ │ │ +00273490: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ +002734a0: 6e66 6f2c 204e 6f64 653a 2067 6574 2c20 nfo, Node: get, │ │ │ │ +002734b0: 4e65 7874 3a20 6765 7463 2c20 5072 6576 Next: getc, Prev │ │ │ │ +002734c0: 3a20 636c 6f73 654f 7574 2c20 5570 3a20 : closeOut, Up: │ │ │ │ +002734d0: 7573 696e 6720 736f 636b 6574 730a 0a67 using sockets..g │ │ │ │ +002734e0: 6574 202d 2d20 6765 7420 7468 6520 636f et -- get the co │ │ │ │ +002734f0: 6e74 656e 7473 206f 6620 6120 6669 6c65 ntents of a file │ │ │ │ +00273500: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +00273510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00273520: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +00273530: 2020 2020 2020 2020 6765 7420 660a 2020 get f. │ │ │ │ +00273540: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +00273550: 2a20 662c 2061 202a 6e6f 7465 2066 696c * f, a *note fil │ │ │ │ +00273560: 653a 2046 696c 652c 206f 7220 6120 2a6e e: File, or a *n │ │ │ │ +00273570: 6f74 6520 7374 7269 6e67 3a20 5374 7269 ote string: Stri │ │ │ │ +00273580: 6e67 2c2e 2020 4966 2066 2069 7320 6120 ng,. If f is a │ │ │ │ +00273590: 7374 7269 6e67 2c0a 2020 2020 2020 2020 string,. │ │ │ │ +002735a0: 7468 656e 2069 7420 6973 206f 7065 6e65 then it is opene │ │ │ │ +002735b0: 642c 2061 7320 7769 7468 202a 6e6f 7465 d, as with *note │ │ │ │ +002735c0: 206f 7065 6e49 6e3a 206f 7065 6e49 6e5f openIn: openIn_ │ │ │ │ +002735d0: 6c70 5374 7269 6e67 5f72 702c 2e0a 2020 lpString_rp,.. │ │ │ │ +002735e0: 2020 2020 2020 4669 6c65 6e61 6d65 7320 Filenames │ │ │ │ +002735f0: 7374 6172 7469 6e67 2077 6974 6820 2120 starting with ! │ │ │ │ +00273600: 6f72 2077 6974 6820 2420 6172 6520 7472 or with $ are tr │ │ │ │ +00273610: 6561 7465 6420 7370 6563 6961 6c6c 792c eated specially, │ │ │ │ +00273620: 2073 6565 202a 6e6f 7465 0a20 2020 2020 see *note. │ │ │ │ +00273630: 2020 206f 7065 6e49 6e4f 7574 3a20 6f70 openInOut: op │ │ │ │ +00273640: 656e 496e 4f75 742c 2e0a 2020 2a20 4f75 enInOut,.. * Ou │ │ │ │ +00273650: 7470 7574 733a 0a20 2020 2020 202a 2061 tputs:. * a │ │ │ │ +00273660: 2073 7472 696e 6720 636f 6e74 6169 6e69 string containi │ │ │ │ +00273670: 6e67 2074 6865 2063 6f6e 7465 6e74 7320 ng the contents │ │ │ │ +00273680: 6f66 2074 6865 2066 696c 652e 2020 4966 of the file. If │ │ │ │ +00273690: 2074 6865 2066 696c 6520 7761 7320 616c the file was al │ │ │ │ +002736a0: 7265 6164 790a 2020 2020 2020 2020 6f70 ready. op │ │ │ │ +002736b0: 656e 2061 6e64 2070 6172 7469 616c 6c79 en and partially │ │ │ │ +002736c0: 2072 6561 642c 2074 6865 2072 656d 6169 read, the remai │ │ │ │ +002736d0: 6e64 6572 206f 6620 7468 6520 636f 6e74 nder of the cont │ │ │ │ +002736e0: 656e 7473 206f 6620 7468 6520 6669 6c65 ents of the file │ │ │ │ +002736f0: 2061 7265 0a20 2020 2020 2020 2072 6574 are. ret │ │ │ │ +00273700: 7572 6e65 642e 0a20 202a 2043 6f6e 7365 urned.. * Conse │ │ │ │ +00273710: 7175 656e 6365 733a 0a20 2020 2020 202a quences:. * │ │ │ │ +00273720: 2049 6e20 7468 6520 6361 7365 2077 6865 In the case whe │ │ │ │ +00273730: 7265 2066 2069 7320 6120 6669 6c65 2c20 re f is a file, │ │ │ │ +00273740: 6974 7320 696e 7075 7420 7369 6465 2069 its input side i │ │ │ │ +00273750: 7320 636c 6f73 6564 2e0a 0a44 6573 6372 s closed...Descr │ │ │ │ +00273760: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00273770: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ==..+----------- │ │ │ │ +00273780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +002737a0: 3120 3a20 2274 6573 742d 6669 6c65 2220 1 : "test-file" │ │ │ │ +002737b0: 3c3c 2022 6869 2074 6865 7265 2220 3c3c << "hi there" << │ │ │ │ +002737c0: 2063 6c6f 7365 7c0a 7c20 2020 2020 2020 close|.| │ │ │ │ +002737d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002737e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002737f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00273800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00273810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00273820: 7c0a 7c6f 3120 3a20 4669 6c65 2020 2020 |.|o1 : File │ │ │ │ +002737f0: 7c0a 7c6f 3120 3d20 7465 7374 2d66 696c |.|o1 = test-fil │ │ │ │ +00273800: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00273810: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00273820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00273830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00273840: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00273850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273870: 2d2d 2d2d 2b0a 7c69 3220 3a20 6765 7420 ----+.|i2 : get │ │ │ │ -00273880: 2274 6573 742d 6669 6c65 2220 2020 2020 "test-file" │ │ │ │ -00273890: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -002738a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00273840: 2020 2020 7c0a 7c6f 3120 3a20 4669 6c65 |.|o1 : File │ │ │ │ +00273850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00273860: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00273870: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00273880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273890: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ +002738a0: 6765 7420 2274 6573 742d 6669 6c65 2220 get "test-file" │ │ │ │ 002738b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002738c0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -002738d0: 6869 2074 6865 7265 2020 2020 2020 2020 hi there │ │ │ │ -002738e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002738f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00273900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00273920: 3320 3a20 7265 6d6f 7665 4669 6c65 2022 3 : removeFile " │ │ │ │ -00273930: 7465 7374 2d66 696c 6522 2020 2020 2020 test-file" │ │ │ │ -00273940: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -00273950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273970: 2b0a 7c69 3420 3a20 6765 7420 2221 6461 +.|i4 : get "!da │ │ │ │ -00273980: 7465 2220 2020 2020 2020 2020 2020 2020 te" │ │ │ │ -00273990: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -002739a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002739b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002739c0: 2020 2020 7c0a 7c6f 3420 3d20 5361 7420 |.|o4 = Sat │ │ │ │ -002739d0: 4f63 7420 3234 2030 333a 3132 3a34 3820 Oct 24 03:12:48 │ │ │ │ -002739e0: 5554 4320 3230 3236 2020 2020 2020 7c0a UTC 2026 |. │ │ │ │ -002739f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00273a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273a10: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -00273a20: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00273a30: 2a20 2a6e 6f74 6520 7265 6164 3a20 7265 * *note read: re │ │ │ │ -00273a40: 6164 2c20 2d2d 2072 6561 6420 6672 6f6d ad, -- read from │ │ │ │ -00273a50: 2061 2066 696c 650a 2020 2a20 2a6e 6f74 a file. * *not │ │ │ │ -00273a60: 6520 7265 6d6f 7665 4669 6c65 3a20 7265 e removeFile: re │ │ │ │ -00273a70: 6d6f 7665 4669 6c65 2c20 2d2d 2072 656d moveFile, -- rem │ │ │ │ -00273a80: 6f76 6520 6120 6669 6c65 0a20 202a 202a ove a file. * * │ │ │ │ -00273a90: 6e6f 7465 2063 6c6f 7365 3a20 636c 6f73 note close: clos │ │ │ │ -00273aa0: 652c 202d 2d20 636c 6f73 6520 6120 6669 e, -- close a fi │ │ │ │ -00273ab0: 6c65 0a20 202a 202a 6e6f 7465 2046 696c le. * *note Fil │ │ │ │ -00273ac0: 6520 3c3c 2054 6869 6e67 3a20 7072 696e e << Thing: prin │ │ │ │ -00273ad0: 7469 6e67 2074 6f20 6120 6669 6c65 2c20 ting to a file, │ │ │ │ -00273ae0: 2d2d 2070 7269 6e74 2074 6f20 6120 6669 -- print to a fi │ │ │ │ -00273af0: 6c65 0a0a 5761 7973 2074 6f20 7573 6520 le..Ways to use │ │ │ │ -00273b00: 6765 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d get:.=========== │ │ │ │ -00273b10: 3d3d 3d3d 3d0a 0a20 202a 2022 6765 7428 =====.. * "get( │ │ │ │ -00273b20: 4669 6c65 2922 0a20 202a 2022 6765 7428 File)". * "get( │ │ │ │ -00273b30: 5374 7269 6e67 2922 0a0a 466f 7220 7468 String)"..For th │ │ │ │ -00273b40: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00273b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00273b60: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00273b70: 6520 6765 743a 2067 6574 2c20 6973 2061 e get: get, is a │ │ │ │ -00273b80: 202a 6e6f 7465 2063 6f6d 7069 6c65 6420 *note compiled │ │ │ │ -00273b90: 6675 6e63 7469 6f6e 3a20 436f 6d70 696c function: Compil │ │ │ │ -00273ba0: 6564 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d edFunction,...-- │ │ │ │ -00273bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002738c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +002738d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002738e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +002738f0: 3220 3d20 6869 2074 6865 7265 2020 2020 2 = hi there │ │ │ │ +00273900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00273910: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00273920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273940: 2b0a 7c69 3320 3a20 7265 6d6f 7665 4669 +.|i3 : removeFi │ │ │ │ +00273950: 6c65 2022 7465 7374 2d66 696c 6522 2020 le "test-file" │ │ │ │ +00273960: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00273970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273990: 2d2d 2d2d 2b0a 7c69 3420 3a20 6765 7420 ----+.|i4 : get │ │ │ │ +002739a0: 2221 6461 7465 2220 2020 2020 2020 2020 "!date" │ │ │ │ +002739b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +002739c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +002739d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002739e0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ +002739f0: 5468 7520 4f63 7420 3239 2031 323a 3036 Thu Oct 29 12:06 │ │ │ │ +00273a00: 3a32 3620 5554 4320 3230 3236 2020 2020 :26 UTC 2026 │ │ │ │ +00273a10: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00273a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +00273a40: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00273a50: 0a0a 2020 2a20 2a6e 6f74 6520 7265 6164 .. * *note read │ │ │ │ +00273a60: 3a20 7265 6164 2c20 2d2d 2072 6561 6420 : read, -- read │ │ │ │ +00273a70: 6672 6f6d 2061 2066 696c 650a 2020 2a20 from a file. * │ │ │ │ +00273a80: 2a6e 6f74 6520 7265 6d6f 7665 4669 6c65 *note removeFile │ │ │ │ +00273a90: 3a20 7265 6d6f 7665 4669 6c65 2c20 2d2d : removeFile, -- │ │ │ │ +00273aa0: 2072 656d 6f76 6520 6120 6669 6c65 0a20 remove a file. │ │ │ │ +00273ab0: 202a 202a 6e6f 7465 2063 6c6f 7365 3a20 * *note close: │ │ │ │ +00273ac0: 636c 6f73 652c 202d 2d20 636c 6f73 6520 close, -- close │ │ │ │ +00273ad0: 6120 6669 6c65 0a20 202a 202a 6e6f 7465 a file. * *note │ │ │ │ +00273ae0: 2046 696c 6520 3c3c 2054 6869 6e67 3a20 File << Thing: │ │ │ │ +00273af0: 7072 696e 7469 6e67 2074 6f20 6120 6669 printing to a fi │ │ │ │ +00273b00: 6c65 2c20 2d2d 2070 7269 6e74 2074 6f20 le, -- print to │ │ │ │ +00273b10: 6120 6669 6c65 0a0a 5761 7973 2074 6f20 a file..Ways to │ │ │ │ +00273b20: 7573 6520 6765 743a 0a3d 3d3d 3d3d 3d3d use get:.======= │ │ │ │ +00273b30: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +00273b40: 6765 7428 4669 6c65 2922 0a20 202a 2022 get(File)". * " │ │ │ │ +00273b50: 6765 7428 5374 7269 6e67 2922 0a0a 466f get(String)"..Fo │ │ │ │ +00273b60: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00273b70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00273b80: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00273b90: 2a6e 6f74 6520 6765 743a 2067 6574 2c20 *note get: get, │ │ │ │ +00273ba0: 6973 2061 202a 6e6f 7465 2063 6f6d 7069 is a *note compi │ │ │ │ +00273bb0: 6c65 6420 6675 6e63 7469 6f6e 3a20 436f led function: Co │ │ │ │ +00273bc0: 6d70 696c 6564 4675 6e63 7469 6f6e 2c2e mpiledFunction,. │ │ │ │ +00273bd0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ 00273be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -00273c00: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -00273c10: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -00273c20: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -00273c30: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -00273c40: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ -00273c50: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -00273c60: 6167 6573 2f0a 4d61 6361 756c 6179 3244 ages/.Macaulay2D │ │ │ │ -00273c70: 6f63 2f6f 765f 7379 7374 656d 2e6d 323a oc/ov_system.m2: │ │ │ │ -00273c80: 3334 303a 302e 0a1f 0a46 696c 653a 204d 340:0....File: M │ │ │ │ -00273c90: 6163 6175 6c61 7932 446f 632e 696e 666f acaulay2Doc.info │ │ │ │ -00273ca0: 2c20 4e6f 6465 3a20 6765 7463 2c20 4e65 , Node: getc, Ne │ │ │ │ -00273cb0: 7874 3a20 7265 6164 2c20 5072 6576 3a20 xt: read, Prev: │ │ │ │ -00273cc0: 6765 742c 2055 703a 2075 7369 6e67 2073 get, Up: using s │ │ │ │ -00273cd0: 6f63 6b65 7473 0a0a 6765 7463 202d 2d20 ockets..getc -- │ │ │ │ -00273ce0: 6765 7420 6120 6279 7465 0a2a 2a2a 2a2a get a byte.***** │ │ │ │ -00273cf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 *************..D │ │ │ │ -00273d00: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00273d10: 3d3d 3d3d 3d3d 0a0a 6765 7463 2066 206f ======..getc f o │ │ │ │ -00273d20: 6274 6169 6e73 206f 6e65 2062 7974 6520 btains one byte │ │ │ │ -00273d30: 6672 6f6d 2074 6865 2069 6e70 7574 2066 from the input f │ │ │ │ -00273d40: 696c 6520 6620 616e 6420 7072 6f76 6964 ile f and provid │ │ │ │ -00273d50: 6573 2069 7420 6173 2061 2073 7472 696e es it as a strin │ │ │ │ -00273d60: 6720 6f66 0a6c 656e 6774 6820 312e 2020 g of.length 1. │ │ │ │ -00273d70: 4f6e 2065 6e64 206f 6620 6669 6c65 2061 On end of file a │ │ │ │ -00273d80: 6e20 656d 7074 7920 7374 7269 6e67 206f n empty string o │ │ │ │ -00273d90: 6620 6973 2072 6574 7572 6e65 642e 0a0a f is returned... │ │ │ │ -00273da0: 0a0a 4275 673a 2074 6865 206e 616d 6520 ..Bug: the name │ │ │ │ -00273db0: 6973 2063 7279 7074 6963 2061 6e64 2073 is cryptic and s │ │ │ │ -00273dc0: 686f 756c 6420 6265 2063 6861 6e67 6564 hould be changed │ │ │ │ -00273dd0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -00273de0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -00273df0: 4669 6c65 3a20 4669 6c65 2c20 2d2d 2074 File: File, -- t │ │ │ │ -00273e00: 6865 2063 6c61 7373 206f 6620 616c 6c20 he class of all │ │ │ │ -00273e10: 6669 6c65 730a 0a57 6179 7320 746f 2075 files..Ways to u │ │ │ │ -00273e20: 7365 2067 6574 633a 0a3d 3d3d 3d3d 3d3d se getc:.======= │ │ │ │ -00273e30: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -00273e40: 2267 6574 6328 4669 6c65 2922 0a0a 466f "getc(File)"..Fo │ │ │ │ -00273e50: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -00273e60: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00273e70: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -00273e80: 2a6e 6f74 6520 6765 7463 3a20 6765 7463 *note getc: getc │ │ │ │ -00273e90: 2c20 6973 2061 202a 6e6f 7465 2063 6f6d , is a *note com │ │ │ │ -00273ea0: 7069 6c65 6420 6675 6e63 7469 6f6e 3a20 piled function: │ │ │ │ -00273eb0: 436f 6d70 696c 6564 4675 6e63 7469 6f6e CompiledFunction │ │ │ │ -00273ec0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ -00273ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273c20: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +00273c30: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +00273c40: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +00273c50: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +00273c60: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +00273c70: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +00273c80: 7061 636b 6167 6573 2f0a 4d61 6361 756c packages/.Macaul │ │ │ │ +00273c90: 6179 3244 6f63 2f6f 765f 7379 7374 656d ay2Doc/ov_system │ │ │ │ +00273ca0: 2e6d 323a 3334 303a 302e 0a1f 0a46 696c .m2:340:0....Fil │ │ │ │ +00273cb0: 653a 204d 6163 6175 6c61 7932 446f 632e e: Macaulay2Doc. │ │ │ │ +00273cc0: 696e 666f 2c20 4e6f 6465 3a20 6765 7463 info, Node: getc │ │ │ │ +00273cd0: 2c20 4e65 7874 3a20 7265 6164 2c20 5072 , Next: read, Pr │ │ │ │ +00273ce0: 6576 3a20 6765 742c 2055 703a 2075 7369 ev: get, Up: usi │ │ │ │ +00273cf0: 6e67 2073 6f63 6b65 7473 0a0a 6765 7463 ng sockets..getc │ │ │ │ +00273d00: 202d 2d20 6765 7420 6120 6279 7465 0a2a -- get a byte.* │ │ │ │ +00273d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00273d20: 2a0a 0a44 6573 6372 6970 7469 6f6e 0a3d *..Description.= │ │ │ │ +00273d30: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 6765 7463 ==========..getc │ │ │ │ +00273d40: 2066 206f 6274 6169 6e73 206f 6e65 2062 f obtains one b │ │ │ │ +00273d50: 7974 6520 6672 6f6d 2074 6865 2069 6e70 yte from the inp │ │ │ │ +00273d60: 7574 2066 696c 6520 6620 616e 6420 7072 ut file f and pr │ │ │ │ +00273d70: 6f76 6964 6573 2069 7420 6173 2061 2073 ovides it as a s │ │ │ │ +00273d80: 7472 696e 6720 6f66 0a6c 656e 6774 6820 tring of.length │ │ │ │ +00273d90: 312e 2020 4f6e 2065 6e64 206f 6620 6669 1. On end of fi │ │ │ │ +00273da0: 6c65 2061 6e20 656d 7074 7920 7374 7269 le an empty stri │ │ │ │ +00273db0: 6e67 206f 6620 6973 2072 6574 7572 6e65 ng of is returne │ │ │ │ +00273dc0: 642e 0a0a 0a0a 4275 673a 2074 6865 206e d.....Bug: the n │ │ │ │ +00273dd0: 616d 6520 6973 2063 7279 7074 6963 2061 ame is cryptic a │ │ │ │ +00273de0: 6e64 2073 686f 756c 6420 6265 2063 6861 nd should be cha │ │ │ │ +00273df0: 6e67 6564 2e0a 0a53 6565 2061 6c73 6f0a nged...See also. │ │ │ │ +00273e00: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00273e10: 6f74 6520 4669 6c65 3a20 4669 6c65 2c20 ote File: File, │ │ │ │ +00273e20: 2d2d 2074 6865 2063 6c61 7373 206f 6620 -- the class of │ │ │ │ +00273e30: 616c 6c20 6669 6c65 730a 0a57 6179 7320 all files..Ways │ │ │ │ +00273e40: 746f 2075 7365 2067 6574 633a 0a3d 3d3d to use getc:.=== │ │ │ │ +00273e50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00273e60: 2020 2a20 2267 6574 6328 4669 6c65 2922 * "getc(File)" │ │ │ │ +00273e70: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ +00273e80: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ +00273e90: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ +00273ea0: 6563 7420 2a6e 6f74 6520 6765 7463 3a20 ect *note getc: │ │ │ │ +00273eb0: 6765 7463 2c20 6973 2061 202a 6e6f 7465 getc, is a *note │ │ │ │ +00273ec0: 2063 6f6d 7069 6c65 6420 6675 6e63 7469 compiled functi │ │ │ │ +00273ed0: 6f6e 3a20 436f 6d70 696c 6564 4675 6e63 on: CompiledFunc │ │ │ │ +00273ee0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ 00273ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00273f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00273f10: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -00273f20: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -00273f30: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -00273f40: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -00273f50: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -00273f60: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -00273f70: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -00273f80: 756c 6179 3244 6f63 2f6f 765f 7379 7374 ulay2Doc/ov_syst │ │ │ │ -00273f90: 656d 2e6d 323a 3534 383a 302e 0a1f 0a46 em.m2:548:0....F │ │ │ │ -00273fa0: 696c 653a 204d 6163 6175 6c61 7932 446f ile: Macaulay2Do │ │ │ │ -00273fb0: 632e 696e 666f 2c20 4e6f 6465 3a20 7265 c.info, Node: re │ │ │ │ -00273fc0: 6164 2c20 4e65 7874 3a20 7363 616e 4c69 ad, Next: scanLi │ │ │ │ -00273fd0: 6e65 732c 2050 7265 763a 2067 6574 632c nes, Prev: getc, │ │ │ │ -00273fe0: 2055 703a 2075 7369 6e67 2073 6f63 6b65 Up: using socke │ │ │ │ -00273ff0: 7473 0a0a 7265 6164 202d 2d20 7265 6164 ts..read -- read │ │ │ │ -00274000: 2066 726f 6d20 6120 6669 6c65 0a2a 2a2a from a file.*** │ │ │ │ -00274010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00274020: 2a2a 2a2a 2a0a 0a2a 204d 656e 753a 0a0a *****..* Menu:.. │ │ │ │ -00274030: 2a20 7265 6164 2853 6571 7565 6e63 6529 * read(Sequence) │ │ │ │ -00274040: 3a20 7265 6164 5f6c 7053 6571 7565 6e63 : read_lpSequenc │ │ │ │ -00274050: 655f 7270 2e20 2072 6561 6420 6672 6f6d e_rp. read from │ │ │ │ -00274060: 2061 2066 696c 650a 2a20 7265 6164 2853 a file.* read(S │ │ │ │ -00274070: 7472 696e 6729 3a20 7265 6164 5f6c 7053 tring): read_lpS │ │ │ │ -00274080: 7472 696e 675f 7270 2e20 2072 6561 6420 tring_rp. read │ │ │ │ -00274090: 6672 6f6d 2061 2066 696c 650a 2a20 7265 from a file.* re │ │ │ │ -002740a0: 6164 2846 696c 6529 3a20 7265 6164 5f6c ad(File): read_l │ │ │ │ -002740b0: 7046 696c 655f 7270 2e20 2020 7265 6164 pFile_rp. read │ │ │ │ -002740c0: 2066 726f 6d20 6120 6669 6c65 0a2a 2072 from a file.* r │ │ │ │ -002740d0: 6561 6428 4669 6c65 2c5a 5a29 3a20 7265 ead(File,ZZ): re │ │ │ │ -002740e0: 6164 5f6c 7046 696c 655f 636d 5a5a 5f72 ad_lpFile_cmZZ_r │ │ │ │ -002740f0: 702e 2020 7265 6164 2066 726f 6d20 6120 p. read from a │ │ │ │ -00274100: 6669 6c65 0a0a 5761 7973 2074 6f20 7573 file..Ways to us │ │ │ │ -00274110: 6520 7265 6164 3a0a 3d3d 3d3d 3d3d 3d3d e read:.======== │ │ │ │ -00274120: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a =========.. * * │ │ │ │ -00274130: 6e6f 7465 2072 6561 6428 4669 6c65 293a note read(File): │ │ │ │ -00274140: 2072 6561 645f 6c70 4669 6c65 5f72 702c read_lpFile_rp, │ │ │ │ -00274150: 202d 2d20 7265 6164 2066 726f 6d20 6120 -- read from a │ │ │ │ -00274160: 6669 6c65 0a20 202a 202a 6e6f 7465 2072 file. * *note r │ │ │ │ -00274170: 6561 6428 4669 6c65 2c5a 5a29 3a20 7265 ead(File,ZZ): re │ │ │ │ -00274180: 6164 5f6c 7046 696c 655f 636d 5a5a 5f72 ad_lpFile_cmZZ_r │ │ │ │ -00274190: 702c 202d 2d20 7265 6164 2066 726f 6d20 p, -- read from │ │ │ │ -002741a0: 6120 6669 6c65 0a20 202a 202a 6e6f 7465 a file. * *note │ │ │ │ -002741b0: 2072 6561 6428 5365 7175 656e 6365 293a read(Sequence): │ │ │ │ -002741c0: 2072 6561 645f 6c70 5365 7175 656e 6365 read_lpSequence │ │ │ │ -002741d0: 5f72 702c 202d 2d20 7265 6164 2066 726f _rp, -- read fro │ │ │ │ -002741e0: 6d20 6120 6669 6c65 0a20 202a 202a 6e6f m a file. * *no │ │ │ │ -002741f0: 7465 2072 6561 6428 5374 7269 6e67 293a te read(String): │ │ │ │ -00274200: 2072 6561 645f 6c70 5374 7269 6e67 5f72 read_lpString_r │ │ │ │ -00274210: 702c 202d 2d20 7265 6164 2066 726f 6d20 p, -- read from │ │ │ │ -00274220: 6120 6669 6c65 0a0a 466f 7220 7468 6520 a file..For the │ │ │ │ -00274230: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00274240: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00274250: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00274260: 7265 6164 3a20 7265 6164 2c20 6973 2061 read: read, is a │ │ │ │ -00274270: 202a 6e6f 7465 2063 6f6d 7069 6c65 6420 *note compiled │ │ │ │ -00274280: 6675 6e63 7469 6f6e 3a20 436f 6d70 696c function: Compil │ │ │ │ -00274290: 6564 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d edFunction,...-- │ │ │ │ -002742a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002742b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002742c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00273f30: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00273f40: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00273f50: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00273f60: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00273f70: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00273f80: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +00273f90: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00273fa0: 4d61 6361 756c 6179 3244 6f63 2f6f 765f Macaulay2Doc/ov_ │ │ │ │ +00273fb0: 7379 7374 656d 2e6d 323a 3534 383a 302e system.m2:548:0. │ │ │ │ +00273fc0: 0a1f 0a46 696c 653a 204d 6163 6175 6c61 ...File: Macaula │ │ │ │ +00273fd0: 7932 446f 632e 696e 666f 2c20 4e6f 6465 y2Doc.info, Node │ │ │ │ +00273fe0: 3a20 7265 6164 2c20 4e65 7874 3a20 7363 : read, Next: sc │ │ │ │ +00273ff0: 616e 4c69 6e65 732c 2050 7265 763a 2067 anLines, Prev: g │ │ │ │ +00274000: 6574 632c 2055 703a 2075 7369 6e67 2073 etc, Up: using s │ │ │ │ +00274010: 6f63 6b65 7473 0a0a 7265 6164 202d 2d20 ockets..read -- │ │ │ │ +00274020: 7265 6164 2066 726f 6d20 6120 6669 6c65 read from a file │ │ │ │ +00274030: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +00274040: 2a2a 2a2a 2a2a 2a2a 2a0a 0a2a 204d 656e *********..* Men │ │ │ │ +00274050: 753a 0a0a 2a20 7265 6164 2853 6571 7565 u:..* read(Seque │ │ │ │ +00274060: 6e63 6529 3a20 7265 6164 5f6c 7053 6571 nce): read_lpSeq │ │ │ │ +00274070: 7565 6e63 655f 7270 2e20 2072 6561 6420 uence_rp. read │ │ │ │ +00274080: 6672 6f6d 2061 2066 696c 650a 2a20 7265 from a file.* re │ │ │ │ +00274090: 6164 2853 7472 696e 6729 3a20 7265 6164 ad(String): read │ │ │ │ +002740a0: 5f6c 7053 7472 696e 675f 7270 2e20 2072 _lpString_rp. r │ │ │ │ +002740b0: 6561 6420 6672 6f6d 2061 2066 696c 650a ead from a file. │ │ │ │ +002740c0: 2a20 7265 6164 2846 696c 6529 3a20 7265 * read(File): re │ │ │ │ +002740d0: 6164 5f6c 7046 696c 655f 7270 2e20 2020 ad_lpFile_rp. │ │ │ │ +002740e0: 7265 6164 2066 726f 6d20 6120 6669 6c65 read from a file │ │ │ │ +002740f0: 0a2a 2072 6561 6428 4669 6c65 2c5a 5a29 .* read(File,ZZ) │ │ │ │ +00274100: 3a20 7265 6164 5f6c 7046 696c 655f 636d : read_lpFile_cm │ │ │ │ +00274110: 5a5a 5f72 702e 2020 7265 6164 2066 726f ZZ_rp. read fro │ │ │ │ +00274120: 6d20 6120 6669 6c65 0a0a 5761 7973 2074 m a file..Ways t │ │ │ │ +00274130: 6f20 7573 6520 7265 6164 3a0a 3d3d 3d3d o use read:.==== │ │ │ │ +00274140: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00274150: 202a 202a 6e6f 7465 2072 6561 6428 4669 * *note read(Fi │ │ │ │ +00274160: 6c65 293a 2072 6561 645f 6c70 4669 6c65 le): read_lpFile │ │ │ │ +00274170: 5f72 702c 202d 2d20 7265 6164 2066 726f _rp, -- read fro │ │ │ │ +00274180: 6d20 6120 6669 6c65 0a20 202a 202a 6e6f m a file. * *no │ │ │ │ +00274190: 7465 2072 6561 6428 4669 6c65 2c5a 5a29 te read(File,ZZ) │ │ │ │ +002741a0: 3a20 7265 6164 5f6c 7046 696c 655f 636d : read_lpFile_cm │ │ │ │ +002741b0: 5a5a 5f72 702c 202d 2d20 7265 6164 2066 ZZ_rp, -- read f │ │ │ │ +002741c0: 726f 6d20 6120 6669 6c65 0a20 202a 202a rom a file. * * │ │ │ │ +002741d0: 6e6f 7465 2072 6561 6428 5365 7175 656e note read(Sequen │ │ │ │ +002741e0: 6365 293a 2072 6561 645f 6c70 5365 7175 ce): read_lpSequ │ │ │ │ +002741f0: 656e 6365 5f72 702c 202d 2d20 7265 6164 ence_rp, -- read │ │ │ │ +00274200: 2066 726f 6d20 6120 6669 6c65 0a20 202a from a file. * │ │ │ │ +00274210: 202a 6e6f 7465 2072 6561 6428 5374 7269 *note read(Stri │ │ │ │ +00274220: 6e67 293a 2072 6561 645f 6c70 5374 7269 ng): read_lpStri │ │ │ │ +00274230: 6e67 5f72 702c 202d 2d20 7265 6164 2066 ng_rp, -- read f │ │ │ │ +00274240: 726f 6d20 6120 6669 6c65 0a0a 466f 7220 rom a file..For │ │ │ │ +00274250: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +00274260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00274270: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +00274280: 6f74 6520 7265 6164 3a20 7265 6164 2c20 ote read: read, │ │ │ │ +00274290: 6973 2061 202a 6e6f 7465 2063 6f6d 7069 is a *note compi │ │ │ │ +002742a0: 6c65 6420 6675 6e63 7469 6f6e 3a20 436f led function: Co │ │ │ │ +002742b0: 6d70 696c 6564 4675 6e63 7469 6f6e 2c2e mpiledFunction,. │ │ │ │ +002742c0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ 002742d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002742e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -002742f0: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -00274300: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -00274310: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -00274320: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -00274330: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ -00274340: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -00274350: 6167 6573 2f0a 4d61 6361 756c 6179 3244 ages/.Macaulay2D │ │ │ │ -00274360: 6f63 2f6f 765f 7379 7374 656d 2e6d 323a oc/ov_system.m2: │ │ │ │ -00274370: 3236 323a 302e 0a1f 0a46 696c 653a 204d 262:0....File: M │ │ │ │ -00274380: 6163 6175 6c61 7932 446f 632e 696e 666f acaulay2Doc.info │ │ │ │ -00274390: 2c20 4e6f 6465 3a20 7265 6164 5f6c 7053 , Node: read_lpS │ │ │ │ -002743a0: 6571 7565 6e63 655f 7270 2c20 4e65 7874 equence_rp, Next │ │ │ │ -002743b0: 3a20 7265 6164 5f6c 7053 7472 696e 675f : read_lpString_ │ │ │ │ -002743c0: 7270 2c20 5570 3a20 7265 6164 0a0a 7265 rp, Up: read..re │ │ │ │ -002743d0: 6164 2853 6571 7565 6e63 6529 202d 2d20 ad(Sequence) -- │ │ │ │ -002743e0: 7265 6164 2066 726f 6d20 6120 6669 6c65 read from a file │ │ │ │ -002743f0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -00274400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00274410: 2a2a 2a0a 0a20 202a 2046 756e 6374 696f ***.. * Functio │ │ │ │ -00274420: 6e3a 202a 6e6f 7465 2072 6561 643a 2072 n: *note read: r │ │ │ │ -00274430: 6561 642c 0a20 202a 2055 7361 6765 3a20 ead,. * Usage: │ │ │ │ -00274440: 0a20 2020 2020 2020 2072 6561 6428 290a . read(). │ │ │ │ -00274450: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -00274460: 2020 2a20 6120 2a6e 6f74 6520 7365 7175 * a *note sequ │ │ │ │ -00274470: 656e 6365 3a20 5365 7175 656e 6365 2c2c ence: Sequence,, │ │ │ │ -00274480: 2028 290a 2020 2a20 4f75 7470 7574 733a (). * Outputs: │ │ │ │ -00274490: 0a20 2020 2020 202a 2061 202a 6e6f 7465 . * a *note │ │ │ │ -002744a0: 2073 7472 696e 673a 2053 7472 696e 672c string: String, │ │ │ │ -002744b0: 2c20 6120 7374 7269 6e67 206f 6274 6169 , a string obtai │ │ │ │ -002744c0: 6e65 6420 6279 2072 6561 6469 6e67 2061 ned by reading a │ │ │ │ -002744d0: 206c 696e 6520 6672 6f6d 2074 6865 0a20 line from the. │ │ │ │ -002744e0: 2020 2020 2020 2073 7461 6e64 6172 6420 standard │ │ │ │ -002744f0: 696e 7075 7420 6669 6c65 2c20 2a6e 6f74 input file, *not │ │ │ │ -00274500: 6520 7374 6469 6f3a 2073 7464 696f 2c2e e stdio: stdio,. │ │ │ │ -00274510: 0a0a 5761 7973 2074 6f20 7573 6520 7468 ..Ways to use th │ │ │ │ -00274520: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d is method:.===== │ │ │ │ -00274530: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00274540: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2072 ===.. * *note r │ │ │ │ -00274550: 6561 6428 5365 7175 656e 6365 293a 2072 ead(Sequence): r │ │ │ │ -00274560: 6561 645f 6c70 5365 7175 656e 6365 5f72 ead_lpSequence_r │ │ │ │ -00274570: 702c 202d 2d20 7265 6164 2066 726f 6d20 p, -- read from │ │ │ │ -00274580: 6120 6669 6c65 0a2d 2d2d 2d2d 2d2d 2d2d a file.--------- │ │ │ │ -00274590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002745a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002742e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002742f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274310: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +00274320: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +00274330: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +00274340: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +00274350: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +00274360: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +00274370: 7061 636b 6167 6573 2f0a 4d61 6361 756c packages/.Macaul │ │ │ │ +00274380: 6179 3244 6f63 2f6f 765f 7379 7374 656d ay2Doc/ov_system │ │ │ │ +00274390: 2e6d 323a 3236 323a 302e 0a1f 0a46 696c .m2:262:0....Fil │ │ │ │ +002743a0: 653a 204d 6163 6175 6c61 7932 446f 632e e: Macaulay2Doc. │ │ │ │ +002743b0: 696e 666f 2c20 4e6f 6465 3a20 7265 6164 info, Node: read │ │ │ │ +002743c0: 5f6c 7053 6571 7565 6e63 655f 7270 2c20 _lpSequence_rp, │ │ │ │ +002743d0: 4e65 7874 3a20 7265 6164 5f6c 7053 7472 Next: read_lpStr │ │ │ │ +002743e0: 696e 675f 7270 2c20 5570 3a20 7265 6164 ing_rp, Up: read │ │ │ │ +002743f0: 0a0a 7265 6164 2853 6571 7565 6e63 6529 ..read(Sequence) │ │ │ │ +00274400: 202d 2d20 7265 6164 2066 726f 6d20 6120 -- read from a │ │ │ │ +00274410: 6669 6c65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a file.*********** │ │ │ │ +00274420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00274430: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e *******.. * Fun │ │ │ │ +00274440: 6374 696f 6e3a 202a 6e6f 7465 2072 6561 ction: *note rea │ │ │ │ +00274450: 643a 2072 6561 642c 0a20 202a 2055 7361 d: read,. * Usa │ │ │ │ +00274460: 6765 3a20 0a20 2020 2020 2020 2072 6561 ge: . rea │ │ │ │ +00274470: 6428 290a 2020 2a20 496e 7075 7473 3a0a d(). * Inputs:. │ │ │ │ +00274480: 2020 2020 2020 2a20 6120 2a6e 6f74 6520 * a *note │ │ │ │ +00274490: 7365 7175 656e 6365 3a20 5365 7175 656e sequence: Sequen │ │ │ │ +002744a0: 6365 2c2c 2028 290a 2020 2a20 4f75 7470 ce,, (). * Outp │ │ │ │ +002744b0: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ +002744c0: 6e6f 7465 2073 7472 696e 673a 2053 7472 note string: Str │ │ │ │ +002744d0: 696e 672c 2c20 6120 7374 7269 6e67 206f ing,, a string o │ │ │ │ +002744e0: 6274 6169 6e65 6420 6279 2072 6561 6469 btained by readi │ │ │ │ +002744f0: 6e67 2061 206c 696e 6520 6672 6f6d 2074 ng a line from t │ │ │ │ +00274500: 6865 0a20 2020 2020 2020 2073 7461 6e64 he. stand │ │ │ │ +00274510: 6172 6420 696e 7075 7420 6669 6c65 2c20 ard input file, │ │ │ │ +00274520: 2a6e 6f74 6520 7374 6469 6f3a 2073 7464 *note stdio: std │ │ │ │ +00274530: 696f 2c2e 0a0a 5761 7973 2074 6f20 7573 io,...Ways to us │ │ │ │ +00274540: 6520 7468 6973 206d 6574 686f 643a 0a3d e this method:.= │ │ │ │ +00274550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00274560: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +00274570: 7465 2072 6561 6428 5365 7175 656e 6365 te read(Sequence │ │ │ │ +00274580: 293a 2072 6561 645f 6c70 5365 7175 656e ): read_lpSequen │ │ │ │ +00274590: 6365 5f72 702c 202d 2d20 7265 6164 2066 ce_rp, -- read f │ │ │ │ +002745a0: 726f 6d20 6120 6669 6c65 0a2d 2d2d 2d2d rom a file.----- │ │ │ │ 002745b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002745c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002745d0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -002745e0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -002745f0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00274600: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00274610: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00274620: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ -00274630: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ -00274640: 6163 6175 6c61 7932 446f 632f 6f76 5f73 acaulay2Doc/ov_s │ │ │ │ -00274650: 7973 7465 6d2e 6d32 3a32 3732 3a30 2e0a ystem.m2:272:0.. │ │ │ │ -00274660: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ -00274670: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ -00274680: 2072 6561 645f 6c70 5374 7269 6e67 5f72 read_lpString_r │ │ │ │ -00274690: 702c 204e 6578 743a 2072 6561 645f 6c70 p, Next: read_lp │ │ │ │ -002746a0: 4669 6c65 5f72 702c 2050 7265 763a 2072 File_rp, Prev: r │ │ │ │ -002746b0: 6561 645f 6c70 5365 7175 656e 6365 5f72 ead_lpSequence_r │ │ │ │ -002746c0: 702c 2055 703a 2072 6561 640a 0a72 6561 p, Up: read..rea │ │ │ │ -002746d0: 6428 5374 7269 6e67 2920 2d2d 2072 6561 d(String) -- rea │ │ │ │ -002746e0: 6420 6672 6f6d 2061 2066 696c 650a 2a2a d from a file.** │ │ │ │ -002746f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00274700: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00274710: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e * Function: *n │ │ │ │ -00274720: 6f74 6520 7265 6164 3a20 7265 6164 2c0a ote read: read,. │ │ │ │ -00274730: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00274740: 2020 2020 7265 6164 2070 0a20 202a 2049 read p. * I │ │ │ │ -00274750: 6e70 7574 733a 0a20 2020 2020 202a 2070 nputs:. * p │ │ │ │ -00274760: 2c20 6120 2a6e 6f74 6520 7374 7269 6e67 , a *note string │ │ │ │ -00274770: 3a20 5374 7269 6e67 2c2c 2061 2073 7472 : String,, a str │ │ │ │ -00274780: 696e 6720 636f 6e74 6169 6e69 6e67 2061 ing containing a │ │ │ │ -00274790: 2070 726f 6d70 7420 746f 2062 650a 2020 prompt to be. │ │ │ │ -002747a0: 2020 2020 2020 6469 7370 6c61 7965 6420 displayed │ │ │ │ -002747b0: 666f 7220 7468 6520 7573 6572 0a20 202a for the user. * │ │ │ │ -002747c0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -002747d0: 2a20 6120 2a6e 6f74 6520 7374 7269 6e67 * a *note string │ │ │ │ -002747e0: 3a20 5374 7269 6e67 2c2c 2061 2073 7472 : String,, a str │ │ │ │ -002747f0: 696e 6720 6f62 7461 696e 6564 2062 7920 ing obtained by │ │ │ │ -00274800: 7265 6164 696e 6720 6672 6f6d 2074 6865 reading from the │ │ │ │ -00274810: 2073 7461 6e64 6172 640a 2020 2020 2020 standard. │ │ │ │ -00274820: 2020 696e 7075 7420 6669 6c65 202a 6e6f input file *no │ │ │ │ -00274830: 7465 2073 7464 696f 3a20 7374 6469 6f2c te stdio: stdio, │ │ │ │ -00274840: 0a0a 5761 7973 2074 6f20 7573 6520 7468 ..Ways to use th │ │ │ │ -00274850: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d is method:.===== │ │ │ │ -00274860: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00274870: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2072 ===.. * *note r │ │ │ │ -00274880: 6561 6428 5374 7269 6e67 293a 2072 6561 ead(String): rea │ │ │ │ -00274890: 645f 6c70 5374 7269 6e67 5f72 702c 202d d_lpString_rp, - │ │ │ │ -002748a0: 2d20 7265 6164 2066 726f 6d20 6120 6669 - read from a fi │ │ │ │ -002748b0: 6c65 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d le.------------- │ │ │ │ -002748c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002748d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002745d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002745e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002745f0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +00274600: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +00274610: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +00274620: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00274630: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00274640: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ +00274650: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +00274660: 732f 0a4d 6163 6175 6c61 7932 446f 632f s/.Macaulay2Doc/ │ │ │ │ +00274670: 6f76 5f73 7973 7465 6d2e 6d32 3a32 3732 ov_system.m2:272 │ │ │ │ +00274680: 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 6361 :0....File: Maca │ │ │ │ +00274690: 756c 6179 3244 6f63 2e69 6e66 6f2c 204e ulay2Doc.info, N │ │ │ │ +002746a0: 6f64 653a 2072 6561 645f 6c70 5374 7269 ode: read_lpStri │ │ │ │ +002746b0: 6e67 5f72 702c 204e 6578 743a 2072 6561 ng_rp, Next: rea │ │ │ │ +002746c0: 645f 6c70 4669 6c65 5f72 702c 2050 7265 d_lpFile_rp, Pre │ │ │ │ +002746d0: 763a 2072 6561 645f 6c70 5365 7175 656e v: read_lpSequen │ │ │ │ +002746e0: 6365 5f72 702c 2055 703a 2072 6561 640a ce_rp, Up: read. │ │ │ │ +002746f0: 0a72 6561 6428 5374 7269 6e67 2920 2d2d .read(String) -- │ │ │ │ +00274700: 2072 6561 6420 6672 6f6d 2061 2066 696c read from a fil │ │ │ │ +00274710: 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a e.************** │ │ │ │ +00274720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00274730: 2a2a 0a0a 2020 2a20 4675 6e63 7469 6f6e **.. * Function │ │ │ │ +00274740: 3a20 2a6e 6f74 6520 7265 6164 3a20 7265 : *note read: re │ │ │ │ +00274750: 6164 2c0a 2020 2a20 5573 6167 653a 200a ad,. * Usage: . │ │ │ │ +00274760: 2020 2020 2020 2020 7265 6164 2070 0a20 read p. │ │ │ │ +00274770: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +00274780: 202a 2070 2c20 6120 2a6e 6f74 6520 7374 * p, a *note st │ │ │ │ +00274790: 7269 6e67 3a20 5374 7269 6e67 2c2c 2061 ring: String,, a │ │ │ │ +002747a0: 2073 7472 696e 6720 636f 6e74 6169 6e69 string containi │ │ │ │ +002747b0: 6e67 2061 2070 726f 6d70 7420 746f 2062 ng a prompt to b │ │ │ │ +002747c0: 650a 2020 2020 2020 2020 6469 7370 6c61 e. displa │ │ │ │ +002747d0: 7965 6420 666f 7220 7468 6520 7573 6572 yed for the user │ │ │ │ +002747e0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +002747f0: 2020 2020 2a20 6120 2a6e 6f74 6520 7374 * a *note st │ │ │ │ +00274800: 7269 6e67 3a20 5374 7269 6e67 2c2c 2061 ring: String,, a │ │ │ │ +00274810: 2073 7472 696e 6720 6f62 7461 696e 6564 string obtained │ │ │ │ +00274820: 2062 7920 7265 6164 696e 6720 6672 6f6d by reading from │ │ │ │ +00274830: 2074 6865 2073 7461 6e64 6172 640a 2020 the standard. │ │ │ │ +00274840: 2020 2020 2020 696e 7075 7420 6669 6c65 input file │ │ │ │ +00274850: 202a 6e6f 7465 2073 7464 696f 3a20 7374 *note stdio: st │ │ │ │ +00274860: 6469 6f2c 0a0a 5761 7973 2074 6f20 7573 dio,..Ways to us │ │ │ │ +00274870: 6520 7468 6973 206d 6574 686f 643a 0a3d e this method:.= │ │ │ │ +00274880: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00274890: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +002748a0: 7465 2072 6561 6428 5374 7269 6e67 293a te read(String): │ │ │ │ +002748b0: 2072 6561 645f 6c70 5374 7269 6e67 5f72 read_lpString_r │ │ │ │ +002748c0: 702c 202d 2d20 7265 6164 2066 726f 6d20 p, -- read from │ │ │ │ +002748d0: 6120 6669 6c65 0a2d 2d2d 2d2d 2d2d 2d2d a file.--------- │ │ │ │ 002748e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002748f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274900: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -00274910: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -00274920: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -00274930: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00274940: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -00274950: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -00274960: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ -00274970: 6c61 7932 446f 632f 6f76 5f73 7973 7465 lay2Doc/ov_syste │ │ │ │ -00274980: 6d2e 6d32 3a32 3832 3a30 2e0a 1f0a 4669 m.m2:282:0....Fi │ │ │ │ -00274990: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -002749a0: 2e69 6e66 6f2c 204e 6f64 653a 2072 6561 .info, Node: rea │ │ │ │ -002749b0: 645f 6c70 4669 6c65 5f72 702c 204e 6578 d_lpFile_rp, Nex │ │ │ │ -002749c0: 743a 2072 6561 645f 6c70 4669 6c65 5f63 t: read_lpFile_c │ │ │ │ -002749d0: 6d5a 5a5f 7270 2c20 5072 6576 3a20 7265 mZZ_rp, Prev: re │ │ │ │ -002749e0: 6164 5f6c 7053 7472 696e 675f 7270 2c20 ad_lpString_rp, │ │ │ │ -002749f0: 5570 3a20 7265 6164 0a0a 7265 6164 2846 Up: read..read(F │ │ │ │ -00274a00: 696c 6529 202d 2d20 7265 6164 2066 726f ile) -- read fro │ │ │ │ -00274a10: 6d20 6120 6669 6c65 0a2a 2a2a 2a2a 2a2a m a file.******* │ │ │ │ -00274a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00274a30: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e *******.. * Fun │ │ │ │ -00274a40: 6374 696f 6e3a 202a 6e6f 7465 2072 6561 ction: *note rea │ │ │ │ -00274a50: 643a 2072 6561 642c 0a20 202a 2055 7361 d: read,. * Usa │ │ │ │ -00274a60: 6765 3a20 0a20 2020 2020 2020 2072 6561 ge: . rea │ │ │ │ -00274a70: 6420 660a 2020 2a20 496e 7075 7473 3a0a d f. * Inputs:. │ │ │ │ -00274a80: 2020 2020 2020 2a20 662c 2061 202a 6e6f * f, a *no │ │ │ │ -00274a90: 7465 2066 696c 653a 2046 696c 652c 2c20 te file: File,, │ │ │ │ -00274aa0: 616e 2069 6e70 7574 2066 696c 650a 2020 an input file. │ │ │ │ -00274ab0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -00274ac0: 202a 2061 202a 6e6f 7465 2073 7472 696e * a *note strin │ │ │ │ -00274ad0: 673a 2053 7472 696e 672c 2c20 6120 7374 g: String,, a st │ │ │ │ -00274ae0: 7269 6e67 206f 6274 6169 6e65 6420 6279 ring obtained by │ │ │ │ -00274af0: 2072 6561 6469 6e67 2066 726f 6d20 662e reading from f. │ │ │ │ -00274b00: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -00274b10: 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d =========..+---- │ │ │ │ -00274b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ -00274b40: 3a20 6620 3d20 6f70 656e 496e 4f75 7420 : f = openInOut │ │ │ │ -00274b50: 2221 6361 7422 2020 2020 207c 0a7c 2020 "!cat" |.| │ │ │ │ -00274b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00274b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00274b80: 3120 3d20 2163 6174 2020 2020 2020 2020 1 = !cat │ │ │ │ -00274b90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00274ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00274bb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00274bc0: 7c6f 3120 3a20 4669 6c65 2020 2020 2020 |o1 : File │ │ │ │ -00274bd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00274be0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00274bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274c00: 2b0a 7c69 3220 3a20 6973 5265 6164 7920 +.|i2 : isReady │ │ │ │ -00274c10: 6620 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ -00274c20: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00274c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00274c40: 2020 7c0a 7c6f 3220 3d20 6661 6c73 6520 |.|o2 = false │ │ │ │ +00274900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274920: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +00274930: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +00274940: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +00274950: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +00274960: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +00274970: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +00274980: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ +00274990: 6163 6175 6c61 7932 446f 632f 6f76 5f73 acaulay2Doc/ov_s │ │ │ │ +002749a0: 7973 7465 6d2e 6d32 3a32 3832 3a30 2e0a ystem.m2:282:0.. │ │ │ │ +002749b0: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +002749c0: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +002749d0: 2072 6561 645f 6c70 4669 6c65 5f72 702c read_lpFile_rp, │ │ │ │ +002749e0: 204e 6578 743a 2072 6561 645f 6c70 4669 Next: read_lpFi │ │ │ │ +002749f0: 6c65 5f63 6d5a 5a5f 7270 2c20 5072 6576 le_cmZZ_rp, Prev │ │ │ │ +00274a00: 3a20 7265 6164 5f6c 7053 7472 696e 675f : read_lpString_ │ │ │ │ +00274a10: 7270 2c20 5570 3a20 7265 6164 0a0a 7265 rp, Up: read..re │ │ │ │ +00274a20: 6164 2846 696c 6529 202d 2d20 7265 6164 ad(File) -- read │ │ │ │ +00274a30: 2066 726f 6d20 6120 6669 6c65 0a2a 2a2a from a file.*** │ │ │ │ +00274a40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00274a50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00274a60: 2046 756e 6374 696f 6e3a 202a 6e6f 7465 Function: *note │ │ │ │ +00274a70: 2072 6561 643a 2072 6561 642c 0a20 202a read: read,. * │ │ │ │ +00274a80: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00274a90: 2072 6561 6420 660a 2020 2a20 496e 7075 read f. * Inpu │ │ │ │ +00274aa0: 7473 3a0a 2020 2020 2020 2a20 662c 2061 ts:. * f, a │ │ │ │ +00274ab0: 202a 6e6f 7465 2066 696c 653a 2046 696c *note file: Fil │ │ │ │ +00274ac0: 652c 2c20 616e 2069 6e70 7574 2066 696c e,, an input fil │ │ │ │ +00274ad0: 650a 2020 2a20 4f75 7470 7574 733a 0a20 e. * Outputs:. │ │ │ │ +00274ae0: 2020 2020 202a 2061 202a 6e6f 7465 2073 * a *note s │ │ │ │ +00274af0: 7472 696e 673a 2053 7472 696e 672c 2c20 tring: String,, │ │ │ │ +00274b00: 6120 7374 7269 6e67 206f 6274 6169 6e65 a string obtaine │ │ │ │ +00274b10: 6420 6279 2072 6561 6469 6e67 2066 726f d by reading fro │ │ │ │ +00274b20: 6d20 662e 0a0a 4465 7363 7269 7074 696f m f...Descriptio │ │ │ │ +00274b30: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b n.===========..+ │ │ │ │ +00274b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00274b60: 7c69 3120 3a20 6620 3d20 6f70 656e 496e |i1 : f = openIn │ │ │ │ +00274b70: 4f75 7420 2221 6361 7422 2020 2020 207c Out "!cat" | │ │ │ │ +00274b80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00274b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00274ba0: 7c0a 7c6f 3120 3d20 2163 6174 2020 2020 |.|o1 = !cat │ │ │ │ +00274bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00274bc0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00274bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00274be0: 2020 7c0a 7c6f 3120 3a20 4669 6c65 2020 |.|o1 : File │ │ │ │ +00274bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00274c00: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00274c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274c20: 2d2d 2d2d 2b0a 7c69 3220 3a20 6973 5265 ----+.|i2 : isRe │ │ │ │ +00274c30: 6164 7920 6620 2020 2020 2020 2020 2020 ady f │ │ │ │ +00274c40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00274c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00274c60: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00274c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274c80: 2d2d 2d2d 2b0a 7c69 3320 3a20 6620 3c3c ----+.|i3 : f << │ │ │ │ -00274c90: 2022 6869 2074 6865 7265 2220 3c3c 2066 "hi there" << f │ │ │ │ -00274ca0: 6c75 7368 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d lush;|.+-------- │ │ │ │ -00274cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274cc0: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6973 ------+.|i4 : is │ │ │ │ -00274cd0: 5265 6164 7920 6620 2020 2020 2020 2020 Ready f │ │ │ │ -00274ce0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00274cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00274d00: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -00274d10: 6661 6c73 6520 2020 2020 2020 2020 2020 false │ │ │ │ -00274d20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00274d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -00274d50: 3a20 7265 6164 2066 2020 2020 2020 2020 : read f │ │ │ │ -00274d60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00274d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00274d80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00274d90: 3520 3d20 6869 2074 6865 7265 2020 2020 5 = hi there │ │ │ │ -00274da0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00274db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00274dd0: 7c69 3620 3a20 6973 5265 6164 7920 6620 |i6 : isReady f │ │ │ │ -00274de0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00274df0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00274e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00274e10: 7c0a 7c6f 3620 3d20 6661 6c73 6520 2020 |.|o6 = false │ │ │ │ +00274c60: 2020 2020 2020 7c0a 7c6f 3220 3d20 6661 |.|o2 = fa │ │ │ │ +00274c70: 6c73 6520 2020 2020 2020 2020 2020 2020 lse │ │ │ │ +00274c80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00274c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274ca0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +00274cb0: 6620 3c3c 2022 6869 2074 6865 7265 2220 f << "hi there" │ │ │ │ +00274cc0: 3c3c 2066 6c75 7368 3b7c 0a2b 2d2d 2d2d << flush;|.+---- │ │ │ │ +00274cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +00274cf0: 3a20 6973 5265 6164 7920 6620 2020 2020 : isReady f │ │ │ │ +00274d00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00274d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00274d20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00274d30: 3420 3d20 6661 6c73 6520 2020 2020 2020 4 = false │ │ │ │ +00274d40: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00274d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00274d70: 7c69 3520 3a20 7265 6164 2066 2020 2020 |i5 : read f │ │ │ │ +00274d80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00274d90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00274da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00274db0: 7c0a 7c6f 3520 3d20 6869 2074 6865 7265 |.|o5 = hi there │ │ │ │ +00274dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00274dd0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00274de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274df0: 2d2d 2b0a 7c69 3620 3a20 6973 5265 6164 --+.|i6 : isRead │ │ │ │ +00274e00: 7920 6620 2020 2020 2020 2020 2020 2020 y f │ │ │ │ +00274e10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00274e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00274e30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00274e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274e50: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ -00274e60: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ -00274e70: 6520 6f70 656e 496e 3a20 6f70 656e 496e e openIn: openIn │ │ │ │ -00274e80: 5f6c 7053 7472 696e 675f 7270 2c20 2d2d _lpString_rp, -- │ │ │ │ -00274e90: 206f 7065 6e20 616e 2069 6e70 7574 2066 open an input f │ │ │ │ -00274ea0: 696c 650a 2020 2a20 2a6e 6f74 6520 6765 ile. * *note ge │ │ │ │ -00274eb0: 743a 2067 6574 2c20 2d2d 2067 6574 2074 t: get, -- get t │ │ │ │ -00274ec0: 6865 2063 6f6e 7465 6e74 7320 6f66 2061 he contents of a │ │ │ │ -00274ed0: 2066 696c 650a 2020 2a20 2a6e 6f74 6520 file. * *note │ │ │ │ -00274ee0: 6973 5265 6164 793a 2069 7352 6561 6479 isReady: isReady │ │ │ │ -00274ef0: 5f6c 7046 696c 655f 7270 2c20 2d2d 2077 _lpFile_rp, -- w │ │ │ │ -00274f00: 6865 7468 6572 2061 2066 696c 6520 6861 hether a file ha │ │ │ │ -00274f10: 7320 6461 7461 2061 7661 696c 6162 6c65 s data available │ │ │ │ -00274f20: 2066 6f72 0a20 2020 2072 6561 6469 6e67 for. reading │ │ │ │ -00274f30: 0a0a 5761 7973 2074 6f20 7573 6520 7468 ..Ways to use th │ │ │ │ -00274f40: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d is method:.===== │ │ │ │ -00274f50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00274f60: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2072 ===.. * *note r │ │ │ │ -00274f70: 6561 6428 4669 6c65 293a 2072 6561 645f ead(File): read_ │ │ │ │ -00274f80: 6c70 4669 6c65 5f72 702c 202d 2d20 7265 lpFile_rp, -- re │ │ │ │ -00274f90: 6164 2066 726f 6d20 6120 6669 6c65 0a2d ad from a file.- │ │ │ │ -00274fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274e30: 2020 2020 7c0a 7c6f 3620 3d20 6661 6c73 |.|o6 = fals │ │ │ │ +00274e40: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00274e50: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00274e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274e70: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 ------+..See als │ │ │ │ +00274e80: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +00274e90: 2a6e 6f74 6520 6f70 656e 496e 3a20 6f70 *note openIn: op │ │ │ │ +00274ea0: 656e 496e 5f6c 7053 7472 696e 675f 7270 enIn_lpString_rp │ │ │ │ +00274eb0: 2c20 2d2d 206f 7065 6e20 616e 2069 6e70 , -- open an inp │ │ │ │ +00274ec0: 7574 2066 696c 650a 2020 2a20 2a6e 6f74 ut file. * *not │ │ │ │ +00274ed0: 6520 6765 743a 2067 6574 2c20 2d2d 2067 e get: get, -- g │ │ │ │ +00274ee0: 6574 2074 6865 2063 6f6e 7465 6e74 7320 et the contents │ │ │ │ +00274ef0: 6f66 2061 2066 696c 650a 2020 2a20 2a6e of a file. * *n │ │ │ │ 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2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00274fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -00274ff0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -00275000: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -00275010: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -00275020: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -00275030: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -00275040: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00275050: 6b61 6765 732f 0a4d 6163 6175 6c61 7932 kages/.Macaulay2 │ │ │ │ -00275060: 446f 632f 6f76 5f73 7973 7465 6d2e 6d32 Doc/ov_system.m2 │ │ │ │ -00275070: 3a33 3031 3a30 2e0a 1f0a 4669 6c65 3a20 :301:0....File: │ │ │ │ -00275080: 4d61 6361 756c 6179 3244 6f63 2e69 6e66 Macaulay2Doc.inf │ │ │ │ -00275090: 6f2c 204e 6f64 653a 2072 6561 645f 6c70 o, Node: read_lp │ │ │ │ -002750a0: 4669 6c65 5f63 6d5a 5a5f 7270 2c20 5072 File_cmZZ_rp, Pr │ │ │ │ -002750b0: 6576 3a20 7265 6164 5f6c 7046 696c 655f ev: read_lpFile_ │ │ │ │ -002750c0: 7270 2c20 5570 3a20 7265 6164 0a0a 7265 rp, Up: read..re │ │ │ │ -002750d0: 6164 2846 696c 652c 5a5a 2920 2d2d 2072 ad(File,ZZ) -- r │ │ │ │ -002750e0: 6561 6420 6672 6f6d 2061 2066 696c 650a ead from a file. │ │ │ │ -002750f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00275100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00275110: 2a0a 0a20 202a 2046 756e 6374 696f 6e3a *.. * Function: │ │ │ │ -00275120: 202a 6e6f 7465 2072 6561 643a 2072 6561 *note read: rea │ │ │ │ -00275130: 642c 0a20 202a 2055 7361 6765 3a20 0a20 d,. * Usage: . │ │ │ │ -00275140: 2020 2020 2020 2072 6561 6428 662c 6e29 read(f,n) │ │ │ │ -00275150: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -00275160: 2020 202a 2066 2c20 6120 2a6e 6f74 6520 * f, a *note │ │ │ │ -00275170: 6669 6c65 3a20 4669 6c65 2c2c 2061 2066 file: File,, a f │ │ │ │ -00275180: 696c 650a 2020 2020 2020 2a20 6e2c 2061 ile. * n, a │ │ │ │ -00275190: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ -002751a0: 205a 5a2c 2c20 616e 2069 6e74 6567 6572 ZZ,, an integer │ │ │ │ -002751b0: 2073 7065 6369 6679 696e 6720 7468 6520 specifying the │ │ │ │ -002751c0: 6d61 7869 6d75 6d20 6e75 6d62 6572 206f maximum number o │ │ │ │ -002751d0: 660a 2020 2020 2020 2020 6279 7465 7320 f. bytes │ │ │ │ -002751e0: 746f 2072 6561 640a 2020 2a20 4f75 7470 to read. * Outp │ │ │ │ -002751f0: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ -00275200: 6e6f 7465 2073 7472 696e 673a 2053 7472 note string: Str │ │ │ │ -00275210: 696e 672c 2c20 6120 7374 7269 6e67 206f ing,, a string o │ │ │ │ -00275220: 6274 6169 6e65 6420 6279 2072 6561 6469 btained by readi │ │ │ │ -00275230: 6e67 2066 726f 6d20 660a 0a44 6573 6372 ng from f..Descr │ │ │ │ -00275240: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00275250: 3d3d 0a0a 496e 7075 7420 6669 6c65 7320 ==..Input files │ │ │ │ -00275260: 6172 6520 6275 6666 6572 6564 2c20 736f are buffered, so │ │ │ │ -00275270: 2074 6865 2063 7572 7265 6e74 2063 6f6e the current con │ │ │ │ -00275280: 7465 6e74 7320 6f66 2074 6865 2062 7566 tents of the buf │ │ │ │ -00275290: 6665 7220 6172 6520 7265 7475 726e 6564 fer are returned │ │ │ │ -002752a0: 2069 660a 7468 6520 6275 6666 6572 2069 if.the buffer i │ │ │ │ -002752b0: 7320 6e6f 7420 656d 7074 792c 206f 7468 s not empty, oth │ │ │ │ -002752c0: 6572 7769 7365 2072 6561 6469 6e67 2066 erwise reading f │ │ │ │ -002752d0: 726f 6d20 7468 6520 6669 6c65 2069 7320 rom the file is │ │ │ │ -002752e0: 6174 7465 6d70 7465 6420 6669 7273 742e attempted first. │ │ │ │ -002752f0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00275300: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206f ===.. * *note o │ │ │ │ -00275310: 7065 6e49 6e3a 206f 7065 6e49 6e5f 6c70 penIn: openIn_lp │ │ │ │ -00275320: 5374 7269 6e67 5f72 702c 202d 2d20 6f70 String_rp, -- op │ │ │ │ -00275330: 656e 2061 6e20 696e 7075 7420 6669 6c65 en an input file │ │ │ │ -00275340: 0a20 202a 202a 6e6f 7465 2067 6574 3a20 . * *note get: │ │ │ │ -00275350: 6765 742c 202d 2d20 6765 7420 7468 6520 get, -- get the │ │ │ │ -00275360: 636f 6e74 656e 7473 206f 6620 6120 6669 contents of a fi │ │ │ │ -00275370: 6c65 0a20 202a 202a 6e6f 7465 2069 7352 le. * *note isR │ │ │ │ -00275380: 6561 6479 3a20 6973 5265 6164 795f 6c70 eady: isReady_lp │ │ │ │ -00275390: 4669 6c65 5f72 702c 202d 2d20 7768 6574 File_rp, -- whet │ │ │ │ -002753a0: 6865 7220 6120 6669 6c65 2068 6173 2064 her a file has d │ │ │ │ -002753b0: 6174 6120 6176 6169 6c61 626c 6520 666f ata available fo │ │ │ │ -002753c0: 720a 2020 2020 7265 6164 696e 670a 0a57 r. reading..W │ │ │ │ -002753d0: 6179 7320 746f 2075 7365 2074 6869 7320 ays to use this │ │ │ │ -002753e0: 6d65 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d method:.======== │ │ │ │ -002753f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00275400: 0a0a 2020 2a20 2a6e 6f74 6520 7265 6164 .. * *note read │ │ │ │ -00275410: 2846 696c 652c 5a5a 293a 2072 6561 645f (File,ZZ): read_ │ │ │ │ -00275420: 6c70 4669 6c65 5f63 6d5a 5a5f 7270 2c20 lpFile_cmZZ_rp, │ │ │ │ -00275430: 2d2d 2072 6561 6420 6672 6f6d 2061 2066 -- read from a f │ │ │ │ -00275440: 696c 650a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ile.------------ │ │ │ │ -00275450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00275460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00274ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00275000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00275010: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00275020: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00275030: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00275040: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00275050: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +00275060: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +00275070: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ +00275080: 6c61 7932 446f 632f 6f76 5f73 7973 7465 lay2Doc/ov_syste │ │ │ │ +00275090: 6d2e 6d32 3a33 3031 3a30 2e0a 1f0a 4669 m.m2:301:0....Fi │ │ │ │ +002750a0: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ +002750b0: 2e69 6e66 6f2c 204e 6f64 653a 2072 6561 .info, Node: rea │ │ │ │ +002750c0: 645f 6c70 4669 6c65 5f63 6d5a 5a5f 7270 d_lpFile_cmZZ_rp │ │ │ │ +002750d0: 2c20 5072 6576 3a20 7265 6164 5f6c 7046 , Prev: read_lpF │ │ │ │ +002750e0: 696c 655f 7270 2c20 5570 3a20 7265 6164 ile_rp, Up: read │ │ │ │ +002750f0: 0a0a 7265 6164 2846 696c 652c 5a5a 2920 ..read(File,ZZ) │ │ │ │ +00275100: 2d2d 2072 6561 6420 6672 6f6d 2061 2066 -- read from a f │ │ │ │ +00275110: 696c 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ile.************ │ │ │ │ +00275120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00275130: 2a2a 2a2a 2a0a 0a20 202a 2046 756e 6374 *****.. * Funct │ │ │ │ +00275140: 696f 6e3a 202a 6e6f 7465 2072 6561 643a ion: *note read: │ │ │ │ +00275150: 2072 6561 642c 0a20 202a 2055 7361 6765 read,. * Usage │ │ │ │ +00275160: 3a20 0a20 2020 2020 2020 2072 6561 6428 : . read( │ │ │ │ +00275170: 662c 6e29 0a20 202a 2049 6e70 7574 733a f,n). * Inputs: │ │ │ │ +00275180: 0a20 2020 2020 202a 2066 2c20 6120 2a6e . * f, a *n │ │ │ │ +00275190: 6f74 6520 6669 6c65 3a20 4669 6c65 2c2c ote file: File,, │ │ │ │ +002751a0: 2061 2066 696c 650a 2020 2020 2020 2a20 a file. * │ │ │ │ +002751b0: 6e2c 2061 6e20 2a6e 6f74 6520 696e 7465 n, an *note inte │ │ │ │ +002751c0: 6765 723a 205a 5a2c 2c20 616e 2069 6e74 ger: ZZ,, an int │ │ │ │ +002751d0: 6567 6572 2073 7065 6369 6679 696e 6720 eger specifying │ │ │ │ +002751e0: 7468 6520 6d61 7869 6d75 6d20 6e75 6d62 the maximum numb │ │ │ │ +002751f0: 6572 206f 660a 2020 2020 2020 2020 6279 er of. by │ │ │ │ +00275200: 7465 7320 746f 2072 6561 640a 2020 2a20 tes to read. * │ │ │ │ +00275210: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ +00275220: 2061 202a 6e6f 7465 2073 7472 696e 673a a *note string: │ │ │ │ +00275230: 2053 7472 696e 672c 2c20 6120 7374 7269 String,, a stri │ │ │ │ +00275240: 6e67 206f 6274 6169 6e65 6420 6279 2072 ng obtained by r │ │ │ │ +00275250: 6561 6469 6e67 2066 726f 6d20 660a 0a44 eading from f..D │ │ │ │ +00275260: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00275270: 3d3d 3d3d 3d3d 0a0a 496e 7075 7420 6669 ======..Input fi │ │ │ │ +00275280: 6c65 7320 6172 6520 6275 6666 6572 6564 les are buffered │ │ │ │ +00275290: 2c20 736f 2074 6865 2063 7572 7265 6e74 , so the current │ │ │ │ +002752a0: 2063 6f6e 7465 6e74 7320 6f66 2074 6865 contents of the │ │ │ │ +002752b0: 2062 7566 6665 7220 6172 6520 7265 7475 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│ │ +00275390: 6120 6669 6c65 0a20 202a 202a 6e6f 7465 a file. * *note │ │ │ │ +002753a0: 2069 7352 6561 6479 3a20 6973 5265 6164 isReady: isRead │ │ │ │ +002753b0: 795f 6c70 4669 6c65 5f72 702c 202d 2d20 y_lpFile_rp, -- │ │ │ │ +002753c0: 7768 6574 6865 7220 6120 6669 6c65 2068 whether a file h │ │ │ │ +002753d0: 6173 2064 6174 6120 6176 6169 6c61 626c as data availabl │ │ │ │ +002753e0: 6520 666f 720a 2020 2020 7265 6164 696e e for. readin │ │ │ │ +002753f0: 670a 0a57 6179 7320 746f 2075 7365 2074 g..Ways to use t │ │ │ │ +00275400: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ +00275410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00275420: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +00275430: 7265 6164 2846 696c 652c 5a5a 293a 2072 read(File,ZZ): r │ │ │ │ +00275440: 6561 645f 6c70 4669 6c65 5f63 6d5a 5a5f ead_lpFile_cmZZ_ │ │ │ │ +00275450: 7270 2c20 2d2d 2072 6561 6420 6672 6f6d rp, -- read from │ │ │ │ +00275460: 2061 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es(f,fn). * Inp │ │ │ │ -00275610: 7574 733a 0a20 2020 2020 202a 2066 0a20 uts:. * f. │ │ │ │ -00275620: 2020 2020 202a 2066 6e2c 2074 6865 206e * fn, the n │ │ │ │ -00275630: 616d 6520 6f66 2061 2066 696c 652c 206f ame of a file, o │ │ │ │ -00275640: 7220 6120 6c69 7374 206f 6620 6e61 6d65 r a list of name │ │ │ │ -00275650: 7320 6f66 2066 696c 6573 0a20 202a 204f s of files. * O │ │ │ │ -00275660: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -00275670: 7265 7475 726e 7320 2a6e 6f74 6520 6e75 returns *note nu │ │ │ │ -00275680: 6c6c 3a20 6e75 6c6c 2c20 756e 6c65 7373 ll: null, unless │ │ │ │ -00275690: 2074 6865 2066 756e 6374 696f 6e20 7573 the function us │ │ │ │ -002756a0: 6573 2062 7265 616b 2078 2077 6974 6820 es break x with │ │ │ │ -002756b0: 610a 2020 2020 2020 2020 6e6f 6e2d 6e75 a. non-nu │ │ │ │ -002756c0: 6c6c 2076 616c 7565 2066 6f72 2078 2c20 ll value for x, │ │ │ │ -002756d0: 696e 2077 6869 6368 2063 6173 6520 7363 in which case sc │ │ │ │ -002756e0: 616e 6e69 6e67 2073 746f 7073 2061 6e64 anning stops and │ │ │ │ -002756f0: 2078 2069 7320 7265 7475 726e 6564 0a20 x is returned. │ │ │ │ -00275700: 2020 2020 2020 2069 6d6d 6564 6961 7465 immediate │ │ │ │ -00275710: 6c79 0a20 202a 2043 6f6e 7365 7175 656e ly. * Consequen │ │ │ │ -00275720: 6365 733a 0a20 2020 2020 202a 2061 7070 ces:. * app │ │ │ │ -00275730: 6c69 6573 2066 2074 6f20 6561 6368 206c lies f to each l │ │ │ │ -00275740: 696e 6520 6f66 2074 6865 2066 696c 6528 ine of the file( │ │ │ │ -00275750: 7329 0a0a 4465 7363 7269 7074 696f 6e0a s)..Description. │ │ │ │ -00275760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -00275770: 2066 696c 6520 6973 2072 6561 6420 616e file is read an │ │ │ │ -00275780: 6420 7072 6f63 6573 7365 6420 6f6e 6520 d processed one │ │ │ │ -00275790: 626c 6f63 6b20 6174 2061 2074 696d 652c block at a time, │ │ │ │ -002757a0: 206d 616b 696e 6720 7468 6973 2070 726f making this pro │ │ │ │ -002757b0: 6365 6475 7265 0a70 6f74 656e 7469 616c cedure.potential │ │ │ │ -002757c0: 6c79 206d 7563 6820 6265 7474 6572 2061 ly much better a │ │ │ │ -002757d0: 7420 636f 6e73 6572 7669 6e67 206d 656d t conserving mem │ │ │ │ -002757e0: 6f72 7920 7468 616e 2073 6361 6e28 6c69 ory than scan(li │ │ │ │ -002757f0: 6e65 7320 6765 7420 666e 2c66 2920 7768 nes get fn,f) wh │ │ │ │ -00275800: 656e 2074 6865 0a66 696c 6520 6973 2076 en the.file is v │ │ │ │ -00275810: 6572 7920 6c61 7267 652e 0a0a 5761 7973 ery large...Ways │ │ │ │ -00275820: 2074 6f20 7573 6520 7363 616e 4c69 6e65 to use scanLine │ │ │ │ -00275830: 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d s:.============= │ │ │ │ -00275840: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ -00275850: 7363 616e 4c69 6e65 7328 4675 6e63 7469 scanLines(Functi │ │ │ │ -00275860: 6f6e 2c4c 6973 7429 220a 2020 2a20 2273 on,List)". * "s │ │ │ │ -00275870: 6361 6e4c 696e 6573 2846 756e 6374 696f canLines(Functio │ │ │ │ -00275880: 6e2c 5374 7269 6e67 2922 0a0a 466f 7220 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2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00275ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00275f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +00275f10: 3a20 6669 6c65 6e61 6d65 203d 2074 6f53 : filename = toS │ │ │ │ +00275f20: 7472 696e 6720 6620 2020 2020 2020 2020 tring f │ │ │ │ +00275f30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00275f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00275f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00275f60: 2020 2020 7c0a 7c6f 3420 3d20 2f74 6d70 |.|o4 = /tmp │ │ │ │ +00275f70: 2f4d 322d 3135 3339 382d 302f 3020 2020 /M2-15398-0/0 │ │ │ │ +00275f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00275f90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00275fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00275fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00275fc0: 7c69 3520 3a20 6669 6c65 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3d20 3820 2020 2020 2020 2020 2020 7 = 8 │ │ │ │ +00276150: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00276160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00276170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00276180: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00276190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002761a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002761b0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 7265 ------+.|i8 : re │ │ │ │ -002761c0: 6d6f 7665 4669 6c65 2066 696c 656e 616d moveFile filenam │ │ │ │ -002761d0: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -002761e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -002761f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276210: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d +..See also.==== │ │ │ │ -00276220: 3d3d 3d3d 0a0a 2020 2a20 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---------------- │ │ │ │ -00276300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276180: 7c0a 7c6f 3720 3d20 3820 2020 2020 2020 |.|o7 = 8 │ │ │ │ +00276190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002761a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +002761b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002761c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002761d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 ----------+.|i8 │ │ │ │ +002761e0: 3a20 7265 6d6f 7665 4669 6c65 2066 696c : removeFile fil │ │ │ │ +002761f0: 656e 616d 6520 2020 2020 2020 2020 2020 ename │ │ │ │ +00276200: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00276210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276230: 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 6f0a ----+..See also. │ │ │ │ +00276240: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 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---------------- │ │ │ │ 00276320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276330: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00276340: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00276350: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00276360: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00276370: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00276380: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ -00276390: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -002763a0: 4d61 6361 756c 6179 3244 6f63 2f6f 765f Macaulay2Doc/ov_ │ │ │ │ -002763b0: 7379 7374 656d 2e6d 323a 3138 3638 3a30 system.m2:1868:0 │ │ │ │ -002763c0: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ -002763d0: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ -002763e0: 653a 2068 6569 6768 745f 6c70 4669 6c65 e: 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-002764c0: 733a 0a20 2020 2020 202a 2066 2c20 6120 s:. * f, a │ │ │ │ -002764d0: 2a6e 6f74 6520 6669 6c65 3a20 4669 6c65 *note file: File │ │ │ │ -002764e0: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20 ,. * Outputs:. │ │ │ │ -002764f0: 2020 2020 202a 2061 6e20 2a6e 6f74 6520 * an *note │ │ │ │ -00276500: 696e 7465 6765 723a 205a 5a2c 2c20 7468 integer: ZZ,, th │ │ │ │ -00276510: 6520 6865 6967 6874 206f 6620 7468 6520 e height of the │ │ │ │ -00276520: 7769 6e64 6f77 206f 7220 7465 726d 696e window or termin │ │ │ │ -00276530: 616c 2061 7474 6163 6865 6420 746f 0a20 al attached to. │ │ │ │ -00276540: 2020 2020 2020 2074 6865 2066 696c 6520 the file │ │ │ │ -00276550: 662c 2069 6620 616e 792c 2065 6c73 6520 f, if any, else │ │ │ │ -00276560: 300a 0a57 6179 7320 746f 2075 7365 2074 0..Ways to use t │ │ │ │ -00276570: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ -00276580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00276590: 3d3d 3d3d 0a0a 2020 2a20 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window │ │ │ │ +00276470: 6865 6967 6874 0a2a 2a2a 2a2a 2a2a 2a2a height.********* │ │ │ │ +00276480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00276490: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 ********.. * Fu │ │ │ │ +002764a0: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 6865 nction: *note he │ │ │ │ +002764b0: 6967 6874 3a20 6865 6967 6874 2c0a 2020 ight: height,. │ │ │ │ +002764c0: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +002764d0: 2020 6865 6967 6874 2066 0a20 202a 2049 height f. * I │ │ │ │ +002764e0: 6e70 7574 733a 0a20 2020 2020 202a 2066 nputs:. * f │ │ │ │ +002764f0: 2c20 6120 2a6e 6f74 6520 6669 6c65 3a20 , a *note file: │ │ │ │ +00276500: 4669 6c65 2c0a 2020 2a20 4f75 7470 7574 File,. * Output │ │ │ │ +00276510: 733a 0a20 2020 2020 202a 2061 6e20 2a6e s:. * an *n │ │ │ │ +00276520: 6f74 6520 696e 7465 6765 723a 205a 5a2c ote integer: ZZ, │ │ │ │ +00276530: 2c20 7468 6520 6865 6967 6874 206f 6620 , the height of │ │ │ │ +00276540: 7468 6520 7769 6e64 6f77 206f 7220 7465 the window or te │ │ │ │ +00276550: 726d 696e 616c 2061 7474 6163 6865 6420 rminal attached │ │ │ │ +00276560: 746f 0a20 2020 2020 2020 2074 6865 2066 to. the f │ │ │ │ +00276570: 696c 6520 662c 2069 6620 616e 792c 2065 ile f, if any, e │ │ │ │ +00276580: 6c73 6520 300a 0a57 6179 7320 746f 2075 lse 0..Ways to u │ │ │ │ +00276590: 7365 2074 6869 7320 6d65 7468 6f64 3a0a se this method:. │ │ │ │ +002765a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002765b0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +002765c0: 6f74 6520 6865 6967 6874 2846 696c 6529 ote height(File) │ │ │ │ +002765d0: 3a20 6865 6967 6874 5f6c 7046 696c 655f : height_lpFile_ │ │ │ │ +002765e0: 7270 2c20 2d2d 2067 6574 2077 696e 646f rp, -- get windo │ │ │ │ +002765f0: 7720 6865 6967 6874 0a2d 2d2d 2d2d 2d2d w height.------- │ │ │ │ 00276600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00276610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276620: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00276630: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00276640: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00276650: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00276660: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00276670: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00276680: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -00276690: 6175 6c61 7932 446f 632f 6f76 5f73 7472 aulay2Doc/ov_str │ │ │ │ -002766a0: 696e 6773 2e6d 323a 3331 333a 302e 0a1f ings.m2:313:0... │ │ │ │ -002766b0: 0a46 696c 653a 204d 6163 6175 6c61 7932 .File: Macaulay2 │ │ │ │ -002766c0: 446f 632e 696e 666f 2c20 4e6f 6465 3a20 Doc.info, Node: │ │ │ │ -002766d0: 7769 6474 685f 6c70 4669 6c65 5f72 702c width_lpFile_rp, │ │ │ │ -002766e0: 204e 6578 743a 2061 7445 6e64 4f66 4669 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-002767c0: 202a 6e6f 7465 2066 696c 653a 2046 696c *note file: Fil │ │ │ │ -002767d0: 652c 0a20 202a 204f 7574 7075 7473 3a0a e,. * Outputs:. │ │ │ │ -002767e0: 2020 2020 2020 2a20 616e 202a 6e6f 7465 * an *note │ │ │ │ -002767f0: 2069 6e74 6567 6572 3a20 5a5a 2c2c 2074 integer: ZZ,, t │ │ │ │ -00276800: 6865 2077 6964 7468 206f 6620 7468 6520 he width of the │ │ │ │ -00276810: 7769 6e64 6f77 206f 7220 7465 726d 696e window or termin │ │ │ │ -00276820: 616c 2061 7474 6163 6865 6420 746f 0a20 al attached to. │ │ │ │ -00276830: 2020 2020 2020 2074 6865 2066 696c 6520 the file │ │ │ │ -00276840: 662c 2069 6620 616e 792c 2065 6c73 6520 f, if any, else │ │ │ │ -00276850: 300a 0a57 6179 7320 746f 2075 7365 2074 0..Ways to use t │ │ │ │ -00276860: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ -00276870: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00276880: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -00276890: 7769 6474 6828 4669 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696e 616c 2061 7474 6163 6865 6420 rminal attached │ │ │ │ +00276850: 746f 0a20 2020 2020 2020 2074 6865 2066 to. the f │ │ │ │ +00276860: 696c 6520 662c 2069 6620 616e 792c 2065 ile f, if any, e │ │ │ │ +00276870: 6c73 6520 300a 0a57 6179 7320 746f 2075 lse 0..Ways to u │ │ │ │ +00276880: 7365 2074 6869 7320 6d65 7468 6f64 3a0a se this method:. │ │ │ │ +00276890: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +002768a0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +002768b0: 6f74 6520 7769 6474 6828 4669 6c65 293a ote width(File): │ │ │ │ +002768c0: 2077 6964 7468 5f6c 7046 696c 655f 7270 width_lpFile_rp │ │ │ │ +002768d0: 2c20 2d2d 2067 6574 2077 696e 646f 7720 , -- get window │ │ │ │ +002768e0: 7769 6474 680a 2d2d 2d2d 2d2d 2d2d 2d2d width.---------- │ │ │ │ 002768f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00276900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276910: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -00276920: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -00276930: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -00276940: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -00276950: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ -00276960: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -00276970: 7061 636b 6167 6573 2f0a 4d61 6361 756c packages/.Macaul │ │ │ │ -00276980: 6179 3244 6f63 2f6f 765f 7374 7269 6e67 ay2Doc/ov_string │ │ │ │ -00276990: 732e 6d32 3a33 3138 3a30 2e0a 1f0a 4669 s.m2:318:0....Fi │ │ │ │ -002769a0: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -002769b0: 2e69 6e66 6f2c 204e 6f64 653a 2061 7445 .info, Node: atE │ │ │ │ -002769c0: 6e64 4f66 4669 6c65 5f6c 7046 696c 655f ndOfFile_lpFile_ │ │ │ │ -002769d0: 7270 2c20 4e65 7874 3a20 6563 686f 4f6e rp, Next: echoOn │ │ │ │ -002769e0: 2c20 5072 6576 3a20 7769 6474 685f 6c70 , 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-00276ac0: 6e70 7574 733a 0a20 2020 2020 202a 2066 nputs:. * f │ │ │ │ -00276ad0: 2c20 6120 2a6e 6f74 6520 6669 6c65 3a20 , a *note file: │ │ │ │ -00276ae0: 4669 6c65 2c0a 2020 2a20 4f75 7470 7574 File,. * Output │ │ │ │ -00276af0: 733a 0a20 2020 2020 202a 2061 202a 6e6f s:. * a *no │ │ │ │ -00276b00: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ -00276b10: 3a20 426f 6f6c 6561 6e2c 2c20 7768 6574 : Boolean,, whet │ │ │ │ -00276b20: 6865 7220 7468 6520 696e 7075 7420 6669 her the input fi │ │ │ │ -00276b30: 6c65 2066 2069 7320 6174 2074 6865 2065 le f is at the e │ │ │ │ -00276b40: 6e64 0a0a 4465 7363 7269 7074 696f 6e0a nd..Description. │ │ │ │ -00276b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d ===========..+-- │ │ │ │ -00276b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00276b80: 0a7c 6931 203a 2066 203d 206f 7065 6e49 .|i1 : f = openI │ │ │ │ -00276b90: 6e4f 7574 2022 2163 6174 2220 2020 2020 nOut "!cat" │ │ │ │ -00276ba0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00276bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00276bc0: 2020 2020 2020 207c 0a7c 6f31 203d 2021 |.|o1 = ! │ │ │ │ -00276bd0: 6361 7420 2020 2020 2020 2020 2020 2020 cat │ │ │ │ -00276be0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00276bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00276910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276930: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +00276940: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +00276950: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +00276960: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +00276970: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +00276980: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +00276990: 6179 322f 7061 636b 6167 6573 2f0a 4d61 ay2/packages/.Ma │ │ │ │ +002769a0: 6361 756c 6179 3244 6f63 2f6f 765f 7374 caulay2Doc/ov_st │ │ │ │ +002769b0: 7269 6e67 732e 6d32 3a33 3138 3a30 2e0a rings.m2:318:0.. │ │ │ │ +002769c0: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +002769d0: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +002769e0: 2061 7445 6e64 4f66 4669 6c65 5f6c 7046 atEndOfFile_lpF │ │ │ │ +002769f0: 696c 655f 7270 2c20 4e65 7874 3a20 6563 ile_rp, Next: ec │ │ │ │ +00276a00: 686f 4f6e 2c20 5072 6576 3a20 7769 6474 hoOn, Prev: widt │ │ │ │ +00276a10: 685f 6c70 4669 6c65 5f72 702c 2055 703a h_lpFile_rp, Up: │ │ │ │ +00276a20: 2075 7369 6e67 2073 6f63 6b65 7473 0a0a using sockets.. │ │ │ │ +00276a30: 6174 456e 644f 6646 696c 6528 4669 6c65 atEndOfFile(File │ │ │ │ +00276a40: 2920 2d2d 2074 6573 7420 666f 7220 656e ) -- test for en │ │ │ │ +00276a50: 6420 6f66 2066 696c 650a 2a2a 2a2a 2a2a d of file.****** │ │ │ │ +00276a60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00276a70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00276a80: 2a2a 2a0a 0a20 202a 2046 756e 6374 696f ***.. * Functio │ │ │ │ +00276a90: 6e3a 202a 6e6f 7465 2061 7445 6e64 4f66 n: *note atEndOf │ │ │ │ +00276aa0: 4669 6c65 3a20 6174 456e 644f 6646 696c File: atEndOfFil │ │ │ │ +00276ab0: 655f 6c70 4669 6c65 5f72 702c 0a20 202a e_lpFile_rp,. * │ │ │ │ +00276ac0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00276ad0: 2061 7445 6e64 4f66 4669 6c65 2066 0a20 atEndOfFile f. │ │ │ │ +00276ae0: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +00276af0: 202a 2066 2c20 6120 2a6e 6f74 6520 6669 * f, a *note fi │ │ │ │ +00276b00: 6c65 3a20 4669 6c65 2c0a 2020 2a20 4f75 le: File,. * Ou │ │ │ │ +00276b10: 7470 7574 733a 0a20 2020 2020 202a 2061 tputs:. * a │ │ │ │ +00276b20: 202a 6e6f 7465 2042 6f6f 6c65 616e 2076 *note Boolean v │ │ │ │ +00276b30: 616c 7565 3a20 426f 6f6c 6561 6e2c 2c20 alue: Boolean,, │ │ │ │ +00276b40: 7768 6574 6865 7220 7468 6520 696e 7075 whether the inpu │ │ │ │ +00276b50: 7420 6669 6c65 2066 2069 7320 6174 2074 t file f is at t │ │ │ │ +00276b60: 6865 2065 6e64 0a0a 4465 7363 7269 7074 he end..Descript │ │ │ │ +00276b70: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +00276b80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00276b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276ba0: 2d2d 2d2b 0a7c 6931 203a 2066 203d 206f ---+.|i1 : f = o │ │ │ │ +00276bb0: 7065 6e49 6e4f 7574 2022 2163 6174 2220 penInOut "!cat" │ │ │ │ +00276bc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00276bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00276be0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +00276bf0: 203d 2021 6361 7420 2020 2020 2020 2020 = !cat │ │ │ │ 00276c00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00276c10: 0a7c 6f31 203a 2046 696c 6520 2020 2020 .|o1 : File │ │ │ │ +00276c10: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00276c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00276c30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -00276c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276c50: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2066 -------+.|i2 : f │ │ │ │ -00276c60: 203c 3c20 2268 6920 7468 6572 6522 203c << "hi there" < │ │ │ │ -00276c70: 3c20 636c 6f73 654f 7574 3b7c 0a2b 2d2d < closeOut;|.+-- │ │ │ │ -00276c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00276ca0: 0a7c 6933 203a 2061 7445 6e64 4f66 4669 .|i3 : atEndOfFi │ │ │ │ -00276cb0: 6c65 2066 2020 2020 2020 2020 2020 2020 le f │ │ │ │ -00276cc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00276cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00276ce0: 2020 2020 2020 207c 0a7c 6f33 203d 2066 |.|o3 = f │ │ │ │ -00276cf0: 616c 7365 2020 2020 2020 2020 2020 2020 alse │ │ │ │ -00276d00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00276d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00276d30: 0a7c 6934 203a 2070 6565 6b20 7265 6164 .|i4 : peek read │ │ │ │ -00276d40: 2066 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ -00276d50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00276d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00276d70: 2020 2020 2020 207c 0a7c 6f34 203d 2022 |.|o4 = " │ │ │ │ -00276d80: 6869 2074 6865 7265 2220 2020 2020 2020 hi there" │ │ │ │ -00276d90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00276da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00276dc0: 0a7c 6935 203a 2061 7445 6e64 4f66 4669 .|i5 : atEndOfFi │ │ │ │ -00276dd0: 6c65 2066 2020 2020 2020 2020 2020 2020 le f │ │ │ │ -00276de0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00276df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00276e00: 2020 2020 2020 207c 0a7c 6f35 203d 2066 |.|o5 = f │ │ │ │ -00276e10: 616c 7365 2020 2020 2020 2020 2020 2020 alse │ │ │ │ -00276e20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00276e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00276e50: 0a0a 5761 7973 2074 6f20 7573 6520 7468 ..Ways to use th │ │ │ │ -00276e60: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d is method:.===== │ │ │ │ -00276e70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00276e80: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2061 ===.. * *note a │ │ │ │ -00276e90: 7445 6e64 4f66 4669 6c65 2846 696c 6529 tEndOfFile(File) │ │ │ │ -00276ea0: 3a20 6174 456e 644f 6646 696c 655f 6c70 : atEndOfFile_lp │ │ │ │ -00276eb0: 4669 6c65 5f72 702c 202d 2d20 7465 7374 File_rp, -- test │ │ │ │ -00276ec0: 2066 6f72 2065 6e64 206f 6620 6669 6c65 for end of file │ │ │ │ -00276ed0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ -00276ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276c30: 2020 207c 0a7c 6f31 203a 2046 696c 6520 |.|o1 : File │ │ │ │ +00276c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00276c50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00276c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +00276c80: 203a 2066 203c 3c20 2268 6920 7468 6572 : f << "hi ther │ │ │ │ +00276c90: 6522 203c 3c20 636c 6f73 654f 7574 3b7c e" << closeOut;| │ │ │ │ +00276ca0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00276cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276cc0: 2d2d 2d2b 0a7c 6933 203a 2061 7445 6e64 ---+.|i3 : atEnd │ │ │ │ +00276cd0: 4f66 4669 6c65 2066 2020 2020 2020 2020 OfFile f │ │ │ │ +00276ce0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00276cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00276d00: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +00276d10: 203d 2066 616c 7365 2020 2020 2020 2020 = false │ │ │ │ +00276d20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00276d30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00276d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276d50: 2d2d 2d2b 0a7c 6934 203a 2070 6565 6b20 ---+.|i4 : peek │ │ │ │ +00276d60: 7265 6164 2066 2020 2020 2020 2020 2020 read f │ │ │ │ +00276d70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00276d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00276d90: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +00276da0: 203d 2022 6869 2074 6865 7265 2220 2020 = "hi there" │ │ │ │ +00276db0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00276dc0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00276dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276de0: 2d2d 2d2b 0a7c 6935 203a 2061 7445 6e64 ---+.|i5 : atEnd │ │ │ │ +00276df0: 4f66 4669 6c65 2066 2020 2020 2020 2020 OfFile f │ │ │ │ +00276e00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00276e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00276e20: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ +00276e30: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ +00276e40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00276e50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00276e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276e70: 2d2d 2d2b 0a0a 5761 7973 2074 6f20 7573 ---+..Ways to us │ │ │ │ +00276e80: 6520 7468 6973 206d 6574 686f 643a 0a3d e this method:.= │ │ │ │ +00276e90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00276ea0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +00276eb0: 7465 2061 7445 6e64 4f66 4669 6c65 2846 te atEndOfFile(F │ │ │ │ +00276ec0: 696c 6529 3a20 6174 456e 644f 6646 696c ile): atEndOfFil │ │ │ │ +00276ed0: 655f 6c70 4669 6c65 5f72 702c 202d 2d20 e_lpFile_rp, -- │ │ │ │ +00276ee0: 7465 7374 2066 6f72 2065 6e64 206f 6620 test for end of │ │ │ │ +00276ef0: 6669 6c65 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d file.----------- │ │ │ │ 00276f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00276f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00276f20: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00276f30: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00276f40: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -00276f50: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -00276f60: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ -00276f70: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -00276f80: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ -00276f90: 7932 446f 632f 6f76 5f66 696c 6573 2e6d y2Doc/ov_files.m │ │ │ │ -00276fa0: 323a 3337 343a 302e 0a1f 0a46 696c 653a 2:374:0....File: │ │ │ │ -00276fb0: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ -00276fc0: 666f 2c20 4e6f 6465 3a20 6563 686f 4f6e fo, Node: echoOn │ │ │ │ -00276fd0: 2c20 4e65 7874 3a20 6563 686f 4f66 662c , Next: echoOff, │ │ │ │ -00276fe0: 2050 7265 763a 2061 7445 6e64 4f66 4669 Prev: atEndOfFi │ │ │ │ -00276ff0: 6c65 5f6c 7046 696c 655f 7270 2c20 5570 le_lpFile_rp, Up │ │ │ │ -00277000: 3a20 7573 696e 6720 736f 636b 6574 730a : using sockets. │ │ │ │ -00277010: 0a65 6368 6f4f 6e20 2d2d 2074 7572 6e20 .echoOn -- turn │ │ │ │ -00277020: 6f6e 2065 6368 6f69 6e67 0a2a 2a2a 2a2a on echoing.***** │ │ │ │ -00277030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00277040: 2a2a 2a2a 0a0a 4465 7363 7269 7074 696f ****..Descriptio │ │ │ │ -00277050: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a65 n.===========..e │ │ │ │ -00277060: 6368 6f4f 6e20 6620 2d2d 2074 7572 6e73 choOn f -- turns │ │ │ │ -00277070: 206f 6e20 6563 686f 696e 6720 666f 7220 on echoing for │ │ │ │ -00277080: 7468 6520 6669 6c65 2066 2e0a 0a57 6179 the file f...Way │ │ │ │ -00277090: 7320 746f 2075 7365 2065 6368 6f4f 6e3a s to use echoOn: │ │ │ │ -002770a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -002770b0: 3d3d 3d3d 0a0a 2020 2a20 2265 6368 6f4f ====.. * "echoO │ │ │ │ -002770c0: 6e28 4669 6c65 2922 0a0a 466f 7220 7468 n(File)"..For th │ │ │ │ -002770d0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -002770e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -002770f0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00277100: 6520 6563 686f 4f6e 3a20 6563 686f 4f6e e echoOn: echoOn │ │ │ │ -00277110: 2c20 6973 2061 202a 6e6f 7465 2063 6f6d , is a *note com │ │ │ │ -00277120: 7069 6c65 6420 6675 6e63 7469 6f6e 3a0a piled function:. │ │ │ │ -00277130: 436f 6d70 696c 6564 4675 6e63 7469 6f6e CompiledFunction │ │ │ │ -00277140: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ -00277150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00276f40: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +00276f50: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +00276f60: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +00276f70: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +00276f80: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +00276f90: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +00276fa0: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ +00276fb0: 6175 6c61 7932 446f 632f 6f76 5f66 696c aulay2Doc/ov_fil │ │ │ │ +00276fc0: 6573 2e6d 323a 3337 343a 302e 0a1f 0a46 es.m2:374:0....F │ │ │ │ +00276fd0: 696c 653a 204d 6163 6175 6c61 7932 446f ile: Macaulay2Do │ │ │ │ +00276fe0: 632e 696e 666f 2c20 4e6f 6465 3a20 6563 c.info, Node: ec │ │ │ │ +00276ff0: 686f 4f6e 2c20 4e65 7874 3a20 6563 686f hoOn, Next: echo │ │ │ │ +00277000: 4f66 662c 2050 7265 763a 2061 7445 6e64 Off, Prev: atEnd │ │ │ │ +00277010: 4f66 4669 6c65 5f6c 7046 696c 655f 7270 OfFile_lpFile_rp │ │ │ │ +00277020: 2c20 5570 3a20 7573 696e 6720 736f 636b , Up: using sock │ │ │ │ +00277030: 6574 730a 0a65 6368 6f4f 6e20 2d2d 2074 ets..echoOn -- t │ │ │ │ +00277040: 7572 6e20 6f6e 2065 6368 6f69 6e67 0a2a urn on echoing.* │ │ │ │ +00277050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00277060: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 ********..Descri │ │ │ │ +00277070: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +00277080: 3d0a 0a65 6368 6f4f 6e20 6620 2d2d 2074 =..echoOn f -- t │ │ │ │ +00277090: 7572 6e73 206f 6e20 6563 686f 696e 6720 urns on echoing │ │ │ │ +002770a0: 666f 7220 7468 6520 6669 6c65 2066 2e0a for the file f.. │ │ │ │ +002770b0: 0a57 6179 7320 746f 2075 7365 2065 6368 .Ways to use ech │ │ │ │ +002770c0: 6f4f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d oOn:.=========== │ │ │ │ +002770d0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 ========.. * "e │ │ │ │ +002770e0: 6368 6f4f 6e28 4669 6c65 2922 0a0a 466f choOn(File)"..Fo │ │ │ │ +002770f0: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00277100: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00277110: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00277120: 2a6e 6f74 6520 6563 686f 4f6e 3a20 6563 *note echoOn: ec │ │ │ │ +00277130: 686f 4f6e 2c20 6973 2061 202a 6e6f 7465 hoOn, is a *note │ │ │ │ +00277140: 2063 6f6d 7069 6c65 6420 6675 6e63 7469 compiled functi │ │ │ │ +00277150: 6f6e 3a0a 436f 6d70 696c 6564 4675 6e63 on:.CompiledFunc │ │ │ │ +00277160: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ 00277170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00277180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277190: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -002771a0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -002771b0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -002771c0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -002771d0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -002771e0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -002771f0: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -00277200: 756c 6179 3244 6f63 2f6f 765f 7379 7374 ulay2Doc/ov_syst │ │ │ │ -00277210: 656d 2e6d 323a 3130 3435 3a30 2e0a 1f0a em.m2:1045:0.... │ │ │ │ -00277220: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -00277230: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2065 oc.info, Node: e │ │ │ │ -00277240: 6368 6f4f 6666 2c20 4e65 7874 3a20 6973 choOff, Next: is │ │ │ │ -00277250: 4f70 656e 2c20 5072 6576 3a20 6563 686f Open, Prev: echo │ │ │ │ -00277260: 4f6e 2c20 5570 3a20 7573 696e 6720 736f On, Up: using so │ │ │ │ -00277270: 636b 6574 730a 0a65 6368 6f4f 6666 202d ckets..echoOff - │ │ │ │ -00277280: 2d20 7475 726e 206f 6666 2065 6368 6f69 - turn off echoi │ │ │ │ -00277290: 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ng.************* │ │ │ │ -002772a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -002772b0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -002772c0: 3d3d 3d3d 3d3d 3d0a 0a65 6368 6f4f 6666 =======..echoOff │ │ │ │ -002772d0: 2066 202d 2d20 7475 726e 7320 6f66 6620 f -- turns off │ │ │ │ -002772e0: 6563 686f 696e 6720 666f 7220 7468 6520 echoing for the │ │ │ │ -002772f0: 6669 6c65 2066 2e0a 0a57 6179 7320 746f file f...Ways to │ │ │ │ -00277300: 2075 7365 2065 6368 6f4f 6666 3a0a 3d3d use echoOff:.== │ │ │ │ -00277310: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00277320: 3d3d 0a0a 2020 2a20 2265 6368 6f4f 6666 ==.. * "echoOff │ │ │ │ -00277330: 2846 696c 6529 220a 0a46 6f72 2074 6865 (File)"..For the │ │ │ │ -00277340: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00277350: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00277360: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00277370: 2065 6368 6f4f 6666 3a20 6563 686f 4f66 echoOff: echoOf │ │ │ │ -00277380: 662c 2069 7320 6120 2a6e 6f74 6520 636f f, is a *note co │ │ │ │ -00277390: 6d70 696c 6564 2066 756e 6374 696f 6e3a mpiled function: │ │ │ │ -002773a0: 0a43 6f6d 7069 6c65 6446 756e 6374 696f .CompiledFunctio │ │ │ │ -002773b0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ -002773c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002773d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002771a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002771b0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +002771c0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +002771d0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +002771e0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +002771f0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00277200: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +00277210: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00277220: 4d61 6361 756c 6179 3244 6f63 2f6f 765f Macaulay2Doc/ov_ │ │ │ │ +00277230: 7379 7374 656d 2e6d 323a 3130 3435 3a30 system.m2:1045:0 │ │ │ │ +00277240: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ +00277250: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ +00277260: 653a 2065 6368 6f4f 6666 2c20 4e65 7874 e: echoOff, Next │ │ │ │ +00277270: 3a20 6973 4f70 656e 2c20 5072 6576 3a20 : isOpen, Prev: │ │ │ │ +00277280: 6563 686f 4f6e 2c20 5570 3a20 7573 696e echoOn, Up: usin │ │ │ │ +00277290: 6720 736f 636b 6574 730a 0a65 6368 6f4f g sockets..echoO │ │ │ │ +002772a0: 6666 202d 2d20 7475 726e 206f 6666 2065 ff -- turn off e │ │ │ │ +002772b0: 6368 6f69 6e67 0a2a 2a2a 2a2a 2a2a 2a2a choing.********* │ │ │ │ +002772c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002772d0: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a **..Description. │ │ │ │ +002772e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a65 6368 ===========..ech │ │ │ │ +002772f0: 6f4f 6666 2066 202d 2d20 7475 726e 7320 oOff f -- turns │ │ │ │ +00277300: 6f66 6620 6563 686f 696e 6720 666f 7220 off echoing for │ │ │ │ +00277310: 7468 6520 6669 6c65 2066 2e0a 0a57 6179 the file f...Way │ │ │ │ +00277320: 7320 746f 2075 7365 2065 6368 6f4f 6666 s to use echoOff │ │ │ │ +00277330: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +00277340: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 6368 ======.. * "ech │ │ │ │ +00277350: 6f4f 6666 2846 696c 6529 220a 0a46 6f72 oOff(File)"..For │ │ │ │ +00277360: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00277370: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00277380: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00277390: 6e6f 7465 2065 6368 6f4f 6666 3a20 6563 note echoOff: ec │ │ │ │ +002773a0: 686f 4f66 662c 2069 7320 6120 2a6e 6f74 hoOff, is a *not │ │ │ │ +002773b0: 6520 636f 6d70 696c 6564 2066 756e 6374 e compiled funct │ │ │ │ +002773c0: 696f 6e3a 0a43 6f6d 7069 6c65 6446 756e ion:.CompiledFun │ │ │ │ +002773d0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ 002773e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002773f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277400: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00277410: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00277420: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00277430: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00277440: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00277450: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00277460: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -00277470: 6175 6c61 7932 446f 632f 6f76 5f73 7973 aulay2Doc/ov_sys │ │ │ │ -00277480: 7465 6d2e 6d32 3a31 3035 313a 302e 0a1f tem.m2:1051:0... │ │ │ │ -00277490: 0a46 696c 653a 204d 6163 6175 6c61 7932 .File: Macaulay2 │ │ │ │ -002774a0: 446f 632e 696e 666f 2c20 4e6f 6465 3a20 Doc.info, Node: │ │ │ │ -002774b0: 6973 4f70 656e 2c20 4e65 7874 3a20 6973 isOpen, Next: is │ │ │ │ -002774c0: 5265 6164 795f 6c70 4669 6c65 5f72 702c Ready_lpFile_rp, │ │ │ │ -002774d0: 2050 7265 763a 2065 6368 6f4f 6666 2c20 Prev: echoOff, │ │ │ │ -002774e0: 5570 3a20 7573 696e 6720 736f 636b 6574 Up: using socket │ │ │ │ -002774f0: 730a 0a69 734f 7065 6e20 2d2d 2077 6865 s..isOpen -- whe │ │ │ │ -00277500: 7468 6572 2061 2066 696c 6520 6f72 2064 ther a file or d │ │ │ │ -00277510: 6174 6162 6173 6520 6973 206f 7065 6e0a atabase is open. │ │ │ │ -00277520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00277530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00277540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -00277550: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -00277560: 2020 6973 4f70 656e 2066 0a20 202a 2049 isOpen f. * I │ │ │ │ -00277570: 6e70 7574 733a 0a20 2020 2020 202a 2066 nputs:. * f │ │ │ │ -00277580: 2c20 6120 2a6e 6f74 6520 6669 6c65 3a20 , a *note file: │ │ │ │ -00277590: 4669 6c65 2c20 6f72 2061 202a 6e6f 7465 File, or a *note │ │ │ │ -002775a0: 2064 6174 6162 6173 653a 2044 6174 6162 database: Datab │ │ │ │ -002775b0: 6173 652c 0a20 202a 204f 7574 7075 7473 ase,. * Outputs │ │ │ │ -002775c0: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ -002775d0: 6520 426f 6f6c 6561 6e20 7661 6c75 653a e Boolean value: │ │ │ │ -002775e0: 2042 6f6f 6c65 616e 2c2c 2020 7768 6574 Boolean,, whet │ │ │ │ -002775f0: 6865 7220 6620 6973 2061 6e20 6f70 656e her f is an open │ │ │ │ -00277600: 2066 696c 6520 6f72 206f 7065 6e0a 2020 file or open. │ │ │ │ -00277610: 2020 2020 2020 6461 7461 6261 7365 0a0a database.. │ │ │ │ -00277620: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00277630: 3d3d 3d3d 3d3d 3d0a 0a41 6e20 6f70 656e =======..An open │ │ │ │ -00277640: 2066 696c 6520 6973 2065 6974 6865 7220 file is either │ │ │ │ -00277650: 616e 2069 6e70 7574 2066 696c 652c 2061 an input file, a │ │ │ │ -00277660: 6e20 6f75 7470 7574 2066 696c 652c 2061 n output file, a │ │ │ │ -00277670: 6e20 696e 7075 7420 6f75 7470 7574 2066 n input output f │ │ │ │ -00277680: 696c 652c 206f 720a 6120 6c69 7374 656e ile, or.a listen │ │ │ │ -00277690: 6572 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d er...+---------- │ │ │ │ -002776a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002776b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -002776c0: 6620 3d20 2274 6573 742d 6669 6c65 2220 f = "test-file" │ │ │ │ -002776d0: 3c3c 2022 6869 2074 6865 7265 227c 0a7c << "hi there"|.| │ │ │ │ -002776e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002776f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00277700: 2020 7c0a 7c6f 3120 3d20 7465 7374 2d66 |.|o1 = test-f │ │ │ │ -00277710: 696c 6520 2020 2020 2020 2020 2020 2020 ile │ │ │ │ -00277720: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00277730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00277740: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00277750: 3120 3a20 4669 6c65 2020 2020 2020 2020 1 : File │ │ │ │ +00277400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277420: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00277430: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00277440: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00277450: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00277460: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00277470: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00277480: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00277490: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ +002774a0: 5f73 7973 7465 6d2e 6d32 3a31 3035 313a _system.m2:1051: │ │ │ │ +002774b0: 302e 0a1f 0a46 696c 653a 204d 6163 6175 0....File: Macau │ │ │ │ +002774c0: 6c61 7932 446f 632e 696e 666f 2c20 4e6f lay2Doc.info, No │ │ │ │ +002774d0: 6465 3a20 6973 4f70 656e 2c20 4e65 7874 de: isOpen, Next │ │ │ │ +002774e0: 3a20 6973 5265 6164 795f 6c70 4669 6c65 : isReady_lpFile │ │ │ │ +002774f0: 5f72 702c 2050 7265 763a 2065 6368 6f4f _rp, Prev: echoO │ │ │ │ +00277500: 6666 2c20 5570 3a20 7573 696e 6720 736f ff, Up: using so │ │ │ │ +00277510: 636b 6574 730a 0a69 734f 7065 6e20 2d2d ckets..isOpen -- │ │ │ │ +00277520: 2077 6865 7468 6572 2061 2066 696c 6520 whether a file │ │ │ │ +00277530: 6f72 2064 6174 6162 6173 6520 6973 206f or database is o │ │ │ │ +00277540: 7065 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a pen.************ │ │ │ │ +00277550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00277560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00277570: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +00277580: 2020 2020 2020 6973 4f70 656e 2066 0a20 isOpen f. │ │ │ │ +00277590: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +002775a0: 202a 2066 2c20 6120 2a6e 6f74 6520 6669 * f, a *note fi │ │ │ │ +002775b0: 6c65 3a20 4669 6c65 2c20 6f72 2061 202a le: File, or a * │ │ │ │ +002775c0: 6e6f 7465 2064 6174 6162 6173 653a 2044 note database: D │ │ │ │ +002775d0: 6174 6162 6173 652c 0a20 202a 204f 7574 atabase,. * Out │ │ │ │ +002775e0: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ +002775f0: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661 *note Boolean va │ │ │ │ +00277600: 6c75 653a 2042 6f6f 6c65 616e 2c2c 2020 lue: Boolean,, │ │ │ │ +00277610: 7768 6574 6865 7220 6620 6973 2061 6e20 whether f is an │ │ │ │ +00277620: 6f70 656e 2066 696c 6520 6f72 206f 7065 open file or ope │ │ │ │ +00277630: 6e0a 2020 2020 2020 2020 6461 7461 6261 n. databa │ │ │ │ +00277640: 7365 0a0a 4465 7363 7269 7074 696f 6e0a se..Description. │ │ │ │ +00277650: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 6e20 ===========..An │ │ │ │ +00277660: 6f70 656e 2066 696c 6520 6973 2065 6974 open file is eit │ │ │ │ +00277670: 6865 7220 616e 2069 6e70 7574 2066 696c her an input fil │ │ │ │ +00277680: 652c 2061 6e20 6f75 7470 7574 2066 696c e, an output fil │ │ │ │ +00277690: 652c 2061 6e20 696e 7075 7420 6f75 7470 e, an input outp │ │ │ │ +002776a0: 7574 2066 696c 652c 206f 720a 6120 6c69 ut file, or.a li │ │ │ │ +002776b0: 7374 656e 6572 2e0a 0a2b 2d2d 2d2d 2d2d stener...+------ │ │ │ │ +002776c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002776d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +002776e0: 3120 3a20 6620 3d20 2274 6573 742d 6669 1 : f = "test-fi │ │ │ │ +002776f0: 6c65 2220 3c3c 2022 6869 2074 6865 7265 le" << "hi there │ │ │ │ +00277700: 227c 0a7c 2020 2020 2020 2020 2020 2020 "|.| │ │ │ │ +00277710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00277720: 2020 2020 2020 7c0a 7c6f 3120 3d20 7465 |.|o1 = te │ │ │ │ +00277730: 7374 2d66 696c 6520 2020 2020 2020 2020 st-file │ │ │ │ +00277740: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00277750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00277760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00277770: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00277780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277790: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 6973 ------+.|i2 : is │ │ │ │ -002777a0: 4f70 656e 2066 2020 2020 2020 2020 2020 Open f │ │ │ │ -002777b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -002777c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002777d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002777e0: 7c0a 7c6f 3220 3d20 7472 7565 2020 2020 |.|o2 = true │ │ │ │ +00277770: 7c0a 7c6f 3120 3a20 4669 6c65 2020 2020 |.|o1 : File │ │ │ │ +00277780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00277790: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +002777a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002777b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +002777c0: 3a20 6973 4f70 656e 2066 2020 2020 2020 : isOpen f │ │ │ │ +002777d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +002777e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 002777f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00277800: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00277810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277820: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -00277830: 3a20 636c 6f73 6520 6620 2020 2020 2020 : close f │ │ │ │ -00277840: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00277850: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00277800: 2020 2020 7c0a 7c6f 3220 3d20 7472 7565 |.|o2 = true │ │ │ │ +00277810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00277820: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00277830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00277850: 7c69 3320 3a20 636c 6f73 6520 6620 2020 |i3 : close f │ │ │ │ 00277860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00277870: 2020 2020 7c0a 7c6f 3320 3d20 7465 7374 |.|o3 = test │ │ │ │ -00277880: 2d66 696c 6520 2020 2020 2020 2020 2020 -file │ │ │ │ -00277890: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -002778a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002778b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -002778c0: 7c6f 3320 3a20 4669 6c65 2020 2020 2020 |o3 : File │ │ │ │ +00277870: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00277880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00277890: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +002778a0: 7465 7374 2d66 696c 6520 2020 2020 2020 test-file │ │ │ │ +002778b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +002778c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002778d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002778e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -002778f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277900: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00277910: 6973 4f70 656e 2066 2020 2020 2020 2020 isOpen f │ │ │ │ -00277920: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00277930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002778e0: 2020 7c0a 7c6f 3320 3a20 4669 6c65 2020 |.|o3 : File │ │ │ │ +002778f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00277900: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00277910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00277930: 3420 3a20 6973 4f70 656e 2066 2020 2020 4 : isOpen f │ │ │ │ 00277940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00277950: 2020 7c0a 7c6f 3420 3d20 6661 6c73 6520 |.|o4 = false │ │ │ │ +00277950: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00277960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00277970: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00277980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -002779a0: 3520 3a20 6765 7420 2274 6573 742d 6669 5 : get "test-fi │ │ │ │ -002779b0: 6c65 2220 2020 2020 2020 2020 2020 2020 le" │ │ │ │ -002779c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -002779d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002779e0: 2020 2020 2020 7c0a 7c6f 3520 3d20 6869 |.|o5 = hi │ │ │ │ -002779f0: 2074 6865 7265 2020 2020 2020 2020 2020 there │ │ │ │ -00277a00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00277a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277a30: 2b0a 7c69 3620 3a20 7265 6d6f 7665 4669 +.|i6 : removeFi │ │ │ │ -00277a40: 6c65 2022 7465 7374 2d66 696c 6522 2020 le "test-file" │ │ │ │ -00277a50: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00277a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 ----------+..See │ │ │ │ -00277a80: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -00277a90: 2020 2a20 2a6e 6f74 6520 6f70 656e 496e * *note openIn │ │ │ │ -00277aa0: 3a20 6f70 656e 496e 5f6c 7053 7472 696e : openIn_lpStrin │ │ │ │ -00277ab0: 675f 7270 2c20 2d2d 206f 7065 6e20 616e g_rp, -- open an │ │ │ │ -00277ac0: 2069 6e70 7574 2066 696c 650a 2020 2a20 input file. * │ │ │ │ -00277ad0: 2a6e 6f74 6520 6f70 656e 496e 4f75 743a *note openInOut: │ │ │ │ -00277ae0: 206f 7065 6e49 6e4f 7574 2c20 2d2d 206f openInOut, -- o │ │ │ │ -00277af0: 7065 6e20 616e 2069 6e70 7574 206f 7574 pen an input out │ │ │ │ -00277b00: 7075 7420 6669 6c65 0a20 202a 202a 6e6f put file. * *no │ │ │ │ -00277b10: 7465 206f 7065 6e4c 6973 7465 6e65 723a te openListener: │ │ │ │ -00277b20: 206f 7065 6e4c 6973 7465 6e65 725f 6c70 openListener_lp │ │ │ │ -00277b30: 5374 7269 6e67 5f72 702c 202d 2d20 6f70 String_rp, -- op │ │ │ │ -00277b40: 656e 2061 2070 6f72 7420 666f 7220 6c69 en a port for li │ │ │ │ -00277b50: 7374 656e 696e 670a 2020 2a20 2a6e 6f74 stening. * *not │ │ │ │ -00277b60: 6520 636c 6f73 653a 2063 6c6f 7365 2c20 e close: close, │ │ │ │ -00277b70: 2d2d 2063 6c6f 7365 2061 2066 696c 650a -- close a file. │ │ │ │ -00277b80: 2020 2a20 2a6e 6f74 6520 6765 743a 2067 * *note get: g │ │ │ │ -00277b90: 6574 2c20 2d2d 2067 6574 2074 6865 2063 et, -- get the c │ │ │ │ -00277ba0: 6f6e 7465 6e74 7320 6f66 2061 2066 696c ontents of a fil │ │ │ │ -00277bb0: 650a 2020 2a20 2a6e 6f74 6520 7265 6d6f e. * *note remo │ │ │ │ -00277bc0: 7665 4669 6c65 3a20 7265 6d6f 7665 4669 veFile: removeFi │ │ │ │ -00277bd0: 6c65 2c20 2d2d 2072 656d 6f76 6520 6120 le, -- remove a │ │ │ │ -00277be0: 6669 6c65 0a0a 5761 7973 2074 6f20 7573 file..Ways to us │ │ │ │ -00277bf0: 6520 6973 4f70 656e 3a0a 3d3d 3d3d 3d3d e isOpen:.====== │ │ │ │ -00277c00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -00277c10: 202a 2022 6973 4f70 656e 2844 6174 6162 * "isOpen(Datab │ │ │ │ -00277c20: 6173 6529 220a 2020 2a20 2269 734f 7065 ase)". * "isOpe │ │ │ │ -00277c30: 6e28 4669 6c65 2922 0a0a 466f 7220 7468 n(File)"..For th │ │ │ │ -00277c40: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00277c50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00277c60: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00277c70: 6520 6973 4f70 656e 3a20 6973 4f70 656e e isOpen: isOpen │ │ │ │ -00277c80: 2c20 6973 2061 202a 6e6f 7465 2063 6f6d , is a *note com │ │ │ │ -00277c90: 7069 6c65 6420 6675 6e63 7469 6f6e 3a0a piled function:. │ │ │ │ -00277ca0: 436f 6d70 696c 6564 4675 6e63 7469 6f6e CompiledFunction │ │ │ │ -00277cb0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ -00277cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277970: 2020 2020 2020 7c0a 7c6f 3420 3d20 6661 |.|o4 = fa │ │ │ │ +00277980: 6c73 6520 2020 2020 2020 2020 2020 2020 lse │ │ │ │ +00277990: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +002779a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002779b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002779c0: 2b0a 7c69 3520 3a20 6765 7420 2274 6573 +.|i5 : get "tes │ │ │ │ +002779d0: 742d 6669 6c65 2220 2020 2020 2020 2020 t-file" │ │ │ │ +002779e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +002779f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00277a00: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +00277a10: 3d20 6869 2074 6865 7265 2020 2020 2020 = hi there │ │ │ │ +00277a20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00277a30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00277a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277a50: 2d2d 2d2d 2b0a 7c69 3620 3a20 7265 6d6f ----+.|i6 : remo │ │ │ │ +00277a60: 7665 4669 6c65 2022 7465 7374 2d66 696c veFile "test-fil │ │ │ │ +00277a70: 6522 2020 2020 2020 207c 0a2b 2d2d 2d2d e" |.+---- │ │ │ │ +00277a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00277aa0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +00277ab0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6f70 ==.. * *note op │ │ │ │ +00277ac0: 656e 496e 3a20 6f70 656e 496e 5f6c 7053 enIn: openIn_lpS │ │ │ │ +00277ad0: 7472 696e 675f 7270 2c20 2d2d 206f 7065 tring_rp, -- ope │ │ │ │ +00277ae0: 6e20 616e 2069 6e70 7574 2066 696c 650a n an input file. │ │ │ │ +00277af0: 2020 2a20 2a6e 6f74 6520 6f70 656e 496e * *note openIn │ │ │ │ +00277b00: 4f75 743a 206f 7065 6e49 6e4f 7574 2c20 Out: openInOut, │ │ │ │ +00277b10: 2d2d 206f 7065 6e20 616e 2069 6e70 7574 -- open an input │ │ │ │ +00277b20: 206f 7574 7075 7420 6669 6c65 0a20 202a output file. * │ │ │ │ +00277b30: 202a 6e6f 7465 206f 7065 6e4c 6973 7465 *note openListe │ │ │ │ +00277b40: 6e65 723a 206f 7065 6e4c 6973 7465 6e65 ner: openListene │ │ │ │ +00277b50: 725f 6c70 5374 7269 6e67 5f72 702c 202d r_lpString_rp, - │ │ │ │ +00277b60: 2d20 6f70 656e 2061 2070 6f72 7420 666f - open a port fo │ │ │ │ +00277b70: 7220 6c69 7374 656e 696e 670a 2020 2a20 r listening. * │ │ │ │ +00277b80: 2a6e 6f74 6520 636c 6f73 653a 2063 6c6f *note close: clo │ │ │ │ +00277b90: 7365 2c20 2d2d 2063 6c6f 7365 2061 2066 se, -- close a f │ │ │ │ +00277ba0: 696c 650a 2020 2a20 2a6e 6f74 6520 6765 ile. * *note ge │ │ │ │ +00277bb0: 743a 2067 6574 2c20 2d2d 2067 6574 2074 t: get, -- get t │ │ │ │ +00277bc0: 6865 2063 6f6e 7465 6e74 7320 6f66 2061 he contents of a │ │ │ │ +00277bd0: 2066 696c 650a 2020 2a20 2a6e 6f74 6520 file. * *note │ │ │ │ +00277be0: 7265 6d6f 7665 4669 6c65 3a20 7265 6d6f removeFile: remo │ │ │ │ +00277bf0: 7665 4669 6c65 2c20 2d2d 2072 656d 6f76 veFile, -- remov │ │ │ │ +00277c00: 6520 6120 6669 6c65 0a0a 5761 7973 2074 e a file..Ways t │ │ │ │ +00277c10: 6f20 7573 6520 6973 4f70 656e 3a0a 3d3d o use isOpen:.== │ │ │ │ +00277c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00277c30: 3d0a 0a20 202a 2022 6973 4f70 656e 2844 =.. * "isOpen(D │ │ │ │ +00277c40: 6174 6162 6173 6529 220a 2020 2a20 2269 atabase)". * "i │ │ │ │ +00277c50: 734f 7065 6e28 4669 6c65 2922 0a0a 466f sOpen(File)"..Fo │ │ │ │ +00277c60: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00277c70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00277c80: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00277c90: 2a6e 6f74 6520 6973 4f70 656e 3a20 6973 *note isOpen: is │ │ │ │ +00277ca0: 4f70 656e 2c20 6973 2061 202a 6e6f 7465 Open, is a *note │ │ │ │ +00277cb0: 2063 6f6d 7069 6c65 6420 6675 6e63 7469 compiled functi │ │ │ │ +00277cc0: 6f6e 3a0a 436f 6d70 696c 6564 4675 6e63 on:.CompiledFunc │ │ │ │ +00277cd0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ 00277ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00277cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277d00: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -00277d10: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -00277d20: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -00277d30: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -00277d40: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -00277d50: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -00277d60: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -00277d70: 756c 6179 3244 6f63 2f6f 765f 7379 7374 ulay2Doc/ov_syst │ │ │ │ -00277d80: 656d 2e6d 323a 3631 313a 302e 0a1f 0a46 em.m2:611:0....F │ │ │ │ -00277d90: 696c 653a 204d 6163 6175 6c61 7932 446f ile: Macaulay2Do │ │ │ │ -00277da0: 632e 696e 666f 2c20 4e6f 6465 3a20 6973 c.info, Node: is │ │ │ │ -00277db0: 5265 6164 795f 6c70 4669 6c65 5f72 702c Ready_lpFile_rp, │ │ │ │ -00277dc0: 204e 6578 743a 2069 734f 7574 7075 7446 Next: isOutputF │ │ │ │ -00277dd0: 696c 655f 6c70 4669 6c65 5f72 702c 2050 ile_lpFile_rp, P │ │ │ │ -00277de0: 7265 763a 2069 734f 7065 6e2c 2055 703a rev: isOpen, Up: │ │ │ │ -00277df0: 2075 7369 6e67 2073 6f63 6b65 7473 0a0a using sockets.. │ │ │ │ -00277e00: 6973 5265 6164 7928 4669 6c65 2920 2d2d isReady(File) -- │ │ │ │ -00277e10: 2077 6865 7468 6572 2061 2066 696c 6520 whether a file │ │ │ │ -00277e20: 6861 7320 6461 7461 2061 7661 696c 6162 has data availab │ │ │ │ -00277e30: 6c65 2066 6f72 2072 6561 6469 6e67 0a2a le for reading.* │ │ │ │ -00277e40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00277e50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00277e60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00277e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -00277e80: 202a 2046 756e 6374 696f 6e3a 202a 6e6f * Function: *no │ │ │ │ -00277e90: 7465 2069 7352 6561 6479 3a20 6973 5265 te isReady: isRe │ │ │ │ -00277ea0: 6164 795f 6c70 4669 6c65 5f72 702c 0a20 ady_lpFile_rp,. │ │ │ │ -00277eb0: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -00277ec0: 2020 2069 7352 6561 6479 2066 0a20 202a isReady f. * │ │ │ │ -00277ed0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00277ee0: 2061 202a 6e6f 7465 2066 696c 653a 2046 a *note file: F │ │ │ │ -00277ef0: 696c 652c 0a20 202a 204f 7574 7075 7473 ile,. * Outputs │ │ │ │ -00277f00: 3a0a 2020 2020 2020 2a20 6120 2a6e 6f74 :. * a *not │ │ │ │ -00277f10: 6520 426f 6f6c 6561 6e20 7661 6c75 653a e Boolean value: │ │ │ │ -00277f20: 2042 6f6f 6c65 616e 2c2c 2077 6865 7468 Boolean,, wheth │ │ │ │ -00277f30: 6572 2074 6865 2069 6e70 7574 2066 696c er the input fil │ │ │ │ -00277f40: 6520 6620 6861 7320 6461 7461 0a20 2020 e f has data. │ │ │ │ -00277f50: 2020 2020 2061 7661 696c 6162 6c65 2066 available f │ │ │ │ -00277f60: 6f72 2072 6561 6469 6e67 0a0a 4465 7363 or reading..Desc │ │ │ │ -00277f70: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -00277f80: 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ===..+---------- │ │ │ │ -00277f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00277fa0: 2d2d 2d2d 2b0a 7c69 3120 3a20 6620 3d20 ----+.|i1 : f = │ │ │ │ -00277fb0: 6f70 656e 496e 4f75 7420 2221 6361 7422 openInOut "!cat" │ │ │ │ -00277fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00277fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00277fe0: 2020 2020 2020 7c0a 7c6f 3120 3d20 2163 |.|o1 = !c │ │ │ │ -00277ff0: 6174 2020 2020 2020 2020 2020 2020 2020 at │ │ │ │ -00278000: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00278010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278020: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -00278030: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ -00278040: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00278050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278060: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ -00278070: 3a20 6973 5265 6164 7920 6620 2020 2020 : isReady f │ │ │ │ -00278080: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00278090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002780a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -002780b0: 3220 3d20 6661 6c73 6520 2020 2020 2020 2 = false │ │ │ │ -002780c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -002780d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002780e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -002780f0: 7c69 3320 3a20 6620 3c3c 2022 6869 2074 |i3 : f << "hi t │ │ │ │ -00278100: 6865 7265 2220 3c3c 2066 6c75 7368 3b7c here" << flush;| │ │ │ │ -00278110: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00278120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278130: 2b0a 7c69 3420 3a20 6973 5265 6164 7920 +.|i4 : isReady │ │ │ │ -00278140: 6620 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ -00278150: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00278160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278170: 2020 7c0a 7c6f 3420 3d20 6661 6c73 6520 |.|o4 = false │ │ │ │ +00277d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277d20: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00277d30: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00277d40: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00277d50: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00277d60: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00277d70: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +00277d80: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00277d90: 4d61 6361 756c 6179 3244 6f63 2f6f 765f Macaulay2Doc/ov_ │ │ │ │ +00277da0: 7379 7374 656d 2e6d 323a 3631 313a 302e system.m2:611:0. │ │ │ │ +00277db0: 0a1f 0a46 696c 653a 204d 6163 6175 6c61 ...File: Macaula │ │ │ │ +00277dc0: 7932 446f 632e 696e 666f 2c20 4e6f 6465 y2Doc.info, Node │ │ │ │ +00277dd0: 3a20 6973 5265 6164 795f 6c70 4669 6c65 : isReady_lpFile │ │ │ │ +00277de0: 5f72 702c 204e 6578 743a 2069 734f 7574 _rp, Next: isOut │ │ │ │ +00277df0: 7075 7446 696c 655f 6c70 4669 6c65 5f72 putFile_lpFile_r │ │ │ │ +00277e00: 702c 2050 7265 763a 2069 734f 7065 6e2c p, Prev: isOpen, │ │ │ │ +00277e10: 2055 703a 2075 7369 6e67 2073 6f63 6b65 Up: using socke │ │ │ │ +00277e20: 7473 0a0a 6973 5265 6164 7928 4669 6c65 ts..isReady(File │ │ │ │ +00277e30: 2920 2d2d 2077 6865 7468 6572 2061 2066 ) -- whether a f │ │ │ │ +00277e40: 696c 6520 6861 7320 6461 7461 2061 7661 ile has data ava │ │ │ │ +00277e50: 696c 6162 6c65 2066 6f72 2072 6561 6469 ilable for readi │ │ │ │ +00277e60: 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ng.************* │ │ │ │ +00277e70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00277e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00277e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00277ea0: 2a0a 0a20 202a 2046 756e 6374 696f 6e3a *.. * Function: │ │ │ │ +00277eb0: 202a 6e6f 7465 2069 7352 6561 6479 3a20 *note isReady: │ │ │ │ +00277ec0: 6973 5265 6164 795f 6c70 4669 6c65 5f72 isReady_lpFile_r │ │ │ │ +00277ed0: 702c 0a20 202a 2055 7361 6765 3a20 0a20 p,. * Usage: . │ │ │ │ +00277ee0: 2020 2020 2020 2069 7352 6561 6479 2066 isReady f │ │ │ │ +00277ef0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +00277f00: 2020 202a 2061 202a 6e6f 7465 2066 696c * a *note fil │ │ │ │ +00277f10: 653a 2046 696c 652c 0a20 202a 204f 7574 e: File,. * Out │ │ │ │ +00277f20: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ +00277f30: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661 *note Boolean va │ │ │ │ +00277f40: 6c75 653a 2042 6f6f 6c65 616e 2c2c 2077 lue: Boolean,, w │ │ │ │ +00277f50: 6865 7468 6572 2074 6865 2069 6e70 7574 hether the input │ │ │ │ +00277f60: 2066 696c 6520 6620 6861 7320 6461 7461 file f has data │ │ │ │ +00277f70: 0a20 2020 2020 2020 2061 7661 696c 6162 . availab │ │ │ │ +00277f80: 6c65 2066 6f72 2072 6561 6469 6e67 0a0a le for reading.. │ │ │ │ +00277f90: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00277fa0: 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d =======..+------ │ │ │ │ +00277fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00277fc0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +00277fd0: 6620 3d20 6f70 656e 496e 4f75 7420 2221 f = openInOut "! │ │ │ │ +00277fe0: 6361 7422 2020 2020 207c 0a7c 2020 2020 cat" |.| │ │ │ │ +00277ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278000: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +00278010: 3d20 2163 6174 2020 2020 2020 2020 2020 = !cat │ │ │ │ +00278020: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00278030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278040: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00278050: 3120 3a20 4669 6c65 2020 2020 2020 2020 1 : File │ │ │ │ +00278060: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00278070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00278090: 7c69 3220 3a20 6973 5265 6164 7920 6620 |i2 : isReady f │ │ │ │ +002780a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +002780b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +002780c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002780d0: 7c0a 7c6f 3220 3d20 6661 6c73 6520 2020 |.|o2 = false │ │ │ │ +002780e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002780f0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00278100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278110: 2d2d 2b0a 7c69 3320 3a20 6620 3c3c 2022 --+.|i3 : f << " │ │ │ │ +00278120: 6869 2074 6865 7265 2220 3c3c 2066 6c75 hi there" << flu │ │ │ │ +00278130: 7368 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d sh;|.+---------- │ │ │ │ +00278140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278150: 2d2d 2d2d 2b0a 7c69 3420 3a20 6973 5265 ----+.|i4 : isRe │ │ │ │ +00278160: 6164 7920 6620 2020 2020 2020 2020 2020 ady f │ │ │ │ +00278170: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00278180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278190: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ -002781a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002781b0: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075 ----+..Ways to u │ │ │ │ -002781c0: 7365 2074 6869 7320 6d65 7468 6f64 3a0a se this method:. │ │ │ │ -002781d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -002781e0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -002781f0: 6f74 6520 6973 5265 6164 7928 4669 6c65 ote isReady(File │ │ │ │ -00278200: 293a 2069 7352 6561 6479 5f6c 7046 696c ): isReady_lpFil │ │ │ │ -00278210: 655f 7270 2c20 2d2d 2077 6865 7468 6572 e_rp, -- whether │ │ │ │ -00278220: 2061 2066 696c 6520 6861 7320 6461 7461 a file has data │ │ │ │ -00278230: 0a20 2020 2061 7661 696c 6162 6c65 2066 . available f │ │ │ │ -00278240: 6f72 2072 6561 6469 6e67 0a2d 2d2d 2d2d or reading.----- │ │ │ │ -00278250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278190: 2020 2020 2020 7c0a 7c6f 3420 3d20 6661 |.|o4 = fa │ │ │ │ +002781a0: 6c73 6520 2020 2020 2020 2020 2020 2020 lse │ │ │ │ +002781b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +002781c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002781d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 --------+..Ways │ │ │ │ +002781e0: 746f 2075 7365 2074 6869 7320 6d65 7468 to use this meth │ │ │ │ +002781f0: 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d od:.============ │ │ │ │ +00278200: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +00278210: 2a20 2a6e 6f74 6520 6973 5265 6164 7928 * *note isReady( │ │ │ │ +00278220: 4669 6c65 293a 2069 7352 6561 6479 5f6c File): isReady_l │ │ │ │ +00278230: 7046 696c 655f 7270 2c20 2d2d 2077 6865 pFile_rp, -- whe │ │ │ │ +00278240: 7468 6572 2061 2066 696c 6520 6861 7320 ther a file has │ │ │ │ +00278250: 6461 7461 0a20 2020 2061 7661 696c 6162 data. availab │ │ │ │ +00278260: 6c65 2066 6f72 2072 6561 6469 6e67 0a2d le for reading.- │ │ │ │ 00278270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00278280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278290: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -002782a0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -002782b0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -002782c0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -002782d0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -002782e0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -002782f0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00278300: 732f 0a4d 6163 6175 6c61 7932 446f 632f s/.Macaulay2Doc/ │ │ │ │ -00278310: 6f76 5f66 696c 6573 2e6d 323a 3335 393a ov_files.m2:359: │ │ │ │ -00278320: 302e 0a1f 0a46 696c 653a 204d 6163 6175 0....File: Macau │ │ │ │ -00278330: 6c61 7932 446f 632e 696e 666f 2c20 4e6f lay2Doc.info, No │ │ │ │ -00278340: 6465 3a20 6973 4f75 7470 7574 4669 6c65 de: isOutputFile │ │ │ │ -00278350: 5f6c 7046 696c 655f 7270 2c20 4e65 7874 _lpFile_rp, Next │ │ │ │ -00278360: 3a20 6973 496e 7075 7446 696c 655f 6c70 : isInputFile_lp │ │ │ │ -00278370: 4669 6c65 5f72 702c 2050 7265 763a 2069 File_rp, Prev: i │ │ │ │ -00278380: 7352 6561 6479 5f6c 7046 696c 655f 7270 sReady_lpFile_rp │ │ │ │ -00278390: 2c20 5570 3a20 7573 696e 6720 736f 636b , Up: using sock │ │ │ │ -002783a0: 6574 730a 0a69 734f 7574 7075 7446 696c ets..isOutputFil │ │ │ │ -002783b0: 6528 4669 6c65 2920 2d2d 2077 6865 7468 e(File) -- wheth │ │ │ │ -002783c0: 6572 2061 2066 696c 6520 6973 206f 7065 er a file is ope │ │ │ │ -002783d0: 6e20 666f 7220 6f75 7470 7574 0a2a 2a2a n for output.*** │ │ │ │ -002783e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002783f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00278400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00278410: 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ -00278420: 6f6e 3a20 2a6e 6f74 6520 6973 4f75 7470 on: *note isOutp │ │ │ │ -00278430: 7574 4669 6c65 3a20 6973 4f75 7470 7574 utFile: isOutput │ │ │ │ -00278440: 4669 6c65 5f6c 7046 696c 655f 7270 2c0a File_lpFile_rp,. │ │ │ │ -00278450: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00278460: 2020 2020 6973 4f75 7470 7574 4669 6c65 isOutputFile │ │ │ │ -00278470: 2066 0a20 202a 2049 6e70 7574 733a 0a20 f. * Inputs:. │ │ │ │ -00278480: 2020 2020 202a 2066 2c20 6120 2a6e 6f74 * f, a *not │ │ │ │ -00278490: 6520 6669 6c65 3a20 4669 6c65 2c0a 2020 e file: File,. │ │ │ │ -002784a0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -002784b0: 202a 2061 202a 6e6f 7465 2042 6f6f 6c65 * a *note Boole │ │ │ │ -002784c0: 616e 2076 616c 7565 3a20 426f 6f6c 6561 an value: Boolea │ │ │ │ -002784d0: 6e2c 2c20 2077 6865 7468 6572 2066 2069 n,, whether f i │ │ │ │ -002784e0: 7320 616e 206f 7065 6e20 6f75 7470 7574 s an open output │ │ │ │ -002784f0: 2066 696c 650a 0a44 6573 6372 6970 7469 file..Descripti │ │ │ │ -00278500: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00278510: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00278520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278530: 2d2d 2d2b 0a7c 6931 203a 2066 203d 2022 ---+.|i1 : f = " │ │ │ │ -00278540: 7465 7374 2d66 696c 6522 203c 3c20 2268 test-file" << "h │ │ │ │ -00278550: 6920 7468 6572 6522 7c0a 7c20 2020 2020 i there"|.| │ │ │ │ -00278560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278570: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00278580: 6f31 203d 2074 6573 742d 6669 6c65 2020 o1 = test-file │ │ │ │ +00278290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002782a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002782b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +002782c0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +002782d0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +002782e0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +002782f0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +00278300: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +00278310: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +00278320: 6b61 6765 732f 0a4d 6163 6175 6c61 7932 kages/.Macaulay2 │ │ │ │ +00278330: 446f 632f 6f76 5f66 696c 6573 2e6d 323a Doc/ov_files.m2: │ │ │ │ +00278340: 3335 393a 302e 0a1f 0a46 696c 653a 204d 359:0....File: M │ │ │ │ +00278350: 6163 6175 6c61 7932 446f 632e 696e 666f acaulay2Doc.info │ │ │ │ +00278360: 2c20 4e6f 6465 3a20 6973 4f75 7470 7574 , Node: isOutput │ │ │ │ +00278370: 4669 6c65 5f6c 7046 696c 655f 7270 2c20 File_lpFile_rp, │ │ │ │ +00278380: 4e65 7874 3a20 6973 496e 7075 7446 696c Next: isInputFil │ │ │ │ +00278390: 655f 6c70 4669 6c65 5f72 702c 2050 7265 e_lpFile_rp, Pre │ │ │ │ +002783a0: 763a 2069 7352 6561 6479 5f6c 7046 696c v: isReady_lpFil │ │ │ │ +002783b0: 655f 7270 2c20 5570 3a20 7573 696e 6720 e_rp, Up: using │ │ │ │ +002783c0: 736f 636b 6574 730a 0a69 734f 7574 7075 sockets..isOutpu │ │ │ │ +002783d0: 7446 696c 6528 4669 6c65 2920 2d2d 2077 tFile(File) -- w │ │ │ │ +002783e0: 6865 7468 6572 2061 2066 696c 6520 6973 hether a file is │ │ │ │ +002783f0: 206f 7065 6e20 666f 7220 6f75 7470 7574 open for output │ │ │ │ +00278400: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +00278410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00278420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00278430: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 ********.. * Fu │ │ │ │ +00278440: 6e63 7469 6f6e 3a20 2a6e 6f74 6520 6973 nction: *note is │ │ │ │ +00278450: 4f75 7470 7574 4669 6c65 3a20 6973 4f75 OutputFile: isOu │ │ │ │ +00278460: 7470 7574 4669 6c65 5f6c 7046 696c 655f tputFile_lpFile_ │ │ │ │ +00278470: 7270 2c0a 2020 2a20 5573 6167 653a 200a rp,. * Usage: . │ │ │ │ +00278480: 2020 2020 2020 2020 6973 4f75 7470 7574 isOutput │ │ │ │ +00278490: 4669 6c65 2066 0a20 202a 2049 6e70 7574 File f. * Input │ │ │ │ +002784a0: 733a 0a20 2020 2020 202a 2066 2c20 6120 s:. * f, a │ │ │ │ +002784b0: 2a6e 6f74 6520 6669 6c65 3a20 4669 6c65 *note file: File │ │ │ │ +002784c0: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20 ,. * Outputs:. │ │ │ │ +002784d0: 2020 2020 202a 2061 202a 6e6f 7465 2042 * a *note B │ │ │ │ +002784e0: 6f6f 6c65 616e 2076 616c 7565 3a20 426f oolean value: Bo │ │ │ │ +002784f0: 6f6c 6561 6e2c 2c20 2077 6865 7468 6572 olean,, whether │ │ │ │ +00278500: 2066 2069 7320 616e 206f 7065 6e20 6f75 f is an open ou │ │ │ │ +00278510: 7470 7574 2066 696c 650a 0a44 6573 6372 tput file..Descr │ │ │ │ +00278520: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00278530: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ==..+----------- │ │ │ │ +00278540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278550: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2066 -------+.|i1 : f │ │ │ │ +00278560: 203d 2022 7465 7374 2d66 696c 6522 203c = "test-file" < │ │ │ │ +00278570: 3c20 2268 6920 7468 6572 6522 7c0a 7c20 < "hi there"|.| │ │ │ │ +00278580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00278590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002785a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -002785b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002785c0: 2020 2020 2020 207c 0a7c 6f31 203a 2046 |.|o1 : F │ │ │ │ -002785d0: 696c 6520 2020 2020 2020 2020 2020 2020 ile │ │ │ │ -002785e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -002785f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278610: 2d2b 0a7c 6932 203a 2069 734f 7574 7075 -+.|i2 : isOutpu │ │ │ │ -00278620: 7446 696c 6520 6620 2020 2020 2020 2020 tFile f │ │ │ │ -00278630: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00278640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278650: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00278660: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ -00278670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278680: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -00278690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002786a0: 2d2d 2d2d 2d2b 0a7c 6933 203a 2063 6c6f -----+.|i3 : clo │ │ │ │ -002786b0: 7365 2066 2020 2020 2020 2020 2020 2020 se f │ │ │ │ -002786c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -002786d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002786e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -002786f0: 0a7c 6f33 203d 2074 6573 742d 6669 6c65 .|o3 = test-file │ │ │ │ +002785a0: 207c 0a7c 6f31 203d 2074 6573 742d 6669 |.|o1 = test-fi │ │ │ │ +002785b0: 6c65 2020 2020 2020 2020 2020 2020 2020 le │ │ │ │ +002785c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +002785d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002785e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +002785f0: 203a 2046 696c 6520 2020 2020 2020 2020 : File │ │ │ │ +00278600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278610: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00278620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278630: 2d2d 2d2d 2d2b 0a7c 6932 203a 2069 734f -----+.|i2 : isO │ │ │ │ +00278640: 7574 7075 7446 696c 6520 6620 2020 2020 utputFile f │ │ │ │ +00278650: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00278660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00278680: 0a7c 6f32 203d 2074 7275 6520 2020 2020 .|o2 = true │ │ │ │ +00278690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002786a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +002786b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002786c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +002786d0: 2063 6c6f 7365 2066 2020 2020 2020 2020 close f │ │ │ │ +002786e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +002786f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00278700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278710: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00278720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278730: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ -00278740: 2046 696c 6520 2020 2020 2020 2020 2020 File │ │ │ │ -00278750: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00278760: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00278770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278780: 2d2d 2d2b 0a7c 6934 203a 2069 734f 7574 ---+.|i4 : isOut │ │ │ │ -00278790: 7075 7446 696c 6520 6620 2020 2020 2020 putFile f │ │ │ │ -002787a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -002787b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002787c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -002787d0: 6f34 203d 2066 616c 7365 2020 2020 2020 o4 = false │ │ │ │ +00278710: 2020 207c 0a7c 6f33 203d 2074 6573 742d |.|o3 = test- │ │ │ │ +00278720: 6669 6c65 2020 2020 2020 2020 2020 2020 file │ │ │ │ +00278730: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00278740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278750: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00278760: 6f33 203a 2046 696c 6520 2020 2020 2020 o3 : File │ │ │ │ +00278770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278780: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00278790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002787a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2069 -------+.|i4 : i │ │ │ │ +002787b0: 734f 7574 7075 7446 696c 6520 6620 2020 sOutputFile f │ │ │ │ +002787c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +002787d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002787e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002787f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00278800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278810: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2067 -------+.|i5 : g │ │ │ │ -00278820: 6574 2022 7465 7374 2d66 696c 6522 2020 et "test-file" │ │ │ │ -00278830: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00278840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278860: 207c 0a7c 6f35 203d 2068 6920 7468 6572 |.|o5 = hi ther │ │ │ │ -00278870: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -00278880: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -00278890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002788a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ -002788b0: 203a 2072 656d 6f76 6546 696c 6520 2274 : removeFile "t │ │ │ │ -002788c0: 6573 742d 6669 6c65 2220 2020 2020 2020 est-file" │ │ │ │ -002788d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -002788e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002788f0: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ -00278900: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -00278910: 6e6f 7465 206f 7065 6e4f 7574 3a20 6f70 note openOut: op │ │ │ │ -00278920: 656e 4f75 745f 6c70 5374 7269 6e67 5f72 enOut_lpString_r │ │ │ │ -00278930: 702c 202d 2d20 6f70 656e 2061 6e20 6f75 p, -- open an ou │ │ │ │ -00278940: 7470 7574 2066 696c 650a 2020 2a20 2a6e tput file. * *n │ │ │ │ -00278950: 6f74 6520 6f70 656e 496e 4f75 743a 206f ote openInOut: o │ │ │ │ -00278960: 7065 6e49 6e4f 7574 2c20 2d2d 206f 7065 penInOut, -- ope │ │ │ │ -00278970: 6e20 616e 2069 6e70 7574 206f 7574 7075 n an input outpu │ │ │ │ -00278980: 7420 6669 6c65 0a20 202a 202a 6e6f 7465 t file. * *note │ │ │ │ -00278990: 2063 6c6f 7365 3a20 636c 6f73 652c 202d close: close, - │ │ │ │ -002789a0: 2d20 636c 6f73 6520 6120 6669 6c65 0a20 - close a file. │ │ │ │ -002789b0: 202a 202a 6e6f 7465 2067 6574 3a20 6765 * *note get: ge │ │ │ │ -002789c0: 742c 202d 2d20 6765 7420 7468 6520 636f t, -- get the co │ │ │ │ -002789d0: 6e74 656e 7473 206f 6620 6120 6669 6c65 ntents of a file │ │ │ │ -002789e0: 0a20 202a 202a 6e6f 7465 2072 656d 6f76 . * *note remov │ │ │ │ -002789f0: 6546 696c 653a 2072 656d 6f76 6546 696c eFile: removeFil │ │ │ │ -00278a00: 652c 202d 2d20 7265 6d6f 7665 2061 2066 e, -- remove a f │ │ │ │ -00278a10: 696c 650a 0a57 6179 7320 746f 2075 7365 ile..Ways to use │ │ │ │ -00278a20: 2074 6869 7320 6d65 7468 6f64 3a0a 3d3d this method:.== │ │ │ │ -00278a30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00278a40: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ -00278a50: 6520 6973 4f75 7470 7574 4669 6c65 2846 e isOutputFile(F │ │ │ │ -00278a60: 696c 6529 3a20 6973 4f75 7470 7574 4669 ile): isOutputFi │ │ │ │ -00278a70: 6c65 5f6c 7046 696c 655f 7270 2c20 2d2d le_lpFile_rp, -- │ │ │ │ -00278a80: 2077 6865 7468 6572 2061 2066 696c 6520 whether a file │ │ │ │ -00278a90: 6973 206f 7065 6e0a 2020 2020 666f 7220 is open. for │ │ │ │ -00278aa0: 6f75 7470 7574 0a2d 2d2d 2d2d 2d2d 2d2d output.--------- │ │ │ │ -00278ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002787f0: 207c 0a7c 6f34 203d 2066 616c 7365 2020 |.|o4 = false │ │ │ │ +00278800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278810: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00278820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ +00278840: 203a 2067 6574 2022 7465 7374 2d66 696c : get "test-fil │ │ │ │ +00278850: 6522 2020 2020 2020 2020 2020 2020 2020 e" │ │ │ │ +00278860: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00278870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278880: 2020 2020 207c 0a7c 6f35 203d 2068 6920 |.|o5 = hi │ │ │ │ +00278890: 7468 6572 6520 2020 2020 2020 2020 2020 there │ │ │ │ +002788a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +002788b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002788c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +002788d0: 0a7c 6936 203a 2072 656d 6f76 6546 696c .|i6 : removeFil │ │ │ │ +002788e0: 6520 2274 6573 742d 6669 6c65 2220 2020 e "test-file" │ │ │ │ +002788f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00278900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278910: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ +00278920: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ +00278930: 202a 202a 6e6f 7465 206f 7065 6e4f 7574 * *note openOut │ │ │ │ +00278940: 3a20 6f70 656e 4f75 745f 6c70 5374 7269 : openOut_lpStri │ │ │ │ +00278950: 6e67 5f72 702c 202d 2d20 6f70 656e 2061 ng_rp, -- open a │ │ │ │ +00278960: 6e20 6f75 7470 7574 2066 696c 650a 2020 n output file. │ │ │ │ +00278970: 2a20 2a6e 6f74 6520 6f70 656e 496e 4f75 * *note openInOu │ │ │ │ +00278980: 743a 206f 7065 6e49 6e4f 7574 2c20 2d2d t: openInOut, -- │ │ │ │ +00278990: 206f 7065 6e20 616e 2069 6e70 7574 206f open an input o │ │ │ │ +002789a0: 7574 7075 7420 6669 6c65 0a20 202a 202a utput file. * * │ │ │ │ +002789b0: 6e6f 7465 2063 6c6f 7365 3a20 636c 6f73 note close: clos │ │ │ │ +002789c0: 652c 202d 2d20 636c 6f73 6520 6120 6669 e, -- close a fi │ │ │ │ +002789d0: 6c65 0a20 202a 202a 6e6f 7465 2067 6574 le. * *note get │ │ │ │ +002789e0: 3a20 6765 742c 202d 2d20 6765 7420 7468 : get, -- get th │ │ │ │ +002789f0: 6520 636f 6e74 656e 7473 206f 6620 6120 e contents of a │ │ │ │ +00278a00: 6669 6c65 0a20 202a 202a 6e6f 7465 2072 file. * *note r │ │ │ │ +00278a10: 656d 6f76 6546 696c 653a 2072 656d 6f76 emoveFile: remov │ │ │ │ +00278a20: 6546 696c 652c 202d 2d20 7265 6d6f 7665 eFile, -- remove │ │ │ │ +00278a30: 2061 2066 696c 650a 0a57 6179 7320 746f a file..Ways to │ │ │ │ +00278a40: 2075 7365 2074 6869 7320 6d65 7468 6f64 use this method │ │ │ │ +00278a50: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +00278a60: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00278a70: 2a6e 6f74 6520 6973 4f75 7470 7574 4669 *note isOutputFi │ │ │ │ +00278a80: 6c65 2846 696c 6529 3a20 6973 4f75 7470 le(File): isOutp │ │ │ │ +00278a90: 7574 4669 6c65 5f6c 7046 696c 655f 7270 utFile_lpFile_rp │ │ │ │ +00278aa0: 2c20 2d2d 2077 6865 7468 6572 2061 2066 , -- whether a f │ │ │ │ +00278ab0: 696c 6520 6973 206f 7065 6e0a 2020 2020 ile is open. │ │ │ │ +00278ac0: 666f 7220 6f75 7470 7574 0a2d 2d2d 2d2d for output.----- │ │ │ │ 00278ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00278ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278af0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00278b00: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00278b10: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00278b20: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00278b30: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00278b40: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ -00278b50: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ -00278b60: 6163 6175 6c61 7932 446f 632f 6f76 5f73 acaulay2Doc/ov_s │ │ │ │ -00278b70: 7973 7465 6d2e 6d32 3a35 3839 3a30 2e0a ystem.m2:589:0.. │ │ │ │ -00278b80: 1f0a 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isInputFil │ │ │ │ -00278c90: 653a 2069 7349 6e70 7574 4669 6c65 5f6c e: isInputFile_l │ │ │ │ -00278ca0: 7046 696c 655f 7270 2c0a 2020 2a20 5573 pFile_rp,. * Us │ │ │ │ -00278cb0: 6167 653a 200a 2020 2020 2020 2020 6973 age: . is │ │ │ │ -00278cc0: 496e 7075 7446 696c 6520 660a 2020 2a20 InputFile f. * │ │ │ │ -00278cd0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -00278ce0: 662c 2061 202a 6e6f 7465 2066 696c 653a f, a *note file: │ │ │ │ -00278cf0: 2046 696c 652c 0a20 202a 204f 7574 7075 File,. * Outpu │ │ │ │ -00278d00: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e ts:. * a *n │ │ │ │ -00278d10: 6f74 6520 426f 6f6c 6561 6e20 7661 6c75 ote Boolean valu │ │ │ │ -00278d20: 653a 2042 6f6f 6c65 616e 2c2c 2020 7768 e: Boolean,, wh │ │ │ │ -00278d30: 6574 6865 7220 6620 6973 2061 6e20 6f70 ether f is an op │ │ │ │ -00278d40: 656e 2069 6e70 7574 2066 696c 650a 0a44 en input file..D │ │ │ │ -00278d50: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00278d60: 3d3d 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2a20 4675 6e63 7469 ****.. * Functi │ │ │ │ +00278ca0: 6f6e 3a20 2a6e 6f74 6520 6973 496e 7075 on: *note isInpu │ │ │ │ +00278cb0: 7446 696c 653a 2069 7349 6e70 7574 4669 tFile: isInputFi │ │ │ │ +00278cc0: 6c65 5f6c 7046 696c 655f 7270 2c0a 2020 le_lpFile_rp,. │ │ │ │ +00278cd0: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +00278ce0: 2020 6973 496e 7075 7446 696c 6520 660a isInputFile f. │ │ │ │ +00278cf0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +00278d00: 2020 2a20 662c 2061 202a 6e6f 7465 2066 * f, a *note f │ │ │ │ +00278d10: 696c 653a 2046 696c 652c 0a20 202a 204f ile: File,. * O │ │ │ │ +00278d20: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +00278d30: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20 a *note Boolean │ │ │ │ +00278d40: 7661 6c75 653a 2042 6f6f 6c65 616e 2c2c value: Boolean,, │ │ │ │ +00278d50: 2020 7768 6574 6865 7220 6620 6973 2061 whether f is a │ │ │ │ +00278d60: 6e20 6f70 656e 2069 6e70 7574 2066 696c n open input fil │ │ │ │ +00278d70: 650a 0a44 6573 6372 6970 7469 6f6e 0a3d e..Description.= │ │ │ │ +00278d80: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d ==========..+--- │ │ │ │ +00278d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278db0: 2d2d 2d2d 2b0a 7c69 3120 3a20 2274 6573 ----+.|i1 : "tes │ │ │ │ +00278dc0: 742d 6669 6c65 2220 3c3c 2022 6869 2074 t-file" << "hi t │ │ │ │ +00278dd0: 6865 7265 2220 3c3c 2063 6c6f 7365 7c0a here" << close|. │ │ │ │ +00278de0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00278df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278e00: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ +00278e10: 7465 7374 2d66 696c 6520 2020 2020 2020 test-file │ │ │ │ 00278e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278e30: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -00278e40: 4669 6c65 2020 2020 2020 2020 2020 2020 File │ │ │ │ -00278e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278e60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -00278e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00278e90: 3220 3a20 6973 496e 7075 7446 696c 6520 2 : isInputFile │ │ │ │ -00278ea0: 6f6f 2020 2020 2020 2020 2020 2020 2020 oo │ │ │ │ -00278eb0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00278ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278ee0: 7c0a 7c6f 3220 3d20 6661 6c73 6520 2020 |.|o2 = false │ │ │ │ +00278e30: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00278e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278e50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00278e60: 3120 3a20 4669 6c65 2020 2020 2020 2020 1 : File │ │ │ │ +00278e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278e80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00278e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278eb0: 2b0a 7c69 3220 3a20 6973 496e 7075 7446 +.|i2 : isInputF │ │ │ │ +00278ec0: 696c 6520 6f6f 2020 2020 2020 2020 2020 ile oo │ │ │ │ +00278ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00278ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00278ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278f00: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00278f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00278f30: 2d2d 2d2d 2b0a 7c69 3320 3a20 6620 3d20 ----+.|i3 : f = │ │ │ │ -00278f40: 6f70 656e 496e 2022 7465 7374 2d66 696c openIn "test-fil │ │ │ │ -00278f50: 6522 2020 2020 2020 2020 2020 2020 7c0a e" |. │ │ │ │ -00278f60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00278f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278f80: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00278f90: 7465 7374 2d66 696c 6520 2020 2020 2020 test-file │ │ │ │ -00278fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278fb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00278f00: 2020 2020 7c0a 7c6f 3220 3d20 6661 6c73 |.|o2 = fals │ │ │ │ +00278f10: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00278f20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00278f30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00278f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00278f50: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +00278f60: 6620 3d20 6f70 656e 496e 2022 7465 7374 f = openIn "test │ │ │ │ +00278f70: 2d66 696c 6522 2020 2020 2020 2020 2020 -file" │ │ │ │ +00278f80: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00278f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00278fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00278fb0: 3320 3d20 7465 7374 2d66 696c 6520 2020 3 = test-file │ │ │ │ 00278fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00278fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00278fe0: 3320 3a20 4669 6c65 2020 2020 2020 2020 3 : File │ │ │ │ +00278fd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00278fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00278ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279000: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -00279010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279030: 2b0a 7c69 3420 3a20 6973 496e 7075 7446 +.|i4 : isInputF │ │ │ │ -00279040: 696c 6520 6620 2020 2020 2020 2020 2020 ile f │ │ │ │ -00279050: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00279060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279080: 2020 2020 7c0a 7c6f 3420 3d20 7472 7565 |.|o4 = true │ │ │ │ +00279000: 7c0a 7c6f 3320 3a20 4669 6c65 2020 2020 |.|o3 : File │ │ │ │ +00279010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00279020: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00279030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279050: 2d2d 2d2d 2b0a 7c69 3420 3a20 6973 496e ----+.|i4 : isIn │ │ │ │ +00279060: 7075 7446 696c 6520 6620 2020 2020 2020 putFile f │ │ │ │ +00279070: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00279080: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00279090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002790a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -002790b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -002790c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002790d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ -002790e0: 6973 4f70 656e 2066 2020 2020 2020 2020 isOpen f │ │ │ │ -002790f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279100: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +002790a0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ +002790b0: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ +002790c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002790d0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +002790e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002790f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00279100: 3520 3a20 6973 4f70 656e 2066 2020 2020 5 : isOpen f │ │ │ │ 00279110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279120: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00279130: 3520 3d20 7472 7565 2020 2020 2020 2020 5 = true │ │ │ │ +00279120: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00279130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00279140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279150: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -00279160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279180: 2b0a 7c69 3620 3a20 6765 7420 6620 2020 +.|i6 : get f │ │ │ │ -00279190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002791a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -002791b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002791c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002791d0: 2020 2020 7c0a 7c6f 3620 3d20 6869 2074 |.|o6 = hi t │ │ │ │ -002791e0: 6865 7265 2020 2020 2020 2020 2020 2020 here │ │ │ │ -002791f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00279200: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00279210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279220: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 --------+.|i7 : │ │ │ │ -00279230: 6973 496e 7075 7446 696c 6520 6620 2020 isInputFile f │ │ │ │ -00279240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279250: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00279260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279270: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00279280: 3720 3d20 6661 6c73 6520 2020 2020 2020 7 = false │ │ │ │ +00279150: 7c0a 7c6f 3520 3d20 7472 7565 2020 2020 |.|o5 = true │ │ │ │ +00279160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00279170: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00279180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002791a0: 2d2d 2d2d 2b0a 7c69 3620 3a20 6765 7420 ----+.|i6 : get │ │ │ │ +002791b0: 6620 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ +002791c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +002791d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +002791e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002791f0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ +00279200: 6869 2074 6865 7265 2020 2020 2020 2020 hi there │ │ │ │ +00279210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00279220: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00279230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00279250: 3720 3a20 6973 496e 7075 7446 696c 6520 7 : isInputFile │ │ │ │ +00279260: 6620 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ +00279270: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00279280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00279290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002792a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -002792b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002792c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002792d0: 2b0a 7c69 3820 3a20 6973 4f70 656e 2066 +.|i8 : isOpen f │ │ │ │ -002792e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002792f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00279300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00279320: 2020 2020 7c0a 7c6f 3820 3d20 6661 6c73 |.|o8 = fals │ │ │ │ -00279330: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ -00279340: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00279350: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00279360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279370: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 --------+.|i9 : │ │ │ │ -00279380: 7265 6d6f 7665 4669 6c65 2022 7465 7374 removeFile "test │ │ │ │ -00279390: 2d66 696c 6522 2020 2020 2020 2020 2020 -file" │ │ │ │ -002793a0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -002793b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002793c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ -002793d0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -002793e0: 0a0a 2020 2a20 2a6e 6f74 6520 6f70 656e .. * *note open │ │ │ │ -002793f0: 496e 3a20 6f70 656e 496e 5f6c 7053 7472 In: openIn_lpStr │ │ │ │ -00279400: 696e 675f 7270 2c20 2d2d 206f 7065 6e20 ing_rp, -- open │ │ │ │ -00279410: 616e 2069 6e70 7574 2066 696c 650a 2020 an input file. │ │ │ │ -00279420: 2a20 2a6e 6f74 6520 6f70 656e 496e 4f75 * *note openInOu │ │ │ │ -00279430: 743a 206f 7065 6e49 6e4f 7574 2c20 2d2d t: openInOut, -- │ │ │ │ -00279440: 206f 7065 6e20 616e 2069 6e70 7574 206f open an input o │ │ │ │ -00279450: 7574 7075 7420 6669 6c65 0a20 202a 202a utput file. * * │ │ │ │ -00279460: 6e6f 7465 2067 6574 3a20 6765 742c 202d note get: get, - │ │ │ │ -00279470: 2d20 6765 7420 7468 6520 636f 6e74 656e - get the conten │ │ │ │ -00279480: 7473 206f 6620 6120 6669 6c65 0a20 202a ts of a file. * │ │ │ │ -00279490: 202a 6e6f 7465 2069 734f 7065 6e3a 2069 *note isOpen: i │ │ │ │ -002794a0: 734f 7065 6e2c 202d 2d20 7768 6574 6865 sOpen, -- whethe │ │ │ │ -002794b0: 7220 6120 6669 6c65 206f 7220 6461 7461 r a file or data │ │ │ │ -002794c0: 6261 7365 2069 7320 6f70 656e 0a20 202a base is open. * │ │ │ │ -002794d0: 202a 6e6f 7465 2063 6c6f 7365 3a20 636c *note close: cl │ │ │ │ -002794e0: 6f73 652c 202d 2d20 636c 6f73 6520 6120 ose, -- close a │ │ │ │ -002794f0: 6669 6c65 0a0a 5761 7973 2074 6f20 7573 file..Ways to us │ │ │ │ -00279500: 6520 7468 6973 206d 6574 686f 643a 0a3d e this method:.= │ │ │ │ -00279510: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00279520: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -00279530: 7465 2069 7349 6e70 7574 4669 6c65 2846 te isInputFile(F │ │ │ │ -00279540: 696c 6529 3a20 6973 496e 7075 7446 696c ile): isInputFil │ │ │ │ -00279550: 655f 6c70 4669 6c65 5f72 702c 202d 2d20 e_lpFile_rp, -- │ │ │ │ -00279560: 7768 6574 6865 7220 6120 6669 6c65 2069 whether a file i │ │ │ │ -00279570: 7320 6f70 656e 0a20 2020 2066 6f72 2069 s open. for i │ │ │ │ -00279580: 6e70 7574 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d nput.----------- │ │ │ │ -00279590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002795a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002792a0: 7c0a 7c6f 3720 3d20 6661 6c73 6520 2020 |.|o7 = false │ │ │ │ +002792b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002792c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +002792d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002792e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002792f0: 2d2d 2d2d 2b0a 7c69 3820 3a20 6973 4f70 ----+.|i8 : isOp │ │ │ │ +00279300: 656e 2066 2020 2020 2020 2020 2020 2020 en f │ │ │ │ +00279310: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00279320: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00279330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00279340: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20 |.|o8 = │ │ │ │ +00279350: 6661 6c73 6520 2020 2020 2020 2020 2020 false │ │ │ │ +00279360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00279370: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00279380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +002793a0: 3920 3a20 7265 6d6f 7665 4669 6c65 2022 9 : removeFile " │ │ │ │ +002793b0: 7465 7374 2d66 696c 6522 2020 2020 2020 test-file" │ │ │ │ +002793c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +002793d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002793e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002793f0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d +..See also.==== │ │ │ │ +00279400: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +00279410: 6f70 656e 496e 3a20 6f70 656e 496e 5f6c openIn: openIn_l │ │ │ │ +00279420: 7053 7472 696e 675f 7270 2c20 2d2d 206f pString_rp, -- o │ │ │ │ +00279430: 7065 6e20 616e 2069 6e70 7574 2066 696c pen an input fil │ │ │ │ +00279440: 650a 2020 2a20 2a6e 6f74 6520 6f70 656e e. * *note open │ │ │ │ +00279450: 496e 4f75 743a 206f 7065 6e49 6e4f 7574 InOut: openInOut │ │ │ │ +00279460: 2c20 2d2d 206f 7065 6e20 616e 2069 6e70 , -- open an inp │ │ │ │ +00279470: 7574 206f 7574 7075 7420 6669 6c65 0a20 ut output file. │ │ │ │ +00279480: 202a 202a 6e6f 7465 2067 6574 3a20 6765 * *note get: ge │ │ │ │ +00279490: 742c 202d 2d20 6765 7420 7468 6520 636f t, -- get the co │ │ │ │ +002794a0: 6e74 656e 7473 206f 6620 6120 6669 6c65 ntents of a file │ │ │ │ +002794b0: 0a20 202a 202a 6e6f 7465 2069 734f 7065 . * *note isOpe │ │ │ │ +002794c0: 6e3a 2069 734f 7065 6e2c 202d 2d20 7768 n: isOpen, -- wh │ │ │ │ +002794d0: 6574 6865 7220 6120 6669 6c65 206f 7220 ether a file or │ │ │ │ +002794e0: 6461 7461 6261 7365 2069 7320 6f70 656e database is open │ │ │ │ +002794f0: 0a20 202a 202a 6e6f 7465 2063 6c6f 7365 . * *note close │ │ │ │ +00279500: 3a20 636c 6f73 652c 202d 2d20 636c 6f73 : close, -- clos │ │ │ │ +00279510: 6520 6120 6669 6c65 0a0a 5761 7973 2074 e a file..Ways t │ │ │ │ +00279520: 6f20 7573 6520 7468 6973 206d 6574 686f o use this metho │ │ │ │ +00279530: 643a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d d:.============= │ │ │ │ +00279540: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00279550: 202a 6e6f 7465 2069 7349 6e70 7574 4669 *note isInputFi │ │ │ │ +00279560: 6c65 2846 696c 6529 3a20 6973 496e 7075 le(File): isInpu │ │ │ │ +00279570: 7446 696c 655f 6c70 4669 6c65 5f72 702c tFile_lpFile_rp, │ │ │ │ +00279580: 202d 2d20 7768 6574 6865 7220 6120 6669 -- whether a fi │ │ │ │ +00279590: 6c65 2069 7320 6f70 656e 0a20 2020 2066 le is open. f │ │ │ │ +002795a0: 6f72 2069 6e70 7574 0a2d 2d2d 2d2d 2d2d or input.------- │ │ │ │ 002795b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002795c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002795d0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -002795e0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -002795f0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00279600: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00279610: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00279620: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00279630: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -00279640: 6175 6c61 7932 446f 632f 6f76 5f73 7973 aulay2Doc/ov_sys │ │ │ │ -00279650: 7465 6d2e 6d32 3a35 3730 3a30 2e0a 1f0a tem.m2:570:0.... │ │ │ │ -00279660: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -00279670: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2069 oc.info, Node: i │ │ │ │ -00279680: 734c 6973 7465 6e65 725f 6c70 4669 6c65 sListener_lpFile │ │ │ │ -00279690: 5f72 702c 204e 6578 743a 206f 7065 6e46 _rp, Next: openF │ │ │ │ -002796a0: 696c 6573 2c20 5072 6576 3a20 6973 496e iles, Prev: isIn │ │ │ │ -002796b0: 7075 7446 696c 655f 6c70 4669 6c65 5f72 putFile_lpFile_r │ │ │ │ -002796c0: 702c 2055 703a 2075 7369 6e67 2073 6f63 p, Up: using soc │ │ │ │ -002796d0: 6b65 7473 0a0a 6973 4c69 7374 656e 6572 kets..isListener │ │ │ │ -002796e0: 2846 696c 6529 202d 2d20 7768 6574 6865 (File) -- whethe │ │ │ │ -002796f0: 7220 6120 6669 6c65 2069 7320 6f70 656e r a file is open │ │ │ │ -00279700: 2066 6f72 206c 6973 7465 6e69 6e67 0a2a for listening.* │ │ │ │ -00279710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00279720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00279730: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00279740: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2046 756e *******.. * Fun │ │ │ │ -00279750: 6374 696f 6e3a 202a 6e6f 7465 2069 734c ction: *note isL │ │ │ │ -00279760: 6973 7465 6e65 723a 2069 734c 6973 7465 istener: isListe │ │ │ │ -00279770: 6e65 725f 6c70 4669 6c65 5f72 702c 0a20 ner_lpFile_rp,. │ │ │ │ -00279780: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -00279790: 2020 2069 734c 6973 7465 6e65 7220 660a isListener f. │ │ │ │ -002797a0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -002797b0: 2020 2a20 662c 2061 202a 6e6f 7465 2066 * f, a *note f │ │ │ │ -002797c0: 696c 653a 2046 696c 652c 0a20 202a 204f ile: File,. * O │ │ │ │ -002797d0: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -002797e0: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20 a *note Boolean │ │ │ │ -002797f0: 7661 6c75 653a 2042 6f6f 6c65 616e 2c2c value: Boolean,, │ │ │ │ -00279800: 2020 7768 6574 6865 7220 6620 6973 2061 whether f is a │ │ │ │ -00279810: 6e20 6f70 656e 206c 6973 7465 6e65 720a n open listener. │ │ │ │ -00279820: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -00279830: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6f70 ==.. * *note op │ │ │ │ -00279840: 656e 4c69 7374 656e 6572 3a20 6f70 656e enListener: open │ │ │ │ -00279850: 4c69 7374 656e 6572 5f6c 7053 7472 696e Listener_lpStrin │ │ │ │ -00279860: 675f 7270 2c20 2d2d 206f 7065 6e20 6120 g_rp, -- open a │ │ │ │ -00279870: 706f 7274 2066 6f72 206c 6973 7465 6e69 port for listeni │ │ │ │ -00279880: 6e67 0a0a 5761 7973 2074 6f20 7573 6520 ng..Ways to use │ │ │ │ -00279890: 7468 6973 206d 6574 686f 643a 0a3d 3d3d this method:.=== │ │ │ │ -002798a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -002798b0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -002798c0: 2069 734c 6973 7465 6e65 7228 4669 6c65 isListener(File │ │ │ │ -002798d0: 293a 2069 734c 6973 7465 6e65 725f 6c70 ): isListener_lp │ │ │ │ -002798e0: 4669 6c65 5f72 702c 202d 2d20 7768 6574 File_rp, -- whet │ │ │ │ -002798f0: 6865 7220 6120 6669 6c65 2069 7320 6f70 her a file is op │ │ │ │ -00279900: 656e 2066 6f72 0a20 2020 206c 6973 7465 en for. liste │ │ │ │ -00279910: 6e69 6e67 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d ning.----------- │ │ │ │ -00279920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002795d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002795e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002795f0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00279600: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00279610: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00279620: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00279630: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00279640: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00279650: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00279660: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ +00279670: 5f73 7973 7465 6d2e 6d32 3a35 3730 3a30 _system.m2:570:0 │ │ │ │ +00279680: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ +00279690: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ +002796a0: 653a 2069 734c 6973 7465 6e65 725f 6c70 e: isListener_lp │ │ │ │ +002796b0: 4669 6c65 5f72 702c 204e 6578 743a 206f File_rp, Next: o │ │ │ │ +002796c0: 7065 6e46 696c 6573 2c20 5072 6576 3a20 penFiles, Prev: │ │ │ │ +002796d0: 6973 496e 7075 7446 696c 655f 6c70 4669 isInputFile_lpFi │ │ │ │ +002796e0: 6c65 5f72 702c 2055 703a 2075 7369 6e67 le_rp, Up: using │ │ │ │ +002796f0: 2073 6f63 6b65 7473 0a0a 6973 4c69 7374 sockets..isList │ │ │ │ +00279700: 656e 6572 2846 696c 6529 202d 2d20 7768 ener(File) -- wh │ │ │ │ +00279710: 6574 6865 7220 6120 6669 6c65 2069 7320 ether a file is │ │ │ │ +00279720: 6f70 656e 2066 6f72 206c 6973 7465 6e69 open for listeni │ │ │ │ +00279730: 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ng.************* │ │ │ │ +00279740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00279750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00279760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00279770: 2046 756e 6374 696f 6e3a 202a 6e6f 7465 Function: *note │ │ │ │ +00279780: 2069 734c 6973 7465 6e65 723a 2069 734c isListener: isL │ │ │ │ +00279790: 6973 7465 6e65 725f 6c70 4669 6c65 5f72 istener_lpFile_r │ │ │ │ +002797a0: 702c 0a20 202a 2055 7361 6765 3a20 0a20 p,. * Usage: . │ │ │ │ +002797b0: 2020 2020 2020 2069 734c 6973 7465 6e65 isListene │ │ │ │ +002797c0: 7220 660a 2020 2a20 496e 7075 7473 3a0a r f. * Inputs:. │ │ │ │ +002797d0: 2020 2020 2020 2a20 662c 2061 202a 6e6f * f, a *no │ │ │ │ +002797e0: 7465 2066 696c 653a 2046 696c 652c 0a20 te file: File,. │ │ │ │ +002797f0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +00279800: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c * a *note Bool │ │ │ │ +00279810: 6561 6e20 7661 6c75 653a 2042 6f6f 6c65 ean value: Boole │ │ │ │ +00279820: 616e 2c2c 2020 7768 6574 6865 7220 6620 an,, whether f │ │ │ │ +00279830: 6973 2061 6e20 6f70 656e 206c 6973 7465 is an open liste │ │ │ │ +00279840: 6e65 720a 0a53 6565 2061 6c73 6f0a 3d3d ner..See also.== │ │ │ │ +00279850: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +00279860: 6520 6f70 656e 4c69 7374 656e 6572 3a20 e openListener: │ │ │ │ +00279870: 6f70 656e 4c69 7374 656e 6572 5f6c 7053 openListener_lpS │ │ │ │ +00279880: 7472 696e 675f 7270 2c20 2d2d 206f 7065 tring_rp, -- ope │ │ │ │ +00279890: 6e20 6120 706f 7274 2066 6f72 206c 6973 n a port for lis │ │ │ │ +002798a0: 7465 6e69 6e67 0a0a 5761 7973 2074 6f20 tening..Ways to │ │ │ │ +002798b0: 7573 6520 7468 6973 206d 6574 686f 643a use this method: │ │ │ │ +002798c0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +002798d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a =========.. * * │ │ │ │ +002798e0: 6e6f 7465 2069 734c 6973 7465 6e65 7228 note isListener( │ │ │ │ +002798f0: 4669 6c65 293a 2069 734c 6973 7465 6e65 File): isListene │ │ │ │ +00279900: 725f 6c70 4669 6c65 5f72 702c 202d 2d20 r_lpFile_rp, -- │ │ │ │ +00279910: 7768 6574 6865 7220 6120 6669 6c65 2069 whether a file i │ │ │ │ +00279920: 7320 6f70 656e 2066 6f72 0a20 2020 206c s open for. l │ │ │ │ +00279930: 6973 7465 6e69 6e67 0a2d 2d2d 2d2d 2d2d istening.------- │ │ │ │ 00279940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00279950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279960: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00279970: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00279980: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00279990: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -002799a0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -002799b0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -002799c0: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -002799d0: 6175 6c61 7932 446f 632f 6f76 5f73 7973 aulay2Doc/ov_sys │ │ │ │ -002799e0: 7465 6d2e 6d32 3a36 3232 3a30 2e0a 1f0a tem.m2:622:0.... │ │ │ │ -002799f0: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -00279a00: 6f63 2e69 6e66 6f2c 204e 6f64 653a 206f oc.info, Node: o │ │ │ │ -00279a10: 7065 6e46 696c 6573 2c20 4e65 7874 3a20 penFiles, Next: │ │ │ │ -00279a20: 636f 6e6e 6563 7469 6f6e 436f 756e 742c connectionCount, │ │ │ │ -00279a30: 2050 7265 763a 2069 734c 6973 7465 6e65 Prev: isListene │ │ │ │ -00279a40: 725f 6c70 4669 6c65 5f72 702c 2055 703a r_lpFile_rp, Up: │ │ │ │ -00279a50: 2075 7369 6e67 2073 6f63 6b65 7473 0a0a using sockets.. │ │ │ │ -00279a60: 6f70 656e 4669 6c65 7320 2d2d 206c 6973 openFiles -- lis │ │ │ │ -00279a70: 7420 7468 6520 6f70 656e 2066 696c 6573 t the open files │ │ │ │ -00279a80: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -00279a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00279aa0: 2a0a 0a44 6573 6372 6970 7469 6f6e 0a3d *..Description.= │ │ │ │ -00279ab0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 6f70 656e ==========..open │ │ │ │ -00279ac0: 4669 6c65 7328 2920 7072 6f64 7563 6573 Files() produces │ │ │ │ -00279ad0: 2061 206c 6973 7420 6f66 2061 6c6c 2063 a list of all c │ │ │ │ -00279ae0: 7572 7265 6e74 6c79 206f 7065 6e20 6669 urrently open fi │ │ │ │ -00279af0: 6c65 732e 0a0a 0a53 6565 2061 6c73 6f0a les....See also. │ │ │ │ -00279b00: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ -00279b10: 6f74 6520 4669 6c65 3a20 4669 6c65 2c20 ote File: File, │ │ │ │ -00279b20: 2d2d 2074 6865 2063 6c61 7373 206f 6620 -- the class of │ │ │ │ -00279b30: 616c 6c20 6669 6c65 730a 0a46 6f72 2074 all files..For t │ │ │ │ -00279b40: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00279b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00279b60: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00279b70: 7465 206f 7065 6e46 696c 6573 3a20 6f70 te openFiles: op │ │ │ │ -00279b80: 656e 4669 6c65 732c 2069 7320 6120 2a6e enFiles, is a *n │ │ │ │ -00279b90: 6f74 6520 636f 6d70 696c 6564 2066 756e ote compiled fun │ │ │ │ -00279ba0: 6374 696f 6e3a 0a43 6f6d 7069 6c65 6446 ction:.CompiledF │ │ │ │ -00279bb0: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ -00279bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279980: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00279990: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +002799a0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +002799b0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +002799c0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +002799d0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +002799e0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +002799f0: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ +00279a00: 5f73 7973 7465 6d2e 6d32 3a36 3232 3a30 _system.m2:622:0 │ │ │ │ +00279a10: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ +00279a20: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ +00279a30: 653a 206f 7065 6e46 696c 6573 2c20 4e65 e: openFiles, Ne │ │ │ │ +00279a40: 7874 3a20 636f 6e6e 6563 7469 6f6e 436f xt: connectionCo │ │ │ │ +00279a50: 756e 742c 2050 7265 763a 2069 734c 6973 unt, Prev: isLis │ │ │ │ +00279a60: 7465 6e65 725f 6c70 4669 6c65 5f72 702c tener_lpFile_rp, │ │ │ │ +00279a70: 2055 703a 2075 7369 6e67 2073 6f63 6b65 Up: using socke │ │ │ │ +00279a80: 7473 0a0a 6f70 656e 4669 6c65 7320 2d2d ts..openFiles -- │ │ │ │ +00279a90: 206c 6973 7420 7468 6520 6f70 656e 2066 list the open f │ │ │ │ +00279aa0: 696c 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a iles.*********** │ │ │ │ +00279ab0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00279ac0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ +00279ad0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +00279ae0: 6f70 656e 4669 6c65 7328 2920 7072 6f64 openFiles() prod │ │ │ │ +00279af0: 7563 6573 2061 206c 6973 7420 6f66 2061 uces a list of a │ │ │ │ +00279b00: 6c6c 2063 7572 7265 6e74 6c79 206f 7065 ll currently ope │ │ │ │ +00279b10: 6e20 6669 6c65 732e 0a0a 0a53 6565 2061 n files....See a │ │ │ │ +00279b20: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +00279b30: 2a20 2a6e 6f74 6520 4669 6c65 3a20 4669 * *note File: Fi │ │ │ │ +00279b40: 6c65 2c20 2d2d 2074 6865 2063 6c61 7373 le, -- the class │ │ │ │ +00279b50: 206f 6620 616c 6c20 6669 6c65 730a 0a46 of all files..F │ │ │ │ +00279b60: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +00279b70: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00279b80: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00279b90: 202a 6e6f 7465 206f 7065 6e46 696c 6573 *note openFiles │ │ │ │ +00279ba0: 3a20 6f70 656e 4669 6c65 732c 2069 7320 : openFiles, is │ │ │ │ +00279bb0: 6120 2a6e 6f74 6520 636f 6d70 696c 6564 a *note compiled │ │ │ │ +00279bc0: 2066 756e 6374 696f 6e3a 0a43 6f6d 7069 function:.Compi │ │ │ │ +00279bd0: 6c65 6446 756e 6374 696f 6e2c 2e0a 0a2d ledFunction,...- │ │ │ │ 00279be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00279bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279c00: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -00279c10: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00279c20: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00279c30: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00279c40: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00279c50: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -00279c60: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00279c70: 732f 0a4d 6163 6175 6c61 7932 446f 632f s/.Macaulay2Doc/ │ │ │ │ -00279c80: 6f76 5f73 7973 7465 6d2e 6d32 3a33 3638 ov_system.m2:368 │ │ │ │ -00279c90: 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 6361 :0....File: Maca │ │ │ │ -00279ca0: 756c 6179 3244 6f63 2e69 6e66 6f2c 204e ulay2Doc.info, N │ │ │ │ -00279cb0: 6f64 653a 2063 6f6e 6e65 6374 696f 6e43 ode: connectionC │ │ │ │ -00279cc0: 6f75 6e74 2c20 5072 6576 3a20 6f70 656e ount, Prev: open │ │ │ │ -00279cd0: 4669 6c65 732c 2055 703a 2075 7369 6e67 Files, Up: using │ │ │ │ -00279ce0: 2073 6f63 6b65 7473 0a0a 636f 6e6e 6563 sockets..connec │ │ │ │ -00279cf0: 7469 6f6e 436f 756e 7420 2d2d 2074 6865 tionCount -- the │ │ │ │ -00279d00: 206e 756d 6265 7220 6f66 2063 6f6e 6e65 number of conne │ │ │ │ -00279d10: 6374 696f 6e73 0a2a 2a2a 2a2a 2a2a 2a2a ctions.********* │ │ │ │ -00279d20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00279d30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00279d40: 2a2a 2a0a 0a44 6573 6372 6970 7469 6f6e ***..Description │ │ │ │ -00279d50: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 636f .===========..co │ │ │ │ -00279d60: 6e6e 6563 7469 6f6e 436f 756e 7420 6620 nnectionCount f │ │ │ │ -00279d70: 2d2d 2072 6574 7572 6e73 2074 6865 206e -- returns the n │ │ │ │ -00279d80: 756d 6265 7220 6f66 2063 6f6e 6e65 6374 umber of connect │ │ │ │ -00279d90: 696f 6e73 2061 6363 6570 7465 6420 6279 ions accepted by │ │ │ │ -00279da0: 2074 6865 206c 6973 7465 6e65 720a 6620 the listener.f │ │ │ │ -00279db0: 736f 2066 6172 2e0a 0a46 6f72 2074 6865 so far...For the │ │ │ │ -00279dc0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00279dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00279de0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00279df0: 2063 6f6e 6e65 6374 696f 6e43 6f75 6e74 connectionCount │ │ │ │ -00279e00: 3a20 636f 6e6e 6563 7469 6f6e 436f 756e : connectionCoun │ │ │ │ -00279e10: 742c 2069 7320 6120 2a6e 6f74 6520 636f t, is a *note co │ │ │ │ -00279e20: 6d70 696c 6564 0a66 756e 6374 696f 6e3a mpiled.function: │ │ │ │ -00279e30: 2043 6f6d 7069 6c65 6446 756e 6374 696f CompiledFunctio │ │ │ │ -00279e40: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ -00279e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00279e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00279c30: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00279c40: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00279c50: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +00279c60: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +00279c70: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +00279c80: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +00279c90: 6b61 6765 732f 0a4d 6163 6175 6c61 7932 kages/.Macaulay2 │ │ │ │ +00279ca0: 446f 632f 6f76 5f73 7973 7465 6d2e 6d32 Doc/ov_system.m2 │ │ │ │ +00279cb0: 3a33 3638 3a30 2e0a 1f0a 4669 6c65 3a20 :368:0....File: │ │ │ │ +00279cc0: 4d61 6361 756c 6179 3244 6f63 2e69 6e66 Macaulay2Doc.inf │ │ │ │ +00279cd0: 6f2c 204e 6f64 653a 2063 6f6e 6e65 6374 o, Node: connect │ │ │ │ +00279ce0: 696f 6e43 6f75 6e74 2c20 5072 6576 3a20 ionCount, 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2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00279ea0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00279eb0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00279ec0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00279ed0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00279ee0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00279ef0: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -00279f00: 6175 6c61 7932 446f 632f 6f76 5f73 7973 aulay2Doc/ov_sys │ │ │ │ -00279f10: 7465 6d2e 6d32 3a31 3033 393a 302e 0a1f tem.m2:1039:0... │ │ │ │ -00279f20: 0a46 696c 653a 204d 6163 6175 6c61 7932 .File: Macaulay2 │ │ │ │ -00279f30: 446f 632e 696e 666f 2c20 4e6f 6465 3a20 Doc.info, Node: │ │ │ │ -00279f40: 6e6f 7469 6679 2c20 4e65 7874 3a20 6175 notify, Next: au │ │ │ │ -00279f50: 746f 6c6f 6164 2c20 5072 6576 3a20 7573 toload, Prev: us │ │ │ │ -00279f60: 696e 6720 736f 636b 6574 732c 2055 703a ing sockets, Up: │ │ │ │ -00279f70: 2073 7973 7465 6d20 6661 6369 6c69 7469 system faciliti │ │ │ │ -00279f80: 6573 0a0a 6e6f 7469 6679 202d 2d20 7768 es..notify -- wh │ │ │ │ -00279f90: 6574 6865 7220 746f 206e 6f74 6966 7920 ether to notify │ │ │ │ -00279fa0: 7468 6520 7573 6572 2077 6865 6e20 6120 the user when a │ │ │ │ -00279fb0: 6669 6c65 2069 7320 6c6f 6164 6564 0a2a file is loaded.* │ │ │ │ -00279fc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00279fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00279fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00279ff0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0027a000: 7361 6765 3a20 0a20 2020 2020 2020 206e sage: . n │ │ │ │ -0027a010: 6f74 6966 7920 3d20 7472 7565 0a20 2020 otify = true. │ │ │ │ -0027a020: 2020 2020 206e 6f74 6966 7920 3d20 6661 notify = fa │ │ │ │ -0027a030: 6c73 650a 2020 2a20 436f 6e73 6571 7565 lse. * Conseque │ │ │ │ -0027a040: 6e63 6573 3a0a 2020 2020 2020 2a20 4966 nces:. * If │ │ │ │ -0027a050: 202a 6e6f 7465 206e 6f74 6966 793a 206e *note notify: n │ │ │ │ -0027a060: 6f74 6966 792c 2069 7320 7365 7420 746f otify, is set to │ │ │ │ -0027a070: 202a 6e6f 7465 2074 7275 653a 2074 7275 *note true: tru │ │ │ │ -0027a080: 652c 2c20 7468 656e 2065 6163 6820 7469 e,, then each ti │ │ │ │ -0027a090: 6d65 2061 0a20 2020 2020 2020 2066 696c me a. fil │ │ │ │ -0027a0a0: 6520 6f72 2061 2070 6163 6b61 6765 2069 e or a package i │ │ │ │ -0027a0b0: 7320 6c6f 6164 6564 2c20 6120 6d65 7373 s loaded, a mess │ │ │ │ -0027a0c0: 6167 6520 7769 6c6c 2062 6520 6469 7370 age will be disp │ │ │ │ -0027a0d0: 6c61 7965 642e 0a0a 4465 7363 7269 7074 layed...Descript │ │ │ │ -0027a0e0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -0027a0f0: 0a2b 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 ---------------- │ │ │ │ +00279e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00279eb0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00279ec0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00279ed0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00279ee0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00279ef0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00279f00: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00279f10: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00279f20: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ +00279f30: 5f73 7973 7465 6d2e 6d32 3a31 3033 393a _system.m2:1039: │ │ │ │ +00279f40: 302e 0a1f 0a46 696c 653a 204d 6163 6175 0....File: Macau │ │ │ │ +00279f50: 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6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ +0027a030: 2020 206e 6f74 6966 7920 3d20 7472 7565 notify = true │ │ │ │ +0027a040: 0a20 2020 2020 2020 206e 6f74 6966 7920 . notify │ │ │ │ +0027a050: 3d20 6661 6c73 650a 2020 2a20 436f 6e73 = false. * Cons │ │ │ │ +0027a060: 6571 7565 6e63 6573 3a0a 2020 2020 2020 equences:. │ │ │ │ +0027a070: 2a20 4966 202a 6e6f 7465 206e 6f74 6966 * If *note notif │ │ │ │ +0027a080: 793a 206e 6f74 6966 792c 2069 7320 7365 y: notify, is se │ │ │ │ +0027a090: 7420 746f 202a 6e6f 7465 2074 7275 653a t to *note true: │ │ │ │ +0027a0a0: 2074 7275 652c 2c20 7468 656e 2065 6163 true,, then eac │ │ │ │ +0027a0b0: 6820 7469 6d65 2061 0a20 2020 2020 2020 h time a. │ │ │ │ +0027a0c0: 2066 696c 6520 6f72 2061 2070 6163 6b61 file or a packa │ │ │ │ +0027a0d0: 6765 2069 7320 6c6f 6164 6564 2c20 6120 ge is loaded, a │ │ │ │ +0027a0e0: 6d65 7373 6167 6520 7769 6c6c 2062 6520 message will be │ │ │ │ +0027a0f0: 6469 7370 6c61 7965 642e 0a0a 4465 7363 displayed...Desc │ │ │ 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2020 7c0a 2b2d |.+- │ │ │ │ -0027aea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027aeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027aec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027ae90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027aea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027aeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027aec0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0027aed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027aee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0027aef0: 3320 3a20 6175 746f 6c6f 6164 2866 2c66 3 : autoload(f,f │ │ │ │ -0027af00: 6e29 2020 2020 2020 2020 2020 2020 2020 n) │ │ │ │ -0027af10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027af20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027af30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0027af40: 2d2d 2d2d 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|.|o │ │ │ │ -0027b030: 3420 3d20 2f75 7372 2f73 6861 7265 2f4d 4 = /usr/share/M │ │ │ │ -0027b040: 6163 6175 6c61 7932 2f43 6f72 652f 6175 acaulay2/Core/au │ │ │ │ -0027b050: 746f 6c6f 6164 2e6d 323a 383a 3139 2d31 toload.m2:8:19-1 │ │ │ │ -0027b060: 323a 3232 3a20 2d2d 736f 7572 6365 2063 2:22: --source c │ │ │ │ -0027b070: 6f64 653a 2020 2020 2020 2020 7c0a 7c20 ode: |.| │ │ │ │ -0027b080: 2020 2020 2020 2020 2073 796d 203c 2d20 sym <- │ │ │ │ -0027b090: 6620 3a3d 2078 202d 3e20 2820 2020 2020 f := x -> ( │ │ │ │ -0027b0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0027b0d0: 2020 2020 2020 2020 2020 2020 2020 6c6f lo │ │ │ │ -0027b0e0: 6164 2066 696c 656e 616d 653b 2020 2020 ad filename; │ │ │ │ -0027b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b110: 2020 2020 2020 2020 2020 2020 7c0a 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│ 0027b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b520: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0027b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027b550: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0027b560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b570: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0027b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027b5a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0027b5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b5c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0027b5c0: 2020 2020 2020 2020 2020 2020 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2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0027b8f0: 3620 3a20 7265 6d6f 7665 4669 6c65 2066 6 : removeFile f │ │ │ │ -0027b900: 6e20 2020 2020 2020 2020 2020 2020 2020 n │ │ │ │ -0027b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027b930: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0027b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027b8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027b900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027b910: 2b0a 7c69 3620 3a20 7265 6d6f 7665 4669 +.|i6 : removeFi │ │ │ │ +0027b920: 6c65 2066 6e20 2020 2020 2020 2020 2020 le fn │ │ │ │ +0027b930: 2020 2020 2020 2020 2020 2020 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6f61 643a 2061 7574 te autoload: aut │ │ │ │ -0027ba20: 6f6c 6f61 642c 2069 7320 6120 2a6e 6f74 oload, is a *not │ │ │ │ -0027ba30: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ -0027ba40: 6e3a 0a4d 6574 686f 6446 756e 6374 696f n:.MethodFunctio │ │ │ │ -0027ba50: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ -0027ba60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027ba70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027b980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027b990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027b9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027b9b0: 2b0a 0a57 6179 7320 746f 2075 7365 2061 +..Ways to use a │ │ │ │ +0027b9c0: 7574 6f6c 6f61 643a 0a3d 3d3d 3d3d 3d3d utoload:.======= │ │ │ │ +0027b9d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0027b9e0: 2020 2a20 2261 7574 6f6c 6f61 6428 5379 * "autoload(Sy │ │ │ │ +0027b9f0: 6d62 6f6c 2c53 7472 696e 6729 220a 0a46 mbol,String)"..F │ │ │ │ +0027ba00: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0027ba10: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0027ba20: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0027ba30: 202a 6e6f 7465 2061 7574 6f6c 6f61 643a *note autoload: │ │ │ │ +0027ba40: 2061 7574 6f6c 6f61 642c 2069 7320 6120 autoload, is a │ │ │ │ +0027ba50: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ +0027ba60: 6374 696f 6e3a 0a4d 6574 686f 6446 756e ction:.MethodFun │ │ │ │ +0027ba70: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ 0027ba80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027baa0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -0027bab0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -0027bac0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -0027bad0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -0027bae0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -0027baf0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -0027bb00: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -0027bb10: 6175 6c61 7932 446f 632f 6f76 5f73 7973 aulay2Doc/ov_sys │ │ │ │ -0027bb20: 7465 6d2e 6d32 3a31 3338 3a30 2e0a 1f0a tem.m2:138:0.... │ │ │ │ -0027bb30: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -0027bb40: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2069 oc.info, Node: i │ │ │ │ -0027bb50: 6e69 7469 616c 697a 6174 696f 6e20 6669 nitialization fi │ │ │ │ -0027bb60: 6c65 2c20 4e65 7874 3a20 696e 7075 742c le, Next: input, │ │ │ │ -0027bb70: 2050 7265 763a 2061 7574 6f6c 6f61 642c Prev: autoload, │ │ │ │ -0027bb80: 2055 703a 2073 7973 7465 6d20 6661 6369 Up: system faci │ │ │ │ -0027bb90: 6c69 7469 6573 0a0a 696e 6974 6961 6c69 lities..initiali │ │ │ │ -0027bba0: 7a61 7469 6f6e 2066 696c 650a 2a2a 2a2a zation file.**** │ │ │ │ -0027bbb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -0027bbc0: 0a54 6865 2066 696c 6520 696e 6974 2e6d .The file init.m │ │ │ │ -0027bbd0: 3220 6973 206c 6f61 6465 6420 6175 746f 2 is loaded auto │ │ │ │ -0027bbe0: 6d61 7469 6361 6c6c 7920 7768 656e 2074 matically when t │ │ │ │ -0027bbf0: 6865 2070 726f 6772 616d 2069 7320 7374 he program is st │ │ │ │ -0027bc00: 6172 7465 642c 2069 6620 6974 0a65 7869 arted, if it.exi │ │ │ │ -0027bc10: 7374 732e 0a0a 4f6e 206d 6f73 7420 7379 sts...On most sy │ │ │ │ -0027bc20: 7374 656d 7320 7468 6520 6669 6c65 2069 stems the file i │ │ │ │ -0027bc30: 7320 736f 7567 6874 2069 6e20 7468 6520 s sought in the │ │ │ │ -0027bc40: 6469 7265 6374 6f72 7920 2448 4f4d 452f directory $HOME/ │ │ │ │ -0027bc50: 2e4d 6163 6175 6c61 7932 2f2c 2077 6865 .Macaulay2/, whe │ │ │ │ -0027bc60: 7265 0a24 484f 4d45 2069 7320 7265 706c re.$HOME is repl │ │ │ │ -0027bc70: 6163 6564 2062 7920 7468 6520 7061 7468 aced by the path │ │ │ │ -0027bc80: 2074 6f20 7468 6520 7573 6572 2773 2068 to the user's h │ │ │ │ -0027bc90: 6f6d 6520 6469 7265 6374 6f72 792e 0a0a ome directory... │ │ │ │ -0027bca0: 556e 6465 7220 4d61 6320 4f53 2058 2c20 Under Mac OS X, │ │ │ │ -0027bcb0: 7468 6520 6669 6c65 2069 7320 736f 7567 the file is soug │ │ │ │ -0027bcc0: 6874 2069 6e73 7465 6164 2069 6e20 7468 ht instead in th │ │ │ │ -0027bcd0: 6520 6469 7265 6374 6f72 790a 2448 4f4d e directory.$HOM │ │ │ │ -0027bce0: 452f 4c69 6272 6172 792f 4170 706c 6963 E/Library/Applic │ │ │ │ -0027bcf0: 6174 696f 6e20 5375 7070 6f72 742f 4d61 ation Support/Ma │ │ │ │ -0027bd00: 6361 756c 6179 322f 2e0a 0a49 6620 7468 caulay2/...If th │ │ │ │ -0027bd10: 6520 7573 6572 2077 616e 7473 2061 2066 e user wants a f │ │ │ │ -0027bd20: 696c 6520 6361 6c6c 6564 2c20 7361 792c ile called, say, │ │ │ │ -0027bd30: 2073 7461 7274 2e6d 3220 696e 2074 6865 start.m2 in the │ │ │ │ -0027bd40: 2063 7572 7265 6e74 2064 6972 6563 746f current directo │ │ │ │ -0027bd50: 7279 2074 6f20 6265 0a6c 6f61 6465 6420 ry to be.loaded │ │ │ │ -0027bd60: 6175 746f 6d61 7469 6361 6c6c 7920 7768 automatically wh │ │ │ │ -0027bd70: 656e 2074 6865 2070 726f 6772 616d 2069 en the program i │ │ │ │ -0027bd80: 7320 7374 6172 7465 642c 2074 6865 6e20 s started, then │ │ │ │ -0027bd90: 7468 6520 666f 6c6c 6f77 696e 6720 6c69 the following li │ │ │ │ -0027bda0: 6e65 206f 660a 636f 6465 2063 616e 2062 ne of.code can b │ │ │ │ -0027bdb0: 6520 706c 6163 6564 2069 6e20 7468 6520 e placed in the │ │ │ │ -0027bdc0: 6669 6c65 2069 6e69 742e 6d32 2e0a 0a69 file init.m2...i │ │ │ │ -0027bdd0: 6620 6669 6c65 4578 6973 7473 2022 7374 f fileExists "st │ │ │ │ -0027bde0: 6172 742e 6d32 2220 7468 656e 206c 6f61 art.m2" then loa │ │ │ │ -0027bdf0: 6428 6375 7272 656e 7444 6972 6563 746f d(currentDirecto │ │ │ │ -0027be00: 7279 2829 7c22 7374 6172 742e 6d32 2229 ry()|"start.m2") │ │ │ │ -0027be10: 0a0a 5761 726e 696e 673a 2066 6f72 6d65 ..Warning: forme │ │ │ │ -0027be20: 7220 7665 7273 696f 6e73 206f 6620 7468 r versions of th │ │ │ │ -0027be30: 6520 7072 6f67 7261 6d20 776f 756c 6420 e program would │ │ │ │ -0027be40: 616c 736f 206c 6f61 6420 6120 6669 6c65 also load a file │ │ │ │ -0027be50: 206e 616d 6564 2069 6e69 742e 6d32 0a66 named init.m2.f │ │ │ │ -0027be60: 6f75 6e64 2069 6e20 7468 6520 6375 7272 ound in the curr │ │ │ │ -0027be70: 656e 7420 6469 7265 6374 6f72 792e 0a0a ent directory... │ │ │ │ -0027be80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027be90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027bea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027bac0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0027bad0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0027bae0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0027baf0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0027bb00: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0027bb10: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +0027bb20: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0027bb30: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ +0027bb40: 5f73 7973 7465 6d2e 6d32 3a31 3338 3a30 _system.m2:138:0 │ │ │ │ +0027bb50: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ +0027bb60: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ +0027bb70: 653a 2069 6e69 7469 616c 697a 6174 696f e: initializatio │ │ │ │ +0027bb80: 6e20 6669 6c65 2c20 4e65 7874 3a20 696e n file, Next: in │ │ │ │ +0027bb90: 7075 742c 2050 7265 763a 2061 7574 6f6c put, Prev: autol │ │ │ │ +0027bba0: 6f61 642c 2055 703a 2073 7973 7465 6d20 oad, Up: system │ │ │ │ +0027bbb0: 6661 6369 6c69 7469 6573 0a0a 696e 6974 facilities..init │ │ │ │ +0027bbc0: 6961 6c69 7a61 7469 6f6e 2066 696c 650a ialization file. │ │ │ │ +0027bbd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0027bbe0: 2a2a 2a0a 0a54 6865 2066 696c 6520 696e ***..The file in │ │ │ │ +0027bbf0: 6974 2e6d 3220 6973 206c 6f61 6465 6420 it.m2 is loaded │ │ │ │ +0027bc00: 6175 746f 6d61 7469 6361 6c6c 7920 7768 automatically wh │ │ │ │ +0027bc10: 656e 2074 6865 2070 726f 6772 616d 2069 en the program i │ │ │ │ +0027bc20: 7320 7374 6172 7465 642c 2069 6620 6974 s started, if it │ │ │ │ +0027bc30: 0a65 7869 7374 732e 0a0a 4f6e 206d 6f73 .exists...On mos │ │ │ │ +0027bc40: 7420 7379 7374 656d 7320 7468 6520 6669 t systems the fi │ │ │ │ +0027bc50: 6c65 2069 7320 736f 7567 6874 2069 6e20 le is sought in │ │ │ │ +0027bc60: 7468 6520 6469 7265 6374 6f72 7920 2448 the directory $H │ │ │ │ +0027bc70: 4f4d 452f 2e4d 6163 6175 6c61 7932 2f2c OME/.Macaulay2/, │ │ │ │ +0027bc80: 2077 6865 7265 0a24 484f 4d45 2069 7320 where.$HOME is │ │ │ │ +0027bc90: 7265 706c 6163 6564 2062 7920 7468 6520 replaced by the │ │ │ │ +0027bca0: 7061 7468 2074 6f20 7468 6520 7573 6572 path to the user │ │ │ │ +0027bcb0: 2773 2068 6f6d 6520 6469 7265 6374 6f72 's home director │ │ │ │ +0027bcc0: 792e 0a0a 556e 6465 7220 4d61 6320 4f53 y...Under Mac OS │ │ │ │ +0027bcd0: 2058 2c20 7468 6520 6669 6c65 2069 7320 X, the file is │ │ │ │ +0027bce0: 736f 7567 6874 2069 6e73 7465 6164 2069 sought instead i │ │ │ │ +0027bcf0: 6e20 7468 6520 6469 7265 6374 6f72 790a n the directory. │ │ │ │ +0027bd00: 2448 4f4d 452f 4c69 6272 6172 792f 4170 $HOME/Library/Ap │ │ │ │ +0027bd10: 706c 6963 6174 696f 6e20 5375 7070 6f72 plication Suppor │ │ │ │ +0027bd20: 742f 4d61 6361 756c 6179 322f 2e0a 0a49 t/Macaulay2/...I │ │ │ │ +0027bd30: 6620 7468 6520 7573 6572 2077 616e 7473 f the user wants │ │ │ │ +0027bd40: 2061 2066 696c 6520 6361 6c6c 6564 2c20 a file called, │ │ │ │ +0027bd50: 7361 792c 2073 7461 7274 2e6d 3220 696e say, start.m2 in │ │ │ │ +0027bd60: 2074 6865 2063 7572 7265 6e74 2064 6972 the current dir │ │ │ │ +0027bd70: 6563 746f 7279 2074 6f20 6265 0a6c 6f61 ectory to be.loa │ │ │ │ +0027bd80: 6465 6420 6175 746f 6d61 7469 6361 6c6c ded automaticall │ │ │ │ +0027bd90: 7920 7768 656e 2074 6865 2070 726f 6772 y when the progr │ │ │ │ +0027bda0: 616d 2069 7320 7374 6172 7465 642c 2074 am is started, t │ │ │ │ +0027bdb0: 6865 6e20 7468 6520 666f 6c6c 6f77 696e hen the followin │ │ │ │ +0027bdc0: 6720 6c69 6e65 206f 660a 636f 6465 2063 g line of.code c │ │ │ │ +0027bdd0: 616e 2062 6520 706c 6163 6564 2069 6e20 an be placed in │ │ │ │ +0027bde0: 7468 6520 6669 6c65 2069 6e69 742e 6d32 the file init.m2 │ │ │ │ +0027bdf0: 2e0a 0a69 6620 6669 6c65 4578 6973 7473 ...if fileExists │ │ │ │ +0027be00: 2022 7374 6172 742e 6d32 2220 7468 656e "start.m2" then │ │ │ │ +0027be10: 206c 6f61 6428 6375 7272 656e 7444 6972 load(currentDir │ │ │ │ +0027be20: 6563 746f 7279 2829 7c22 7374 6172 742e ectory()|"start. │ │ │ │ +0027be30: 6d32 2229 0a0a 5761 726e 696e 673a 2066 m2")..Warning: f │ │ │ │ +0027be40: 6f72 6d65 7220 7665 7273 696f 6e73 206f ormer versions o │ │ │ │ +0027be50: 6620 7468 6520 7072 6f67 7261 6d20 776f f the program wo │ │ │ │ +0027be60: 756c 6420 616c 736f 206c 6f61 6420 6120 uld also load a │ │ │ │ +0027be70: 6669 6c65 206e 616d 6564 2069 6e69 742e file named init. │ │ │ │ +0027be80: 6d32 0a66 6f75 6e64 2069 6e20 7468 6520 m2.found in the │ │ │ │ +0027be90: 6375 7272 656e 7420 6469 7265 6374 6f72 current director │ │ │ │ +0027bea0: 792e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d y...------------ │ │ │ │ 0027beb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0027bed0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0027bee0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0027bef0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0027bf00: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -0027bf10: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -0027bf20: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -0027bf30: 636b 6167 6573 2f0a 4d61 6361 756c 6179 ckages/.Macaulay │ │ │ │ -0027bf40: 3244 6f63 2f6f 765f 7379 7374 656d 2e6d 2Doc/ov_system.m │ │ │ │ -0027bf50: 323a 3136 333a 302e 0a1f 0a46 696c 653a 2:163:0....File: │ │ │ │ -0027bf60: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ -0027bf70: 666f 2c20 4e6f 6465 3a20 696e 7075 742c fo, Node: input, │ │ │ │ -0027bf80: 204e 6578 743a 206c 6f61 642c 2050 7265 Next: load, Pre │ │ │ │ -0027bf90: 763a 2069 6e69 7469 616c 697a 6174 696f v: initializatio │ │ │ │ -0027bfa0: 6e20 6669 6c65 2c20 5570 3a20 7379 7374 n file, Up: syst │ │ │ │ -0027bfb0: 656d 2066 6163 696c 6974 6965 730a 0a69 em facilities..i │ │ │ │ -0027bfc0: 6e70 7574 202d 2d20 7265 6164 204d 6163 nput -- read Mac │ │ │ │ -0027bfd0: 6175 6c61 7932 2063 6f6d 6d61 6e64 7320 aulay2 commands │ │ │ │ -0027bfe0: 616e 6420 6563 686f 0a2a 2a2a 2a2a 2a2a and echo.******* │ │ │ │ -0027bff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0027c000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0027c010: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ -0027c020: 2020 2020 2020 2020 696e 7075 7420 666e input fn │ │ │ │ -0027c030: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -0027c040: 2020 202a 2066 6e2c 2061 202a 6e6f 7465 * fn, a *note │ │ │ │ -0027c050: 2073 7472 696e 673a 2053 7472 696e 672c string: String, │ │ │ │ -0027c060: 0a20 202a 2043 6f6e 7365 7175 656e 6365 . * Consequence │ │ │ │ -0027c070: 733a 0a20 2020 2020 202a 2072 6561 6473 s:. * reads │ │ │ │ -0027c080: 2061 6e64 2065 7865 6375 7465 7320 7468 and executes th │ │ │ │ -0027c090: 6520 636f 6d6d 616e 6473 2066 6f75 6e64 e commands found │ │ │ │ -0027c0a0: 2069 6e20 7468 6520 6669 6c65 2077 686f in the file who │ │ │ │ -0027c0b0: 7365 206e 616d 6520 6973 0a20 2020 2020 se name is. │ │ │ │ -0027c0c0: 2020 2063 6f6e 7461 696e 6564 2069 6e20 contained in │ │ │ │ -0027c0d0: 7468 6520 7374 7269 6e67 2066 6e2c 2065 the string fn, e │ │ │ │ -0027c0e0: 6368 6f69 6e67 2074 6865 2069 6e70 7574 choing the input │ │ │ │ -0027c0f0: 2c20 7072 696e 7469 6e67 2074 6865 2076 , printing the v │ │ │ │ -0027c100: 616c 7565 732c 2061 6e64 0a20 2020 2020 alues, and. │ │ │ │ -0027c110: 2020 2069 6e63 7265 6d65 6e74 696e 6720 incrementing │ │ │ │ -0027c120: 7468 6520 6c69 6e65 206e 756d 6265 720a the line number. │ │ │ │ -0027c130: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ -0027c140: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6669 ========..The fi │ │ │ │ -0027c150: 6c65 2069 7320 736f 7567 6874 2069 6e20 le is sought in │ │ │ │ -0027c160: 7468 6520 6469 7265 6374 6f72 7920 636f the directory co │ │ │ │ -0027c170: 6e74 6169 6e69 6e67 2074 6865 2066 696c ntaining the fil │ │ │ │ -0027c180: 6520 6375 7272 656e 746c 7920 6265 696e e currently bein │ │ │ │ -0027c190: 6720 6c6f 6164 6564 2c0a 6966 2061 6e79 g loaded,.if any │ │ │ │ -0027c1a0: 2c20 616e 6420 7468 656e 2061 6c6f 6e67 , and then along │ │ │ │ -0027c1b0: 2074 6865 202a 6e6f 7465 2070 6174 683a the *note path: │ │ │ │ -0027c1c0: 2070 6174 682c 2c20 756e 6c65 7373 2074 path,, unless t │ │ │ │ -0027c1d0: 6865 206e 616d 6520 6f66 2074 6865 2066 he name of the f │ │ │ │ -0027c1e0: 696c 650a 6265 6769 6e73 2077 6974 6820 ile.begins with │ │ │ │ -0027c1f0: 2f2c 207e 2f2c 2024 2c20 6f72 2021 2e20 /, ~/, $, or !. │ │ │ │ -0027c200: 4966 2074 6865 2066 696c 6520 6265 6769 If the file begi │ │ │ │ -0027c210: 6e73 2077 6974 6820 2e2f 206f 7220 2e2e ns with ./ or .. │ │ │ │ -0027c220: 2f2c 2074 6865 6e20 6974 206c 6f6f 6b73 /, then it looks │ │ │ │ -0027c230: 0a69 6e73 7465 6164 2069 6e20 7468 6520 .instead in the │ │ │ │ -0027c240: 6469 7265 6374 6f72 7920 6f66 2074 6865 directory of the │ │ │ │ -0027c250: 2066 696c 6520 6375 7272 656e 746c 7920 file currently │ │ │ │ -0027c260: 6265 696e 6720 6c6f 6164 6564 2028 7365 being loaded (se │ │ │ │ -0027c270: 6520 2a6e 6f74 650a 6375 7272 656e 7446 e *note.currentF │ │ │ │ -0027c280: 696c 6544 6972 6563 746f 7279 3a20 6375 ileDirectory: cu │ │ │ │ -0027c290: 7272 656e 7446 696c 6544 6972 6563 746f rrentFileDirecto │ │ │ │ -0027c2a0: 7279 2c29 2e20 2049 6620 6e6f 2066 696c ry,). If no fil │ │ │ │ -0027c2b0: 6520 6973 2062 6569 6e67 206c 6f61 6465 e is being loade │ │ │ │ -0027c2c0: 6420 6974 0a77 696c 6c20 6c6f 6f6b 2069 d it.will look i │ │ │ │ -0027c2d0: 6e20 7468 6520 6375 7272 656e 7420 6469 n the current di │ │ │ │ -0027c2e0: 7265 6374 6f72 7920 2873 6565 202a 6e6f rectory (see *no │ │ │ │ -0027c2f0: 7465 2063 7572 7265 6e74 4469 7265 6374 te currentDirect │ │ │ │ -0027c300: 6f72 793a 0a63 7572 7265 6e74 4469 7265 ory:.currentDire │ │ │ │ -0027c310: 6374 6f72 792c 292e 0a0a 4966 206f 6e65 ctory,)...If one │ │ │ │ -0027c320: 206f 6620 7468 6520 746f 7020 6c65 7665 of the top leve │ │ │ │ -0027c330: 6c20 6578 7072 6573 7369 6f6e 7320 696e l expressions in │ │ │ │ -0027c340: 2074 6865 2066 696c 6520 6576 616c 7561 the file evalua │ │ │ │ -0027c350: 7465 7320 746f 2074 6865 2073 796d 626f tes to the symbo │ │ │ │ -0027c360: 6c20 2a6e 6f74 650a 656e 643a 2065 6e64 l *note.end: end │ │ │ │ -0027c370: 2c20 7468 6520 7265 6164 696e 6720 6f66 , the reading of │ │ │ │ -0027c380: 2074 6865 2066 696c 6520 6973 2073 746f the file is sto │ │ │ │ -0027c390: 7070 6564 2061 7420 7468 6174 2070 6f69 pped at that poi │ │ │ │ -0027c3a0: 6e74 2e0a 0a49 6620 616e 2065 7272 6f72 nt...If an error │ │ │ │ -0027c3b0: 206f 6363 7572 7320 7768 696c 6520 6576 occurs while ev │ │ │ │ -0027c3c0: 616c 7561 7469 6e67 2074 6865 2065 7870 aluating the exp │ │ │ │ -0027c3d0: 7265 7373 696f 6e73 2069 6e20 7468 6520 ressions in the │ │ │ │ -0027c3e0: 6669 6c65 2c20 7265 6164 696e 6720 6973 file, reading is │ │ │ │ -0027c3f0: 0a73 746f 7070 6564 2e0a 0a53 6565 2061 .stopped...See a │ │ │ │ -0027c400: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -0027c410: 2a20 2a6e 6f74 6520 7061 7468 3a20 7061 * *note path: pa │ │ │ │ -0027c420: 7468 2c20 2d2d 206c 6973 7420 6f66 2064 th, -- list of d │ │ │ │ -0027c430: 6972 6563 746f 7269 6573 2074 6f20 6c6f irectories to lo │ │ │ │ -0027c440: 6f6b 2069 6e0a 2020 2a20 2a6e 6f74 6520 ok in. * *note │ │ │ │ -0027c450: 6e65 6564 733a 206e 6565 6473 2c20 2d2d needs: needs, -- │ │ │ │ -0027c460: 2072 6561 6420 4d61 6361 756c 6179 3220 read Macaulay2 │ │ │ │ -0027c470: 636f 6d6d 616e 6473 2069 6620 6e65 6365 commands if nece │ │ │ │ -0027c480: 7373 6172 790a 2020 2a20 2a6e 6f74 6520 ssary. * *note │ │ │ │ -0027c490: 6c6f 6164 3a20 6c6f 6164 2c20 2d2d 2072 load: load, -- r │ │ │ │ -0027c4a0: 6561 6420 4d61 6361 756c 6179 3220 636f ead Macaulay2 co │ │ │ │ -0027c4b0: 6d6d 616e 6473 0a0a 466f 7220 7468 6520 mmands..For the │ │ │ │ -0027c4c0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0027c4d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0027c4e0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0027c4f0: 696e 7075 743a 2069 6e70 7574 2c20 6973 input: input, is │ │ │ │ -0027c500: 2061 202a 6e6f 7465 2066 756e 6374 696f a *note functio │ │ │ │ -0027c510: 6e20 636c 6f73 7572 653a 2046 756e 6374 n closure: Funct │ │ │ │ -0027c520: 696f 6e43 6c6f 7375 7265 2c2e 0a0a 2d2d ionClosure,...-- │ │ │ │ -0027c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027bec0: 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 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +0027bf00: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +0027bf10: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +0027bf20: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +0027bf30: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +0027bf40: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +0027bf50: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ +0027bf60: 756c 6179 3244 6f63 2f6f 765f 7379 7374 ulay2Doc/ov_syst │ │ │ │ +0027bf70: 656d 2e6d 323a 3136 333a 302e 0a1f 0a46 em.m2:163:0....F │ │ │ │ +0027bf80: 696c 653a 204d 6163 6175 6c61 7932 446f ile: Macaulay2Do │ │ │ │ +0027bf90: 632e 696e 666f 2c20 4e6f 6465 3a20 696e c.info, Node: in │ │ │ │ +0027bfa0: 7075 742c 204e 6578 743a 206c 6f61 642c put, Next: load, │ │ │ │ +0027bfb0: 2050 7265 763a 2069 6e69 7469 616c 697a Prev: initializ │ │ │ │ +0027bfc0: 6174 696f 6e20 6669 6c65 2c20 5570 3a20 ation file, Up: │ │ │ │ +0027bfd0: 7379 7374 656d 2066 6163 696c 6974 6965 system facilitie │ │ │ │ +0027bfe0: 730a 0a69 6e70 7574 202d 2d20 7265 6164 s..input -- read │ │ │ │ +0027bff0: 204d 6163 6175 6c61 7932 2063 6f6d 6d61 Macaulay2 comma │ │ │ │ +0027c000: 6e64 7320 616e 6420 6563 686f 0a2a 2a2a nds and echo.*** │ │ │ │ +0027c010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0027c020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0027c030: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ +0027c040: 653a 200a 2020 2020 2020 2020 696e 7075 e: . inpu │ │ │ │ +0027c050: 7420 666e 0a20 202a 2049 6e70 7574 733a t fn. * Inputs: │ │ │ │ +0027c060: 0a20 2020 2020 202a 2066 6e2c 2061 202a . * fn, a * │ │ │ │ +0027c070: 6e6f 7465 2073 7472 696e 673a 2053 7472 note string: Str │ │ │ │ +0027c080: 696e 672c 0a20 202a 2043 6f6e 7365 7175 ing,. * Consequ │ │ │ │ +0027c090: 656e 6365 733a 0a20 2020 2020 202a 2072 ences:. * r │ │ │ │ +0027c0a0: 6561 6473 2061 6e64 2065 7865 6375 7465 eads and execute │ │ │ │ +0027c0b0: 7320 7468 6520 636f 6d6d 616e 6473 2066 s the commands f │ │ │ │ +0027c0c0: 6f75 6e64 2069 6e20 7468 6520 6669 6c65 ound in the file │ │ │ │ +0027c0d0: 2077 686f 7365 206e 616d 6520 6973 0a20 whose name is. │ │ │ │ +0027c0e0: 2020 2020 2020 2063 6f6e 7461 696e 6564 contained │ │ │ │ +0027c0f0: 2069 6e20 7468 6520 7374 7269 6e67 2066 in the string f │ │ │ │ +0027c100: 6e2c 2065 6368 6f69 6e67 2074 6865 2069 n, echoing the i │ │ │ │ +0027c110: 6e70 7574 2c20 7072 696e 7469 6e67 2074 nput, printing t │ │ │ │ +0027c120: 6865 2076 616c 7565 732c 2061 6e64 0a20 he values, and. │ │ │ │ +0027c130: 2020 2020 2020 2069 6e63 7265 6d65 6e74 increment │ │ │ │ +0027c140: 696e 6720 7468 6520 6c69 6e65 206e 756d ing the line num │ │ │ │ +0027c150: 6265 720a 0a44 6573 6372 6970 7469 6f6e ber..Description │ │ │ │ +0027c160: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ +0027c170: 6520 6669 6c65 2069 7320 736f 7567 6874 e file is sought │ │ │ │ +0027c180: 2069 6e20 7468 6520 6469 7265 6374 6f72 in the director │ │ │ │ +0027c190: 7920 636f 6e74 6169 6e69 6e67 2074 6865 y containing the │ │ │ │ +0027c1a0: 2066 696c 6520 6375 7272 656e 746c 7920 file currently │ │ │ │ +0027c1b0: 6265 696e 6720 6c6f 6164 6564 2c0a 6966 being loaded,.if │ │ │ │ +0027c1c0: 2061 6e79 2c20 616e 6420 7468 656e 2061 any, and then a │ │ │ │ +0027c1d0: 6c6f 6e67 2074 6865 202a 6e6f 7465 2070 long the *note p │ │ │ │ +0027c1e0: 6174 683a 2070 6174 682c 2c20 756e 6c65 ath: path,, unle │ │ │ │ +0027c1f0: 7373 2074 6865 206e 616d 6520 6f66 2074 ss the name of t │ │ │ │ +0027c200: 6865 2066 696c 650a 6265 6769 6e73 2077 he file.begins w │ │ │ │ +0027c210: 6974 6820 2f2c 207e 2f2c 2024 2c20 6f72 ith /, ~/, $, or │ │ │ │ +0027c220: 2021 2e20 4966 2074 6865 2066 696c 6520 !. If the file │ │ │ │ +0027c230: 6265 6769 6e73 2077 6974 6820 2e2f 206f begins with ./ o │ │ │ │ +0027c240: 7220 2e2e 2f2c 2074 6865 6e20 6974 206c r ../, then it l │ │ │ │ +0027c250: 6f6f 6b73 0a69 6e73 7465 6164 2069 6e20 ooks.instead in │ │ │ │ +0027c260: 7468 6520 6469 7265 6374 6f72 7920 6f66 the directory of │ │ │ │ +0027c270: 2074 6865 2066 696c 6520 6375 7272 656e the file curren │ │ │ │ +0027c280: 746c 7920 6265 696e 6720 6c6f 6164 6564 tly being loaded │ │ │ │ +0027c290: 2028 7365 6520 2a6e 6f74 650a 6375 7272 (see *note.curr │ │ │ │ +0027c2a0: 656e 7446 696c 6544 6972 6563 746f 7279 entFileDirectory │ │ │ │ +0027c2b0: 3a20 6375 7272 656e 7446 696c 6544 6972 : currentFileDir │ │ │ │ +0027c2c0: 6563 746f 7279 2c29 2e20 2049 6620 6e6f ectory,). If no │ │ │ │ +0027c2d0: 2066 696c 6520 6973 2062 6569 6e67 206c file is being l │ │ │ │ +0027c2e0: 6f61 6465 6420 6974 0a77 696c 6c20 6c6f oaded it.will lo │ │ │ │ +0027c2f0: 6f6b 2069 6e20 7468 6520 6375 7272 656e ok in the curren │ │ │ │ +0027c300: 7420 6469 7265 6374 6f72 7920 2873 6565 t directory (see │ │ │ │ +0027c310: 202a 6e6f 7465 2063 7572 7265 6e74 4469 *note currentDi │ │ │ │ +0027c320: 7265 6374 6f72 793a 0a63 7572 7265 6e74 rectory:.current │ │ │ │ +0027c330: 4469 7265 6374 6f72 792c 292e 0a0a 4966 Directory,)...If │ │ │ │ +0027c340: 206f 6e65 206f 6620 7468 6520 746f 7020 one of the top │ │ │ │ +0027c350: 6c65 7665 6c20 6578 7072 6573 7369 6f6e level expression │ │ │ │ +0027c360: 7320 696e 2074 6865 2066 696c 6520 6576 s in the file ev │ │ │ │ +0027c370: 616c 7561 7465 7320 746f 2074 6865 2073 aluates to the s │ │ │ │ +0027c380: 796d 626f 6c20 2a6e 6f74 650a 656e 643a ymbol *note.end: │ │ │ │ +0027c390: 2065 6e64 2c20 7468 6520 7265 6164 696e end, the readin │ │ │ │ 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6520 6e65 6564 733a 206e 6565 6473 ote needs: needs │ │ │ │ +0027c480: 2c20 2d2d 2072 6561 6420 4d61 6361 756c , -- read Macaul │ │ │ │ +0027c490: 6179 3220 636f 6d6d 616e 6473 2069 6620 ay2 commands if │ │ │ │ +0027c4a0: 6e65 6365 7373 6172 790a 2020 2a20 2a6e necessary. * *n │ │ │ │ +0027c4b0: 6f74 6520 6c6f 6164 3a20 6c6f 6164 2c20 ote load: load, │ │ │ │ +0027c4c0: 2d2d 2072 6561 6420 4d61 6361 756c 6179 -- read Macaulay │ │ │ │ +0027c4d0: 3220 636f 6d6d 616e 6473 0a0a 466f 7220 2 commands..For │ │ │ │ +0027c4e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +0027c4f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0027c500: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +0027c510: 6f74 6520 696e 7075 743a 2069 6e70 7574 ote input: input │ │ │ │ +0027c520: 2c20 6973 2061 202a 6e6f 7465 2066 756e , is a *note fun │ │ │ │ +0027c530: 6374 696f 6e20 636c 6f73 7572 653a 2046 ction closure: F │ │ │ │ +0027c540: 756e 6374 696f 6e43 6c6f 7375 7265 2c2e unctionClosure,. │ │ │ │ +0027c550: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ 0027c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0027c580: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0027c590: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0027c5a0: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0027c5b0: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0027c5c0: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ -0027c5d0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0027c5e0: 6167 6573 2f0a 4d61 6361 756c 6179 3244 ages/.Macaulay2D │ │ │ │ -0027c5f0: 6f63 2f6f 765f 7379 7374 656d 2e6d 323a oc/ov_system.m2: │ │ │ │ -0027c600: 3139 303a 302e 0a1f 0a46 696c 653a 204d 190:0....File: M │ │ │ │ -0027c610: 6163 6175 6c61 7932 446f 632e 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expressions fo │ │ │ │ -0027c6f0: 756e 6420 696e 2074 6865 2066 696c 6520 und in the file │ │ │ │ -0027c700: 6e61 6d65 6420 662e 0a0a 5468 6520 6669 named f...The fi │ │ │ │ -0027c710: 6c65 2069 7320 736f 7567 6874 2069 6e20 le is sought in │ │ │ │ -0027c720: 7468 6520 6469 7265 6374 6f72 7920 636f the directory co │ │ │ │ -0027c730: 6e74 6169 6e69 6e67 2074 6865 2066 696c ntaining the fil │ │ │ │ -0027c740: 6520 6375 7272 656e 746c 7920 6265 696e e currently bein │ │ │ │ -0027c750: 6720 6c6f 6164 6564 2c0a 6966 2061 6e79 g loaded,.if any │ │ │ │ -0027c760: 2c20 616e 6420 7468 656e 2061 6c6f 6e67 , and then along │ │ │ │ -0027c770: 2074 6865 202a 6e6f 7465 2070 6174 683a the *note path: │ │ │ │ -0027c780: 2070 6174 682c 2c20 756e 6c65 7373 2074 path,, unless t │ │ │ │ -0027c790: 6865 206e 616d 6520 6f66 2074 6865 2066 he name of the f │ │ │ │ -0027c7a0: 696c 650a 6265 6769 6e73 2077 6974 6820 ile.begins with │ │ │ │ -0027c7b0: 2f2c 2024 2c20 6f72 2021 2e20 4966 2074 /, $, or !. If t │ │ │ │ -0027c7c0: 6865 2066 696c 6520 6265 6769 6e73 2077 he file begins w │ │ │ │ -0027c7d0: 6974 6820 2e2f 206f 7220 2e2e 2f2c 2074 ith ./ or ../, t │ │ │ │ -0027c7e0: 6865 6e20 6974 206c 6f6f 6b73 0a69 6e73 hen it looks.ins │ │ │ │ -0027c7f0: 7465 6164 2069 6e20 7468 6520 6469 7265 tead in the dire │ │ │ │ -0027c800: 6374 6f72 7920 6f66 2074 6865 2066 696c ctory of the fil │ │ │ │ -0027c810: 6520 6375 7272 656e 746c 7920 6265 696e e currently bein │ │ │ │ -0027c820: 6720 6c6f 6164 6564 2028 7365 6520 2a6e g loaded (see *n │ │ │ │ -0027c830: 6f74 650a 6375 7272 656e 7446 696c 6544 ote.currentFileD │ │ │ │ -0027c840: 6972 6563 746f 7279 3a20 6375 7272 656e irectory: curren │ │ │ │ -0027c850: 7446 696c 6544 6972 6563 746f 7279 2c29 tFileDirectory,) │ │ │ │ -0027c860: 2e20 2049 6620 6e6f 2066 696c 6520 6973 . If no file is │ │ │ │ -0027c870: 2062 6569 6e67 206c 6f61 6465 6420 6974 being loaded it │ │ │ │ -0027c880: 0a77 696c 6c20 6c6f 6f6b 2069 6e20 7468 .will look in th │ │ │ │ -0027c890: 6520 6375 7272 656e 7420 6469 7265 6374 e current direct │ │ │ │ -0027c8a0: 6f72 7920 2873 6565 202a 6e6f 7465 2063 ory (see *note c │ │ │ │ -0027c8b0: 7572 7265 6e74 4469 7265 6374 6f72 793a urrentDirectory: │ │ │ │ -0027c8c0: 0a63 7572 7265 6e74 4469 7265 6374 6f72 .currentDirector │ │ │ │ -0027c8d0: 792c 292e 0a0a 5468 6520 6669 6c65 2069 y,)...The file i │ │ │ │ -0027c8e0: 7320 7265 6164 2077 6974 686f 7574 2065 s read without e │ │ │ │ -0027c8f0: 6368 6f69 6e67 2074 6865 2069 6e70 7574 choing the input │ │ │ │ -0027c900: 2c20 7072 696e 7469 6e67 2074 6865 2076 , printing the v │ │ │ │ -0027c910: 616c 7565 732c 206f 720a 696e 6372 656d alues, or.increm │ │ │ │ -0027c920: 656e 7469 6e67 2074 6865 206c 696e 6520 enting the line │ │ │ │ -0027c930: 6e75 6d62 6572 2e0a 0a49 6620 6f6e 6520 number...If one │ │ │ │ -0027c940: 6f66 2074 6865 2074 6f70 206c 6576 656c of the top level │ │ │ │ -0027c950: 2065 7870 7265 7373 696f 6e73 2069 6e20 expressions in │ │ │ │ -0027c960: 7468 6520 6669 6c65 2065 7661 6c75 6174 the file evaluat │ │ │ │ -0027c970: 6573 2074 6f20 7468 6520 7379 6d62 6f6c es to the symbol │ │ │ │ -0027c980: 202a 6e6f 7465 0a65 6e64 3a20 656e 642c *note.end: end, │ │ │ │ -0027c990: 2074 6865 2072 6561 6469 6e67 206f 6620 the reading of │ │ │ │ -0027c9a0: 7468 6520 6669 6c65 2069 7320 7374 6f70 the file is stop │ │ │ │ -0027c9b0: 7065 6420 6174 2074 6861 7420 706f 696e ped at that poin │ │ │ │ -0027c9c0: 742e 0a0a 4966 2074 6865 2076 6172 6961 t...If the varia │ │ │ │ -0027c9d0: 626c 6520 2a6e 6f74 6520 6e6f 7469 6679 ble *note notify │ │ │ │ -0027c9e0: 3a20 6e6f 7469 6679 2c20 6973 2073 6574 : notify, is set │ │ │ │ -0027c9f0: 2074 6f20 7472 7565 2c20 7468 656e 2061 to true, then a │ │ │ │ -0027ca00: 6e20 696e 666f 726d 6174 696f 6e61 6c0a n informational. │ │ │ │ -0027ca10: 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7265 6164 204d 6163 6175 t, -- read Macau │ │ │ │ -0027caf0: 6c61 7932 2063 6f6d 6d61 6e64 7320 616e lay2 commands an │ │ │ │ -0027cb00: 6420 6563 686f 0a20 202a 202a 6e6f 7465 d echo. * *note │ │ │ │ -0027cb10: 206e 6f74 6966 793a 206e 6f74 6966 792c notify: notify, │ │ │ │ -0027cb20: 202d 2d20 7768 6574 6865 7220 746f 206e -- whether to n │ │ │ │ -0027cb30: 6f74 6966 7920 7468 6520 7573 6572 2077 otify the user w │ │ │ │ -0027cb40: 6865 6e20 6120 6669 6c65 2069 7320 6c6f hen a file is lo │ │ │ │ -0027cb50: 6164 6564 0a0a 466f 7220 7468 6520 7072 aded..For the pr │ │ │ │ -0027cb60: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -0027cb70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -0027cb80: 206f 626a 6563 7420 2a6e 6f74 6520 6c6f object *note lo │ │ │ │ -0027cb90: 6164 3a20 6c6f 6164 2c20 6973 2061 202a ad: load, is a * │ │ │ │ -0027cba0: 6e6f 7465 2066 756e 6374 696f 6e20 636c note function cl │ │ │ │ -0027cbb0: 6f73 7572 653a 2046 756e 6374 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packages/.Macaul │ │ │ │ +0027c610: 6179 3244 6f63 2f6f 765f 7379 7374 656d ay2Doc/ov_system │ │ │ │ +0027c620: 2e6d 323a 3139 303a 302e 0a1f 0a46 696c .m2:190:0....Fil │ │ │ │ +0027c630: 653a 204d 6163 6175 6c61 7932 446f 632e e: Macaulay2Doc. │ │ │ │ +0027c640: 696e 666f 2c20 4e6f 6465 3a20 6c6f 6164 info, Node: load │ │ │ │ +0027c650: 2c20 4e65 7874 3a20 6e65 6564 732c 2050 , Next: needs, P │ │ │ │ +0027c660: 7265 763a 2069 6e70 7574 2c20 5570 3a20 rev: input, Up: │ │ │ │ +0027c670: 7379 7374 656d 2066 6163 696c 6974 6965 system facilitie │ │ │ │ +0027c680: 730a 0a6c 6f61 6420 2d2d 2072 6561 6420 s..load -- read │ │ │ │ +0027c690: 4d61 6361 756c 6179 3220 636f 6d6d 616e Macaulay2 comman │ │ │ │ +0027c6a0: 6473 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ds.************* │ │ │ │ +0027c6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0027c6c0: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a **..Description. │ │ │ │ +0027c6d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a6c 6f61 ===========..loa │ │ │ │ +0027c6e0: 6420 2266 2220 2d2d 2072 6561 6473 2061 d "f" -- reads a │ │ │ │ +0027c6f0: 6e64 2065 7865 6375 7465 7320 4d61 6361 nd executes Maca │ │ │ │ +0027c700: 756c 6179 3220 6578 7072 6573 7369 6f6e ulay2 expression │ │ │ │ +0027c710: 7320 666f 756e 6420 696e 2074 6865 2066 s found in the f │ │ │ │ +0027c720: 696c 6520 6e61 6d65 6420 662e 0a0a 5468 ile named f...Th │ │ │ │ +0027c730: 6520 6669 6c65 2069 7320 736f 7567 6874 e file is sought │ │ │ │ +0027c740: 2069 6e20 7468 6520 6469 7265 6374 6f72 in the director │ │ │ │ +0027c750: 7920 636f 6e74 6169 6e69 6e67 2074 6865 y containing the │ │ │ │ +0027c760: 2066 696c 6520 6375 7272 656e 746c 7920 file currently │ │ │ │ +0027c770: 6265 696e 6720 6c6f 6164 6564 2c0a 6966 being loaded,.if │ │ │ │ +0027c780: 2061 6e79 2c20 616e 6420 7468 656e 2061 any, and then a │ │ │ │ +0027c790: 6c6f 6e67 2074 6865 202a 6e6f 7465 2070 long the *note p │ │ │ │ +0027c7a0: 6174 683a 2070 6174 682c 2c20 756e 6c65 ath: path,, unle 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If no fil │ │ │ │ +0027c890: 6520 6973 2062 6569 6e67 206c 6f61 6465 e is being loade │ │ │ │ +0027c8a0: 6420 6974 0a77 696c 6c20 6c6f 6f6b 2069 d it.will look i │ │ │ │ +0027c8b0: 6e20 7468 6520 6375 7272 656e 7420 6469 n the current di │ │ │ │ +0027c8c0: 7265 6374 6f72 7920 2873 6565 202a 6e6f rectory (see *no │ │ │ │ +0027c8d0: 7465 2063 7572 7265 6e74 4469 7265 6374 te currentDirect │ │ │ │ +0027c8e0: 6f72 793a 0a63 7572 7265 6e74 4469 7265 ory:.currentDire │ │ │ │ +0027c8f0: 6374 6f72 792c 292e 0a0a 5468 6520 6669 ctory,)...The fi │ │ │ │ +0027c900: 6c65 2069 7320 7265 6164 2077 6974 686f le is read witho │ │ │ │ +0027c910: 7574 2065 6368 6f69 6e67 2074 6865 2069 ut echoing the i │ │ │ │ +0027c920: 6e70 7574 2c20 7072 696e 7469 6e67 2074 nput, printing t │ │ │ │ +0027c930: 6865 2076 616c 7565 732c 206f 720a 696e he values, or.in │ │ │ │ +0027c940: 6372 656d 656e 7469 6e67 2074 6865 206c crementing the l │ │ │ │ +0027c950: 696e 6520 6e75 6d62 6572 2e0a 0a49 6620 ine number...If │ │ 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2d20 7265 6164 204d input, -- read M │ │ │ │ +0027cb10: 6163 6175 6c61 7932 2063 6f6d 6d61 6e64 acaulay2 command │ │ │ │ +0027cb20: 7320 616e 6420 6563 686f 0a20 202a 202a s and echo. * * │ │ │ │ +0027cb30: 6e6f 7465 206e 6f74 6966 793a 206e 6f74 note notify: not │ │ │ │ +0027cb40: 6966 792c 202d 2d20 7768 6574 6865 7220 ify, -- whether │ │ │ │ +0027cb50: 746f 206e 6f74 6966 7920 7468 6520 7573 to notify the us │ │ │ │ +0027cb60: 6572 2077 6865 6e20 6120 6669 6c65 2069 er when a file i │ │ │ │ +0027cb70: 7320 6c6f 6164 6564 0a0a 466f 7220 7468 s loaded..For th │ │ │ │ +0027cb80: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0027cb90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0027cba0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0027cbb0: 6520 6c6f 6164 3a20 6c6f 6164 2c20 6973 e load: load, is │ │ │ │ +0027cbc0: 2061 202a 6e6f 7465 2066 756e 6374 696f a *note functio │ │ │ │ +0027cbd0: 6e20 636c 6f73 7572 653a 2046 756e 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│ │ +0027cc40: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +0027cc50: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +0027cc60: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +0027cc70: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +0027cc80: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ +0027cc90: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +0027cca0: 6167 6573 2f0a 4d61 6361 756c 6179 3244 ages/.Macaulay2D │ │ │ │ +0027ccb0: 6f63 2f6f 765f 7379 7374 656d 2e6d 323a oc/ov_system.m2: │ │ │ │ +0027ccc0: 3231 343a 302e 0a1f 0a46 696c 653a 204d 214:0....File: M │ │ │ │ +0027ccd0: 6163 6175 6c61 7932 446f 632e 696e 666f acaulay2Doc.info │ │ │ │ +0027cce0: 2c20 4e6f 6465 3a20 6e65 6564 732c 204e , Node: needs, N │ │ │ │ +0027ccf0: 6578 743a 2065 6e64 2c20 5072 6576 3a20 ext: end, Prev: │ │ │ │ +0027cd00: 6c6f 6164 2c20 5570 3a20 7379 7374 656d load, Up: system │ │ │ │ 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656e 6365 733a * Consequences: │ │ │ │ -0027d140: 0a20 2020 2020 202a 2054 6869 7320 7379 . * This sy │ │ │ │ -0027d150: 6d62 6f6c 2c20 656e 636f 756e 7465 7265 mbol, encountere │ │ │ │ -0027d160: 6420 6174 2074 6f70 206c 6576 656c 2c20 d at top level, │ │ │ │ -0027d170: 6361 7573 6573 206c 6f61 6469 6e67 206f causes loading o │ │ │ │ -0027d180: 6620 7468 6520 6375 7272 656e 740a 2020 f the current. │ │ │ │ -0027d190: 2020 2020 2020 696e 7075 7420 6669 6c65 input file │ │ │ │ -0027d1a0: 2074 6f20 6265 2073 746f 7070 6564 2e0a to be stopped.. │ │ │ │ -0027d1b0: 2020 2020 2020 2a20 416c 7465 726e 6174 * Alternat │ │ │ │ -0027d1c0: 6976 656c 792c 2069 6e20 7468 6520 6465 ively, in the de │ │ │ │ -0027d1d0: 6275 6767 6572 2069 7420 6361 7573 6573 bugger it causes │ │ │ │ -0027d1e0: 2074 6865 2063 7572 7265 6e74 2063 6f64 the current cod │ │ │ │ -0027d1f0: 6520 746f 2062 650a 2020 2020 2020 2020 e to be. │ │ │ │ -0027d200: 6162 616e 646f 6e65 642c 2061 6e64 2074 abandoned, and t │ │ │ │ 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If. t │ │ │ │ -0027d250: 6865 7265 2061 7265 206e 6f20 6d6f 7265 here are no more │ │ │ │ -0027d260: 2073 7573 7065 6e64 6564 206c 6576 656c suspended level │ │ │ │ -0027d270: 7320 6f66 2065 7865 6375 7469 6f6e 2c20 s of execution, │ │ │ │ -0027d280: 7468 656e 2063 6f6e 7472 6f6c 2069 730a then control is. │ │ │ │ -0027d290: 2020 2020 2020 2020 7265 7475 726e 6564 returned │ │ │ │ -0027d2a0: 2074 6f20 7468 6520 746f 7020 6c65 7665 to the top leve │ │ │ │ -0027d2b0: 6c2e 0a0a 4465 7363 7269 7074 696f 6e0a l...Description. │ │ │ │ -0027d2c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d ===========..+-- │ │ │ │ -0027d2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d2f0: 2d2d 2d2d 2d2b 0a7c 6931 203a 206c 6f61 -----+.|i1 : loa │ │ │ │ -0027d300: 6420 224d 6163 6175 6c61 7932 446f 632f d "Macaulay2Doc/ │ │ │ │ -0027d310: 6465 6d6f 732f 6465 6d6f 332e 6d32 227c demos/demo3.m2"| │ │ │ │ -0027d320: 0a7c 6869 2020 2020 2020 2020 2020 2020 .|hi │ │ │ │ -0027d330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d340: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0027d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d370: 2d2d 2d2b 0a7c 6932 203a 2067 6574 206c ---+.|i2 : get l │ │ │ │ -0027d380: 6f61 6465 6446 696c 6573 2328 236c 6f61 oadedFiles#(#loa │ │ │ │ -0027d390: 6465 6446 696c 6573 2d31 2920 207c 0a7c dedFiles-1) |.| │ │ │ │ -0027d3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d3c0: 2020 2020 2020 207c 0a7c 6f32 203d 2070 |.|o2 = p │ │ │ │ -0027d3d0: 7269 6e74 2068 6920 2020 2020 2020 2020 rint hi │ │ │ │ -0027d3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d3f0: 207c 0a7c 2020 2020 2065 6e64 2020 2020 |.| end │ │ │ │ +0027cff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +0027d020: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +0027d030: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +0027d040: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +0027d050: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +0027d060: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ +0027d070: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +0027d080: 6167 6573 2f0a 4d61 6361 756c 6179 3244 ages/.Macaulay2D │ │ │ │ +0027d090: 6f63 2f6f 765f 7379 7374 656d 2e6d 323a oc/ov_system.m2: │ │ │ │ +0027d0a0: 3232 353a 302e 0a1f 0a46 696c 653a 204d 225:0....File: M │ │ │ │ +0027d0b0: 6163 6175 6c61 7932 446f 632e 696e 666f acaulay2Doc.info │ │ │ │ +0027d0c0: 2c20 4e6f 6465 3a20 656e 642c 204e 6578 , Node: end, Nex │ │ │ │ +0027d0d0: 743a 206c 6f61 6465 6446 696c 6573 2c20 t: loadedFiles, │ │ │ │ +0027d0e0: 5072 6576 3a20 6e65 6564 732c 2055 703a Prev: needs, Up: │ │ │ │ +0027d0f0: 2073 7973 7465 6d20 6661 6369 6c69 7469 system faciliti │ │ │ │ +0027d100: 6573 0a0a 656e 6420 2d2d 2073 746f 7020 es..end -- stop │ │ │ │ +0027d110: 6c6f 6164 696e 6720 6120 6669 6c65 0a2a loading a file.* │ │ │ │ +0027d120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0027d130: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +0027d140: 7361 6765 3a20 0a20 2020 2020 2020 2065 sage: . e │ │ │ │ +0027d150: 6e64 0a20 202a 2043 6f6e 7365 7175 656e nd. * Consequen │ │ │ │ +0027d160: 6365 733a 0a20 2020 2020 202a 2054 6869 ces:. * Thi │ │ │ │ +0027d170: 7320 7379 6d62 6f6c 2c20 656e 636f 756e s symbol, encoun │ │ │ │ +0027d180: 7465 7265 6420 6174 2074 6f70 206c 6576 tered at top lev │ │ │ │ +0027d190: 656c 2c20 6361 7573 6573 206c 6f61 6469 el, causes loadi │ │ │ │ +0027d1a0: 6e67 206f 6620 7468 6520 6375 7272 656e ng of the curren │ │ │ │ +0027d1b0: 740a 2020 2020 2020 2020 696e 7075 7420 t. input │ │ │ │ +0027d1c0: 6669 6c65 2074 6f20 6265 2073 746f 7070 file to be stopp │ │ │ │ +0027d1d0: 6564 2e0a 2020 2020 2020 2a20 416c 7465 ed.. * Alte │ │ │ │ +0027d1e0: 726e 6174 6976 656c 792c 2069 6e20 7468 rnatively, in th │ │ │ │ +0027d1f0: 6520 6465 6275 6767 6572 2069 7420 6361 e debugger it ca │ │ │ │ +0027d200: 7573 6573 2074 6865 2063 7572 7265 6e74 uses the current │ │ │ │ +0027d210: 2063 6f64 6520 746f 2062 650a 2020 2020 code to be. │ │ │ │ +0027d220: 2020 2020 6162 616e 646f 6e65 642c 2061 abandoned, a │ │ │ │ +0027d230: 6e64 2074 6865 2064 6562 7567 6765 7220 nd the debugger │ │ │ │ +0027d240: 746f 2062 6520 7265 2d65 6e74 6572 6564 to be re-entered │ │ │ │ +0027d250: 206f 6e65 206c 6576 656c 2066 7572 7468 one level furth │ │ │ │ +0027d260: 6572 2075 702e 2020 4966 0a20 2020 2020 er up. If. │ │ │ │ +0027d270: 2020 2074 6865 7265 2061 7265 206e 6f20 there are no │ │ │ │ +0027d280: 6d6f 7265 2073 7573 7065 6e64 6564 206c more suspended l │ │ │ │ +0027d290: 6576 656c 7320 6f66 2065 7865 6375 7469 evels of executi │ │ │ │ +0027d2a0: 6f6e 2c20 7468 656e 2063 6f6e 7472 6f6c on, then control │ │ │ │ +0027d2b0: 2069 730a 2020 2020 2020 2020 7265 7475 is. retu │ │ │ │ +0027d2c0: 726e 6564 2074 6f20 7468 6520 746f 7020 rned to the top │ │ │ │ +0027d2d0: 6c65 7665 6c2e 0a0a 4465 7363 7269 7074 level...Descript │ │ │ │ +0027d2e0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0027d2f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0027d300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d310: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ +0027d320: 206c 6f61 6420 224d 6163 6175 6c61 7932 load "Macaulay2 │ │ │ │ +0027d330: 446f 632f 6465 6d6f 732f 6465 6d6f 332e Doc/demos/demo3. │ │ │ │ +0027d340: 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│ │ -0027d430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d440: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0027d450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0027d470: 0a0a 4865 7265 2069 7320 616e 2065 7861 ..Here is an exa │ │ │ │ -0027d480: 6d70 6c65 206f 6620 6974 7320 7573 6520 mple of its use │ │ │ │ -0027d490: 696e 2074 6865 2064 6562 7567 6765 722e in the debugger. │ │ │ │ -0027d4a0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ -0027d4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d410: 2020 2020 207c 0a7c 2020 2020 2065 6e64 |.| end │ │ │ │ +0027d420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d430: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0027d440: 0a7c 2020 2020 2070 7269 6e74 2068 6f20 .| print ho │ │ │ │ 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---------------- │ │ │ │ -0027d590: 2b0a 7c69 3420 3a20 6720 3220 2020 2020 +.|i4 : g 2 │ │ │ │ -0027d5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d5b0: 2d2d 2d2d 2b0a 7c69 3420 3a20 6720 3220 ----+.|i4 : g 2 │ │ │ │ 0027d5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027d5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d5e0: 7c0a 7c2f 7573 722f 7368 6172 652f 4d61 |.|/usr/share/Ma │ │ │ │ -0027d5f0: 6361 756c 6179 322f 4d61 6361 756c 6179 caulay2/Macaulay │ │ │ │ -0027d600: 3244 6f63 2f64 656d 6f73 2f64 656d 6f31 2Doc/demos/demo1 │ │ │ │ -0027d610: 2e6d 323a 383a 3131 3a28 3329 3a5b 335d .m2:8:11:(3):[3] │ │ │ │ -0027d620: 3a20 6572 726f 7220 2020 2020 2020 2020 : error │ │ │ │ -0027d630: 7c0a 7c2f 7573 722f 7368 6172 652f 4d61 |.|/usr/share/Ma │ │ │ │ -0027d640: 6361 756c 6179 322f 4d61 6361 756c 6179 caulay2/Macaulay │ │ │ │ -0027d650: 3244 6f63 2f64 656d 6f73 2f64 656d 6f31 2Doc/demos/demo1 │ │ │ │ -0027d660: 2e6d 323a 383a 3131 3a28 3329 3a20 656e .m2:8:11:(3): en │ │ │ │ -0027d670: 7465 7269 6e67 2020 2020 2020 2020 2020 tering │ │ │ │ -0027d680: 7c0a 7c2f 7573 722f 7368 6172 652f 4d61 |.|/usr/share/Ma │ │ │ │ -0027d690: 6361 756c 6179 322f 2020 2020 2020 2020 caulay2/ │ │ │ │ -0027d6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d600: 2020 2020 7c0a 7c2f 7573 722f 7368 6172 |.|/usr/shar │ │ │ │ +0027d610: 652f 4d61 6361 756c 6179 322f 4d61 6361 e/Macaulay2/Maca │ │ │ │ +0027d620: 756c 6179 3244 6f63 2f64 656d 6f73 2f64 ulay2Doc/demos/d │ │ │ │ +0027d630: 656d 6f31 2e6d 323a 383a 3131 3a28 3329 emo1.m2:8:11:(3) │ │ │ │ +0027d640: 3a5b 335d 3a20 6572 726f 7220 2020 2020 :[3]: error │ │ │ │ +0027d650: 2020 2020 7c0a 7c2f 7573 722f 7368 6172 |.|/usr/shar │ │ │ │ +0027d660: 652f 4d61 6361 756c 6179 322f 4d61 6361 e/Macaulay2/Maca │ │ │ │ +0027d670: 756c 6179 3244 6f63 2f64 656d 6f73 2f64 ulay2Doc/demos/d │ │ │ │ +0027d680: 656d 6f31 2e6d 323a 383a 3131 3a28 3329 emo1.m2:8:11:(3) │ │ │ │ +0027d690: 3a20 656e 7465 7269 6e67 2020 2020 2020 : entering │ │ │ │ +0027d6a0: 2020 2020 7c0a 7c2f 7573 722f 7368 6172 |.|/usr/shar │ │ │ │ +0027d6b0: 652f 4d61 6361 756c 6179 322f 2020 2020 e/Macaulay2/ │ │ │ │ 0027d6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d6d0: 7c0a 7c4d 6163 6175 6c61 7932 446f 632f |.|Macaulay2Doc/ │ │ │ │ -0027d6e0: 6465 6d6f 732f 6465 6d6f 312e 6d32 3a38 demos/demo1.m2:8 │ │ │ │ -0027d6f0: 3a31 302d 383a 3133 3a20 2d2d 736f 7572 :10-8:13: --sour │ │ │ │ -0027d700: 6365 2063 6f64 653a 2020 2020 2020 2020 ce code: │ │ │ │ -0027d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d720: 7c0a 7c20 2020 2020 6220 3a3d 2031 2f78 |.| b := 1/x │ │ │ │ -0027d730: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -0027d740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d6f0: 2020 2020 7c0a 7c4d 6163 6175 6c61 7932 |.|Macaulay2 │ │ │ │ +0027d700: 446f 632f 6465 6d6f 732f 6465 6d6f 312e Doc/demos/demo1. │ │ │ │ +0027d710: 6d32 3a38 3a31 302d 383a 3133 3a20 2d2d m2:8:10-8:13: -- │ │ │ │ +0027d720: 736f 7572 6365 2063 6f64 653a 2020 2020 source code: │ │ │ │ +0027d730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d740: 2020 2020 7c0a 7c20 2020 2020 6220 3a3d |.| b := │ │ │ │ +0027d750: 2031 2f78 3b20 2020 2020 2020 2020 2020 1/x; │ │ │ │ 0027d760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d770: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------- │ │ │ │ -0027d780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d790: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ 0027d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d7c0: 7c0a 7c3a 2064 6976 6973 696f 6e20 6279 |.|: division by │ │ │ │ -0027d7d0: 207a 6572 6f20 2020 2020 2020 2020 2020 zero │ │ │ │ -0027d7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d7e0: 2d2d 2d2d 7c0a 7c3a 2064 6976 6973 696f ----|.|: divisio │ │ │ │ +0027d7f0: 6e20 6279 207a 6572 6f20 2020 2020 2020 n by zero │ │ │ │ 0027d800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d810: 7c0a 7c64 6562 7567 6765 7220 2865 6e74 |.|debugger (ent │ │ │ │ -0027d820: 6572 2027 6865 6c70 2720 746f 2073 6565 er 'help' to see │ │ │ │ -0027d830: 2063 6f6d 6d61 6e64 7329 2020 2020 2020 commands) │ │ │ │ -0027d840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d860: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0027d870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d830: 2020 2020 7c0a 7c64 6562 7567 6765 7220 |.|debugger │ │ │ │ +0027d840: 2865 6e74 6572 2027 6865 6c70 2720 746f (enter 'help' to │ │ │ │ +0027d850: 2073 6565 2063 6f6d 6d61 6e64 7329 2020 see commands) │ │ │ │ +0027d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d880: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0027d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027d8b0: 2b0a 7c69 6935 203a 2065 6e64 2020 2020 +.|ii5 : end │ │ │ │ -0027d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027d8d0: 2d2d 2d2d 2b0a 7c69 6935 203a 2065 6e64 ----+.|ii5 : end │ │ │ │ 0027d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027d8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d900: 7c0a 7c2f 7573 722f 7368 6172 652f 4d61 |.|/usr/share/Ma │ │ │ │ -0027d910: 6361 756c 6179 322f 4d61 6361 756c 6179 caulay2/Macaulay │ │ │ │ -0027d920: 3244 6f63 2f64 656d 6f73 2f64 656d 6f31 2Doc/demos/demo1 │ │ │ │ -0027d930: 2e6d 323a 3133 3a31 313a 2833 293a 2065 .m2:13:11:(3): e │ │ │ │ -0027d940: 6e74 6572 696e 6720 6465 6275 6767 6572 ntering debugger │ │ │ │ -0027d950: 7c0a 7c2f 7573 722f 7368 6172 652f 4d61 |.|/usr/share/Ma │ │ │ │ -0027d960: 6361 756c 6179 322f 2020 2020 2020 2020 caulay2/ │ │ │ │ -0027d970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d920: 2020 2020 7c0a 7c2f 7573 722f 7368 6172 |.|/usr/shar │ │ │ │ +0027d930: 652f 4d61 6361 756c 6179 322f 4d61 6361 e/Macaulay2/Maca │ │ │ │ +0027d940: 756c 6179 3244 6f63 2f64 656d 6f73 2f64 ulay2Doc/demos/d │ │ │ │ +0027d950: 656d 6f31 2e6d 323a 3133 3a31 313a 2833 emo1.m2:13:11:(3 │ │ │ │ +0027d960: 293a 2065 6e74 6572 696e 6720 6465 6275 ): entering debu │ │ │ │ +0027d970: 6767 6572 7c0a 7c2f 7573 722f 7368 6172 gger|.|/usr/shar │ │ │ │ +0027d980: 652f 4d61 6361 756c 6179 322f 2020 2020 e/Macaulay2/ │ │ │ │ 0027d990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d9a0: 7c0a 7c4d 6163 6175 6c61 7932 446f 632f |.|Macaulay2Doc/ │ │ │ │ -0027d9b0: 6465 6d6f 732f 6465 6d6f 312e 6d32 3a31 demos/demo1.m2:1 │ │ │ │ -0027d9c0: 333a 3130 2d31 333a 3136 3a20 2d2d 736f 3:10-13:16: --so │ │ │ │ -0027d9d0: 7572 6365 2063 6f64 653a 2020 2020 2020 urce code: │ │ │ │ -0027d9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027d9f0: 7c0a 7c20 2020 2020 6420 3a3d 2066 2879 |.| d := f(y │ │ │ │ -0027da00: 2d32 293b 2020 2020 2020 2020 2020 2020 -2); │ │ │ │ -0027da10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027da20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027d9c0: 2020 2020 7c0a 7c4d 6163 6175 6c61 7932 |.|Macaulay2 │ │ │ │ +0027d9d0: 446f 632f 6465 6d6f 732f 6465 6d6f 312e Doc/demos/demo1. │ │ │ │ +0027d9e0: 6d32 3a31 333a 3130 2d31 333a 3136 3a20 m2:13:10-13:16: │ │ │ │ +0027d9f0: 2d2d 736f 7572 6365 2063 6f64 653a 2020 --source code: │ │ │ │ +0027da00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027da10: 2020 2020 7c0a 7c20 2020 2020 6420 3a3d |.| d := │ │ │ │ +0027da20: 2066 2879 2d32 293b 2020 2020 2020 2020 f(y-2); │ │ │ │ 0027da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027da40: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------- │ │ │ │ -0027da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027da40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027da60: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ 0027da70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027da80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027da90: 7c0a 7c28 656e 7465 7220 2768 656c 7027 |.|(enter 'help' │ │ │ │ -0027daa0: 2074 6f20 7365 6520 636f 6d6d 616e 6473 to see commands │ │ │ │ -0027dab0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0027dac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dae0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0027daf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027db00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027dab0: 2d2d 2d2d 7c0a 7c28 656e 7465 7220 2768 ----|.|(enter 'h │ │ │ │ +0027dac0: 656c 7027 2074 6f20 7365 6520 636f 6d6d elp' to see comm │ │ │ │ +0027dad0: 616e 6473 2920 2020 2020 2020 2020 2020 ands) │ │ │ │ +0027dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027db00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0027db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027db30: 2b0a 7c69 6936 203a 2065 6e64 2020 2020 +.|ii6 : end │ │ │ │ -0027db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027db40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027db50: 2d2d 2d2d 2b0a 7c69 6936 203a 2065 6e64 ----+.|ii6 : end │ │ │ │ 0027db60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027db70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027db80: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0027db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027dba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027db90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027dba0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0027dbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027dbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027dbd0: 2b0a 7c69 3720 3a20 6720 3320 2020 2020 +.|i7 : g 3 │ │ │ │ -0027dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027dbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027dbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027dbf0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6720 3320 ----+.|i7 : g 3 │ │ │ │ 0027dc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dc20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0027dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027dc40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0027dc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dc70: 7c0a 7c20 2020 2020 3720 2020 2020 2020 |.| 7 │ │ │ │ +0027dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027dc90: 2020 2020 7c0a 7c20 2020 2020 3720 2020 |.| 7 │ │ │ │ 0027dca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dcc0: 7c0a 7c6f 3720 3d20 2d20 2020 2020 2020 |.|o7 = - │ │ │ │ +0027dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027dce0: 2020 2020 7c0a 7c6f 3720 3d20 2d20 2020 |.|o7 = - │ │ │ │ 0027dcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dd10: 7c0a 7c20 2020 2020 3220 2020 2020 2020 |.| 2 │ │ │ │ +0027dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027dd30: 2020 2020 7c0a 7c20 2020 2020 3220 2020 |.| 2 │ │ │ │ 0027dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dd60: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0027dd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027dd80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0027dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027ddb0: 7c0a 7c6f 3720 3a20 5151 2020 2020 2020 |.|o7 : QQ │ │ │ │ +0027ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027ddd0: 2020 2020 7c0a 7c6f 3720 3a20 5151 2020 |.|o7 : QQ │ │ │ │ 0027dde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027de00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0027de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027de00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027de20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0027de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027de40: 2d2d 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6520 7072 6f67 ho..For the prog │ │ │ │ -0027df20: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -0027df30: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -0027df40: 626a 6563 7420 2a6e 6f74 6520 656e 643a bject *note end: │ │ │ │ -0027df50: 2065 6e64 2c20 6973 2061 202a 6e6f 7465 end, is a *note │ │ │ │ -0027df60: 2073 796d 626f 6c3a 2053 796d 626f 6c2c symbol: Symbol, │ │ │ │ -0027df70: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ -0027df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027df90: 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 2b0a 0a53 6565 2061 6c73 6f0a ----+..See also. │ │ │ │ +0027de80: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +0027de90: 6f74 6520 6e65 6564 733a 206e 6565 6473 ote needs: needs │ │ │ │ +0027dea0: 2c20 2d2d 2072 6561 6420 4d61 6361 756c , -- read Macaul │ │ │ │ +0027deb0: 6179 3220 636f 6d6d 616e 6473 2069 6620 ay2 commands if │ │ │ │ +0027dec0: 6e65 6365 7373 6172 790a 2020 2a20 2a6e necessary. * *n │ │ │ │ +0027ded0: 6f74 6520 6c6f 6164 3a20 6c6f 6164 2c20 ote load: load, │ │ │ │ +0027dee0: 2d2d 2072 6561 6420 4d61 6361 756c 6179 -- read Macaulay │ │ │ │ +0027def0: 3220 636f 6d6d 616e 6473 0a20 202a 202a 2 commands. * * │ │ │ │ +0027df00: 6e6f 7465 2069 6e70 7574 3a20 696e 7075 note input: inpu │ │ │ │ +0027df10: 742c 202d 2d20 7265 6164 204d 6163 6175 t, -- read Macau │ │ │ │ +0027df20: 6c61 7932 2063 6f6d 6d61 6e64 7320 616e lay2 commands an │ │ │ │ +0027df30: 6420 6563 686f 0a0a 466f 7220 7468 6520 d echo..For the │ │ │ │ +0027df40: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +0027df50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +0027df60: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +0027df70: 656e 643a 2065 6e64 2c20 6973 2061 202a end: end, is a * │ │ │ │ +0027df80: 6e6f 7465 2073 796d 626f 6c3a 2053 796d note symbol: Sym │ │ │ │ +0027df90: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d bol,...--------- │ │ │ │ 0027dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027dfc0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0027dfd0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0027dfe0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0027dff0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0027e000: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -0027e010: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0027e020: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ -0027e030: 6c61 7932 446f 632f 6f76 5f73 7973 7465 lay2Doc/ov_syste │ │ │ │ -0027e040: 6d2e 6d32 3a32 3532 3a30 2e0a 1f0a 4669 m.m2:252:0....Fi │ │ │ │ -0027e050: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -0027e060: 2e69 6e66 6f2c 204e 6f64 653a 206c 6f61 .info, Node: loa │ │ │ │ -0027e070: 6465 6446 696c 6573 2c20 4e65 7874 3a20 dedFiles, Next: │ │ │ │ -0027e080: 6669 6c65 4578 6974 486f 6f6b 732c 2050 fileExitHooks, P │ │ │ │ -0027e090: 7265 763a 2065 6e64 2c20 5570 3a20 7379 rev: end, Up: sy │ │ │ │ -0027e0a0: 7374 656d 2066 6163 696c 6974 6965 730a stem facilities. │ │ │ │ -0027e0b0: 0a6c 6f61 6465 6446 696c 6573 0a2a 2a2a .loadedFiles.*** │ │ │ │ -0027e0c0: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 ********..Descri │ │ │ │ -0027e0d0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0027e0e0: 3d0a 0a41 6674 6572 2065 6163 6820 736f =..After each so │ │ │ │ -0027e0f0: 7572 6365 2066 696c 6520 6973 2073 7563 urce file is suc │ │ │ │ -0027e100: 6365 7373 6675 6c6c 7920 6c6f 6164 6564 cessfully loaded │ │ │ │ -0027e110: 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It is │ │ │ │ -0027e170: 7374 6f72 6564 2061 7320 7468 650a 7661 stored as the.va │ │ │ │ -0027e180: 6c75 652c 2077 6974 6820 7468 6520 636f lue, with the co │ │ │ │ -0027e190: 7272 6573 706f 6e64 696e 6720 6b65 7920 rresponding key │ │ │ │ -0027e1a0: 6265 696e 6720 6120 736d 616c 6c20 696e being a small in │ │ │ │ -0027e1b0: 7465 6765 722c 2063 6f6e 7365 6375 7469 teger, consecuti │ │ │ │ -0027e1c0: 7665 6c79 0a61 7373 6967 6e65 642c 2073 vely.assigned, s │ │ │ │ -0027e1d0: 7461 7274 696e 6720 6174 2030 2e0a 0a2b tarting at 0...+ │ │ │ │ -0027e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0027e200: 7c69 3120 3a20 6c6f 6164 6564 4669 6c65 |i1 : loadedFile │ │ │ │ -0027e210: 7323 3020 2020 2020 2020 2020 2020 207c s#0 | │ │ │ │ -0027e220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0027e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027e240: 7c0a 7c6f 3120 3d20 2f75 7372 2f73 6861 |.|o1 = /usr/sha │ │ │ │ -0027e250: 7265 2f20 2020 2020 2020 2020 2020 2020 re/ │ │ │ │ -0027e260: 207c 0a7c 2020 2020 204d 6163 6175 6c61 |.| Macaula │ │ │ │ -0027e270: 7932 2f43 6f72 652f 7665 7273 696f 6e2e y2/Core/version. │ │ │ │ -0027e280: 6d32 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d m2|.+----------- │ │ │ │ -0027e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027e2a0: 2d2d 2d2b 0a7c 6932 203a 2023 6c6f 6164 ---+.|i2 : #load │ │ │ │ -0027e2b0: 6564 4669 6c65 7320 2020 2020 2020 2020 edFiles │ │ │ │ -0027e2c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0027e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027e2e0: 2020 2020 207c 0a7c 6f32 203d 2031 3531 |.|o2 = 151 │ │ │ │ -0027e2f0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -0027e300: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ -0027e310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027e320: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -0027e330: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -0027e340: 202a 6e6f 7465 206c 6f61 643a 206c 6f61 *note load: loa │ │ │ │ -0027e350: 642c 202d 2d20 7265 6164 204d 6163 6175 d, -- read Macau │ │ │ │ -0027e360: 6c61 7932 2063 6f6d 6d61 6e64 730a 0a46 lay2 commands..F │ │ │ │ -0027e370: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -0027e380: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -0027e390: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -0027e3a0: 202a 6e6f 7465 206c 6f61 6465 6446 696c *note loadedFil │ │ │ │ -0027e3b0: 6573 3a20 6c6f 6164 6564 4669 6c65 732c es: loadedFiles, │ │ │ │ -0027e3c0: 2069 7320 6120 2a6e 6f74 6520 6d75 7461 is a *note muta │ │ │ │ -0027e3d0: 626c 6520 6861 7368 2074 6162 6c65 3a0a ble hash table:. │ │ │ │ -0027e3e0: 4d75 7461 626c 6548 6173 6854 6162 6c65 MutableHashTable │ │ │ │ -0027e3f0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ -0027e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027dfe0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0027dff0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0027e000: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0027e010: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0027e020: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0027e030: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +0027e040: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ +0027e050: 6163 6175 6c61 7932 446f 632f 6f76 5f73 acaulay2Doc/ov_s │ │ │ │ +0027e060: 7973 7465 6d2e 6d32 3a32 3532 3a30 2e0a ystem.m2:252:0.. │ │ │ │ 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6174 6820 746f 2074 6865 2066 696c 6520 ath to the file │ │ │ │ +0027e150: 6973 0a73 746f 7265 6420 696e 2074 6865 is.stored in the │ │ │ │ +0027e160: 2068 6173 6820 7461 626c 6520 2a6e 6f74 hash table *not │ │ │ │ +0027e170: 6520 6c6f 6164 6564 4669 6c65 733a 206c e loadedFiles: l │ │ │ │ +0027e180: 6f61 6465 6446 696c 6573 2c2e 2020 4974 oadedFiles,. It │ │ │ │ +0027e190: 2069 7320 7374 6f72 6564 2061 7320 7468 is stored as th │ │ │ │ +0027e1a0: 650a 7661 6c75 652c 2077 6974 6820 7468 e.value, with th │ │ │ │ +0027e1b0: 6520 636f 7272 6573 706f 6e64 696e 6720 e corresponding │ │ │ │ +0027e1c0: 6b65 7920 6265 696e 6720 6120 736d 616c key being a smal │ │ │ │ +0027e1d0: 6c20 696e 7465 6765 722c 2063 6f6e 7365 l integer, conse │ │ │ │ +0027e1e0: 6375 7469 7665 6c79 0a61 7373 6967 6e65 cutively.assigne │ │ │ │ +0027e1f0: 642c 2073 7461 7274 696e 6720 6174 2030 d, starting at 0 │ │ │ │ +0027e200: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ +0027e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027e220: 2d2d 2b0a 7c69 3120 3a20 6c6f 6164 6564 --+.|i1 : loaded │ │ │ │ +0027e230: 4669 6c65 7323 3020 2020 2020 2020 2020 Files#0 │ │ │ │ +0027e240: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0027e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027e260: 2020 2020 7c0a 7c6f 3120 3d20 2f75 7372 |.|o1 = /usr │ │ │ │ +0027e270: 2f73 6861 7265 2f20 2020 2020 2020 2020 /share/ │ │ │ │ +0027e280: 2020 2020 207c 0a7c 2020 2020 204d 6163 |.| Mac │ │ │ │ +0027e290: 6175 6c61 7932 2f43 6f72 652f 7665 7273 aulay2/Core/vers │ │ │ │ +0027e2a0: 696f 6e2e 6d32 7c0a 2b2d 2d2d 2d2d 2d2d ion.m2|.+------- │ │ │ │ +0027e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027e2c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2023 -------+.|i2 : # │ │ │ │ +0027e2d0: 6c6f 6164 6564 4669 6c65 7320 2020 2020 loadedFiles │ │ │ │ +0027e2e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0027e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027e300: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ +0027e310: 2031 3531 3320 2020 2020 2020 2020 2020 1513 │ │ │ │ +0027e320: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0027e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 -----------+..Se │ │ │ │ +0027e350: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +0027e360: 0a20 202a 202a 6e6f 7465 206c 6f61 643a . * *note load: │ │ │ │ +0027e370: 206c 6f61 642c 202d 2d20 7265 6164 204d load, -- read M │ │ │ │ +0027e380: 6163 6175 6c61 7932 2063 6f6d 6d61 6e64 acaulay2 command │ │ │ │ +0027e390: 730a 0a46 6f72 2074 6865 2070 726f 6772 s..For the progr │ │ │ │ +0027e3a0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +0027e3b0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +0027e3c0: 6a65 6374 202a 6e6f 7465 206c 6f61 6465 ject *note loade │ │ │ │ +0027e3d0: 6446 696c 6573 3a20 6c6f 6164 6564 4669 dFiles: loadedFi │ │ │ │ +0027e3e0: 6c65 732c 2069 7320 6120 2a6e 6f74 6520 les, is a *note │ │ │ │ +0027e3f0: 6d75 7461 626c 6520 6861 7368 2074 6162 mutable hash tab │ │ │ │ +0027e400: 6c65 3a0a 4d75 7461 626c 6548 6173 6854 le:.MutableHashT │ │ │ │ +0027e410: 6162 6c65 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d able,...-------- │ │ │ │ 0027e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027e440: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0027e450: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0027e460: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0027e470: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0027e480: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0027e490: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -0027e4a0: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -0027e4b0: 756c 6179 3244 6f63 2f6f 765f 7379 7374 ulay2Doc/ov_syst │ │ │ │ -0027e4c0: 656d 2e6d 323a 3138 3739 3a30 2e0a 1f0a em.m2:1879:0.... │ │ │ │ -0027e4d0: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -0027e4e0: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2066 oc.info, Node: f 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2a2a 2a2a 2a2a **************** │ │ │ │ +0027e610: 2a2a 2a0a 0a46 6f72 2074 6865 2070 726f ***..For the pro │ │ │ │ +0027e620: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +0027e630: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0027e640: 6f62 6a65 6374 202a 6e6f 7465 2066 696c object *note fil │ │ │ │ +0027e650: 6545 7869 7448 6f6f 6b73 3a20 6669 6c65 eExitHooks: file │ │ │ │ +0027e660: 4578 6974 486f 6f6b 732c 2069 7320 6120 ExitHooks, is a │ │ │ │ +0027e670: 2a6e 6f74 6520 6c69 7374 3a20 4c69 7374 *note list: List │ │ │ │ +0027e680: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ 0027e690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0027e6b0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0027e6c0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0027e6d0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 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│ │ +0027e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027e6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027e6d0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +0027e6e0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +0027e6f0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +0027e700: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +0027e710: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +0027e720: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +0027e730: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ +0027e740: 756c 6179 3244 6f63 2f6f 765f 7379 7374 ulay2Doc/ov_syst │ │ │ │ +0027e750: 656d 2e6d 323a 3134 3337 3a30 2e0a 1f0a em.m2:1437:0.... │ │ │ │ +0027e760: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ +0027e770: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2070 oc.info, Node: p │ │ │ │ +0027e780: 6174 682c 204e 6578 743a 2072 6f6f 7455 ath, Next: rootU │ │ │ │ +0027e790: 5249 2c20 5072 6576 3a20 6669 6c65 4578 RI, Prev: fileEx │ │ │ │ +0027e7a0: 6974 486f 6f6b 732c 2055 703a 2073 7973 itHooks, Up: sys │ │ │ │ +0027e7b0: 7465 6d20 6661 6369 6c69 7469 6573 0a0a tem facilities.. │ │ │ │ +0027e7c0: 7061 7468 202d 2d20 6c69 7374 206f 6620 path -- list of │ │ │ │ +0027e7d0: 6469 7265 6374 6f72 6965 7320 746f 206c directories to l │ │ │ │ +0027e7e0: 6f6f 6b20 696e 0a2a 2a2a 2a2a 2a2a 2a2a ook in.********* │ │ │ │ +0027e7f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0027e800: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 *************..D │ │ │ │ +0027e810: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0027e820: 3d3d 3d3d 3d3d 0a0a 4120 6c69 7374 206f ======..A list o │ │ │ │ +0027e830: 6620 7374 7269 6e67 7320 636f 6e74 6169 f strings contai │ │ │ │ +0027e840: 6e69 6e67 206e 616d 6573 206f 6620 6469 ning names of di │ │ │ │ +0027e850: 7265 6374 6f72 6965 7320 696e 2077 6869 rectories in whi │ │ │ │ +0027e860: 6368 202a 6e6f 7465 206c 6f61 643a 206c ch *note load: l │ │ │ │ +0027e870: 6f61 642c 2c0a 2a6e 6f74 6520 696e 7075 oad,,.*note inpu │ │ │ │ +0027e880: 743a 2069 6e70 7574 2c2c 202a 6e6f 7465 t: input,, *note │ │ │ │ +0027e890: 206c 6f61 6450 6163 6b61 6765 3a20 6c6f loadPackage: lo │ │ │ │ +0027e8a0: 6164 5061 636b 6167 652c 2c20 2a6e 6f74 adPackage,, *not │ │ │ │ +0027e8b0: 6520 6e65 6564 7350 6163 6b61 6765 3a0a e needsPackage:. │ │ │ │ +0027e8c0: 6e65 6564 7350 6163 6b61 6765 2c2c 2061 needsPackage,, a │ │ │ │ +0027e8d0: 6e64 202a 6e6f 7465 2069 6e73 7461 6c6c nd *note install │ │ │ │ +0027e8e0: 5061 636b 6167 653a 2069 6e73 7461 6c6c Package: install │ │ │ │ +0027e8f0: 5061 636b 6167 652c 2073 686f 756c 6420 Package, should │ │ │ │ +0027e900: 7365 656b 2066 696c 6573 2e0a 5468 6573 seek files..Thes │ │ │ │ +0027e910: 6520 7374 7269 6e67 7320 6172 6520 7369 e strings are si │ │ │ │ +0027e920: 6d70 6c79 2063 6f6e 6361 7465 6e61 7465 mply concatenate │ │ │ │ +0027e930: 6420 7769 7468 2074 6865 2066 696c 656e d with the filen │ │ │ │ +0027e940: 616d 6520 6265 696e 6720 736f 7567 6874 ame being sought │ │ │ │ +0027e950: 2c20 736f 2073 686f 756c 640a 696e 636c , so should.incl │ │ │ │ +0027e960: 7564 6520 6120 7465 726d 696e 616c 2073 ude a terminal s │ │ │ │ +0027e970: 6c61 7368 2e20 204f 6e65 2066 7572 7468 lash. One furth │ │ │ │ +0027e980: 6572 2064 6972 6563 746f 7279 2069 7320 er directory is │ │ │ │ +0027e990: 696d 706c 6963 6974 6c79 2073 6561 7263 implicitly searc │ │ │ │ +0027e9a0: 6865 6420 6669 7273 743a 0a74 6865 2064 hed first:.the d │ │ │ │ +0027e9b0: 6972 6563 746f 7279 2063 6f6e 7461 696e irectory contain │ │ │ │ +0027e9c0: 696e 6720 7468 6520 6375 7272 656e 7420 ing the current │ │ │ │ +0027e9d0: 696e 7075 7420 6669 6c65 3b20 7768 656e input file; when │ │ │ │ +0027e9e0: 2069 6e70 7574 2069 7320 636f 6d69 6e67 input is coming │ │ │ │ +0027e9f0: 2066 726f 6d20 7468 650a 7374 616e 6461 from the.standa │ │ │ │ +0027ea00: 7264 2069 6e70 7574 2c20 7468 6174 2064 rd input, that d │ │ │ │ +0027ea10: 6972 6563 746f 7279 2069 7320 7468 6520 irectory is the │ │ │ │ +0027ea20: 6375 7272 656e 7420 6469 7265 6374 6f72 current director │ │ │ │ +0027ea30: 7920 6f66 2074 6865 2070 726f 6365 7373 y of the process │ │ │ │ +0027ea40: 2e0a 0a41 6674 6572 2074 6865 2063 6f72 ...After the cor │ │ │ │ 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============..Th │ │ │ │ +0027f300: 6520 6f62 6a65 6374 202a 6e6f 7465 2072 e object *note r │ │ │ │ +0027f310: 6f6f 7455 5249 3a20 726f 6f74 5552 492c ootURI: rootURI, │ │ │ │ +0027f320: 2069 7320 6120 2a6e 6f74 6520 7374 7269 is a *note stri │ │ │ │ +0027f330: 6e67 3a20 5374 7269 6e67 2c2e 0a0a 2d2d ng: String,...-- │ │ │ │ 0027f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027f360: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -0027f370: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0027f380: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0027f390: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0027f3a0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0027f3b0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -0027f3c0: 6361 756c 6179 322f 7061 636b 6167 6573 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756e 6465 7273 7461 6e64 6162 one understandab │ │ │ │ -0027f580: 6c65 0a62 7920 6578 7465 726e 616c 2070 le.by external p │ │ │ │ -0027f590: 726f 6772 616d 732e 0a0a 2b2d 2d2d 2d2d rograms...+----- │ │ │ │ -0027f5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027f5b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -0027f5c0: 666e 203d 2074 656d 706f 7261 7279 4669 fn = temporaryFi │ │ │ │ -0027f5d0: 6c65 4e61 6d65 2829 7c0a 7c20 2020 2020 leName()|.| │ │ │ │ -0027f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027f5f0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -0027f600: 2f74 6d70 2f4d 322d 3131 3131 362d 302f /tmp/M2-11116-0/ │ │ │ │ -0027f610: 3020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 0 |.+----- │ │ │ │ -0027f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027f630: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -0027f640: 726f 6f74 5061 7468 207c 2066 6e20 2020 rootPath | fn │ │ │ │ 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rootPath: roo │ │ │ │ -0027f730: 7450 6174 682c 2069 7320 6120 2a6e 6f74 tPath, is a *not │ │ │ │ -0027f740: 6520 7374 7269 6e67 3a20 5374 7269 6e67 e string: String │ │ │ │ -0027f750: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ -0027f760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027f770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +0027f390: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +0027f3a0: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +0027f3b0: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +0027f3c0: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +0027f3d0: 6179 322d 312e 3236 2e30 362b 6473 2f4d 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6520 7061 7468 2074 6f20 bsolute path to │ │ │ │ +0027f590: 6765 7420 6f6e 6520 756e 6465 7273 7461 get one understa │ │ │ │ +0027f5a0: 6e64 6162 6c65 0a62 7920 6578 7465 726e ndable.by extern │ │ │ │ +0027f5b0: 616c 2070 726f 6772 616d 732e 0a0a 2b2d al programs...+- │ │ │ │ +0027f5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027f5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0027f5e0: 3120 3a20 666e 203d 2074 656d 706f 7261 1 : fn = tempora │ │ │ │ +0027f5f0: 7279 4669 6c65 4e61 6d65 2829 7c0a 7c20 ryFileName()|.| │ │ │ │ +0027f600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0027f610: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0027f620: 3120 3d20 2f74 6d70 2f4d 322d 3131 3430 1 = /tmp/M2-1140 │ │ │ │ +0027f630: 352d 302f 3020 2020 2020 2020 7c0a 2b2d 5-0/0 |.+- │ │ │ │ +0027f640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027f650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 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====..The object │ │ │ │ +0027f740: 202a 6e6f 7465 2072 6f6f 7450 6174 683a *note rootPath: │ │ │ │ +0027f750: 2072 6f6f 7450 6174 682c 2069 7320 6120 rootPath, is a │ │ │ │ +0027f760: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374 *note string: St │ │ │ │ +0027f770: 7269 6e67 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ring,...-------- │ │ │ │ 0027f780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027f790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027f7a0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0027f7b0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0027f7c0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0027f7d0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0027f7e0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0027f7f0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -0027f800: 322f 7061 636b 6167 6573 2f0a 4d61 6361 2/packages/.Maca │ │ │ │ -0027f810: 756c 6179 3244 6f63 2f6f 765f 7379 7374 ulay2Doc/ov_syst │ │ │ │ -0027f820: 656d 2e6d 323a 3230 3235 3a30 2e0a 1f0a em.m2:2025:0.... │ │ │ │ -0027f830: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ -0027f840: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2068 oc.info, Node: h │ │ │ │ -0027f850: 6f6d 6544 6972 6563 746f 7279 2c20 4e65 omeDirectory, Ne │ │ │ │ -0027f860: 7874 3a20 6d69 6e69 6d69 7a65 4669 6c65 xt: minimizeFile │ │ │ │ -0027f870: 6e61 6d65 2c20 5072 6576 3a20 726f 6f74 name, Prev: root │ │ │ │ -0027f880: 5061 7468 2c20 5570 3a20 7379 7374 656d Path, Up: system │ │ │ │ -0027f890: 2066 6163 696c 6974 6965 730a 0a68 6f6d facilities..hom │ │ │ │ -0027f8a0: 6544 6972 6563 746f 7279 202d 2d20 7468 eDirectory -- th │ │ │ │ -0027f8b0: 6520 686f 6d65 2064 6972 6563 746f 7279 e home directory │ │ │ │ -0027f8c0: 206f 6620 7468 6520 7573 6572 0a2a 2a2a of the user.*** │ │ │ │ -0027f8d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0027f8e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0027f8f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -0027f900: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -0027f910: 2020 686f 6d65 4469 7265 6374 6f72 790a homeDirectory. │ │ │ │ -0027f920: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -0027f930: 2020 202a 2061 202a 6e6f 7465 2073 7472 * a *note str │ │ │ │ -0027f940: 696e 673a 2053 7472 696e 672c 2c20 7468 ing: String,, th │ │ │ │ -0027f950: 6520 686f 6d65 2064 6972 6563 746f 7279 e home directory │ │ │ │ -0027f960: 206f 6620 7468 6520 7573 6572 0a0a 4465 of the user..De │ │ │ │ -0027f970: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -0027f980: 3d3d 3d3d 3d0a 0a49 6e20 6669 6c65 206f =====..In file o │ │ │ │ -0027f990: 7065 7261 7469 6f6e 732c 2066 696c 6520 perations, file │ │ │ │ -0027f9a0: 6e61 6d65 7320 6265 6769 6e6e 696e 6720 names beginning │ │ │ │ -0027f9b0: 7769 7468 207e 2f20 7769 6c6c 2068 6176 with ~/ will hav │ │ │ │ -0027f9c0: 6520 6974 2072 6570 6c61 6365 6420 7769 e it replaced wi │ │ │ │ -0027f9d0: 7468 2074 6865 0a68 6f6d 6520 6469 7265 th the.home dire │ │ │ │ -0027f9e0: 6374 6f72 792e 0a0a 2b2d 2d2d 2d2d 2d2d ctory...+------- │ │ │ │ -0027f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0027fa00: 203a 2068 6f6d 6544 6972 6563 746f 7279 : homeDirectory │ │ │ │ -0027fa10: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0027fa20: 2020 2020 207c 0a7c 6f31 203d 202f 686f |.|o1 = /ho │ │ │ │ -0027fa30: 6d65 2f6d 3275 7365 722f 7c0a 2b2d 2d2d me/m2user/|.+--- │ │ │ │ -0027fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0027fa50: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0027fa60: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0027fa70: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0027fa80: 6563 7420 2a6e 6f74 6520 686f 6d65 4469 ect *note homeDi │ │ │ │ -0027fa90: 7265 6374 6f72 793a 2068 6f6d 6544 6972 rectory: homeDir │ │ │ │ -0027faa0: 6563 746f 7279 2c20 6973 2061 202a 6e6f ectory, is a *no │ │ │ │ -0027fab0: 7465 2073 7472 696e 673a 2053 7472 696e te string: Strin │ │ │ │ -0027fac0: 672c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d g,...----------- │ │ │ │ -0027fad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027f7c0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +0027f7d0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +0027f7e0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +0027f7f0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +0027f800: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +0027f810: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +0027f820: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +0027f830: 4d61 6361 756c 6179 3244 6f63 2f6f 765f Macaulay2Doc/ov_ │ │ │ │ +0027f840: 7379 7374 656d 2e6d 323a 3230 3235 3a30 system.m2:2025:0 │ │ │ │ +0027f850: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ +0027f860: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ +0027f870: 653a 2068 6f6d 6544 6972 6563 746f 7279 e: homeDirectory │ │ │ │ +0027f880: 2c20 4e65 7874 3a20 6d69 6e69 6d69 7a65 , Next: minimize │ │ │ │ +0027f890: 4669 6c65 6e61 6d65 2c20 5072 6576 3a20 Filename, Prev: │ │ │ │ +0027f8a0: 726f 6f74 5061 7468 2c20 5570 3a20 7379 rootPath, Up: sy │ │ │ │ +0027f8b0: 7374 656d 2066 6163 696c 6974 6965 730a stem facilities. │ │ │ │ +0027f8c0: 0a68 6f6d 6544 6972 6563 746f 7279 202d .homeDirectory - │ │ │ │ +0027f8d0: 2d20 7468 6520 686f 6d65 2064 6972 6563 - the home direc │ │ │ │ +0027f8e0: 746f 7279 206f 6620 7468 6520 7573 6572 tory of the user │ │ │ │ +0027f8f0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +0027f900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0027f910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0027f920: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0027f930: 2020 2020 2020 686f 6d65 4469 7265 6374 homeDirect │ │ │ │ +0027f940: 6f72 790a 2020 2a20 4f75 7470 7574 733a ory. * Outputs: │ │ │ │ +0027f950: 0a20 2020 2020 202a 2061 202a 6e6f 7465 . * a *note │ │ │ │ +0027f960: 2073 7472 696e 673a 2053 7472 696e 672c string: String, │ │ │ │ +0027f970: 2c20 7468 6520 686f 6d65 2064 6972 6563 , the home direc │ │ │ │ +0027f980: 746f 7279 206f 6620 7468 6520 7573 6572 tory of the user │ │ │ │ +0027f990: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0027f9a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6e20 6669 =========..In fi │ │ │ │ +0027f9b0: 6c65 206f 7065 7261 7469 6f6e 732c 2066 le operations, f │ │ │ │ +0027f9c0: 696c 6520 6e61 6d65 7320 6265 6769 6e6e ile names beginn │ │ │ │ +0027f9d0: 696e 6720 7769 7468 207e 2f20 7769 6c6c ing with ~/ will │ │ │ │ +0027f9e0: 2068 6176 6520 6974 2072 6570 6c61 6365 have it replace │ │ │ │ +0027f9f0: 6420 7769 7468 2074 6865 0a68 6f6d 6520 d with the.home │ │ │ │ +0027fa00: 6469 7265 6374 6f72 792e 0a0a 2b2d 2d2d directory...+--- │ │ │ │ +0027fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0027fa20: 0a7c 6931 203a 2068 6f6d 6544 6972 6563 .|i1 : homeDirec │ │ │ │ +0027fa30: 746f 7279 7c0a 7c20 2020 2020 2020 2020 tory|.| │ │ │ │ +0027fa40: 2020 2020 2020 2020 207c 0a7c 6f31 203d |.|o1 = │ │ │ │ +0027fa50: 202f 686f 6d65 2f6d 3275 7365 722f 7c0a /home/m2user/|. │ │ │ │ +0027fa60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0027fa70: 2d2d 2d2b 0a0a 466f 7220 7468 6520 7072 ---+..For the pr │ │ │ │ +0027fa80: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +0027fa90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +0027faa0: 206f 626a 6563 7420 2a6e 6f74 6520 686f object *note ho │ │ │ │ +0027fab0: 6d65 4469 7265 6374 6f72 793a 2068 6f6d meDirectory: hom │ │ │ │ +0027fac0: 6544 6972 6563 746f 7279 2c20 6973 2061 eDirectory, is a │ │ │ │ +0027fad0: 202a 6e6f 7465 2073 7472 696e 673a 2053 *note string: S │ │ │ │ +0027fae0: 7472 696e 672c 2e0a 0a2d 2d2d 2d2d 2d2d tring,...------- │ │ │ │ 0027faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0027fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027fb10: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -0027fb20: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -0027fb30: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -0027fb40: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -0027fb50: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -0027fb60: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -0027fb70: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -0027fb80: 6175 6c61 7932 446f 632f 6f76 5f73 7973 aulay2Doc/ov_sys │ │ │ │ -0027fb90: 7465 6d2e 6d32 3a31 3838 393a 302e 0a1f tem.m2:1889:0... │ │ │ │ -0027fba0: 0a46 696c 653a 204d 6163 6175 6c61 7932 .File: Macaulay2 │ │ │ │ -0027fbb0: 446f 632e 696e 666f 2c20 4e6f 6465 3a20 Doc.info, Node: │ │ │ │ -0027fbc0: 6d69 6e69 6d69 7a65 4669 6c65 6e61 6d65 minimizeFilename │ │ │ │ -0027fbd0: 2c20 4e65 7874 3a20 7265 6c61 7469 7669 , Next: relativi │ │ │ │ -0027fbe0: 7a65 4669 6c65 6e61 6d65 2c20 5072 6576 zeFilename, Prev │ │ │ │ -0027fbf0: 3a20 686f 6d65 4469 7265 6374 6f72 792c : homeDirectory, │ │ │ │ -0027fc00: 2055 703a 2073 7973 7465 6d20 6661 6369 Up: system faci │ │ │ │ -0027fc10: 6c69 7469 6573 0a0a 6d69 6e69 6d69 7a65 lities..minimize │ │ │ │ -0027fc20: 4669 6c65 6e61 6d65 202d 2d20 6d69 6e69 Filename -- mini │ │ │ │ -0027fc30: 6d69 7a65 2061 2066 696c 6520 6e61 6d65 mize a file name │ │ │ │ -0027fc40: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -0027fc50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0027fc60: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0027fc70: 7361 6765 3a20 0a20 2020 2020 2020 206d sage: . m │ │ │ │ -0027fc80: 696e 696d 697a 6546 696c 656e 616d 6520 inimizeFilename │ │ │ │ -0027fc90: 666e 0a20 202a 2049 6e70 7574 733a 0a20 fn. * Inputs:. │ │ │ │ -0027fca0: 2020 2020 202a 2066 6e2c 2061 2070 6174 * fn, a pat │ │ │ │ -0027fcb0: 6820 746f 2061 2066 696c 650a 2020 2a20 h to a file. * │ │ │ │ -0027fcc0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -0027fcd0: 2061 206d 696e 696d 697a 6564 2070 6174 a minimized pat │ │ │ │ -0027fce0: 682c 2065 7175 6976 616c 656e 7420 746f h, equivalent to │ │ │ │ -0027fcf0: 2066 6e0a 0a44 6573 6372 6970 7469 6f6e fn..Description │ │ │ │ -0027fd00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d .===========..+- │ │ │ │ -0027fd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027fd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027fd30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ -0027fd40: 206d 696e 696d 697a 6546 696c 656e 616d minimizeFilenam │ │ │ │ -0027fd50: 6520 2261 2f62 2f63 2f2e 2e2f 6422 2020 e "a/b/c/../d" │ │ │ │ -0027fd60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0027fd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027fd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027fd90: 2020 207c 0a7c 6f31 203d 2061 2f62 2f64 |.|o1 = a/b/d │ │ │ │ +0027fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027fb30: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0027fb40: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0027fb50: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0027fb60: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0027fb70: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0027fb80: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +0027fb90: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0027fba0: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ +0027fbb0: 5f73 7973 7465 6d2e 6d32 3a31 3838 393a _system.m2:1889: │ │ │ │ +0027fbc0: 302e 0a1f 0a46 696c 653a 204d 6163 6175 0....File: Macau │ │ │ │ +0027fbd0: 6c61 7932 446f 632e 696e 666f 2c20 4e6f lay2Doc.info, No │ │ │ │ +0027fbe0: 6465 3a20 6d69 6e69 6d69 7a65 4669 6c65 de: minimizeFile │ │ │ │ +0027fbf0: 6e61 6d65 2c20 4e65 7874 3a20 7265 6c61 name, Next: rela │ │ │ │ +0027fc00: 7469 7669 7a65 4669 6c65 6e61 6d65 2c20 tivizeFilename, │ │ │ │ +0027fc10: 5072 6576 3a20 686f 6d65 4469 7265 6374 Prev: homeDirect │ │ │ │ +0027fc20: 6f72 792c 2055 703a 2073 7973 7465 6d20 ory, Up: system │ │ │ │ +0027fc30: 6661 6369 6c69 7469 6573 0a0a 6d69 6e69 facilities..mini │ │ │ │ +0027fc40: 6d69 7a65 4669 6c65 6e61 6d65 202d 2d20 mizeFilename -- │ │ │ │ +0027fc50: 6d69 6e69 6d69 7a65 2061 2066 696c 6520 minimize a file │ │ │ │ +0027fc60: 6e61 6d65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a name.*********** │ │ │ │ +0027fc70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0027fc80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ +0027fc90: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ +0027fca0: 2020 206d 696e 696d 697a 6546 696c 656e minimizeFilen │ │ │ │ +0027fcb0: 616d 6520 666e 0a20 202a 2049 6e70 7574 ame fn. * Input │ │ │ │ +0027fcc0: 733a 0a20 2020 2020 202a 2066 6e2c 2061 s:. * fn, a │ │ │ │ +0027fcd0: 2070 6174 6820 746f 2061 2066 696c 650a path to a file. │ │ │ │ +0027fce0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +0027fcf0: 2020 202a 2061 206d 696e 696d 697a 6564 * a minimized │ │ │ │ +0027fd00: 2070 6174 682c 2065 7175 6976 616c 656e path, equivalen │ │ │ │ +0027fd10: 7420 746f 2066 6e0a 0a44 6573 6372 6970 t to fn..Descrip │ │ │ │ +0027fd20: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +0027fd30: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +0027fd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0027fd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0027fd60: 6931 203a 206d 696e 696d 697a 6546 696c i1 : minimizeFil │ │ │ │ +0027fd70: 656e 616d 6520 2261 2f62 2f63 2f2e 2e2f ename "a/b/c/../ │ │ │ │ +0027fd80: 6422 2020 2020 2020 2020 7c0a 7c20 2020 d" |.| │ │ │ │ +0027fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0027fda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027fdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027fdc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0027fdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027fde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0027fdf0: 6932 203a 206d 696e 696d 697a 6546 696c i2 : minimizeFil │ │ │ │ -0027fe00: 656e 616d 6520 222e 2e2f 2e2e 2f2e 2e2f ename "../../../ │ │ │ │ -0027fe10: 2e2e 2f2e 2e2f 2e2e 2f22 7c0a 7c20 2020 ../../../"|.| │ │ │ │ -0027fe20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027fe40: 2020 2020 2020 207c 0a7c 6f32 203d 202f |.|o2 = / │ │ │ │ -0027fe50: 6275 696c 642f 2020 2020 2020 2020 2020 build/ │ │ │ │ -0027fe60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0027fe70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0027fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0027fea0: 2d2b 0a0a 5061 7468 7320 6f66 2074 6865 -+..Paths of the │ │ │ │ -0027feb0: 2066 6f72 6d20 666f 6f2f 782f 2e2e 2f62 form foo/x/../b │ │ │ │ -0027fec0: 6172 2c20 6172 6520 7368 6f72 7465 6e65 ar, are shortene │ │ │ │ -0027fed0: 6420 746f 2066 6f6f 2f62 6172 2077 6974 d to foo/bar wit │ │ │ │ -0027fee0: 686f 7574 2063 6865 636b 696e 6720 7468 hout checking th │ │ │ │ -0027fef0: 650a 6669 6c65 2073 7973 7465 6d20 746f e.file system to │ │ │ │ -0027ff00: 2073 6565 2077 6865 7468 6572 2078 2069 see whether x i │ │ │ │ -0027ff10: 7320 6120 7379 6d62 6f6c 6963 206c 696e s a symbolic lin │ │ │ │ -0027ff20: 6b2e 2020 466f 7220 7468 6520 6f74 6865 k. For the othe │ │ │ │ -0027ff30: 7220 6265 6861 7669 6f72 2c20 7365 650a r behavior, see. │ │ │ │ -0027ff40: 2a6e 6f74 6520 7265 616c 7061 7468 3a20 *note realpath: │ │ │ │ -0027ff50: 7265 616c 7061 7468 2c2e 0a0a 5365 6520 realpath,...See │ │ │ │ -0027ff60: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ -0027ff70: 202a 202a 6e6f 7465 2046 696c 653a 2046 * *note File: F │ │ │ │ -0027ff80: 696c 652c 202d 2d20 7468 6520 636c 6173 ile, -- the clas │ │ │ │ -0027ff90: 7320 6f66 2061 6c6c 2066 696c 6573 0a20 s of all files. │ │ │ │ -0027ffa0: 202a 202a 6e6f 7465 2072 656c 6174 6976 * *note relativ │ │ │ │ -0027ffb0: 697a 6546 696c 656e 616d 653a 2072 656c izeFilename: rel │ │ │ │ -0027ffc0: 6174 6976 697a 6546 696c 656e 616d 652c ativizeFilename, │ │ │ │ -0027ffd0: 202d 2d20 7265 6c61 7469 7669 7a65 2061 -- relativize a │ │ │ │ -0027ffe0: 2066 696c 6520 6e61 6d65 0a20 202a 202a file name. * * │ │ │ │ -0027fff0: 6e6f 7465 2062 6173 6546 696c 656e 616d note baseFilenam │ │ │ │ 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│ │ │ │ +002803e0: 2020 2020 202a 2066 6e2c 2061 2070 6174 * fn, a pat │ │ │ │ +002803f0: 6820 746f 2061 2066 696c 650a 2020 2a20 h to a file. * │ │ │ │ +00280400: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ +00280410: 2061 2072 656c 6174 6976 697a 6564 2070 a relativized p │ │ │ │ +00280420: 6174 682c 2065 7175 6976 616c 656e 7420 ath, equivalent │ │ │ │ +00280430: 746f 2066 6e20 7768 656e 2073 7461 7274 to fn when start │ │ │ │ +00280440: 696e 6720 6672 6f6d 2064 6972 0a0a 4465 ing from dir..De │ │ │ │ +00280450: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00280460: 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d 2d2d 2d2d =====..+-------- │ │ │ │ +00280470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00280480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00280490: 2d2d 2d2d 2b0a 7c69 3120 3a20 7265 6c61 ----+.|i1 : rela │ │ │ │ +002804a0: 7469 7669 7a65 4669 6c65 6e61 6d65 2822 tivizeFilename(" │ │ │ │ +002804b0: 612f 622f 222c 2261 2f62 2f63 2f64 2229 a/b/","a/b/c/d") │ │ │ │ +002804c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +002804d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 002804e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002804f0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00280500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00280510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00280520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00280530: 3220 3a20 7265 6c61 7469 7669 7a65 4669 2 : relativizeFi │ │ │ │ -00280540: 6c65 6e61 6d65 2822 612f 622f 632f 6422 lename("a/b/c/d" │ │ │ │ -00280550: 2c22 612f 622f 2229 2020 207c 0a7c 2020 ,"a/b/") |.| │ │ │ │ -00280560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00280570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00280580: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -00280590: 3d20 2e2e 2f2e 2e2f 2020 2020 2020 2020 = ../../ │ │ │ │ -002805a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002805b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -002805c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002805d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002805e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -002805f0: 7265 6c61 7469 7669 7a65 4669 6c65 6e61 relativizeFilena │ │ │ │ -00280600: 6d65 2822 612f 622f 632f 6422 2c22 612f me("a/b/c/d","a/ │ │ │ │ -00280610: 622f 652f 6622 297c 0a7c 2020 2020 2020 b/e/f")|.| │ │ │ │ -00280620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00280630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00280640: 2020 2020 2020 7c0a 7c6f 3320 3d20 2e2e |.|o3 = .. │ │ │ │ -00280650: 2f2e 2e2f 652f 6620 2020 2020 2020 2020 /../e/f │ │ │ │ -00280660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00280670: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -00280680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00280690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002806a0: 2d2d 2d2d 2b0a 0a50 6174 6873 206f 6620 ----+..Paths of │ │ │ │ -002806b0: 7468 6520 666f 726d 2066 6f6f 2f78 2f2e the form foo/x/. │ │ │ │ -002806c0: 2e2f 6261 722c 2061 7265 2073 686f 7274 ./bar, are short │ │ │ │ -002806d0: 656e 6564 2074 6f20 666f 6f2f 6261 7220 ened to foo/bar │ │ │ │ -002806e0: 7769 7468 6f75 7420 6368 6563 6b69 6e67 without checking │ │ │ │ -002806f0: 2074 6865 0a66 696c 6520 7379 7374 656d the.file system │ │ │ │ -00280700: 2074 6f20 7365 6520 7768 6574 6865 7220 to see whether │ │ │ │ -00280710: 7820 6973 2061 2073 796d 626f 6c69 6320 x is a symbolic │ │ │ │ -00280720: 6c69 6e6b 2e20 2046 6f72 2074 6865 206f link. For the o │ │ │ │ -00280730: 7468 6572 2062 6568 6176 696f 722c 2073 ther behavior, s │ │ │ │ -00280740: 6565 0a2a 6e6f 7465 2072 6561 6c70 6174 ee.*note realpat │ │ │ │ -00280750: 683a 2072 6561 6c70 6174 682c 2e0a 0a53 h: realpath,...S │ │ │ │ -00280760: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -00280770: 0a0a 2020 2a20 2a6e 6f74 6520 4669 6c65 .. * *note File │ │ │ │ -00280780: 3a20 4669 6c65 2c20 2d2d 2074 6865 2063 : File, -- the c │ │ │ │ -00280790: 6c61 7373 206f 6620 616c 6c20 6669 6c65 lass of all file │ │ │ │ -002807a0: 730a 2020 2a20 2a6e 6f74 6520 6d69 6e69 s. * *note mini │ │ │ │ -002807b0: 6d69 7a65 4669 6c65 6e61 6d65 3a20 6d69 mizeFilename: mi │ │ │ │ -002807c0: 6e69 6d69 7a65 4669 6c65 6e61 6d65 2c20 nimizeFilename, │ │ │ │ -002807d0: 2d2d 206d 696e 696d 697a 6520 6120 6669 -- minimize a fi │ │ │ │ -002807e0: 6c65 206e 616d 650a 2020 2a20 2a6e 6f74 le name. * *not │ │ │ │ -002807f0: 6520 6261 7365 4669 6c65 6e61 6d65 3a20 e baseFilename: │ │ │ │ -00280800: 6261 7365 4669 6c65 6e61 6d65 2c20 2d2d baseFilename, -- │ │ │ │ -00280810: 2074 6865 2062 6173 6520 7061 7274 206f the base part o │ │ │ │ -00280820: 6620 6120 6669 6c65 6e61 6d65 206f 7220 f a filename or │ │ │ │ -00280830: 7061 7468 0a20 202a 202a 6e6f 7465 2074 path. * *note t │ │ │ │ -00280840: 6f41 6273 6f6c 7574 6550 6174 683a 2074 oAbsolutePath: t │ │ │ │ -00280850: 6f41 6273 6f6c 7574 6550 6174 682c 202d oAbsolutePath, - │ │ │ │ -00280860: 2d20 7468 6520 6162 736f 6c75 7465 2070 - the absolute p │ │ │ │ -00280870: 6174 6820 7665 7273 696f 6e20 6f66 2061 ath version of a │ │ │ │ -00280880: 0a20 2020 2066 696c 6520 6e61 6d65 0a20 . file name. │ │ │ │ -00280890: 202a 202a 6e6f 7465 2073 6561 7263 6850 * *note searchP │ │ │ │ -002808a0: 6174 683a 2073 6561 7263 6850 6174 685f ath: searchPath_ │ │ │ │ -002808b0: 6c70 4c69 7374 5f63 6d53 7472 696e 675f lpList_cmString_ │ │ │ │ -002808c0: 7270 2c20 2d2d 2073 6561 7263 6820 6120 rp, -- search a │ │ │ │ -002808d0: 7061 7468 2066 6f72 2061 0a20 2020 2066 path for a. f │ │ │ │ -002808e0: 696c 650a 2020 2a20 2a6e 6f74 6520 7061 ile. * *note pa │ │ │ │ -002808f0: 7468 3a20 7061 7468 2c20 2d2d 206c 6973 th: path, -- lis │ │ │ │ -00280900: 7420 6f66 2064 6972 6563 746f 7269 6573 t of directories │ │ │ │ -00280910: 2074 6f20 6c6f 6f6b 2069 6e0a 0a46 6f72 to look in..For │ │ │ │ -00280920: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -00280930: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00280940: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -00280950: 6e6f 7465 2072 656c 6174 6976 697a 6546 note relativizeF │ │ │ │ -00280960: 696c 656e 616d 653a 2072 656c 6174 6976 ilename: relativ │ │ │ │ -00280970: 697a 6546 696c 656e 616d 652c 2069 7320 izeFilename, is │ │ │ │ -00280980: 6120 2a6e 6f74 6520 636f 6d70 696c 6564 a *note compiled │ │ │ │ -00280990: 0a66 756e 6374 696f 6e3a 2043 6f6d 7069 .function: Compi │ │ │ │ -002809a0: 6c65 6446 756e 6374 696f 6e2c 2e0a 0a2d ledFunction,...- │ │ │ │ -002809b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002809c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002809d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002804f0: 2020 7c0a 7c6f 3120 3d20 632f 6420 2020 |.|o1 = c/d │ │ │ │ +00280500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00280510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00280520: 207c 0a2b 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: 2b0a 7c69 3220 3a20 7265 6c61 7469 7669 +.|i2 : relativi │ │ │ │ +00280560: 7a65 4669 6c65 6e61 6d65 2822 612f 622f zeFilename("a/b/ │ │ │ │ +00280570: 632f 6422 2c22 612f 622f 2229 2020 207c c/d","a/b/") | │ │ │ │ +00280580: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00280590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002805a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +002805b0: 7c6f 3220 3d20 2e2e 2f2e 2e2f 2020 2020 |o2 = ../../ │ │ │ │ +002805c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002805d0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +002805e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002805f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00280600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00280610: 3320 3a20 7265 6c61 7469 7669 7a65 4669 3 : relativizeFi │ │ │ │ +00280620: 6c65 6e61 6d65 2822 612f 622f 632f 6422 lename("a/b/c/d" │ │ │ │ +00280630: 2c22 612f 622f 652f 6622 297c 0a7c 2020 ,"a/b/e/f")|.| │ │ │ │ +00280640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00280650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00280660: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +00280670: 3d20 2e2e 2f2e 2e2f 652f 6620 2020 2020 = ../../e/f │ │ │ │ +00280680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00280690: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +002806a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002806b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002806c0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a50 6174 6873 --------+..Paths │ │ │ │ +002806d0: 206f 6620 7468 6520 666f 726d 2066 6f6f of the form foo │ │ │ │ +002806e0: 2f78 2f2e 2e2f 6261 722c 2061 7265 2073 /x/../bar, are s │ │ │ │ +002806f0: 686f 7274 656e 6564 2074 6f20 666f 6f2f hortened to foo/ │ │ │ │ +00280700: 6261 7220 7769 7468 6f75 7420 6368 6563 bar without chec │ │ │ │ +00280710: 6b69 6e67 2074 6865 0a66 696c 6520 7379 king the.file sy │ │ │ │ +00280720: 7374 656d 2074 6f20 7365 6520 7768 6574 stem to see whet │ │ │ │ +00280730: 6865 7220 7820 6973 2061 2073 796d 626f her x is a symbo │ │ │ │ +00280740: 6c69 6320 6c69 6e6b 2e20 2046 6f72 2074 lic link. For t │ │ │ │ +00280750: 6865 206f 7468 6572 2062 6568 6176 696f he other behavio │ │ │ │ +00280760: 722c 2073 6565 0a2a 6e6f 7465 2072 6561 r, see.*note rea │ │ │ │ +00280770: 6c70 6174 683a 2072 6561 6c70 6174 682c lpath: realpath, │ │ │ │ +00280780: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ +00280790: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +002807a0: 4669 6c65 3a20 4669 6c65 2c20 2d2d 2074 File: File, -- t │ │ │ │ +002807b0: 6865 2063 6c61 7373 206f 6620 616c 6c20 he class of all │ │ │ │ +002807c0: 6669 6c65 730a 2020 2a20 2a6e 6f74 6520 files. * *note │ │ │ │ +002807d0: 6d69 6e69 6d69 7a65 4669 6c65 6e61 6d65 minimizeFilename │ │ │ │ +002807e0: 3a20 6d69 6e69 6d69 7a65 4669 6c65 6e61 : minimizeFilena │ │ │ │ +002807f0: 6d65 2c20 2d2d 206d 696e 696d 697a 6520 me, -- minimize │ │ │ │ +00280800: 6120 6669 6c65 206e 616d 650a 2020 2a20 a file name. * │ │ │ │ +00280810: 2a6e 6f74 6520 6261 7365 4669 6c65 6e61 *note baseFilena │ │ │ │ +00280820: 6d65 3a20 6261 7365 4669 6c65 6e61 6d65 me: baseFilename │ │ │ │ +00280830: 2c20 2d2d 2074 6865 2062 6173 6520 7061 , -- the base pa │ │ │ │ +00280840: 7274 206f 6620 6120 6669 6c65 6e61 6d65 rt of a filename │ │ │ │ +00280850: 206f 7220 7061 7468 0a20 202a 202a 6e6f or path. * *no │ │ │ │ +00280860: 7465 2074 6f41 6273 6f6c 7574 6550 6174 te toAbsolutePat │ │ │ │ +00280870: 683a 2074 6f41 6273 6f6c 7574 6550 6174 h: toAbsolutePat │ │ │ │ +00280880: 682c 202d 2d20 7468 6520 6162 736f 6c75 h, -- the absolu │ │ │ │ +00280890: 7465 2070 6174 6820 7665 7273 696f 6e20 te path version │ │ │ │ +002808a0: 6f66 2061 0a20 2020 2066 696c 6520 6e61 of a. file na │ │ │ │ +002808b0: 6d65 0a20 202a 202a 6e6f 7465 2073 6561 me. * *note sea │ │ │ │ +002808c0: 7263 6850 6174 683a 2073 6561 7263 6850 rchPath: searchP │ │ │ │ +002808d0: 6174 685f 6c70 4c69 7374 5f63 6d53 7472 ath_lpList_cmStr │ │ │ │ +002808e0: 696e 675f 7270 2c20 2d2d 2073 6561 7263 ing_rp, -- searc │ │ │ │ +002808f0: 6820 6120 7061 7468 2066 6f72 2061 0a20 h a path for a. │ │ │ │ +00280900: 2020 2066 696c 650a 2020 2a20 2a6e 6f74 file. * *not │ │ │ │ +00280910: 6520 7061 7468 3a20 7061 7468 2c20 2d2d e path: path, -- │ │ │ │ +00280920: 206c 6973 7420 6f66 2064 6972 6563 746f list of directo │ │ │ │ +00280930: 7269 6573 2074 6f20 6c6f 6f6b 2069 6e0a ries to look in. │ │ │ │ +00280940: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +00280950: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +00280960: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +00280970: 6374 202a 6e6f 7465 2072 656c 6174 6976 ct *note relativ │ │ │ │ +00280980: 697a 6546 696c 656e 616d 653a 2072 656c izeFilename: rel │ │ │ │ +00280990: 6174 6976 697a 6546 696c 656e 616d 652c ativizeFilename, │ │ │ │ +002809a0: 2069 7320 6120 2a6e 6f74 6520 636f 6d70 is a *note comp │ │ │ │ +002809b0: 696c 6564 0a66 756e 6374 696f 6e3a 2043 iled.function: C │ │ │ │ +002809c0: 6f6d 7069 6c65 6446 756e 6374 696f 6e2c ompiledFunction, │ │ │ │ +002809d0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 002809e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002809f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -00280a00: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -00280a10: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -00280a20: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -00280a30: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -00280a40: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -00280a50: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00280a60: 6b61 6765 732f 0a4d 6163 6175 6c61 7932 kages/.Macaulay2 │ │ │ │ -00280a70: 446f 632f 6f76 5f73 7973 7465 6d2e 6d32 Doc/ov_system.m2 │ │ │ │ -00280a80: 3a31 3430 333a 302e 0a1f 0a46 696c 653a :1403:0....File: │ │ │ │ -00280a90: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ -00280aa0: 666f 2c20 4e6f 6465 3a20 746f 4162 736f fo, Node: toAbso │ │ │ │ -00280ab0: 6c75 7465 5061 7468 2c20 4e65 7874 3a20 lutePath, Next: │ │ │ │ -00280ac0: 7365 6172 6368 5061 7468 5f6c 704c 6973 searchPath_lpLis │ │ │ │ -00280ad0: 745f 636d 5374 7269 6e67 5f72 702c 2050 t_cmString_rp, P │ │ │ │ -00280ae0: 7265 763a 2072 656c 6174 6976 697a 6546 rev: relativizeF │ │ │ │ -00280af0: 696c 656e 616d 652c 2055 703a 2073 7973 ilename, Up: sys │ │ │ │ -00280b00: 7465 6d20 6661 6369 6c69 7469 6573 0a0a tem facilities.. │ │ │ │ -00280b10: 746f 4162 736f 6c75 7465 5061 7468 202d toAbsolutePath - │ │ │ │ -00280b20: 2d20 7468 6520 6162 736f 6c75 7465 2070 - the absolute p │ │ │ │ -00280b30: 6174 6820 7665 7273 696f 6e20 6f66 2061 ath version of a │ │ │ │ -00280b40: 2066 696c 6520 6e61 6d65 0a2a 2a2a 2a2a file name.***** │ │ │ │ -00280b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00280b60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002809f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00280a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00280a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00280a20: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00280a30: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00280a40: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00280a50: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00280a60: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +00280a70: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +00280a80: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ +00280a90: 6c61 7932 446f 632f 6f76 5f73 7973 7465 lay2Doc/ov_syste │ │ │ │ +00280aa0: 6d2e 6d32 3a31 3430 333a 302e 0a1f 0a46 m.m2:1403:0....F │ │ │ │ +00280ab0: 696c 653a 204d 6163 6175 6c61 7932 446f ile: Macaulay2Do │ │ │ │ +00280ac0: 632e 696e 666f 2c20 4e6f 6465 3a20 746f c.info, Node: to │ │ │ │ +00280ad0: 4162 736f 6c75 7465 5061 7468 2c20 4e65 AbsolutePath, Ne │ │ │ │ +00280ae0: 7874 3a20 7365 6172 6368 5061 7468 5f6c xt: searchPath_l │ │ │ │ +00280af0: 704c 6973 745f 636d 5374 7269 6e67 5f72 pList_cmString_r │ │ │ │ +00280b00: 702c 2050 7265 763a 2072 656c 6174 6976 p, Prev: relativ │ │ │ │ +00280b10: 697a 6546 696c 656e 616d 652c 2055 703a izeFilename, Up: │ │ │ │ +00280b20: 2073 7973 7465 6d20 6661 6369 6c69 7469 system faciliti │ │ │ │ +00280b30: 6573 0a0a 746f 4162 736f 6c75 7465 5061 es..toAbsolutePa │ │ │ │ +00280b40: 7468 202d 2d20 7468 6520 6162 736f 6c75 th -- the absolu │ │ │ │ +00280b50: 7465 2070 6174 6820 7665 7273 696f 6e20 te path version │ │ │ │ +00280b60: 6f66 2061 2066 696c 6520 6e61 6d65 0a2a of a file name.* │ │ │ │ 00280b70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00280b80: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -00280b90: 3a20 0a20 2020 2020 2020 2074 6f41 6273 : . toAbs │ │ │ │ -00280ba0: 6f6c 7574 6550 6174 6820 6669 6c65 6e61 olutePath filena │ │ │ │ -00280bb0: 6d65 0a20 202a 2049 6e70 7574 733a 0a20 me. * Inputs:. │ │ │ │ -00280bc0: 2020 2020 202a 2066 696c 656e 616d 652c * filename, │ │ │ │ -00280bd0: 2061 202a 6e6f 7465 2073 7472 696e 673a a *note string: │ │ │ │ -00280be0: 2053 7472 696e 672c 0a20 202a 204f 7574 String,. * Out │ │ │ │ -00280bf0: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ -00280c00: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374 *note string: St │ │ │ │ -00280c10: 7269 6e67 2c2c 2074 6865 2061 6273 6f6c ring,, the absol │ │ │ │ -00280c20: 7574 6520 2872 6561 6c29 2070 6174 6820 ute (real) path │ │ │ │ -00280c30: 6e61 6d65 206f 6620 6669 6c65 6e61 6d65 name of filename │ │ │ │ -00280c40: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -00280c50: 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d 2d2d =========..+---- │ │ │ │ -00280c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00280c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00280c80: 7c69 3120 3a20 746f 4162 736f 6c75 7465 |i1 : toAbsolute │ │ │ │ -00280c90: 5061 7468 2022 612f 622e 6d32 2220 2020 Path "a/b.m2" │ │ │ │ -00280ca0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00280cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00280cc0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -00280cd0: 2f75 7372 2f73 6861 7265 2f4d 6163 6175 /usr/share/Macau │ │ │ │ -00280ce0: 6c61 7932 2f20 2020 2020 2020 207c 0a7c lay2/ |.| │ │ │ │ -00280cf0: 2020 2020 204d 6163 6175 6c61 7932 446f Macaulay2Do │ │ │ │ -00280d00: 632d 7465 6d70 6f72 6172 792f 612f 622e c-temporary/a/b. │ │ │ │ -00280d10: 6d32 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d m2|.+----------- │ │ │ │ -00280d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00280d30: 2d2d 2d2d 2d2d 2d2b 0a0a 5061 7468 7320 -------+..Paths │ │ │ │ -00280d40: 6f66 2074 6865 2066 6f72 6d20 666f 6f2f of the form foo/ │ │ │ │ -00280d50: 782f 2e2e 2f62 6172 2c20 6172 6520 7368 x/../bar, are sh │ │ │ │ -00280d60: 6f72 7465 6e65 6420 746f 2066 6f6f 2f62 ortened to foo/b │ │ │ │ -00280d70: 6172 2077 6974 686f 7574 2063 6865 636b ar without check │ │ │ │ -00280d80: 696e 6720 7468 650a 6669 6c65 2073 7973 ing the.file sys │ │ │ │ -00280d90: 7465 6d20 746f 2073 6565 2077 6865 7468 tem to see wheth │ │ │ │ -00280da0: 6572 2078 2069 7320 6120 7379 6d62 6f6c er x is a symbol │ │ │ │ -00280db0: 6963 206c 696e 6b2e 2020 466f 7220 7468 ic link. For th │ │ │ │ -00280dc0: 6520 6f74 6865 7220 6265 6861 7669 6f72 e other behavior │ │ │ │ -00280dd0: 2c20 7365 650a 2a6e 6f74 6520 7265 616c , see.*note real │ │ │ │ -00280de0: 7061 7468 3a20 7265 616c 7061 7468 2c2e path: realpath,. │ │ │ │ -00280df0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00280e00: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2046 ===.. * *note F │ │ │ │ -00280e10: 696c 653a 2046 696c 652c 202d 2d20 7468 ile: File, -- th │ │ │ │ -00280e20: 6520 636c 6173 7320 6f66 2061 6c6c 2066 e class of all f │ │ │ │ -00280e30: 696c 6573 0a20 202a 202a 6e6f 7465 206d iles. * *note m │ │ │ │ -00280e40: 696e 696d 697a 6546 696c 656e 616d 653a inimizeFilename: │ │ │ │ -00280e50: 206d 696e 696d 697a 6546 696c 656e 616d minimizeFilenam │ │ │ │ -00280e60: 652c 202d 2d20 6d69 6e69 6d69 7a65 2061 e, -- minimize a │ │ │ │ -00280e70: 2066 696c 6520 6e61 6d65 0a20 202a 202a file name. * * │ │ │ │ -00280e80: 6e6f 7465 2072 656c 6174 6976 697a 6546 note relativizeF │ │ │ │ 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5061 7468 2c0a 2020 2a20 2a6e rootPath,. * *n │ │ │ │ -00280f70: 6f74 6520 726f 6f74 5552 493a 2072 6f6f ote rootURI: roo │ │ │ │ -00280f80: 7455 5249 2c0a 0a46 6f72 2074 6865 2070 tURI,..For the p │ │ │ │ -00280f90: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -00280fa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -00280fb0: 6520 6f62 6a65 6374 202a 6e6f 7465 2074 e object *note t │ │ │ │ -00280fc0: 6f41 6273 6f6c 7574 6550 6174 683a 2074 oAbsolutePath: t │ │ │ │ -00280fd0: 6f41 6273 6f6c 7574 6550 6174 682c 2069 oAbsolutePath, i │ │ │ │ -00280fe0: 7320 6120 2a6e 6f74 6520 6675 6e63 7469 s a *note functi │ │ │ │ -00280ff0: 6f6e 2063 6c6f 7375 7265 3a0a 4675 6e63 on closure:.Func │ │ │ │ -00281000: 7469 6f6e 436c 6f73 7572 652c 2e0a 0a2d tionClosure,...- │ │ │ │ -00281010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00281020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00281030: 2d2d 2d2d 2d2d 2d2d 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│ │ │ │ -00281180: 2d20 7365 6172 6368 2061 2070 6174 6820 - search a path │ │ │ │ -00281190: 666f 7220 6120 6669 6c65 0a2a 2a2a 2a2a for a file.***** │ │ │ │ -002811a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002811b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002811c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -002811d0: 2020 2a20 4675 6e63 7469 6f6e 3a20 2a6e * Function: *n │ │ │ │ -002811e0: 6f74 6520 7365 6172 6368 5061 7468 3a20 ote searchPath: │ │ │ │ -002811f0: 7365 6172 6368 5061 7468 5f6c 704c 6973 searchPath_lpLis │ │ │ │ -00281200: 745f 636d 5374 7269 6e67 5f72 702c 0a20 t_cmString_rp,. │ │ │ │ -00281210: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -00281220: 2020 2073 6561 7263 6850 6174 6828 7061 searchPath(pa │ │ │ │ -00281230: 2c66 6e29 0a20 2020 2020 2020 2073 6561 ,fn). sea │ │ │ │ -00281240: 7263 6850 6174 6820 666e 0a20 202a 2049 rchPath fn. * I │ │ │ │ -00281250: 6e70 7574 733a 0a20 2020 2020 202a 2070 nputs:. * p │ │ │ │ -00281260: 612c 2061 202a 6e6f 7465 206c 6973 743a a, a *note list: │ │ │ │ -00281270: 204c 6973 742c 2c20 6120 6c69 7374 206f List,, a list o │ │ │ │ -00281280: 6620 7374 7269 6e67 7320 6769 7669 6e67 f strings giving │ │ │ │ -00281290: 2070 6174 6873 2074 6f20 6469 7265 6374 paths to direct │ │ │ │ -002812a0: 6f72 6965 732e 0a20 2020 2020 2020 2045 ories.. E │ │ │ │ -002812b0: 6163 6820 6f6e 6520 656e 6473 2077 6974 ach one ends wit │ │ │ │ -002812c0: 6820 6120 736c 6173 682e 2049 6620 6f6d h a slash. 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to use this m │ │ │ │ -00281550: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ -00281560: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00281570: 0a20 202a 202a 6e6f 7465 2073 6561 7263 . * *note searc │ │ │ │ -00281580: 6850 6174 6828 4c69 7374 2c53 7472 696e hPath(List,Strin │ │ │ │ -00281590: 6729 3a20 7365 6172 6368 5061 7468 5f6c g): searchPath_l │ │ │ │ -002815a0: 704c 6973 745f 636d 5374 7269 6e67 5f72 pList_cmString_r │ │ │ │ -002815b0: 702c 202d 2d20 7365 6172 6368 2061 0a20 p, -- search a. │ │ │ │ -002815c0: 2020 2070 6174 6820 666f 7220 6120 6669 path for a fi │ │ │ │ -002815d0: 6c65 0a20 202a 2022 7365 6172 6368 5061 le. * "searchPa │ │ │ │ -002815e0: 7468 2853 7472 696e 6729 220a 2d2d 2d2d th(String)".---- │ │ │ │ -002815f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00281600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281080: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00281090: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +002810a0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +002810b0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +002810c0: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +002810d0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +002810e0: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ +002810f0: 6c61 7932 446f 632f 6f76 5f73 7973 7465 lay2Doc/ov_syste │ │ │ │ +00281100: 6d2e 6d32 3a31 3335 313a 302e 0a1f 0a46 m.m2:1351:0....F │ │ │ │ +00281110: 696c 653a 204d 6163 6175 6c61 7932 446f ile: Macaulay2Do │ │ │ │ +00281120: 632e 696e 666f 2c20 4e6f 6465 3a20 7365 c.info, Node: se │ │ │ │ +00281130: 6172 6368 5061 7468 5f6c 704c 6973 745f archPath_lpList_ │ │ │ │ +00281140: 636d 5374 7269 6e67 5f72 702c 204e 6578 cmString_rp, Nex │ │ │ │ +00281150: 743a 2063 6c65 6172 4563 686f 2c20 5072 t: clearEcho, Pr │ │ │ │ +00281160: 6576 3a20 746f 4162 736f 6c75 7465 5061 ev: toAbsolutePa │ │ │ │ +00281170: 7468 2c20 5570 3a20 7379 7374 656d 2066 th, Up: system f │ │ │ │ +00281180: 6163 696c 6974 6965 730a 0a73 6561 7263 acilities..searc │ │ │ │ +00281190: 6850 6174 6828 4c69 7374 2c53 7472 696e hPath(List,Strin │ │ │ │ +002811a0: 6729 202d 2d20 7365 6172 6368 2061 2070 g) -- search a p │ │ │ │ +002811b0: 6174 6820 666f 7220 6120 6669 6c65 0a2a ath for a file.* │ │ │ │ +002811c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002811d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002811e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002811f0: 2a2a 0a0a 2020 2a20 4675 6e63 7469 6f6e **.. * Function │ │ │ │ +00281200: 3a20 2a6e 6f74 6520 7365 6172 6368 5061 : *note searchPa │ │ │ │ +00281210: 7468 3a20 7365 6172 6368 5061 7468 5f6c th: searchPath_l │ │ │ │ +00281220: 704c 6973 745f 636d 5374 7269 6e67 5f72 pList_cmString_r │ │ │ │ +00281230: 702c 0a20 202a 2055 7361 6765 3a20 0a20 p,. * Usage: . │ │ │ │ +00281240: 2020 2020 2020 2073 6561 7263 6850 6174 searchPat │ │ │ │ +00281250: 6828 7061 2c66 6e29 0a20 2020 2020 2020 h(pa,fn). │ │ │ │ +00281260: 2073 6561 7263 6850 6174 6820 666e 0a20 searchPath fn. │ │ │ │ +00281270: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +00281280: 202a 2070 612c 2061 202a 6e6f 7465 206c * pa, a *note l │ │ │ │ +00281290: 6973 743a 204c 6973 742c 2c20 6120 6c69 ist: List,, a li │ │ │ │ +002812a0: 7374 206f 6620 7374 7269 6e67 7320 6769 st of strings gi │ │ │ │ +002812b0: 7669 6e67 2070 6174 6873 2074 6f20 6469 ving paths to di │ │ │ │ +002812c0: 7265 6374 6f72 6965 732e 0a20 2020 2020 rectories.. │ │ │ │ +002812d0: 2020 2045 6163 6820 6f6e 6520 656e 6473 Each one ends │ │ │ │ +002812e0: 2077 6974 6820 6120 736c 6173 682e 2049 with a slash. I │ │ │ │ +002812f0: 6620 6f6d 6974 7465 642c 2074 6865 6e20 f omitted, then │ │ │ │ +00281300: 2a6e 6f74 6520 7061 7468 3a20 7061 7468 *note path: path │ │ │ │ +00281310: 2c20 6973 2075 7365 640a 2020 2020 2020 , is used. │ │ │ │ +00281320: 2a20 666e 2c20 6120 2a6e 6f74 6520 7374 * fn, a *note st │ │ │ │ +00281330: 7269 6e67 3a20 5374 7269 6e67 2c0a 2020 ring: String,. │ │ │ │ +00281340: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00281350: 202a 2061 202a 6e6f 7465 206c 6973 743a * a *note list: │ │ │ │ +00281360: 204c 6973 742c 2c20 6120 6c69 7374 206f List,, a list o │ │ │ │ +00281370: 6620 7468 6f73 6520 6469 7265 6374 6f72 f those director │ │ │ │ +00281380: 6965 7320 696e 2070 6120 636f 6e74 6169 ies in pa contai │ │ │ │ +00281390: 6e69 6e67 2066 696c 6573 0a20 2020 2020 ning files. │ │ │ │ +002813a0: 2020 206e 616d 6564 2066 6e0a 0a53 6565 named fn..See │ │ │ │ +002813b0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ +002813c0: 2020 2a20 2a6e 6f74 6520 4669 6c65 3a20 * *note File: │ │ │ │ +002813d0: 4669 6c65 2c20 2d2d 2074 6865 2063 6c61 File, -- the cla │ │ │ │ +002813e0: 7373 206f 6620 616c 6c20 6669 6c65 730a ss of all files. │ │ │ │ +002813f0: 2020 2a20 2a6e 6f74 6520 6d69 6e69 6d69 * *note minimi │ │ │ │ +00281400: 7a65 4669 6c65 6e61 6d65 3a20 6d69 6e69 zeFilename: mini │ │ │ │ +00281410: 6d69 7a65 4669 6c65 6e61 6d65 2c20 2d2d mizeFilename, -- │ │ │ │ +00281420: 206d 696e 696d 697a 6520 6120 6669 6c65 minimize a file │ │ │ │ +00281430: 206e 616d 650a 2020 2a20 2a6e 6f74 6520 name. * *note │ │ │ │ +00281440: 7265 6c61 7469 7669 7a65 4669 6c65 6e61 relativizeFilena │ │ │ │ +00281450: 6d65 3a20 7265 6c61 7469 7669 7a65 4669 me: relativizeFi │ │ │ │ +00281460: 6c65 6e61 6d65 2c20 2d2d 2072 656c 6174 lename, -- relat │ │ │ │ +00281470: 6976 697a 6520 6120 6669 6c65 206e 616d ivize a file nam │ │ │ │ +00281480: 650a 2020 2a20 2a6e 6f74 6520 6261 7365 e. * *note base │ │ │ │ +00281490: 4669 6c65 6e61 6d65 3a20 6261 7365 4669 Filename: baseFi │ │ │ │ +002814a0: 6c65 6e61 6d65 2c20 2d2d 2074 6865 2062 lename, -- the b │ │ │ │ +002814b0: 6173 6520 7061 7274 206f 6620 6120 6669 ase part of a fi │ │ │ │ +002814c0: 6c65 6e61 6d65 206f 7220 7061 7468 0a20 lename or path. │ │ │ │ +002814d0: 202a 202a 6e6f 7465 2074 6f41 6273 6f6c * *note toAbsol │ │ │ │ +002814e0: 7574 6550 6174 683a 2074 6f41 6273 6f6c utePath: toAbsol │ │ │ │ +002814f0: 7574 6550 6174 682c 202d 2d20 7468 6520 utePath, -- the │ │ │ │ +00281500: 6162 736f 6c75 7465 2070 6174 6820 7665 absolute path ve │ │ │ │ +00281510: 7273 696f 6e20 6f66 2061 0a20 2020 2066 rsion of a. f │ │ │ │ +00281520: 696c 6520 6e61 6d65 0a20 202a 202a 6e6f ile name. * *no │ │ │ │ +00281530: 7465 2070 6174 683a 2070 6174 682c 202d te path: path, - │ │ │ │ +00281540: 2d20 6c69 7374 206f 6620 6469 7265 6374 - list of direct │ │ │ │ +00281550: 6f72 6965 7320 746f 206c 6f6f 6b20 696e ories to look in │ │ │ │ +00281560: 0a0a 5761 7973 2074 6f20 7573 6520 7468 ..Ways to use th │ │ │ │ +00281570: 6973 206d 6574 686f 643a 0a3d 3d3d 3d3d is method:.===== │ │ │ │ +00281580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00281590: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2073 ===.. * *note s │ │ │ │ +002815a0: 6561 7263 6850 6174 6828 4c69 7374 2c53 earchPath(List,S │ │ │ │ +002815b0: 7472 696e 6729 3a20 7365 6172 6368 5061 tring): searchPa │ │ │ │ +002815c0: 7468 5f6c 704c 6973 745f 636d 5374 7269 th_lpList_cmStri │ │ │ │ +002815d0: 6e67 5f72 702c 202d 2d20 7365 6172 6368 ng_rp, -- search │ │ │ │ +002815e0: 2061 0a20 2020 2070 6174 6820 666f 7220 a. path for │ │ │ │ +002815f0: 6120 6669 6c65 0a20 202a 2022 7365 6172 a file. * "sear │ │ │ │ +00281600: 6368 5061 7468 2853 7472 696e 6729 220a chPath(String)". │ │ │ │ 00281610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00281620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00281630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -00281640: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -00281650: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -00281660: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -00281670: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -00281680: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ -00281690: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -002816a0: 6573 2f0a 4d61 6361 756c 6179 3244 6f63 es/.Macaulay2Doc │ │ │ │ -002816b0: 2f6f 765f 7379 7374 656d 2e6d 323a 3133 /ov_system.m2:13 │ │ │ │ -002816c0: 3730 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 70:0....File: Ma │ │ │ │ -002816d0: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ -002816e0: 204e 6f64 653a 2063 6c65 6172 4563 686f Node: clearEcho │ │ │ │ -002816f0: 2c20 4e65 7874 3a20 7365 7445 6368 6f2c , Next: setEcho, │ │ │ │ -00281700: 2050 7265 763a 2073 6561 7263 6850 6174 Prev: searchPat │ │ │ │ -00281710: 685f 6c70 4c69 7374 5f63 6d53 7472 696e h_lpList_cmStrin │ │ │ │ -00281720: 675f 7270 2c20 5570 3a20 7379 7374 656d g_rp, Up: system │ │ │ │ -00281730: 2066 6163 696c 6974 6965 730a 0a63 6c65 facilities..cle │ │ │ │ -00281740: 6172 4563 686f 202d 2d20 7475 726e 206f arEcho -- turn o │ │ │ │ -00281750: 6666 2065 6368 6f69 6e67 0a2a 2a2a 2a2a ff echoing.***** │ │ │ │ -00281760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00281770: 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 ********..Descri │ │ │ │ -00281780: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00281790: 3d0a 0a63 6c65 6172 4563 686f 2073 7464 =..clearEcho std │ │ │ │ -002817a0: 696f 202d 2d20 7475 726e 206f 6666 2065 io -- turn off e │ │ │ │ -002817b0: 6368 6f69 6e67 206f 6620 6368 6172 6163 choing of charac │ │ │ │ -002817c0: 7465 7273 2074 7970 6564 2074 6f20 7468 ters typed to th │ │ │ │ -002817d0: 6520 7374 616e 6461 7264 2069 6e70 7574 e standard input │ │ │ │ -002817e0: 2e0a 0a46 6f72 2074 6865 2070 726f 6772 ...For the progr │ │ │ │ -002817f0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -00281800: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -00281810: 6a65 6374 202a 6e6f 7465 2063 6c65 6172 ject *note clear │ │ │ │ -00281820: 4563 686f 3a20 636c 6561 7245 6368 6f2c Echo: clearEcho, │ │ │ │ -00281830: 2069 7320 6120 2a6e 6f74 6520 636f 6d70 is a *note comp │ │ │ │ -00281840: 696c 6564 2066 756e 6374 696f 6e3a 0a43 iled function:.C │ │ │ │ -00281850: 6f6d 7069 6c65 6446 756e 6374 696f 6e2c ompiledFunction, │ │ │ │ -00281860: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ -00281870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00281880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00281660: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00281670: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00281680: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00281690: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +002816a0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +002816b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +002816c0: 636b 6167 6573 2f0a 4d61 6361 756c 6179 ckages/.Macaulay │ │ │ │ +002816d0: 3244 6f63 2f6f 765f 7379 7374 656d 2e6d 2Doc/ov_system.m │ │ │ │ +002816e0: 323a 3133 3730 3a30 2e0a 1f0a 4669 6c65 2:1370:0....File │ │ │ │ +002816f0: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ +00281700: 6e66 6f2c 204e 6f64 653a 2063 6c65 6172 nfo, Node: clear │ │ │ │ +00281710: 4563 686f 2c20 4e65 7874 3a20 7365 7445 Echo, Next: setE │ │ │ │ +00281720: 6368 6f2c 2050 7265 763a 2073 6561 7263 cho, Prev: searc │ │ │ │ +00281730: 6850 6174 685f 6c70 4c69 7374 5f63 6d53 hPath_lpList_cmS │ │ │ │ +00281740: 7472 696e 675f 7270 2c20 5570 3a20 7379 tring_rp, Up: sy │ │ │ │ +00281750: 7374 656d 2066 6163 696c 6974 6965 730a stem facilities. │ │ │ │ +00281760: 0a63 6c65 6172 4563 686f 202d 2d20 7475 .clearEcho -- tu │ │ │ │ +00281770: 726e 206f 6666 2065 6368 6f69 6e67 0a2a rn off echoing.* │ │ │ │ +00281780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00281790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 ************..De │ │ │ │ +002817a0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +002817b0: 3d3d 3d3d 3d0a 0a63 6c65 6172 4563 686f =====..clearEcho │ │ │ │ +002817c0: 2073 7464 696f 202d 2d20 7475 726e 206f stdio -- turn o │ │ │ │ +002817d0: 6666 2065 6368 6f69 6e67 206f 6620 6368 ff echoing of ch │ │ │ │ +002817e0: 6172 6163 7465 7273 2074 7970 6564 2074 aracters typed t │ │ │ │ +002817f0: 6f20 7468 6520 7374 616e 6461 7264 2069 o the standard i │ │ │ │ +00281800: 6e70 7574 2e0a 0a46 6f72 2074 6865 2070 nput...For the p │ │ │ │ +00281810: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00281820: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00281830: 6520 6f62 6a65 6374 202a 6e6f 7465 2063 e object *note c │ │ │ │ +00281840: 6c65 6172 4563 686f 3a20 636c 6561 7245 learEcho: clearE │ │ │ │ +00281850: 6368 6f2c 2069 7320 6120 2a6e 6f74 6520 cho, is a *note │ │ │ │ +00281860: 636f 6d70 696c 6564 2066 756e 6374 696f compiled functio │ │ │ │ +00281870: 6e3a 0a43 6f6d 7069 6c65 6446 756e 6374 n:.CompiledFunct │ │ │ │ +00281880: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ 00281890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002818a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002818b0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -002818c0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -002818d0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -002818e0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -002818f0: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -00281900: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -00281910: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ -00281920: 6c61 7932 446f 632f 6f76 5f73 7973 7465 lay2Doc/ov_syste │ │ │ │ -00281930: 6d2e 6d32 3a36 3335 3a30 2e0a 1f0a 4669 m.m2:635:0....Fi │ │ │ │ -00281940: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -00281950: 2e69 6e66 6f2c 204e 6f64 653a 2073 6574 .info, Node: set │ │ │ │ -00281960: 4563 686f 2c20 4e65 7874 3a20 746f 7020 Echo, Next: top │ │ │ │ -00281970: 6c65 7665 6c20 6c6f 6f70 2c20 5072 6576 level loop, Prev │ │ │ │ -00281980: 3a20 636c 6561 7245 6368 6f2c 2055 703a : clearEcho, Up: │ │ │ │ -00281990: 2073 7973 7465 6d20 6661 6369 6c69 7469 system faciliti │ │ │ │ -002819a0: 6573 0a0a 7365 7445 6368 6f20 2d2d 2074 es..setEcho -- t │ │ │ │ -002819b0: 7572 6e20 6f6e 2065 6368 6f69 6e67 0a2a urn on echoing.* │ │ │ │ -002819c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002819d0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 *********..Descr │ │ │ │ -002819e0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -002819f0: 3d3d 0a0a 7365 7445 6368 6f20 7374 6469 ==..setEcho stdi │ │ │ │ -00281a00: 6f20 2d2d 2074 7572 6e20 6f6e 2065 6368 o -- turn on ech │ │ │ │ -00281a10: 6f69 6e67 206f 6620 6368 6172 6163 7465 oing of characte │ │ │ │ -00281a20: 7273 2074 7970 6564 2074 6f20 7468 6520 rs typed to the │ │ │ │ -00281a30: 7374 616e 6461 7264 2069 6e70 7574 2e0a standard input.. │ │ │ │ -00281a40: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00281a50: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00281a60: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00281a70: 6374 202a 6e6f 7465 2073 6574 4563 686f ct *note setEcho │ │ │ │ -00281a80: 3a20 7365 7445 6368 6f2c 2069 7320 6120 : setEcho, is a │ │ │ │ -00281a90: 2a6e 6f74 6520 636f 6d70 696c 6564 2066 *note compiled f │ │ │ │ -00281aa0: 756e 6374 696f 6e3a 0a43 6f6d 7069 6c65 unction:.Compile │ │ │ │ -00281ab0: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ -00281ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00281ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00281ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002818b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002818c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002818d0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +002818e0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +002818f0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +00281900: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +00281910: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +00281920: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +00281930: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ +00281940: 6163 6175 6c61 7932 446f 632f 6f76 5f73 acaulay2Doc/ov_s │ │ │ │ +00281950: 7973 7465 6d2e 6d32 3a36 3335 3a30 2e0a ystem.m2:635:0.. │ │ │ │ +00281960: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +00281970: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +00281980: 2073 6574 4563 686f 2c20 4e65 7874 3a20 setEcho, Next: │ │ │ │ +00281990: 746f 7020 6c65 7665 6c20 6c6f 6f70 2c20 top level loop, │ │ │ │ +002819a0: 5072 6576 3a20 636c 6561 7245 6368 6f2c Prev: clearEcho, │ │ │ │ +002819b0: 2055 703a 2073 7973 7465 6d20 6661 6369 Up: system faci │ │ │ │ +002819c0: 6c69 7469 6573 0a0a 7365 7445 6368 6f20 lities..setEcho │ │ │ │ +002819d0: 2d2d 2074 7572 6e20 6f6e 2065 6368 6f69 -- turn on echoi │ │ │ │ +002819e0: 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ng.************* │ │ │ │ +002819f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 *************..D │ │ │ │ +00281a00: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00281a10: 3d3d 3d3d 3d3d 0a0a 7365 7445 6368 6f20 ======..setEcho │ │ │ │ +00281a20: 7374 6469 6f20 2d2d 2074 7572 6e20 6f6e stdio -- turn on │ │ │ │ +00281a30: 2065 6368 6f69 6e67 206f 6620 6368 6172 echoing of char │ │ │ │ +00281a40: 6163 7465 7273 2074 7970 6564 2074 6f20 acters typed to │ │ │ │ +00281a50: 7468 6520 7374 616e 6461 7264 2069 6e70 the standard inp │ │ │ │ +00281a60: 7574 2e0a 0a46 6f72 2074 6865 2070 726f ut...For the pro │ │ │ │ +00281a70: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00281a80: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00281a90: 6f62 6a65 6374 202a 6e6f 7465 2073 6574 object *note set │ │ │ │ +00281aa0: 4563 686f 3a20 7365 7445 6368 6f2c 2069 Echo: setEcho, i │ │ │ │ +00281ab0: 7320 6120 2a6e 6f74 6520 636f 6d70 696c s a *note compil │ │ │ │ +00281ac0: 6564 2066 756e 6374 696f 6e3a 0a43 6f6d ed function:.Com │ │ │ │ +00281ad0: 7069 6c65 6446 756e 6374 696f 6e2c 2e0a piledFunction,.. │ │ │ │ +00281ae0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 00281af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00281b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00281b10: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00281b20: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00281b30: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00281b40: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00281b50: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00281b60: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00281b70: 6765 732f 0a4d 6163 6175 6c61 7932 446f ges/.Macaulay2Do │ │ │ │ -00281b80: 632f 6f76 5f73 7973 7465 6d2e 6d32 3a36 c/ov_system.m2:6 │ │ │ │ -00281b90: 3239 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 29:0....File: Ma │ │ │ │ -00281ba0: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ -00281bb0: 204e 6f64 653a 2074 6f70 206c 6576 656c Node: top level │ │ │ │ -00281bc0: 206c 6f6f 702c 204e 6578 743a 2072 6573 loop, Next: res │ │ │ │ -00281bd0: 7461 7274 2c20 5072 6576 3a20 7365 7445 tart, Prev: setE │ │ │ │ -00281be0: 6368 6f2c 2055 703a 2073 7973 7465 6d20 cho, Up: system │ │ │ │ -00281bf0: 6661 6369 6c69 7469 6573 0a0a 746f 7020 facilities..top │ │ │ │ -00281c00: 6c65 7665 6c20 6c6f 6f70 0a2a 2a2a 2a2a level loop.***** │ │ │ │ -00281c10: 2a2a 2a2a 2a2a 2a2a 2a0a 0a54 6865 2074 *********..The t │ │ │ │ -00281c20: 6f70 206c 6576 656c 2065 7661 6c75 6174 op level evaluat │ │ │ │ -00281c30: 696f 6e20 6c6f 6f70 206f 6620 7468 6520 ion loop of the │ │ │ │ -00281c40: 696e 7465 7270 7265 7465 7220 636f 6e74 interpreter cont │ │ │ │ -00281c50: 6169 6e73 2068 6f6f 6b73 2073 6f20 7468 ains hooks so th │ │ │ │ -00281c60: 6520 7573 6572 2063 616e 0a63 6f6e 7472 e user can.contr │ │ │ │ -00281c70: 6f6c 2068 6f77 2070 7269 6e74 696e 6720 ol how printing │ │ │ │ -00281c80: 6f66 2074 6865 2072 6573 756c 7473 206f of the results o │ │ │ │ -00281c90: 6620 6576 616c 7561 7469 6f6e 2069 7320 f evaluation is │ │ │ │ -00281ca0: 646f 6e65 2e20 2049 6620 7468 6520 7265 done. If the re │ │ │ │ -00281cb0: 7375 6c74 2069 730a 2a6e 6f74 6520 6e75 sult is.*note nu │ │ │ │ -00281cc0: 6c6c 3a20 6e75 6c6c 2c20 7468 656e 206e ll: null, then n │ │ │ │ -00281cd0: 6f74 6869 6e67 2069 7320 7072 696e 7465 othing is printe │ │ │ │ -00281ce0: 642e 2020 4f74 6865 7277 6973 652c 2074 d. Otherwise, t │ │ │ │ -00281cf0: 6865 2061 7070 726f 7072 6961 7465 206d he appropriate m │ │ │ │ -00281d00: 6574 686f 640a 6173 736f 6369 6174 6564 ethod.associated │ │ │ │ -00281d10: 2077 6974 6820 7468 6520 7379 6d62 6f6c with the symbol │ │ │ │ -00281d20: 202a 6e6f 7465 2050 7269 6e74 3a20 5072 *note Print: Pr │ │ │ │ -00281d30: 696e 742c 2069 7320 6170 706c 6965 6420 int, is applied │ │ │ │ -00281d40: 746f 2070 6572 666f 726d 2074 6865 0a70 to perform the.p │ │ │ │ -00281d50: 7269 6e74 696e 672c 2075 6e6c 6573 7320 rinting, unless │ │ │ │ -00281d60: 7468 6520 7072 696e 7469 6e67 2069 7320 the printing is │ │ │ │ -00281d70: 746f 2062 6520 7375 7070 7265 7373 6564 to be suppressed │ │ │ │ -00281d80: 2c20 6173 2069 6e64 6963 6174 6564 2062 , as indicated b │ │ │ │ -00281d90: 7920 6120 7365 6d69 636f 6c6f 6e0a 6174 y a semicolon.at │ │ │ │ -00281da0: 2074 6865 2065 6e64 206f 6620 7468 6520 the end of the │ │ │ │ -00281db0: 7374 6174 656d 656e 742c 2069 6e20 7768 statement, in wh │ │ │ │ -00281dc0: 6963 6820 6361 7365 2074 6865 202a 6e6f ich case the *no │ │ │ │ -00281dd0: 7465 204e 6f50 7269 6e74 3a20 4e6f 5072 te NoPrint: NoPr │ │ │ │ -00281de0: 696e 742c 206d 6574 686f 640a 6973 2061 int, method.is a │ │ │ │ -00281df0: 7070 6c69 6564 2e0a 0a53 6565 2061 6c73 pplied...See als │ │ │ │ -00281e00: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ -00281e10: 2a6e 6f74 6520 5072 696e 743a 2050 7269 *note Print: Pri │ │ │ │ -00281e20: 6e74 2c20 2d2d 2074 6f70 206c 6576 656c nt, -- top level │ │ │ │ -00281e30: 206d 6574 686f 6420 666f 7220 7072 696e method for prin │ │ │ │ -00281e40: 7469 6e67 2072 6573 756c 7473 0a20 202a ting results. * │ │ │ │ -00281e50: 202a 6e6f 7465 204e 6f50 7269 6e74 3a20 *note NoPrint: │ │ │ │ -00281e60: 4e6f 5072 696e 742c 202d 2d20 746f 7020 NoPrint, -- top │ │ │ │ -00281e70: 6c65 7665 6c20 6d65 7468 6f64 2066 6f72 level method for │ │ │ │ -00281e80: 206e 6f6e 2d70 7269 6e74 696e 6720 7265 non-printing re │ │ │ │ -00281e90: 7375 6c74 730a 2020 2a20 2a6e 6f74 6520 sults. * *note │ │ │ │ -00281ea0: 4265 666f 7265 5072 696e 743a 2042 6566 BeforePrint: Bef │ │ │ │ -00281eb0: 6f72 6550 7269 6e74 2c20 2d2d 2074 6f70 orePrint, -- top │ │ │ │ -00281ec0: 206c 6576 656c 206d 6574 686f 6420 6170 level method ap │ │ │ │ -00281ed0: 706c 6965 6420 6265 666f 7265 2070 7269 plied before pri │ │ │ │ -00281ee0: 6e74 696e 670a 2020 2020 7265 7375 6c74 nting. result │ │ │ │ -00281ef0: 730a 2020 2a20 2a6e 6f74 6520 4166 7465 s. * *note Afte │ │ │ │ -00281f00: 7250 7269 6e74 3a20 4166 7465 7250 7269 rPrint: AfterPri │ │ │ │ -00281f10: 6e74 2c20 2d2d 2074 6f70 206c 6576 656c nt, -- top level │ │ │ │ -00281f20: 206d 6574 686f 6420 6170 706c 6965 6420 method applied │ │ │ │ -00281f30: 6166 7465 7220 7072 696e 7469 6e67 0a20 after printing. │ │ │ │ -00281f40: 202a 202a 6e6f 7465 2041 6674 6572 4e6f * *note AfterNo │ │ │ │ -00281f50: 5072 696e 743a 2041 6674 6572 4e6f 5072 Print: AfterNoPr │ │ │ │ -00281f60: 696e 742c 202d 2d20 746f 7020 6c65 7665 int, -- top leve │ │ │ │ -00281f70: 6c20 6d65 7468 6f64 2061 7070 6c69 6564 l method applied │ │ │ │ -00281f80: 2061 6674 6572 206e 6f74 0a20 2020 2070 after not. p │ │ │ │ -00281f90: 7269 6e74 696e 670a 2a20 4d65 6e75 3a0a rinting.* Menu:. │ │ │ │ -00281fa0: 0a2a 2076 616c 7565 3a3a 2020 2020 2020 .* value:: │ │ │ │ -00281fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00281fc0: 2065 7661 6c75 6174 650a 2a20 6361 7074 evaluate.* capt │ │ │ │ -00281fd0: 7572 653a 3a20 2020 2020 2020 2020 2020 ure:: │ │ │ │ -00281fe0: 2020 2020 2020 2020 2020 6576 616c 7561 evalua │ │ │ │ -00281ff0: 7465 204d 6163 6175 6c61 7932 2063 6f64 te Macaulay2 cod │ │ │ │ -00282000: 6520 616e 6420 6361 7074 7572 6520 7468 e and capture th │ │ │ │ -00282010: 6520 6f75 7470 7574 0a2a 206f 6f3a 3a20 e output.* oo:: │ │ │ │ -00282020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282030: 2020 2020 2020 2020 2074 6865 206c 6173 the las │ │ │ │ -00282040: 7420 6f75 7470 7574 2076 616c 7565 0a2a t output value.* │ │ │ │ -00282050: 206f 6f6f 3a3a 2020 2020 2020 2020 2020 ooo:: │ │ │ │ -00282060: 2020 2020 2020 2020 2020 2020 2020 2074 t │ │ │ │ -00282070: 6865 206e 6578 7420 746f 2074 6865 206c he next to the l │ │ │ │ -00282080: 6173 7420 6f75 7470 7574 2076 616c 7565 ast output value │ │ │ │ -00282090: 0a2a 206f 6f6f 6f3a 3a20 2020 2020 2020 .* oooo:: │ │ │ │ -002820a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002820b0: 2074 6865 2074 6869 7264 2074 6f20 7468 the third to th │ │ │ │ -002820c0: 6520 6c61 7374 206f 7574 7075 7420 7661 e last output va │ │ │ │ -002820d0: 6c75 650a 2a20 4f75 7470 7574 4469 6374 lue.* OutputDict │ │ │ │ -002820e0: 696f 6e61 7279 3a3a 2020 2020 2020 2020 ionary:: │ │ │ │ -002820f0: 2020 2020 7468 6520 6469 6374 696f 6e61 the dictiona │ │ │ │ -00282100: 7279 2066 6f72 206f 7574 7075 7420 7661 ry for output va │ │ │ │ -00282110: 6c75 6573 0a2a 2063 6c65 6172 416c 6c3a lues.* clearAll: │ │ │ │ -00282120: 3a20 2020 2020 2020 2020 2020 2020 2020 : │ │ │ │ -00282130: 2020 2020 2066 6f72 6765 7420 6576 6572 forget ever │ │ │ │ -00282140: 7974 6869 6e67 0a2a 2063 6c65 6172 4f75 ything.* clearOu │ │ │ │ -00282150: 7470 7574 3a3a 2020 2020 2020 2020 2020 tput:: │ │ │ │ -00282160: 2020 2020 2020 2066 6f72 6765 7420 6f75 forget ou │ │ │ │ -00282170: 7470 7574 2076 616c 7565 730a 2a20 746f tput values.* to │ │ │ │ -00282180: 704c 6576 656c 4d6f 6465 3a3a 2020 2020 pLevelMode:: │ │ │ │ -00282190: 2020 2020 2020 2020 2020 2020 7468 6520 the │ │ │ │ -002821a0: 6375 7272 656e 7420 746f 7020 6c65 7665 current top leve │ │ │ │ -002821b0: 6c20 6d6f 6465 0a2d 2d2d 2d2d 2d2d 2d2d l mode.--------- │ │ │ │ -002821c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002821d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00281b30: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00281b40: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +00281b50: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +00281b60: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00281b70: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +00281b80: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00281b90: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ +00281ba0: 7932 446f 632f 6f76 5f73 7973 7465 6d2e y2Doc/ov_system. │ │ │ │ +00281bb0: 6d32 3a36 3239 3a30 2e0a 1f0a 4669 6c65 m2:629:0....File │ │ │ │ +00281bc0: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ +00281bd0: 6e66 6f2c 204e 6f64 653a 2074 6f70 206c nfo, Node: top l │ │ │ │ +00281be0: 6576 656c 206c 6f6f 702c 204e 6578 743a evel loop, Next: │ │ │ │ +00281bf0: 2072 6573 7461 7274 2c20 5072 6576 3a20 restart, Prev: │ │ │ │ +00281c00: 7365 7445 6368 6f2c 2055 703a 2073 7973 setEcho, Up: sys │ │ │ │ +00281c10: 7465 6d20 6661 6369 6c69 7469 6573 0a0a tem facilities.. │ │ │ │ +00281c20: 746f 7020 6c65 7665 6c20 6c6f 6f70 0a2a top level loop.* │ │ │ │ +00281c30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a54 *************..T │ │ │ │ +00281c40: 6865 2074 6f70 206c 6576 656c 2065 7661 he top level eva │ │ │ │ +00281c50: 6c75 6174 696f 6e20 6c6f 6f70 206f 6620 luation loop of │ │ │ │ +00281c60: 7468 6520 696e 7465 7270 7265 7465 7220 the interpreter │ │ │ │ +00281c70: 636f 6e74 6169 6e73 2068 6f6f 6b73 2073 contains hooks s │ │ │ │ +00281c80: 6f20 7468 6520 7573 6572 2063 616e 0a63 o the user can.c │ │ │ │ +00281c90: 6f6e 7472 6f6c 2068 6f77 2070 7269 6e74 ontrol how print │ │ │ │ +00281ca0: 696e 6720 6f66 2074 6865 2072 6573 756c ing of the resul │ │ │ │ +00281cb0: 7473 206f 6620 6576 616c 7561 7469 6f6e ts of evaluation │ │ │ │ +00281cc0: 2069 7320 646f 6e65 2e20 2049 6620 7468 is done. If th │ │ │ │ +00281cd0: 6520 7265 7375 6c74 2069 730a 2a6e 6f74 e result is.*not │ │ │ │ +00281ce0: 6520 6e75 6c6c 3a20 6e75 6c6c 2c20 7468 e null: null, th │ │ │ │ +00281cf0: 656e 206e 6f74 6869 6e67 2069 7320 7072 en nothing is pr │ │ │ │ +00281d00: 696e 7465 642e 2020 4f74 6865 7277 6973 inted. Otherwis │ │ │ │ +00281d10: 652c 2074 6865 2061 7070 726f 7072 6961 e, the appropria │ │ │ │ +00281d20: 7465 206d 6574 686f 640a 6173 736f 6369 te method.associ │ │ │ │ +00281d30: 6174 6564 2077 6974 6820 7468 6520 7379 ated with the sy │ │ │ │ +00281d40: 6d62 6f6c 202a 6e6f 7465 2050 7269 6e74 mbol *note Print │ │ │ │ +00281d50: 3a20 5072 696e 742c 2069 7320 6170 706c : Print, is appl │ │ │ │ +00281d60: 6965 6420 746f 2070 6572 666f 726d 2074 ied to perform t │ │ │ │ +00281d70: 6865 0a70 7269 6e74 696e 672c 2075 6e6c he.printing, unl │ │ │ │ +00281d80: 6573 7320 7468 6520 7072 696e 7469 6e67 ess the printing │ │ │ │ +00281d90: 2069 7320 746f 2062 6520 7375 7070 7265 is to be suppre │ │ │ │ +00281da0: 7373 6564 2c20 6173 2069 6e64 6963 6174 ssed, as indicat │ │ │ │ +00281db0: 6564 2062 7920 6120 7365 6d69 636f 6c6f ed by a semicolo │ │ │ │ +00281dc0: 6e0a 6174 2074 6865 2065 6e64 206f 6620 n.at the end of │ │ │ │ +00281dd0: 7468 6520 7374 6174 656d 656e 742c 2069 the statement, i │ │ │ │ +00281de0: 6e20 7768 6963 6820 6361 7365 2074 6865 n which case the │ │ │ │ +00281df0: 202a 6e6f 7465 204e 6f50 7269 6e74 3a20 *note NoPrint: │ │ │ │ +00281e00: 4e6f 5072 696e 742c 206d 6574 686f 640a NoPrint, method. │ │ │ │ +00281e10: 6973 2061 7070 6c69 6564 2e0a 0a53 6565 is applied...See │ │ │ │ +00281e20: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ +00281e30: 2020 2a20 2a6e 6f74 6520 5072 696e 743a * *note Print: │ │ │ │ +00281e40: 2050 7269 6e74 2c20 2d2d 2074 6f70 206c Print, -- top l │ │ │ │ +00281e50: 6576 656c 206d 6574 686f 6420 666f 7220 evel method for │ │ │ │ +00281e60: 7072 696e 7469 6e67 2072 6573 756c 7473 printing results │ │ │ │ +00281e70: 0a20 202a 202a 6e6f 7465 204e 6f50 7269 . * *note NoPri │ │ │ │ +00281e80: 6e74 3a20 4e6f 5072 696e 742c 202d 2d20 nt: NoPrint, -- │ │ │ │ +00281e90: 746f 7020 6c65 7665 6c20 6d65 7468 6f64 top level method │ │ │ │ +00281ea0: 2066 6f72 206e 6f6e 2d70 7269 6e74 696e for non-printin │ │ │ │ +00281eb0: 6720 7265 7375 6c74 730a 2020 2a20 2a6e g results. * *n │ │ │ │ +00281ec0: 6f74 6520 4265 666f 7265 5072 696e 743a ote BeforePrint: │ │ │ │ +00281ed0: 2042 6566 6f72 6550 7269 6e74 2c20 2d2d BeforePrint, -- │ │ │ │ +00281ee0: 2074 6f70 206c 6576 656c 206d 6574 686f top level metho │ │ │ │ +00281ef0: 6420 6170 706c 6965 6420 6265 666f 7265 d applied before │ │ │ │ +00281f00: 2070 7269 6e74 696e 670a 2020 2020 7265 printing. re │ │ │ │ +00281f10: 7375 6c74 730a 2020 2a20 2a6e 6f74 6520 sults. * *note │ │ │ │ +00281f20: 4166 7465 7250 7269 6e74 3a20 4166 7465 AfterPrint: Afte │ │ │ │ +00281f30: 7250 7269 6e74 2c20 2d2d 2074 6f70 206c rPrint, -- top l │ │ │ │ +00281f40: 6576 656c 206d 6574 686f 6420 6170 706c evel method appl │ │ │ │ +00281f50: 6965 6420 6166 7465 7220 7072 696e 7469 ied after printi │ │ │ │ +00281f60: 6e67 0a20 202a 202a 6e6f 7465 2041 6674 ng. * *note Aft │ │ │ │ +00281f70: 6572 4e6f 5072 696e 743a 2041 6674 6572 erNoPrint: After │ │ │ │ +00281f80: 4e6f 5072 696e 742c 202d 2d20 746f 7020 NoPrint, -- top │ │ │ │ +00281f90: 6c65 7665 6c20 6d65 7468 6f64 2061 7070 level method app │ │ │ │ +00281fa0: 6c69 6564 2061 6674 6572 206e 6f74 0a20 lied after not. │ │ │ │ +00281fb0: 2020 2070 7269 6e74 696e 670a 2a20 4d65 printing.* Me │ │ │ │ +00281fc0: 6e75 3a0a 0a2a 2076 616c 7565 3a3a 2020 nu:..* value:: │ │ │ │ +00281fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00281fe0: 2020 2020 2065 7661 6c75 6174 650a 2a20 evaluate.* │ │ │ │ +00281ff0: 6361 7074 7572 653a 3a20 2020 2020 2020 capture:: │ │ │ │ +00282000: 2020 2020 2020 2020 2020 2020 2020 6576 ev │ │ │ │ +00282010: 616c 7561 7465 204d 6163 6175 6c61 7932 aluate Macaulay2 │ │ │ │ +00282020: 2063 6f64 6520 616e 6420 6361 7074 7572 code and captur │ │ │ │ +00282030: 6520 7468 6520 6f75 7470 7574 0a2a 206f e the output.* o │ │ │ │ +00282040: 6f3a 3a20 2020 2020 2020 2020 2020 2020 o:: │ │ │ │ +00282050: 2020 2020 2020 2020 2020 2020 2074 6865 the │ │ │ │ +00282060: 206c 6173 7420 6f75 7470 7574 2076 616c last output val │ │ │ │ +00282070: 7565 0a2a 206f 6f6f 3a3a 2020 2020 2020 ue.* ooo:: │ │ │ │ +00282080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00282090: 2020 2074 6865 206e 6578 7420 746f 2074 the next to t │ │ │ │ +002820a0: 6865 206c 6173 7420 6f75 7470 7574 2076 he last output v │ │ │ │ +002820b0: 616c 7565 0a2a 206f 6f6f 6f3a 3a20 2020 alue.* oooo:: │ │ │ │ +002820c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002820d0: 2020 2020 2074 6865 2074 6869 7264 2074 the third t │ │ │ │ +002820e0: 6f20 7468 6520 6c61 7374 206f 7574 7075 o the last outpu │ │ │ │ +002820f0: 7420 7661 6c75 650a 2a20 4f75 7470 7574 t value.* Output │ │ │ │ +00282100: 4469 6374 696f 6e61 7279 3a3a 2020 2020 Dictionary:: │ │ │ │ +00282110: 2020 2020 2020 2020 7468 6520 6469 6374 the dict │ │ │ │ +00282120: 696f 6e61 7279 2066 6f72 206f 7574 7075 ionary for outpu │ │ │ │ +00282130: 7420 7661 6c75 6573 0a2a 2063 6c65 6172 t values.* clear │ │ │ │ +00282140: 416c 6c3a 3a20 2020 2020 2020 2020 2020 All:: │ │ │ │ +00282150: 2020 2020 2020 2020 2066 6f72 6765 7420 forget │ │ │ │ +00282160: 6576 6572 7974 6869 6e67 0a2a 2063 6c65 everything.* cle │ │ │ │ +00282170: 6172 4f75 7470 7574 3a3a 2020 2020 2020 arOutput:: │ │ │ │ +00282180: 2020 2020 2020 2020 2020 2066 6f72 6765 forge │ │ │ │ +00282190: 7420 6f75 7470 7574 2076 616c 7565 730a t output values. │ │ │ │ +002821a0: 2a20 746f 704c 6576 656c 4d6f 6465 3a3a * topLevelMode:: │ │ │ │ +002821b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +002821c0: 7468 6520 6375 7272 656e 7420 746f 7020 the current top │ │ │ │ +002821d0: 6c65 7665 6c20 6d6f 6465 0a2d 2d2d 2d2d level mode.----- │ │ │ │ 002821e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 002821f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00282200: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00282210: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00282220: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00282230: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00282240: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00282250: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ -00282260: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ -00282270: 6163 6175 6c61 7932 446f 632f 6f76 5f72 acaulay2Doc/ov_r │ │ │ │ -00282280: 6570 6c2e 6d32 3a32 383a 302e 0a1f 0a46 epl.m2:28:0....F │ │ │ │ -00282290: 696c 653a 204d 6163 6175 6c61 7932 446f ile: Macaulay2Do │ │ │ │ -002822a0: 632e 696e 666f 2c20 4e6f 6465 3a20 7661 c.info, Node: va │ │ │ │ -002822b0: 6c75 652c 204e 6578 743a 2063 6170 7475 lue, Next: captu │ │ │ │ -002822c0: 7265 2c20 5570 3a20 746f 7020 6c65 7665 re, Up: top leve │ │ │ │ -002822d0: 6c20 6c6f 6f70 0a0a 7661 6c75 6520 2d2d l loop..value -- │ │ │ │ -002822e0: 2065 7661 6c75 6174 650a 2a2a 2a2a 2a2a evaluate.****** │ │ │ │ -002822f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a57 6179 ***********..Way │ │ │ │ -00282300: 7320 746f 2075 7365 2076 616c 7565 3a0a s to use value:. │ │ │ │ -00282310: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00282320: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7661 ==.. * *note va │ │ │ │ -00282330: 6c75 6528 4578 7072 6573 7369 6f6e 293a lue(Expression): │ │ │ │ -00282340: 2076 616c 7565 5f6c 7045 7870 7265 7373 value_lpExpress │ │ │ │ -00282350: 696f 6e5f 7270 2c20 2d2d 2065 7661 6c75 ion_rp, -- evalu │ │ │ │ -00282360: 6174 650a 2020 2a20 2a6e 6f74 6520 7661 ate. * *note va │ │ │ │ -00282370: 6c75 6528 496e 6465 7865 6456 6172 6961 lue(IndexedVaria │ │ │ │ -00282380: 626c 6529 3a20 7661 6c75 655f 6c70 496e ble): value_lpIn │ │ │ │ -00282390: 6465 7865 6456 6172 6961 626c 655f 7270 dexedVariable_rp │ │ │ │ -002823a0: 2c20 2d2d 2072 6574 7269 6576 6520 7468 , -- retrieve th │ │ │ │ -002823b0: 650a 2020 2020 7661 6c75 6520 6f66 2061 e. value of a │ │ │ │ -002823c0: 6e20 696e 6465 7865 6420 7661 7269 6162 n indexed variab │ │ │ │ -002823d0: 6c65 0a20 202a 202a 6e6f 7465 2076 616c le. * *note val │ │ │ │ -002823e0: 7565 2850 7365 7564 6f63 6f64 6529 3a20 ue(Pseudocode): │ │ │ │ -002823f0: 7661 6c75 655f 6c70 5073 6575 646f 636f value_lpPseudoco │ │ │ │ -00282400: 6465 5f72 702c 202d 2d20 6578 6563 7574 de_rp, -- execut │ │ │ │ -00282410: 6520 7073 6575 646f 636f 6465 0a20 202a e pseudocode. * │ │ │ │ -00282420: 202a 6e6f 7465 2076 616c 7565 2853 7472 *note value(Str │ │ │ │ -00282430: 696e 6729 3a20 7661 6c75 655f 6c70 5374 ing): value_lpSt │ │ │ │ -00282440: 7269 6e67 5f72 702c 202d 2d20 6576 616c ring_rp, -- eval │ │ │ │ -00282450: 7561 7465 2061 2073 7472 696e 670a 2020 uate a string. │ │ │ │ -00282460: 2a20 2a6e 6f74 6520 7661 6c75 6528 5379 * *note value(Sy │ │ │ │ -00282470: 6d62 6f6c 293a 2076 616c 7565 5f6c 7053 mbol): value_lpS │ │ │ │ -00282480: 796d 626f 6c5f 7270 2c20 2d2d 2072 6574 ymbol_rp, -- ret │ │ │ │ -00282490: 7269 6576 6520 7468 6520 7661 6c75 6520 rieve the value │ │ │ │ -002824a0: 6f66 2061 2073 796d 626f 6c0a 0a46 6f72 of a symbol..For │ │ │ │ -002824b0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -002824c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -002824d0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -002824e0: 6e6f 7465 2076 616c 7565 3a20 7661 6c75 note value: valu │ │ │ │ -002824f0: 652c 2069 7320 6120 2a6e 6f74 6520 6d65 e, is a *note me │ │ │ │ -00282500: 7468 6f64 2066 756e 6374 696f 6e3a 204d thod function: M │ │ │ │ -00282510: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ -00282520: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ -00282530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00282540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00282200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00282210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00282220: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +00282230: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +00282240: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +00282250: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00282260: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00282270: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ +00282280: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +00282290: 732f 0a4d 6163 6175 6c61 7932 446f 632f s/.Macaulay2Doc/ │ │ │ │ +002822a0: 6f76 5f72 6570 6c2e 6d32 3a32 383a 302e ov_repl.m2:28:0. │ │ │ │ +002822b0: 0a1f 0a46 696c 653a 204d 6163 6175 6c61 ...File: Macaula │ │ │ │ +002822c0: 7932 446f 632e 696e 666f 2c20 4e6f 6465 y2Doc.info, Node │ │ │ │ +002822d0: 3a20 7661 6c75 652c 204e 6578 743a 2063 : value, Next: c │ │ │ │ +002822e0: 6170 7475 7265 2c20 5570 3a20 746f 7020 apture, Up: top │ │ │ │ +002822f0: 6c65 7665 6c20 6c6f 6f70 0a0a 7661 6c75 level loop..valu │ │ │ │ +00282300: 6520 2d2d 2065 7661 6c75 6174 650a 2a2a e -- evaluate.** │ │ │ │ +00282310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +00282320: 0a57 6179 7320 746f 2075 7365 2076 616c .Ways to use val │ │ │ │ +00282330: 7565 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ue:.============ │ │ │ │ +00282340: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +00282350: 6520 7661 6c75 6528 4578 7072 6573 7369 e value(Expressi │ │ │ │ +00282360: 6f6e 293a 2076 616c 7565 5f6c 7045 7870 on): value_lpExp │ │ │ │ +00282370: 7265 7373 696f 6e5f 7270 2c20 2d2d 2065 ression_rp, -- e │ │ │ │ +00282380: 7661 6c75 6174 650a 2020 2a20 2a6e 6f74 valuate. * *not │ │ │ │ +00282390: 6520 7661 6c75 6528 496e 6465 7865 6456 e value(IndexedV │ │ │ │ +002823a0: 6172 6961 626c 6529 3a20 7661 6c75 655f ariable): value_ │ │ │ │ +002823b0: 6c70 496e 6465 7865 6456 6172 6961 626c lpIndexedVariabl │ │ │ │ +002823c0: 655f 7270 2c20 2d2d 2072 6574 7269 6576 e_rp, -- retriev │ │ │ │ +002823d0: 6520 7468 650a 2020 2020 7661 6c75 6520 e the. value │ │ │ │ +002823e0: 6f66 2061 6e20 696e 6465 7865 6420 7661 of an indexed va │ │ │ │ +002823f0: 7269 6162 6c65 0a20 202a 202a 6e6f 7465 riable. * *note │ │ │ │ +00282400: 2076 616c 7565 2850 7365 7564 6f63 6f64 value(Pseudocod │ │ │ │ +00282410: 6529 3a20 7661 6c75 655f 6c70 5073 6575 e): value_lpPseu │ │ │ │ +00282420: 646f 636f 6465 5f72 702c 202d 2d20 6578 docode_rp, -- ex │ │ │ │ +00282430: 6563 7574 6520 7073 6575 646f 636f 6465 ecute pseudocode │ │ │ │ +00282440: 0a20 202a 202a 6e6f 7465 2076 616c 7565 . * *note value │ │ │ │ +00282450: 2853 7472 696e 6729 3a20 7661 6c75 655f (String): value_ │ │ │ │ +00282460: 6c70 5374 7269 6e67 5f72 702c 202d 2d20 lpString_rp, -- │ │ │ │ +00282470: 6576 616c 7561 7465 2061 2073 7472 696e evaluate a strin │ │ │ │ +00282480: 670a 2020 2a20 2a6e 6f74 6520 7661 6c75 g. * *note valu │ │ │ │ +00282490: 6528 5379 6d62 6f6c 293a 2076 616c 7565 e(Symbol): value │ │ │ │ +002824a0: 5f6c 7053 796d 626f 6c5f 7270 2c20 2d2d _lpSymbol_rp, -- │ │ │ │ +002824b0: 2072 6574 7269 6576 6520 7468 6520 7661 retrieve the va │ │ │ │ +002824c0: 6c75 6520 6f66 2061 2073 796d 626f 6c0a lue of a symbol. │ │ │ │ +002824d0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +002824e0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +002824f0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +00282500: 6374 202a 6e6f 7465 2076 616c 7565 3a20 ct *note value: │ │ │ │ +00282510: 7661 6c75 652c 2069 7320 6120 2a6e 6f74 value, is a *not │ │ │ │ +00282520: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +00282530: 6e3a 204d 6574 686f 6446 756e 6374 696f n: MethodFunctio │ │ │ │ +00282540: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ 00282550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00282560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00282570: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00282580: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00282590: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -002825a0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -002825b0: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ -002825c0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -002825d0: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ -002825e0: 7932 446f 632f 6675 6e63 7469 6f6e 732f y2Doc/functions/ │ │ │ │ -002825f0: 7661 6c75 652d 646f 632e 6d32 3a37 3a30 value-doc.m2:7:0 │ │ │ │ -00282600: 2e0a 1f0a 4669 6c65 3a20 4d61 6361 756c ....File: Macaul │ │ │ │ -00282610: 6179 3244 6f63 2e69 6e66 6f2c 204e 6f64 ay2Doc.info, Nod │ │ │ │ -00282620: 653a 2063 6170 7475 7265 2c20 4e65 7874 e: capture, Next │ │ │ │ -00282630: 3a20 6f6f 2c20 5072 6576 3a20 7661 6c75 : oo, Prev: valu │ │ │ │ -00282640: 652c 2055 703a 2074 6f70 206c 6576 656c e, Up: top level │ │ │ │ -00282650: 206c 6f6f 700a 0a63 6170 7475 7265 202d loop..capture - │ │ │ │ -00282660: 2d20 6576 616c 7561 7465 204d 6163 6175 - evaluate Macau │ │ │ │ -00282670: 6c61 7932 2063 6f64 6520 616e 6420 6361 lay2 code and ca │ │ │ │ -00282680: 7074 7572 6520 7468 6520 6f75 7470 7574 pture the output │ │ │ │ -00282690: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -002826a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002826b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002826c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -002826d0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -002826e0: 2865 7272 2c20 6f75 7470 7574 2920 3a3d (err, output) := │ │ │ │ -002826f0: 2063 6170 7475 7265 2073 7472 0a20 202a capture str. * │ │ │ │ -00282700: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00282710: 2073 7472 2c20 6120 2a6e 6f74 6520 6e65 str, a *note ne │ │ │ │ -00282720: 743a 204e 6574 2c2c 2061 202a 6e6f 7465 t: Net,, a *note │ │ │ │ -00282730: 206c 6973 743a 204c 6973 742c 2c20 6120 list: List,, a │ │ │ │ -00282740: 2a6e 6f74 6520 7374 7269 6e67 3a20 5374 *note string: St │ │ │ │ -00282750: 7269 6e67 2c2c 0a20 2020 2020 2020 206f ring,,. o │ │ │ │ -00282760: 7220 6120 2a6e 6f74 6520 7465 7374 2069 r a *note test i │ │ │ │ -00282770: 6e70 7574 3a20 7465 7374 732c 2c20 7468 nput: tests,, th │ │ │ │ -00282780: 6520 4d61 6361 756c 6179 3220 636f 6465 e Macaulay2 code │ │ │ │ -00282790: 2074 6f20 6265 2065 7661 6c75 6174 6564 to be evaluated │ │ │ │ -002827a0: 0a20 202a 202a 6e6f 7465 204f 7074 696f . * *note Optio │ │ │ │ -002827b0: 6e61 6c20 696e 7075 7473 3a20 7573 696e nal inputs: usin │ │ │ │ -002827c0: 6720 6675 6e63 7469 6f6e 7320 7769 7468 g functions with │ │ │ │ -002827d0: 206f 7074 696f 6e61 6c20 696e 7075 7473 optional inputs │ │ │ │ -002827e0: 2c3a 0a20 2020 2020 202a 2055 7365 724d ,:. * UserM │ │ │ │ -002827f0: 6f64 6520 3d3e 2061 202a 6e6f 7465 2042 ode => a *note B │ │ │ │ -00282800: 6f6f 6c65 616e 2076 616c 7565 3a20 426f oolean value: Bo │ │ │ │ -00282810: 6f6c 6561 6e2c 2c20 6465 6661 756c 7420 olean,, default │ │ │ │ -00282820: 7661 6c75 6520 7472 7565 2c0a 2020 2020 value true,. │ │ │ │ -00282830: 2020 2020 696e 6469 6361 7465 7320 7768 indicates wh │ │ │ │ -00282840: 6574 6865 7220 6375 7272 656e 746c 7920 ether currently │ │ │ │ -00282850: 6c6f 6164 6564 2070 6163 6b61 6765 7320 loaded packages │ │ │ │ -00282860: 616e 6420 6578 706f 7274 6564 2073 796d and exported sym │ │ │ │ -00282870: 626f 6c73 2073 686f 756c 640a 2020 2020 bols should. │ │ │ │ -00282880: 2020 2020 6265 2061 6363 6573 7369 626c be accessibl │ │ │ │ -00282890: 6520 7768 696c 6520 7374 7220 6973 2062 e while str is b │ │ │ │ -002828a0: 6569 6e67 2065 7661 6c75 6174 6564 2e0a eing evaluated.. │ │ │ │ -002828b0: 2020 2020 2020 2a20 5061 636b 6167 6545 * PackageE │ │ │ │ -002828c0: 7870 6f72 7473 203d 3e20 6120 2a6e 6f74 xports => a *not │ │ │ │ -002828d0: 6520 6c69 7374 3a20 4c69 7374 2c2c 2064 e list: List,, d │ │ │ │ -002828e0: 6566 6175 6c74 2076 616c 7565 206e 756c efault value nul │ │ │ │ -002828f0: 6c2c 206f 6620 7061 636b 6167 6573 0a20 l, of packages. │ │ │ │ -00282900: 2020 2020 2020 2074 6f20 6265 206c 6f61 to be loa │ │ │ │ -00282910: 6465 6420 6265 666f 7265 2065 7661 6c75 ded before evalu │ │ │ │ -00282920: 6174 696e 6720 7374 722e 2020 5468 6520 ating str. The │ │ │ │ -00282930: 6375 7272 656e 7420 7061 636b 6167 652c current package, │ │ │ │ -00282940: 2061 7320 6769 7665 6e20 6279 0a20 2020 as given by. │ │ │ │ -00282950: 2020 2020 202a 6e6f 7465 204d 6163 6175 *note Macau │ │ │ │ -00282960: 6c61 7932 446f 633a 2054 6f70 2c2c 2077 lay2Doc: Top,, w │ │ │ │ -00282970: 696c 6c20 6265 2069 6e63 6c75 6465 642e ill be included. │ │ │ │ -00282980: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -00282990: 2020 2020 2a20 6572 722c 2061 202a 6e6f * err, a *no │ │ │ │ -002829a0: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ -002829b0: 3a20 426f 6f6c 6561 6e2c 2c20 7768 6574 : Boolean,, whet │ │ │ │ -002829c0: 6865 7220 7468 6520 6576 616c 7561 7469 her the evaluati │ │ │ │ -002829d0: 6f6e 2077 6173 0a20 2020 2020 2020 2069 on was. i │ │ │ │ -002829e0: 6e74 6572 7275 7074 6564 2062 6563 6175 nterrupted becau │ │ │ │ -002829f0: 7365 206f 6620 616e 2065 7272 6f72 0a20 se of an error. │ │ │ │ -00282a00: 2020 2020 202a 206f 7574 7075 742c 2061 * output, a │ │ │ │ -00282a10: 202a 6e6f 7465 2073 7472 696e 673a 2053 *note string: S │ │ │ │ -00282a20: 7472 696e 672c 2c20 7468 6520 7265 7375 tring,, the resu │ │ │ │ -00282a30: 6c74 206f 6620 7468 6520 6576 616c 7561 lt of the evalua │ │ │ │ -00282a40: 7469 6f6e 0a0a 4465 7363 7269 7074 696f tion..Descriptio │ │ │ │ -00282a50: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 n.===========..T │ │ │ │ -00282a60: 6869 7320 6675 6e63 7469 6f6e 2065 7661 his function eva │ │ │ │ -00282a70: 6c75 6174 6573 2074 6865 2067 6976 656e luates the given │ │ │ │ -00282a80: 204d 6163 6175 6c61 7932 2063 6f64 6520 Macaulay2 code │ │ │ │ -00282a90: 7769 7468 696e 2074 6865 2073 616d 6520 within the same │ │ │ │ -00282aa0: 7072 6f63 6573 7320 696e 0a6f 7264 6572 process in.order │ │ │ │ -00282ab0: 2074 6f20 7370 6565 6420 7570 2074 6173 to speed up tas │ │ │ │ -00282ac0: 6b73 2073 7563 6820 6173 2067 656e 6572 ks such as gener │ │ │ │ -00282ad0: 6174 696e 6720 6578 616d 706c 6520 7265 ating example re │ │ │ │ -00282ae0: 7375 6c74 7320 6f72 2063 6865 636b 696e sults or checkin │ │ │ │ -00282af0: 6720 7468 650a 7061 636b 6167 6520 7465 g the.package te │ │ │ │ -00282b00: 7374 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d sts...+--------- │ │ │ │ -00282b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00282b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00282570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00282580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00282590: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +002825a0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +002825b0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +002825c0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +002825d0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +002825e0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +002825f0: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ +00282600: 6175 6c61 7932 446f 632f 6675 6e63 7469 aulay2Doc/functi │ │ │ │ +00282610: 6f6e 732f 7661 6c75 652d 646f 632e 6d32 ons/value-doc.m2 │ │ │ │ +00282620: 3a37 3a30 2e0a 1f0a 4669 6c65 3a20 4d61 :7:0....File: Ma │ │ │ │ +00282630: 6361 756c 6179 3244 6f63 2e69 6e66 6f2c caulay2Doc.info, │ │ │ │ +00282640: 204e 6f64 653a 2063 6170 7475 7265 2c20 Node: capture, │ │ │ │ +00282650: 4e65 7874 3a20 6f6f 2c20 5072 6576 3a20 Next: oo, Prev: │ │ │ │ +00282660: 7661 6c75 652c 2055 703a 2074 6f70 206c value, Up: top l │ │ │ │ +00282670: 6576 656c 206c 6f6f 700a 0a63 6170 7475 evel loop..captu │ │ │ │ +00282680: 7265 202d 2d20 6576 616c 7561 7465 204d re -- evaluate M │ │ │ │ +00282690: 6163 6175 6c61 7932 2063 6f64 6520 616e acaulay2 code an │ │ │ │ +002826a0: 6420 6361 7074 7572 6520 7468 6520 6f75 d capture the ou │ │ │ │ +002826b0: 7470 7574 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a tput.*********** │ │ │ │ +002826c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002826d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002826e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +002826f0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00282700: 2020 2020 2865 7272 2c20 6f75 7470 7574 (err, output │ │ │ │ +00282710: 2920 3a3d 2063 6170 7475 7265 2073 7472 ) := capture str │ │ │ │ +00282720: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +00282730: 2020 202a 2073 7472 2c20 6120 2a6e 6f74 * str, a *not │ │ │ │ +00282740: 6520 6e65 743a 204e 6574 2c2c 2061 202a e net: Net,, a * │ │ │ │ +00282750: 6e6f 7465 206c 6973 743a 204c 6973 742c note list: List, │ │ │ │ +00282760: 2c20 6120 2a6e 6f74 6520 7374 7269 6e67 , a *note string │ │ │ │ +00282770: 3a20 5374 7269 6e67 2c2c 0a20 2020 2020 : String,,. │ │ │ │ +00282780: 2020 206f 7220 6120 2a6e 6f74 6520 7465 or a *note te │ │ │ │ +00282790: 7374 2069 6e70 7574 3a20 7465 7374 732c st input: tests, │ │ │ │ +002827a0: 2c20 7468 6520 4d61 6361 756c 6179 3220 , the Macaulay2 │ │ │ │ +002827b0: 636f 6465 2074 6f20 6265 2065 7661 6c75 code to be evalu │ │ │ │ +002827c0: 6174 6564 0a20 202a 202a 6e6f 7465 204f ated. * *note O │ │ │ │ +002827d0: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ +002827e0: 7573 696e 6720 6675 6e63 7469 6f6e 7320 using functions │ │ │ │ +002827f0: 7769 7468 206f 7074 696f 6e61 6c20 696e with optional in │ │ │ │ +00282800: 7075 7473 2c3a 0a20 2020 2020 202a 2055 puts,:. * U │ │ │ │ +00282810: 7365 724d 6f64 6520 3d3e 2061 202a 6e6f serMode => a *no │ │ │ │ +00282820: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ +00282830: 3a20 426f 6f6c 6561 6e2c 2c20 6465 6661 : Boolean,, defa │ │ │ │ +00282840: 756c 7420 7661 6c75 6520 7472 7565 2c0a ult value true,. │ │ │ │ +00282850: 2020 2020 2020 2020 696e 6469 6361 7465 indicate │ │ │ │ +00282860: 7320 7768 6574 6865 7220 6375 7272 656e s whether curren │ │ │ │ +00282870: 746c 7920 6c6f 6164 6564 2070 6163 6b61 tly loaded packa │ │ │ │ +00282880: 6765 7320 616e 6420 6578 706f 7274 6564 ges and exported │ │ │ │ +00282890: 2073 796d 626f 6c73 2073 686f 756c 640a symbols should. │ │ │ │ +002828a0: 2020 2020 2020 2020 6265 2061 6363 6573 be acces │ │ │ │ +002828b0: 7369 626c 6520 7768 696c 6520 7374 7220 sible while str │ │ │ │ +002828c0: 6973 2062 6569 6e67 2065 7661 6c75 6174 is being evaluat │ │ │ │ +002828d0: 6564 2e0a 2020 2020 2020 2a20 5061 636b ed.. * Pack │ │ │ │ +002828e0: 6167 6545 7870 6f72 7473 203d 3e20 6120 ageExports => a │ │ │ │ +002828f0: 2a6e 6f74 6520 6c69 7374 3a20 4c69 7374 *note list: List │ │ │ │ +00282900: 2c2c 2064 6566 6175 6c74 2076 616c 7565 ,, default value │ │ │ │ +00282910: 206e 756c 6c2c 206f 6620 7061 636b 6167 null, of packag │ │ │ │ +00282920: 6573 0a20 2020 2020 2020 2074 6f20 6265 es. to be │ │ │ │ +00282930: 206c 6f61 6465 6420 6265 666f 7265 2065 loaded before e │ │ │ │ +00282940: 7661 6c75 6174 696e 6720 7374 722e 2020 valuating str. │ │ │ │ +00282950: 5468 6520 6375 7272 656e 7420 7061 636b The current pack │ │ │ │ +00282960: 6167 652c 2061 7320 6769 7665 6e20 6279 age, as given by │ │ │ │ +00282970: 0a20 2020 2020 2020 202a 6e6f 7465 204d . *note M │ │ │ │ +00282980: 6163 6175 6c61 7932 446f 633a 2054 6f70 acaulay2Doc: Top │ │ │ │ +00282990: 2c2c 2077 696c 6c20 6265 2069 6e63 6c75 ,, will be inclu │ │ │ │ +002829a0: 6465 642e 0a20 202a 204f 7574 7075 7473 ded.. * Outputs │ │ │ │ +002829b0: 3a0a 2020 2020 2020 2a20 6572 722c 2061 :. * err, a │ │ │ │ +002829c0: 202a 6e6f 7465 2042 6f6f 6c65 616e 2076 *note Boolean v │ │ │ │ +002829d0: 616c 7565 3a20 426f 6f6c 6561 6e2c 2c20 alue: Boolean,, │ │ │ │ +002829e0: 7768 6574 6865 7220 7468 6520 6576 616c whether the eval │ │ │ │ +002829f0: 7561 7469 6f6e 2077 6173 0a20 2020 2020 uation was. │ │ │ │ +00282a00: 2020 2069 6e74 6572 7275 7074 6564 2062 interrupted b │ │ │ │ +00282a10: 6563 6175 7365 206f 6620 616e 2065 7272 ecause of an err │ │ │ │ +00282a20: 6f72 0a20 2020 2020 202a 206f 7574 7075 or. * outpu │ │ │ │ +00282a30: 742c 2061 202a 6e6f 7465 2073 7472 696e t, a *note strin │ │ │ │ +00282a40: 673a 2053 7472 696e 672c 2c20 7468 6520 g: String,, the │ │ │ │ +00282a50: 7265 7375 6c74 206f 6620 7468 6520 6576 result of the ev │ │ │ │ +00282a60: 616c 7561 7469 6f6e 0a0a 4465 7363 7269 aluation..Descri │ │ │ │ +00282a70: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +00282a80: 3d0a 0a54 6869 7320 6675 6e63 7469 6f6e =..This function │ │ │ │ +00282a90: 2065 7661 6c75 6174 6573 2074 6865 2067 evaluates the g │ │ │ │ +00282aa0: 6976 656e 204d 6163 6175 6c61 7932 2063 iven Macaulay2 c │ │ │ │ +00282ab0: 6f64 6520 7769 7468 696e 2074 6865 2073 ode within the s │ │ │ │ +00282ac0: 616d 6520 7072 6f63 6573 7320 696e 0a6f ame process in.o │ │ │ │ +00282ad0: 7264 6572 2074 6f20 7370 6565 6420 7570 rder to speed up │ │ │ │ +00282ae0: 2074 6173 6b73 2073 7563 6820 6173 2067 tasks such as g │ │ │ │ +00282af0: 656e 6572 6174 696e 6720 6578 616d 706c enerating exampl │ │ │ │ +00282b00: 6520 7265 7375 6c74 7320 6f72 2063 6865 e results or che │ │ │ │ +00282b10: 636b 696e 6720 7468 650a 7061 636b 6167 cking the.packag │ │ │ │ +00282b20: 6520 7465 7374 732e 0a0a 2b2d 2d2d 2d2d e tests...+----- │ │ │ │ 00282b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00282b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00282b50: 2d2d 2d2d 2b0a 7c69 3120 3a20 2865 7272 ----+.|i1 : (err │ │ │ │ -00282b60: 2c20 6f75 7470 7574 2920 3d20 6361 7074 , output) = capt │ │ │ │ -00282b70: 7572 6520 6578 616d 706c 6573 2872 6573 ure examples(res │ │ │ │ -00282b80: 6f6c 7574 696f 6e2c 2049 6465 616c 2920 olution, Ideal) │ │ │ │ -00282b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282ba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00282bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00282b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00282b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00282b70: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +00282b80: 2865 7272 2c20 6f75 7470 7574 2920 3d20 (err, output) = │ │ │ │ +00282b90: 6361 7074 7572 6520 6578 616d 706c 6573 capture examples │ │ │ │ +00282ba0: 2872 6573 6f6c 7574 696f 6e2c 2049 6465 (resolution, Ide │ │ │ │ +00282bb0: 616c 2920 2020 2020 2020 2020 2020 2020 al) │ │ │ │ +00282bc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00282bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00282be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282bf0: 2020 2020 7c0a 7c6f 3120 3d20 2866 616c |.|o1 = (fal │ │ │ │ -00282c00: 7365 2c20 2020 2020 2020 2020 2020 2020 se, │ │ │ │ -00282c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00282bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00282c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00282c10: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ +00282c20: 2866 616c 7365 2c20 2020 2020 2020 2020 (false, │ │ │ │ 00282c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282c40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00282c50: 2020 2020 6931 203a 202d 2d20 6578 616d i1 : -- exam │ │ │ │ -00282c60: 706c 6573 2066 6f72 2074 6167 3a20 6672 ples for tag: fr │ │ │ │ -00282c70: 6565 5265 736f 6c75 7469 6f6e 2849 6465 eeResolution(Ide │ │ │ │ -00282c80: 616c 2920 2020 2020 2020 2020 2020 2020 al) │ │ │ │ -00282c90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00282ca0: 2020 2020 2020 2020 202d 2d20 2f62 7569 -- /bui │ │ │ │ -00282cb0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00282cc0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00282cd0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -00282ce0: 6175 6c61 7c0a 7c20 2020 2020 2020 2020 aula|.| │ │ │ │ -00282cf0: 2020 2020 2020 2020 2052 203d 2051 515b R = QQ[ │ │ │ │ -00282d00: 612e 2e64 5d3b 2020 2020 2020 2020 2020 a..d]; │ │ │ │ -00282d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282d30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00282c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00282c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00282c60: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00282c70: 2020 2020 2020 2020 6931 203a 202d 2d20 i1 : -- │ │ │ │ +00282c80: 6578 616d 706c 6573 2066 6f72 2074 6167 examples for tag │ │ │ │ +00282c90: 3a20 6672 6565 5265 736f 6c75 7469 6f6e : freeResolution │ │ │ │ +00282ca0: 2849 6465 616c 2920 2020 2020 2020 2020 (Ideal) │ │ │ │ +00282cb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00282cc0: 2020 2020 2020 2020 2020 2020 202d 2d20 -- │ │ │ │ +00282cd0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00282ce0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00282cf0: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ +00282d00: 2f4d 6163 6175 6c61 7c0a 7c20 2020 2020 /Macaula|.| │ │ │ │ +00282d10: 2020 2020 2020 2020 2020 2020 2052 203d R = │ │ │ │ +00282d20: 2051 515b 612e 2e64 5d3b 2020 2020 2020 QQ[a..d]; │ │ │ │ +00282d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00282d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00282d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00282d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00282d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282d80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00282d90: 2020 2020 6932 203a 2049 203d 2069 6465 i2 : I = ide │ │ │ │ -00282da0: 616c 2863 5e32 2d62 2a64 2c20 622a 632d al(c^2-b*d, b*c- │ │ │ │ -00282db0: 612a 642c 2062 5e32 2d61 2a63 2920 2020 a*d, b^2-a*c) │ │ │ │ -00282dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282dd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00282de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282df0: 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 7c0a 7c20 2020 2020 |.| │ │ │ │ +00282db0: 2020 2020 2020 2020 6932 203a 2049 203d i2 : I = │ │ │ │ +00282dc0: 2069 6465 616c 2863 5e32 2d62 2a64 2c20 ideal(c^2-b*d, │ │ │ │ +00282dd0: 622a 632d 612a 642c 2062 5e32 2d61 2a63 b*c-a*d, b^2-a*c │ │ │ │ +00282de0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00282df0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00282e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00282e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00282e20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00282e20: 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c010: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c030: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0028c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c060: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028c060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c080: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0028c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c0b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c300: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0028c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c330: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028c330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c350: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0028c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c380: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c3a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0028c3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ 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2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0028c540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c560: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028c560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c580: 2020 2020 2034 2020 2020 2020 2020 2020 4 │ │ │ │ +0028c580: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0028c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c5b0: 2020 2020 7c0a 7c20 2020 2020 2d2d 2d2d |.| ---- │ │ │ │ -0028c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028c5d0: 2d2d 2d20 5320 203a 2032 2020 2020 2020 --- S : 2 │ │ │ │ -0028c5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c600: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0028c610: 2020 2020 2020 7879 3244 782d 7932 2020 xy2Dx-y2 │ │ │ │ -0028c620: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0028c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c650: 2020 2020 7c0a 7c20 2020 2020 4478 2d32 |.| Dx-2 │ │ │ │ -0028c660: 7944 792b 3220 2d78 3279 4479 2b78 3220 yDy+2 -x2yDy+x2 │ │ │ │ -0028c670: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0028c680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c6a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028c5a0: 2020 2020 2020 2020 2034 2020 2020 2020 4 │ │ │ │ +0028c5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c5d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0028c5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c5f0: 2d2d 2d2d 2d2d 2d20 5320 203a 2032 2020 ------- S : 2 │ │ │ │ +0028c600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c620: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0028c630: 2020 2020 2020 2020 2020 7879 3244 782d xy2Dx- │ │ │ │ +0028c640: 7932 2020 7c20 2020 2020 2020 2020 2020 y2 | │ │ │ │ +0028c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c670: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0028c680: 4478 2d32 7944 792b 3220 2d78 3279 4479 Dx-2yDy+2 -x2yDy │ │ │ │ +0028c690: 2b78 3220 7c20 2020 2020 2020 2020 2020 +x2 | │ │ │ │ +0028c6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c6c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 0028c6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c6f0: 2020 2020 7c0a 7c6f 3120 3a20 5365 7175 |.|o1 : Sequ │ │ │ │ -0028c700: 656e 6365 2020 2020 2020 2020 2020 2020 ence │ │ │ │ -0028c710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c710: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +0028c720: 5365 7175 656e 6365 2020 2020 2020 2020 Sequence │ │ │ │ 0028c730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c740: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0028c750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028c760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c760: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 0028c770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028c790: 2d2d 2d2d 2b0a 7c69 3220 3a20 6173 7365 ----+.|i2 : asse │ │ │ │ -0028c7a0: 7274 206e 6f74 2065 7272 2020 2020 2020 rt not err │ │ │ │ -0028c7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c7b0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ +0028c7c0: 6173 7365 7274 206e 6f74 2065 7272 2020 assert not err │ │ │ │ 0028c7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c7e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0028c7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028c800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c800: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ 0028c810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028c820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028c830: 2d2d 2d2d 2b0a 0a49 6620 5573 6572 4d6f ----+..If UserMo │ │ │ │ -0028c840: 6465 203d 3e20 6661 6c73 6520 6769 7665 de => false give │ │ │ │ -0028c850: 6e2c 2074 6865 2064 6963 7469 6f6e 6172 n, the dictionar │ │ │ │ -0028c860: 6965 7320 6176 6169 6c61 626c 6520 746f ies available to │ │ │ │ -0028c870: 2074 6865 2075 7365 7220 7769 6c6c 206e the user will n │ │ │ │ -0028c880: 6f74 2062 650a 6166 6665 6374 6564 2062 ot be.affected b │ │ │ │ -0028c890: 7920 7468 6520 6576 616c 7561 7469 6f6e y the evaluation │ │ │ │ -0028c8a0: 2061 6e64 202a 6e6f 7465 2063 6f6c 6c65 and *note colle │ │ │ │ -0028c8b0: 6374 4761 7262 6167 653a 2063 6f6c 6c65 ctGarbage: colle │ │ │ │ -0028c8c0: 6374 4761 7262 6167 652c 2069 7320 6361 ctGarbage, is ca │ │ │ │ -0028c8d0: 6c6c 6564 0a61 6674 6572 7761 7264 732e lled.afterwards. │ │ │ │ -0028c8e0: 0a0a 4572 726f 7273 206f 6363 7572 7265 ..Errors occurre │ │ │ │ -0028c8f0: 6420 7768 696c 6520 6576 616c 7561 7469 d while evaluati │ │ │ │ -0028c900: 6e67 2073 7472 2064 6f20 6e6f 7420 6361 ng str do not ca │ │ │ │ -0028c910: 7573 6520 616e 2065 7272 6f72 206f 7574 use an error out │ │ │ │ -0028c920: 7369 6465 206f 6620 6361 7074 7572 652e side of capture. │ │ │ │ -0028c930: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ -0028c940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028c950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c850: 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6620 5573 --------+..If Us │ │ │ │ +0028c860: 6572 4d6f 6465 203d 3e20 6661 6c73 6520 erMode => false │ │ │ │ +0028c870: 6769 7665 6e2c 2074 6865 2064 6963 7469 given, the dicti │ │ │ │ +0028c880: 6f6e 6172 6965 7320 6176 6169 6c61 626c onaries availabl │ │ │ │ +0028c890: 6520 746f 2074 6865 2075 7365 7220 7769 e to the user wi │ │ │ │ +0028c8a0: 6c6c 206e 6f74 2062 650a 6166 6665 6374 ll not be.affect │ │ │ │ +0028c8b0: 6564 2062 7920 7468 6520 6576 616c 7561 ed by the evalua │ │ │ │ +0028c8c0: 7469 6f6e 2061 6e64 202a 6e6f 7465 2063 tion and *note c │ │ │ │ +0028c8d0: 6f6c 6c65 6374 4761 7262 6167 653a 2063 ollectGarbage: c │ │ │ │ +0028c8e0: 6f6c 6c65 6374 4761 7262 6167 652c 2069 ollectGarbage, i │ │ │ │ +0028c8f0: 7320 6361 6c6c 6564 0a61 6674 6572 7761 s called.afterwa │ │ │ │ +0028c900: 7264 732e 0a0a 4572 726f 7273 206f 6363 rds...Errors occ │ │ │ │ +0028c910: 7572 7265 6420 7768 696c 6520 6576 616c urred while eval │ │ │ │ +0028c920: 7561 7469 6e67 2073 7472 2064 6f20 6e6f uating str do no │ │ │ │ +0028c930: 7420 6361 7573 6520 616e 2065 7272 6f72 t cause an error │ │ │ │ +0028c940: 206f 7574 7369 6465 206f 6620 6361 7074 outside of capt │ │ │ │ +0028c950: 7572 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ure...+--------- │ │ │ │ 0028c960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028c970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028c980: 2b0a 7c69 3320 3a20 2865 7272 2c20 6f75 +.|i3 : (err, ou │ │ │ │ -0028c990: 7470 7574 2920 3d20 6361 7074 7572 6520 tput) = capture │ │ │ │ -0028c9a0: 2f2f 2f20 7374 6465 7272 203c 3c20 2243 /// stderr << "C │ │ │ │ -0028c9b0: 6865 636b 696e 6720 6120 6661 6c73 6520 hecking a false │ │ │ │ -0028c9c0: 7374 6174 656d 656e 743a 2220 3c3c 2020 statement:" << │ │ │ │ -0028c9d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0028c9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028c9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028c980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028c9a0: 2d2d 2d2d 2b0a 7c69 3320 3a20 2865 7272 ----+.|i3 : (err │ │ │ │ +0028c9b0: 2c20 6f75 7470 7574 2920 3d20 6361 7074 , output) = capt │ │ │ │ +0028c9c0: 7572 6520 2f2f 2f20 7374 6465 7272 203c ure /// stderr < │ │ │ │ +0028c9d0: 3c20 2243 6865 636b 696e 6720 6120 6661 < "Checking a fa │ │ │ │ +0028c9e0: 6c73 6520 7374 6174 656d 656e 743a 2220 lse statement:" │ │ │ │ +0028c9f0: 3c3c 2020 7c0a 7c20 2020 2020 2020 2020 << |.| │ │ │ │ 0028ca00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028ca10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028ca20: 7c0a 7c6f 3320 3d20 2874 7275 652c 2020 |.|o3 = (true, │ │ │ │ +0028ca20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028ca30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028ca40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028ca50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028ca60: 2020 2020 2020 2020 2020 2920 2020 2020 ) │ │ │ │ -0028ca70: 7c0a 7c20 2020 2020 2020 2020 2020 2069 |.| i │ │ │ │ -0028ca80: 3120 3a20 2073 7464 6572 7220 3c3c 2022 1 : stderr << " │ │ │ │ -0028ca90: 4368 6563 6b69 6e67 2061 2066 616c 7365 Checking a false │ │ │ │ -0028caa0: 2073 7461 7465 6d65 6e74 3a22 203c 3c20 statement:" << │ │ │ │ -0028cab0: 656e 646c 3b20 312f 3020 2020 2020 2020 endl; 1/0 │ │ │ │ -0028cac0: 7c0a 7c20 2020 2020 2020 2020 2020 2043 |.| C │ │ │ │ -0028cad0: 6865 636b 696e 6720 6120 6661 6c73 6520 hecking a false │ │ │ │ -0028cae0: 7374 6174 656d 656e 743a 2020 2020 2020 statement: │ │ │ │ -0028caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cb10: 7c0a 7c20 2020 2020 2020 2020 2020 2063 |.| c │ │ │ │ -0028cb20: 7572 7265 6e74 5374 7269 6e67 3a31 3a35 urrentString:1:5 │ │ │ │ -0028cb30: 313a 2833 293a 5b34 5d3a 2065 7272 6f72 1:(3):[4]: error │ │ │ │ -0028cb40: 3a20 6469 7669 7369 6f6e 2062 7920 7a65 : division by ze │ │ │ │ -0028cb50: 726f 2020 2020 2020 2020 2020 2020 2020 ro │ │ │ │ -0028cb60: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0028cb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028ca40: 2020 2020 7c0a 7c6f 3320 3d20 2874 7275 |.|o3 = (tru │ │ │ │ +0028ca50: 652c 2020 2020 2020 2020 2020 2020 2020 e, │ │ │ │ +0028ca60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028ca70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028ca80: 2020 2020 2020 2020 2020 2020 2020 2920 ) │ │ │ │ +0028ca90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028caa0: 2020 2069 3120 3a20 2073 7464 6572 7220 i1 : stderr │ │ │ │ +0028cab0: 3c3c 2022 4368 6563 6b69 6e67 2061 2066 << "Checking a f │ │ │ │ +0028cac0: 616c 7365 2073 7461 7465 6d65 6e74 3a22 alse statement:" │ │ │ │ +0028cad0: 203c 3c20 656e 646c 3b20 312f 3020 2020 << endl; 1/0 │ │ │ │ +0028cae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028caf0: 2020 2043 6865 636b 696e 6720 6120 6661 Checking a fa │ │ │ │ +0028cb00: 6c73 6520 7374 6174 656d 656e 743a 2020 lse statement: │ │ │ │ +0028cb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cb30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0028cb40: 2020 2063 7572 7265 6e74 5374 7269 6e67 currentString │ │ │ │ +0028cb50: 3a31 3a35 313a 2833 293a 5b34 5d3a 2065 :1:51:(3):[4]: e │ │ │ │ +0028cb60: 7272 6f72 3a20 6469 7669 7369 6f6e 2062 rror: division b │ │ │ │ +0028cb70: 7920 7a65 726f 2020 2020 2020 2020 2020 y zero │ │ │ │ +0028cb80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0028cb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028cba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cbb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0028cbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028cbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cbd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0028cbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028cbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cc00: 7c0a 7c6f 3320 3a20 5365 7175 656e 6365 |.|o3 : Sequence │ │ │ │ +0028cc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cc20: 2020 2020 7c0a 7c6f 3320 3a20 5365 7175 |.|o3 : Sequ │ │ │ │ +0028cc30: 656e 6365 2020 2020 2020 2020 2020 2020 ence │ │ │ │ 0028cc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cc50: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------- │ │ │ │ -0028cc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028cc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028cc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cc70: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ 0028cc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028cc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028cca0: 7c0a 7c65 6e64 6c3b 2031 2f30 202f 2f2f |.|endl; 1/0 /// │ │ │ │ -0028ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028ccc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028ccd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028ccb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028ccc0: 2d2d 2d2d 7c0a 7c65 6e64 6c3b 2031 2f30 ----|.|endl; 1/0 │ │ │ │ +0028ccd0: 202f 2f2f 2020 2020 2020 2020 2020 2020 /// │ │ │ │ 0028cce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028ccf0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0028cd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028cd10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028ccf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cd10: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0028cd20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028cd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028cd40: 2b0a 7c69 3420 3a20 6173 7365 7274 2065 +.|i4 : assert e │ │ │ │ -0028cd50: 7272 2020 2020 2020 2020 2020 2020 2020 rr │ │ │ │ -0028cd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028cd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028cd60: 2d2d 2d2d 2b0a 7c69 3420 3a20 6173 7365 ----+.|i4 : asse │ │ │ │ +0028cd70: 7274 2065 7272 2020 2020 2020 2020 2020 rt err │ │ │ │ 0028cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028cd90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0028cda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028cdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028cd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028cdb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 0028cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028cde0: 2b0a 0a43 6572 7461 696e 2065 7861 6d70 +..Certain examp │ │ │ │ -0028cdf0: 6c65 7320 6f72 2074 6573 7473 206d 6967 les or tests mig │ │ │ │ -0028ce00: 6874 206e 6f74 2062 6520 636f 6d70 6174 ht not be compat │ │ │ │ -0028ce10: 6962 6c65 2077 6974 6820 6361 7074 7572 ible with captur │ │ │ │ -0028ce20: 652e 2049 6e20 7375 6368 2063 6173 6573 e. In such cases │ │ │ │ -0028ce30: 2c0a 7573 6520 7468 6520 7374 7269 6e67 ,.use the string │ │ │ │ -0028ce40: 202d 2a20 6e6f 2d63 6170 7475 7265 2d66 -* no-capture-f │ │ │ │ -0028ce50: 6c61 6720 2a2d 2073 6f6d 6577 6865 7265 lag *- somewhere │ │ │ │ -0028ce60: 2077 6974 6869 6e20 7374 7220 736f 2074 within str so t │ │ │ │ -0028ce70: 6861 7420 7468 6520 636f 6465 2069 730a hat the code is. │ │ │ │ -0028ce80: 7275 6e20 696e 2061 2073 6570 6172 6174 run in a separat │ │ │ │ -0028ce90: 6520 4d61 6361 756c 6179 3220 7375 6270 e Macaulay2 subp │ │ │ │ -0028cea0: 726f 6365 7373 2069 6e73 7465 6164 2e0a rocess instead.. │ │ │ │ -0028ceb0: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -0028cec0: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6368 ==.. * *note ch │ │ │ │ -0028ced0: 6563 6b3a 2063 6865 636b 2c20 2d2d 2070 eck: check, -- p │ │ │ │ -0028cee0: 6572 666f 726d 2074 6573 7473 206f 6620 erform tests of │ │ │ │ -0028cef0: 6120 7061 636b 6167 650a 2020 2a20 2a6e a package. * *n │ │ │ │ -0028cf00: 6f74 6520 6578 616d 706c 6573 3a20 6578 ote examples: ex │ │ │ │ -0028cf10: 616d 706c 6573 2c20 2d2d 206c 6973 7420 amples, -- list │ │ │ │ -0028cf20: 7468 6520 6578 616d 706c 6573 2069 6e20 the examples in │ │ │ │ -0028cf30: 646f 6375 6d65 6e74 6174 696f 6e0a 0a57 documentation..W │ │ │ │ -0028cf40: 6179 7320 746f 2075 7365 2063 6170 7475 ays to use captu │ │ │ │ -0028cf50: 7265 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d re:.============ │ │ │ │ -0028cf60: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2263 ========.. * "c │ │ │ │ -0028cf70: 6170 7475 7265 284c 6973 7429 220a 2020 apture(List)". │ │ │ │ -0028cf80: 2a20 2263 6170 7475 7265 284e 6574 2922 * "capture(Net)" │ │ │ │ -0028cf90: 0a20 202a 2022 6361 7074 7572 6528 5374 . * "capture(St │ │ │ │ -0028cfa0: 7269 6e67 2922 0a20 202a 2022 6361 7074 ring)". * "capt │ │ │ │ -0028cfb0: 7572 6528 5465 7374 496e 7075 7429 220a ure(TestInput)". │ │ │ │ -0028cfc0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -0028cfd0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -0028cfe0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -0028cff0: 6374 202a 6e6f 7465 2063 6170 7475 7265 ct *note capture │ │ │ │ -0028d000: 3a20 6361 7074 7572 652c 2069 7320 6120 : capture, is a │ │ │ │ -0028d010: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ -0028d020: 6374 696f 6e20 7769 7468 206f 7074 696f ction with optio │ │ │ │ -0028d030: 6e73 3a0a 4d65 7468 6f64 4675 6e63 7469 ns:.MethodFuncti │ │ │ │ -0028d040: 6f6e 5769 7468 4f70 7469 6f6e 732c 2e0a onWithOptions,.. │ │ │ │ -0028d050: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ -0028d060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028d070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028ce00: 2d2d 2d2d 2b0a 0a43 6572 7461 696e 2065 ----+..Certain e │ │ │ │ +0028ce10: 7861 6d70 6c65 7320 6f72 2074 6573 7473 xamples or tests │ │ │ │ +0028ce20: 206d 6967 6874 206e 6f74 2062 6520 636f might not be co │ │ │ │ +0028ce30: 6d70 6174 6962 6c65 2077 6974 6820 6361 mpatible with ca │ │ │ │ +0028ce40: 7074 7572 652e 2049 6e20 7375 6368 2063 pture. In such c │ │ │ │ +0028ce50: 6173 6573 2c0a 7573 6520 7468 6520 7374 ases,.use the st │ │ │ │ +0028ce60: 7269 6e67 202d 2a20 6e6f 2d63 6170 7475 ring -* no-captu │ │ │ │ +0028ce70: 7265 2d66 6c61 6720 2a2d 2073 6f6d 6577 re-flag *- somew │ │ │ │ +0028ce80: 6865 7265 2077 6974 6869 6e20 7374 7220 here within str │ │ │ │ +0028ce90: 736f 2074 6861 7420 7468 6520 636f 6465 so that the code │ │ │ │ +0028cea0: 2069 730a 7275 6e20 696e 2061 2073 6570 is.run in a sep │ │ │ │ +0028ceb0: 6172 6174 6520 4d61 6361 756c 6179 3220 arate Macaulay2 │ │ │ │ +0028cec0: 7375 6270 726f 6365 7373 2069 6e73 7465 subprocess inste │ │ │ │ +0028ced0: 6164 2e0a 0a53 6565 2061 6c73 6f0a 3d3d ad...See also.== │ │ │ │ +0028cee0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +0028cef0: 6520 6368 6563 6b3a 2063 6865 636b 2c20 e check: check, │ │ │ │ +0028cf00: 2d2d 2070 6572 666f 726d 2074 6573 7473 -- perform tests │ │ │ │ +0028cf10: 206f 6620 6120 7061 636b 6167 650a 2020 of a package. │ │ │ │ +0028cf20: 2a20 2a6e 6f74 6520 6578 616d 706c 6573 * *note examples │ │ │ │ +0028cf30: 3a20 6578 616d 706c 6573 2c20 2d2d 206c : examples, -- l │ │ │ │ +0028cf40: 6973 7420 7468 6520 6578 616d 706c 6573 ist the examples │ │ │ │ +0028cf50: 2069 6e20 646f 6375 6d65 6e74 6174 696f in documentatio │ │ │ │ +0028cf60: 6e0a 0a57 6179 7320 746f 2075 7365 2063 n..Ways to use c │ │ │ │ +0028cf70: 6170 7475 7265 3a0a 3d3d 3d3d 3d3d 3d3d apture:.======== │ │ │ │ +0028cf80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +0028cf90: 2a20 2263 6170 7475 7265 284c 6973 7429 * "capture(List) │ │ │ │ +0028cfa0: 220a 2020 2a20 2263 6170 7475 7265 284e ". * "capture(N │ │ │ │ +0028cfb0: 6574 2922 0a20 202a 2022 6361 7074 7572 et)". * "captur │ │ │ │ +0028cfc0: 6528 5374 7269 6e67 2922 0a20 202a 2022 e(String)". * " │ │ │ │ +0028cfd0: 6361 7074 7572 6528 5465 7374 496e 7075 capture(TestInpu │ │ │ │ +0028cfe0: 7429 220a 0a46 6f72 2074 6865 2070 726f t)"..For the pro │ │ │ │ +0028cff0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +0028d000: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0028d010: 6f62 6a65 6374 202a 6e6f 7465 2063 6170 object *note cap │ │ │ │ +0028d020: 7475 7265 3a20 6361 7074 7572 652c 2069 ture: capture, i │ │ │ │ +0028d030: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +0028d040: 2066 756e 6374 696f 6e20 7769 7468 206f function with o │ │ │ │ +0028d050: 7074 696f 6e73 3a0a 4d65 7468 6f64 4675 ptions:.MethodFu │ │ │ │ +0028d060: 6e63 7469 6f6e 5769 7468 4f70 7469 6f6e nctionWithOption │ │ │ │ +0028d070: 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d s,...----------- │ │ │ │ 0028d080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028d090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028d0a0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -0028d0b0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -0028d0c0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -0028d0d0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -0028d0e0: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ -0028d0f0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -0028d100: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ -0028d110: 7932 446f 632f 6675 6e63 7469 6f6e 732f y2Doc/functions/ │ │ │ │ -0028d120: 6578 616d 706c 6573 2d64 6f63 2e6d 323a examples-doc.m2: │ │ │ │ -0028d130: 3133 343a 302e 0a1f 0a46 696c 653a 204d 134:0....File: M │ │ │ │ -0028d140: 6163 6175 6c61 7932 446f 632e 696e 666f acaulay2Doc.info │ │ │ │ -0028d150: 2c20 4e6f 6465 3a20 6f6f 2c20 4e65 7874 , Node: oo, Next │ │ │ │ -0028d160: 3a20 6f6f 6f2c 2050 7265 763a 2063 6170 : ooo, Prev: cap │ │ │ │ -0028d170: 7475 7265 2c20 5570 3a20 746f 7020 6c65 ture, Up: top le │ │ │ │ -0028d180: 7665 6c20 6c6f 6f70 0a0a 6f6f 202d 2d20 vel loop..oo -- │ │ │ │ -0028d190: 7468 6520 6c61 7374 206f 7574 7075 7420 the last output │ │ │ │ -0028d1a0: 7661 6c75 650a 2a2a 2a2a 2a2a 2a2a 2a2a value.********** │ │ │ │ -0028d1b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028d1c0: 2a0a 0a44 6573 6372 6970 7469 6f6e 0a3d *..Description.= │ │ │ │ -0028d1d0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 6f6f 202d ==========..oo - │ │ │ │ -0028d1e0: 2d20 6465 6e6f 7465 7320 7468 6520 7661 - denotes the va │ │ │ │ -0028d1f0: 6c75 6520 6f66 2074 6865 2065 7870 7265 lue of the expre │ │ │ │ -0028d200: 7373 696f 6e20 6f6e 2074 6865 2070 7265 ssion on the pre │ │ │ │ -0028d210: 7669 6f75 7320 6f75 7470 7574 206c 696e vious output lin │ │ │ │ -0028d220: 652e 0a0a 5365 6520 616c 736f 0a3d 3d3d e...See also.=== │ │ │ │ -0028d230: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0028d240: 206f 6f3a 206f 6f2c 202d 2d20 7468 6520 oo: oo, -- the │ │ │ │ -0028d250: 6c61 7374 206f 7574 7075 7420 7661 6c75 last output valu │ │ │ │ -0028d260: 650a 2020 2a20 2a6e 6f74 6520 6f6f 6f3a e. * *note ooo: │ │ │ │ -0028d270: 206f 6f6f 2c20 2d2d 2074 6865 206e 6578 ooo, -- the nex │ │ │ │ -0028d280: 7420 746f 2074 6865 206c 6173 7420 6f75 t to the last ou │ │ │ │ -0028d290: 7470 7574 2076 616c 7565 0a20 202a 202a tput value. * * │ │ │ │ -0028d2a0: 6e6f 7465 206f 6f6f 6f3a 206f 6f6f 6f2c note oooo: oooo, │ │ │ │ -0028d2b0: 202d 2d20 7468 6520 7468 6972 6420 746f -- the third to │ │ │ │ -0028d2c0: 2074 6865 206c 6173 7420 6f75 7470 7574 the last output │ │ │ │ -0028d2d0: 2076 616c 7565 0a0a 466f 7220 7468 6520 value..For the │ │ │ │ -0028d2e0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0028d2f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0028d300: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0028d310: 6f6f 3a20 6f6f 2c20 6973 2061 202a 6e6f oo: oo, is a *no │ │ │ │ -0028d320: 7465 2073 796d 626f 6c3a 2053 796d 626f te symbol: Symbo │ │ │ │ -0028d330: 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d l,...----------- │ │ │ │ -0028d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d0c0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +0028d0d0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +0028d0e0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +0028d0f0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +0028d100: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +0028d110: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +0028d120: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ +0028d130: 6175 6c61 7932 446f 632f 6675 6e63 7469 aulay2Doc/functi │ │ │ │ +0028d140: 6f6e 732f 6578 616d 706c 6573 2d64 6f63 ons/examples-doc │ │ │ │ +0028d150: 2e6d 323a 3133 343a 302e 0a1f 0a46 696c .m2:134:0....Fil │ │ │ │ +0028d160: 653a 204d 6163 6175 6c61 7932 446f 632e e: Macaulay2Doc. │ │ │ │ +0028d170: 696e 666f 2c20 4e6f 6465 3a20 6f6f 2c20 info, Node: oo, │ │ │ │ +0028d180: 4e65 7874 3a20 6f6f 6f2c 2050 7265 763a Next: ooo, Prev: │ │ │ │ +0028d190: 2063 6170 7475 7265 2c20 5570 3a20 746f capture, Up: to │ │ │ │ +0028d1a0: 7020 6c65 7665 6c20 6c6f 6f70 0a0a 6f6f p level loop..oo │ │ │ │ +0028d1b0: 202d 2d20 7468 6520 6c61 7374 206f 7574 -- the last out │ │ │ │ +0028d1c0: 7075 7420 7661 6c75 650a 2a2a 2a2a 2a2a put value.****** │ │ │ │ +0028d1d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028d1e0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ +0028d1f0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0028d200: 6f6f 202d 2d20 6465 6e6f 7465 7320 7468 oo -- denotes th │ │ │ │ +0028d210: 6520 7661 6c75 6520 6f66 2074 6865 2065 e value of the e │ │ │ │ +0028d220: 7870 7265 7373 696f 6e20 6f6e 2074 6865 xpression on the │ │ │ │ +0028d230: 2070 7265 7669 6f75 7320 6f75 7470 7574 previous output │ │ │ │ +0028d240: 206c 696e 652e 0a0a 5365 6520 616c 736f line...See also │ │ │ │ +0028d250: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0028d260: 6e6f 7465 206f 6f3a 206f 6f2c 202d 2d20 note oo: oo, -- │ │ │ │ +0028d270: 7468 6520 6c61 7374 206f 7574 7075 7420 the last output │ │ │ │ +0028d280: 7661 6c75 650a 2020 2a20 2a6e 6f74 6520 value. * *note │ │ │ │ +0028d290: 6f6f 6f3a 206f 6f6f 2c20 2d2d 2074 6865 ooo: ooo, -- the │ │ │ │ +0028d2a0: 206e 6578 7420 746f 2074 6865 206c 6173 next to the las │ │ │ │ +0028d2b0: 7420 6f75 7470 7574 2076 616c 7565 0a20 t output value. │ │ │ │ +0028d2c0: 202a 202a 6e6f 7465 206f 6f6f 6f3a 206f * *note oooo: o │ │ │ │ +0028d2d0: 6f6f 6f2c 202d 2d20 7468 6520 7468 6972 ooo, -- the thir │ │ │ │ +0028d2e0: 6420 746f 2074 6865 206c 6173 7420 6f75 d to the last ou │ │ │ │ +0028d2f0: 7470 7574 2076 616c 7565 0a0a 466f 7220 tput value..For │ │ │ │ +0028d300: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +0028d310: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0028d320: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +0028d330: 6f74 6520 6f6f 3a20 6f6f 2c20 6973 2061 ote oo: oo, is a │ │ │ │ +0028d340: 202a 6e6f 7465 2073 796d 626f 6c3a 2053 *note symbol: S │ │ │ │ +0028d350: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d ymbol,...------- │ │ │ │ 0028d360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028d380: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -0028d390: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -0028d3a0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -0028d3b0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -0028d3c0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -0028d3d0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -0028d3e0: 7932 2f70 6163 6b61 6765 732f 0a4d 6163 y2/packages/.Mac │ │ │ │ -0028d3f0: 6175 6c61 7932 446f 632f 6f76 5f72 6570 aulay2Doc/ov_rep │ │ │ │ -0028d400: 6c2e 6d32 3a35 3633 3a30 2e0a 1f0a 4669 l.m2:563:0....Fi │ │ │ │ -0028d410: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ -0028d420: 2e69 6e66 6f2c 204e 6f64 653a 206f 6f6f .info, Node: ooo │ │ │ │ -0028d430: 2c20 4e65 7874 3a20 6f6f 6f6f 2c20 5072 , Next: oooo, Pr │ │ │ │ -0028d440: 6576 3a20 6f6f 2c20 5570 3a20 746f 7020 ev: oo, Up: top │ │ │ │ -0028d450: 6c65 7665 6c20 6c6f 6f70 0a0a 6f6f 6f20 level loop..ooo │ │ │ │ -0028d460: 2d2d 2074 6865 206e 6578 7420 746f 2074 -- the next to t │ │ │ │ -0028d470: 6865 206c 6173 7420 6f75 7470 7574 2076 he last output v │ │ │ │ -0028d480: 616c 7565 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a alue.*********** │ │ │ │ -0028d490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028d4a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 *************..D │ │ │ │ -0028d4b0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0028d4c0: 3d3d 3d3d 3d3d 0a0a 6f6f 6f20 2d2d 2064 ======..ooo -- d │ │ │ │ -0028d4d0: 656e 6f74 6573 2074 6865 2076 616c 7565 enotes the value │ │ │ │ -0028d4e0: 206f 6620 7468 6520 6578 7072 6573 7369 of the expressi │ │ │ │ -0028d4f0: 6f6e 206f 6e20 7468 6520 6f75 7470 7574 on on the output │ │ │ │ -0028d500: 206c 696e 6520 7477 6f20 6c69 6e65 7320 line two lines │ │ │ │ -0028d510: 6162 6f76 652e 0a0a 5365 6520 616c 736f above...See also │ │ │ │ -0028d520: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -0028d530: 6e6f 7465 206f 6f3a 206f 6f2c 202d 2d20 note oo: oo, -- │ │ │ │ -0028d540: 7468 6520 6c61 7374 206f 7574 7075 7420 the last output │ │ │ │ -0028d550: 7661 6c75 650a 2020 2a20 2a6e 6f74 6520 value. * *note │ │ │ │ -0028d560: 6f6f 6f6f 3a20 6f6f 6f6f 2c20 2d2d 2074 oooo: oooo, -- t │ │ │ │ -0028d570: 6865 2074 6869 7264 2074 6f20 7468 6520 he third to the │ │ │ │ -0028d580: 6c61 7374 206f 7574 7075 7420 7661 6c75 last output valu │ │ │ │ -0028d590: 650a 0a46 6f72 2074 6865 2070 726f 6772 e..For the progr │ │ │ │ -0028d5a0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -0028d5b0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -0028d5c0: 6a65 6374 202a 6e6f 7465 206f 6f6f 3a20 ject *note ooo: │ │ │ │ -0028d5d0: 6f6f 6f2c 2069 7320 6120 2a6e 6f74 6520 ooo, is a *note │ │ │ │ -0028d5e0: 7379 6d62 6f6c 3a20 5379 6d62 6f6c 2c2e symbol: Symbol,. │ │ │ │ -0028d5f0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ -0028d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d3a0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0028d3b0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0028d3c0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0028d3d0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0028d3e0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0028d3f0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +0028d400: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0028d410: 0a4d 6163 6175 6c61 7932 446f 632f 6f76 .Macaulay2Doc/ov │ │ │ │ +0028d420: 5f72 6570 6c2e 6d32 3a35 3633 3a30 2e0a _repl.m2:563:0.. │ │ │ │ +0028d430: 1f0a 4669 6c65 3a20 4d61 6361 756c 6179 ..File: Macaulay │ │ │ │ +0028d440: 3244 6f63 2e69 6e66 6f2c 204e 6f64 653a 2Doc.info, Node: │ │ │ │ +0028d450: 206f 6f6f 2c20 4e65 7874 3a20 6f6f 6f6f ooo, Next: oooo │ │ │ │ +0028d460: 2c20 5072 6576 3a20 6f6f 2c20 5570 3a20 , Prev: oo, Up: │ │ │ │ +0028d470: 746f 7020 6c65 7665 6c20 6c6f 6f70 0a0a top level loop.. │ │ │ │ +0028d480: 6f6f 6f20 2d2d 2074 6865 206e 6578 7420 ooo -- the next │ │ │ │ +0028d490: 746f 2074 6865 206c 6173 7420 6f75 7470 to the last outp │ │ │ │ +0028d4a0: 7574 2076 616c 7565 0a2a 2a2a 2a2a 2a2a ut value.******* │ │ │ │ +0028d4b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028d4c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028d4d0: 2a0a 0a44 6573 6372 6970 7469 6f6e 0a3d *..Description.= │ │ │ │ +0028d4e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 6f6f 6f20 ==========..ooo │ │ │ │ +0028d4f0: 2d2d 2064 656e 6f74 6573 2074 6865 2076 -- denotes the v │ │ │ │ +0028d500: 616c 7565 206f 6620 7468 6520 6578 7072 alue of the expr │ │ │ │ +0028d510: 6573 7369 6f6e 206f 6e20 7468 6520 6f75 ession on the ou │ │ │ │ +0028d520: 7470 7574 206c 696e 6520 7477 6f20 6c69 tput line two li │ │ │ │ +0028d530: 6e65 7320 6162 6f76 652e 0a0a 5365 6520 nes above...See │ │ │ │ +0028d540: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ +0028d550: 202a 202a 6e6f 7465 206f 6f3a 206f 6f2c * *note oo: oo, │ │ │ │ +0028d560: 202d 2d20 7468 6520 6c61 7374 206f 7574 -- the last out │ │ │ │ +0028d570: 7075 7420 7661 6c75 650a 2020 2a20 2a6e put value. * *n │ │ │ │ +0028d580: 6f74 6520 6f6f 6f6f 3a20 6f6f 6f6f 2c20 ote oooo: oooo, │ │ │ │ +0028d590: 2d2d 2074 6865 2074 6869 7264 2074 6f20 -- the third to │ │ │ │ +0028d5a0: 7468 6520 6c61 7374 206f 7574 7075 7420 the last output │ │ │ │ +0028d5b0: 7661 6c75 650a 0a46 6f72 2074 6865 2070 value..For the p │ │ │ │ +0028d5c0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +0028d5d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +0028d5e0: 6520 6f62 6a65 6374 202a 6e6f 7465 206f e object *note o │ │ │ │ +0028d5f0: 6f6f 3a20 6f6f 6f2c 2069 7320 6120 2a6e oo: ooo, is a *n │ │ │ │ +0028d600: 6f74 6520 7379 6d62 6f6c 3a20 5379 6d62 ote symbol: Symb │ │ │ │ +0028d610: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ol,...---------- │ │ │ │ 0028d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028d640: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0028d650: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0028d660: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0028d670: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0028d680: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ -0028d690: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0028d6a0: 7061 636b 6167 6573 2f0a 4d61 6361 756c packages/.Macaul │ │ │ │ -0028d6b0: 6179 3244 6f63 2f6f 765f 7265 706c 2e6d ay2Doc/ov_repl.m │ │ │ │ -0028d6c0: 323a 3537 313a 302e 0a1f 0a46 696c 653a 2:571:0....File: │ │ │ │ -0028d6d0: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ -0028d6e0: 666f 2c20 4e6f 6465 3a20 6f6f 6f6f 2c20 fo, Node: oooo, │ │ │ │ -0028d6f0: 4e65 7874 3a20 4f75 7470 7574 4469 6374 Next: OutputDict │ │ │ │ -0028d700: 696f 6e61 7279 2c20 5072 6576 3a20 6f6f ionary, Prev: oo │ │ │ │ -0028d710: 6f2c 2055 703a 2074 6f70 206c 6576 656c o, Up: top level │ │ │ │ -0028d720: 206c 6f6f 700a 0a6f 6f6f 6f20 2d2d 2074 loop..oooo -- t │ │ │ │ -0028d730: 6865 2074 6869 7264 2074 6f20 7468 6520 he third to the │ │ │ │ -0028d740: 6c61 7374 206f 7574 7075 7420 7661 6c75 last output valu │ │ │ │ -0028d750: 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a e.************** │ │ │ │ -0028d760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028d770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 ************..De │ │ │ │ -0028d780: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -0028d790: 3d3d 3d3d 3d0a 0a6f 6f6f 6f20 2d2d 2064 =====..oooo -- d │ │ │ │ -0028d7a0: 656e 6f74 6573 2074 6865 2076 616c 7565 enotes the value │ │ │ │ -0028d7b0: 206f 6620 7468 6520 6578 7072 6573 7369 of the expressi │ │ │ │ -0028d7c0: 6f6e 206f 6e20 7468 6520 6f75 7470 7574 on on the output │ │ │ │ -0028d7d0: 206c 696e 6520 7468 7265 6520 6c69 6e65 line three line │ │ │ │ -0028d7e0: 730a 6162 6f76 652e 0a0a 5365 6520 616c s.above...See al │ │ │ │ -0028d7f0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -0028d800: 202a 6e6f 7465 206f 6f3a 206f 6f2c 202d *note oo: oo, - │ │ │ │ -0028d810: 2d20 7468 6520 6c61 7374 206f 7574 7075 - the last outpu │ │ │ │ -0028d820: 7420 7661 6c75 650a 2020 2a20 2a6e 6f74 t value. * *not │ │ │ │ -0028d830: 6520 6f6f 6f3a 206f 6f6f 2c20 2d2d 2074 e ooo: ooo, -- t │ │ │ │ -0028d840: 6865 206e 6578 7420 746f 2074 6865 206c he next to the l │ │ │ │ -0028d850: 6173 7420 6f75 7470 7574 2076 616c 7565 ast output value │ │ │ │ -0028d860: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0028d870: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0028d880: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0028d890: 6563 7420 2a6e 6f74 6520 6f6f 6f6f 3a20 ect *note oooo: │ │ │ │ -0028d8a0: 6f6f 6f6f 2c20 6973 2061 202a 6e6f 7465 oooo, is a *note │ │ │ │ -0028d8b0: 2073 796d 626f 6c3a 2053 796d 626f 6c2c symbol: Symbol, │ │ │ │ -0028d8c0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ -0028d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d660: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0028d670: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0028d680: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0028d690: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0028d6a0: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +0028d6b0: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +0028d6c0: 6179 322f 7061 636b 6167 6573 2f0a 4d61 ay2/packages/.Ma │ │ │ │ +0028d6d0: 6361 756c 6179 3244 6f63 2f6f 765f 7265 caulay2Doc/ov_re │ │ │ │ +0028d6e0: 706c 2e6d 323a 3537 313a 302e 0a1f 0a46 pl.m2:571:0....F │ │ │ │ +0028d6f0: 696c 653a 204d 6163 6175 6c61 7932 446f ile: Macaulay2Do │ │ │ │ +0028d700: 632e 696e 666f 2c20 4e6f 6465 3a20 6f6f c.info, Node: oo │ │ │ │ +0028d710: 6f6f 2c20 4e65 7874 3a20 4f75 7470 7574 oo, Next: Output │ │ │ │ +0028d720: 4469 6374 696f 6e61 7279 2c20 5072 6576 Dictionary, Prev │ │ │ │ +0028d730: 3a20 6f6f 6f2c 2055 703a 2074 6f70 206c : ooo, Up: top l │ │ │ │ +0028d740: 6576 656c 206c 6f6f 700a 0a6f 6f6f 6f20 evel loop..oooo │ │ │ │ +0028d750: 2d2d 2074 6865 2074 6869 7264 2074 6f20 -- the third to │ │ │ │ +0028d760: 7468 6520 6c61 7374 206f 7574 7075 7420 the last output │ │ │ │ +0028d770: 7661 6c75 650a 2a2a 2a2a 2a2a 2a2a 2a2a value.********** │ │ │ │ +0028d780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028d790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028d7a0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0028d7b0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a6f 6f6f 6f20 =========..oooo │ │ │ │ +0028d7c0: 2d2d 2064 656e 6f74 6573 2074 6865 2076 -- denotes the v │ │ │ │ +0028d7d0: 616c 7565 206f 6620 7468 6520 6578 7072 alue of the expr │ │ │ │ +0028d7e0: 6573 7369 6f6e 206f 6e20 7468 6520 6f75 ession on the ou │ │ │ │ +0028d7f0: 7470 7574 206c 696e 6520 7468 7265 6520 tput line three │ │ │ │ +0028d800: 6c69 6e65 730a 6162 6f76 652e 0a0a 5365 lines.above...Se │ │ │ │ +0028d810: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +0028d820: 0a20 202a 202a 6e6f 7465 206f 6f3a 206f . * *note oo: o │ │ │ │ +0028d830: 6f2c 202d 2d20 7468 6520 6c61 7374 206f o, -- the last o │ │ │ │ +0028d840: 7574 7075 7420 7661 6c75 650a 2020 2a20 utput value. * │ │ │ │ +0028d850: 2a6e 6f74 6520 6f6f 6f3a 206f 6f6f 2c20 *note ooo: ooo, │ │ │ │ +0028d860: 2d2d 2074 6865 206e 6578 7420 746f 2074 -- the next to t │ │ │ │ +0028d870: 6865 206c 6173 7420 6f75 7470 7574 2076 he last output v │ │ │ │ +0028d880: 616c 7565 0a0a 466f 7220 7468 6520 7072 alue..For the pr │ │ │ │ +0028d890: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +0028d8a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +0028d8b0: 206f 626a 6563 7420 2a6e 6f74 6520 6f6f object *note oo │ │ │ │ +0028d8c0: 6f6f 3a20 6f6f 6f6f 2c20 6973 2061 202a oo: oooo, is a * │ │ │ │ +0028d8d0: 6e6f 7465 2073 796d 626f 6c3a 2053 796d note symbol: Sym │ │ │ │ +0028d8e0: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d bol,...--------- │ │ │ │ 0028d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028d910: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0028d920: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0028d930: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0028d940: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0028d950: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -0028d960: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0028d970: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ -0028d980: 6c61 7932 446f 632f 6f76 5f72 6570 6c2e lay2Doc/ov_repl. │ │ │ │ -0028d990: 6d32 3a35 3739 3a30 2e0a 1f0a 4669 6c65 m2:579:0....File │ │ │ │ -0028d9a0: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ -0028d9b0: 6e66 6f2c 204e 6f64 653a 204f 7574 7075 nfo, Node: Outpu │ │ │ │ -0028d9c0: 7444 6963 7469 6f6e 6172 792c 204e 6578 tDictionary, Nex │ │ │ │ -0028d9d0: 743a 2063 6c65 6172 416c 6c2c 2050 7265 t: clearAll, Pre │ │ │ │ -0028d9e0: 763a 206f 6f6f 6f2c 2055 703a 2074 6f70 v: oooo, Up: top │ │ │ │ -0028d9f0: 206c 6576 656c 206c 6f6f 700a 0a4f 7574 level loop..Out │ │ │ │ -0028da00: 7075 7444 6963 7469 6f6e 6172 7920 2d2d putDictionary -- │ │ │ │ -0028da10: 2074 6865 2064 6963 7469 6f6e 6172 7920 the dictionary │ │ │ │ -0028da20: 666f 7220 6f75 7470 7574 2076 616c 7565 for output value │ │ │ │ -0028da30: 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a s.************** │ │ │ │ -0028da40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028da50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028da60: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074 ******..Descript │ │ │ │ -0028da70: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -0028da80: 0a54 6865 2073 796d 626f 6c73 206f 312c .The symbols o1, │ │ │ │ -0028da90: 206f 322c 206f 332c 2065 7463 2e2c 2061 o2, o3, etc., a │ │ │ │ -0028daa0: 7265 2075 7365 6420 746f 2073 746f 7265 re used to store │ │ │ │ -0028dab0: 2074 6865 206f 7574 7075 7420 7661 6c75 the output valu │ │ │ │ -0028dac0: 6573 2061 7269 7369 6e67 2066 726f 6d0a es arising from. │ │ │ │ -0028dad0: 696e 7465 7261 6374 696f 6e20 7769 7468 interaction with │ │ │ │ -0028dae0: 2074 6865 2075 7365 722c 206f 6e65 206c the user, one l │ │ │ │ -0028daf0: 696e 6520 6174 2061 2074 696d 652e 2020 ine at a time. │ │ │ │ -0028db00: 5468 6520 6469 6374 696f 6e61 7279 204f The dictionary O │ │ │ │ -0028db10: 7574 7075 7444 6963 7469 6f6e 6172 790a utputDictionary. │ │ │ │ -0028db20: 6973 2074 6865 2064 6963 7469 6f6e 6172 is the dictionar │ │ │ │ -0028db30: 7920 696e 2077 6869 6368 2074 686f 7365 y in which those │ │ │ │ -0028db40: 2073 796d 626f 6c73 2072 6573 6964 652e symbols reside. │ │ │ │ -0028db50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0028db60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028d930: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0028d940: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0028d950: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0028d960: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0028d970: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0028d980: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +0028d990: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ +0028d9a0: 6163 6175 6c61 7932 446f 632f 6f76 5f72 acaulay2Doc/ov_r │ │ │ │ +0028d9b0: 6570 6c2e 6d32 3a35 3739 3a30 2e0a 1f0a epl.m2:579:0.... │ │ │ │ +0028d9c0: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ +0028d9d0: 6f63 2e69 6e66 6f2c 204e 6f64 653a 204f oc.info, Node: O │ │ │ │ +0028d9e0: 7574 7075 7444 6963 7469 6f6e 6172 792c utputDictionary, │ │ │ │ +0028d9f0: 204e 6578 743a 2063 6c65 6172 416c 6c2c Next: clearAll, │ │ │ │ +0028da00: 2050 7265 763a 206f 6f6f 6f2c 2055 703a Prev: oooo, Up: │ │ │ │ +0028da10: 2074 6f70 206c 6576 656c 206c 6f6f 700a top level loop. │ │ │ │ +0028da20: 0a4f 7574 7075 7444 6963 7469 6f6e 6172 .OutputDictionar │ │ │ │ +0028da30: 7920 2d2d 2074 6865 2064 6963 7469 6f6e y -- the diction │ │ │ │ +0028da40: 6172 7920 666f 7220 6f75 7470 7574 2076 ary for output v │ │ │ │ +0028da50: 616c 7565 730a 2a2a 2a2a 2a2a 2a2a 2a2a alues.********** │ │ │ │ +0028da60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028da70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028da80: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 7363 **********..Desc │ │ │ │ +0028da90: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +0028daa0: 3d3d 3d0a 0a54 6865 2073 796d 626f 6c73 ===..The symbols │ │ │ │ +0028dab0: 206f 312c 206f 322c 206f 332c 2065 7463 o1, o2, o3, etc │ │ │ │ +0028dac0: 2e2c 2061 7265 2075 7365 6420 746f 2073 ., are used to s │ │ │ │ +0028dad0: 746f 7265 2074 6865 206f 7574 7075 7420 tore the output │ │ │ │ +0028dae0: 7661 6c75 6573 2061 7269 7369 6e67 2066 values arising f │ │ │ │ +0028daf0: 726f 6d0a 696e 7465 7261 6374 696f 6e20 rom.interaction │ │ │ │ +0028db00: 7769 7468 2074 6865 2075 7365 722c 206f with the user, o │ │ │ │ +0028db10: 6e65 206c 696e 6520 6174 2061 2074 696d ne line at a tim │ │ │ │ +0028db20: 652e 2020 5468 6520 6469 6374 696f 6e61 e. The dictiona │ │ │ │ +0028db30: 7279 204f 7574 7075 7444 6963 7469 6f6e ry OutputDiction │ │ │ │ +0028db40: 6172 790a 6973 2074 6865 2064 6963 7469 ary.is the dicti │ │ │ │ +0028db50: 6f6e 6172 7920 696e 2077 6869 6368 2074 onary in which t │ │ │ │ +0028db60: 686f 7365 2073 796d 626f 6c73 2072 6573 hose symbols res │ │ │ │ +0028db70: 6964 652e 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ide..+---------- │ │ │ │ 0028db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0028dba0: 0a7c 6931 203a 2032 2b32 2020 2020 2020 .|i1 : 2+2 │ │ │ │ -0028dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028db90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028dba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028dbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028dbc0: 2d2d 2d2b 0a7c 6931 203a 2032 2b32 2020 ---+.|i1 : 2+2 │ │ │ │ 0028dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dbe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028dbf0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0028dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028dc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dc10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0028dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dc30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028dc40: 0a7c 6f31 203d 2034 2020 2020 2020 2020 .|o1 = 4 │ │ │ │ +0028dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028dc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dc60: 2020 207c 0a7c 6f31 203d 2034 2020 2020 |.|o1 = 4 │ │ │ │ 0028dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dc80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028dc90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0028dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028dc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dcb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0028dcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0028dce0: 0a7c 6932 203a 2022 6173 6466 2220 7c20 .|i2 : "asdf" | │ │ │ │ -0028dcf0: 2271 7765 7222 2020 2020 2020 2020 2020 "qwer" │ │ │ │ -0028dd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dd20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028dd30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0028dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028dce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028dd00: 2d2d 2d2b 0a7c 6932 203a 2022 6173 6466 ---+.|i2 : "asdf │ │ │ │ +0028dd10: 2220 7c20 2271 7765 7222 2020 2020 2020 " | "qwer" │ │ │ │ +0028dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dd50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0028dd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dd70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028dd80: 0a7c 6f32 203d 2061 7364 6671 7765 7220 .|o2 = asdfqwer │ │ │ │ +0028dd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028ddc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028ddd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0028dde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028ddf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028dda0: 2020 207c 0a7c 6f32 203d 2061 7364 6671 |.|o2 = asdfq │ │ │ │ +0028ddb0: 7765 7220 2020 2020 2020 2020 2020 2020 wer │ │ │ │ +0028ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028ddf0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0028de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0028de20: 0a7c 6933 203a 2076 616c 7565 205c 2076 .|i3 : value \ v │ │ │ │ -0028de30: 616c 7565 7320 4f75 7470 7574 4469 6374 alues OutputDict │ │ │ │ -0028de40: 696f 6e61 7279 2020 2020 2020 2020 2020 ionary │ │ │ │ -0028de50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028de60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028de70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0028de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028de40: 2d2d 2d2b 0a7c 6933 203a 2076 616c 7565 ---+.|i3 : value │ │ │ │ +0028de50: 205c 2076 616c 7565 7320 4f75 7470 7574 \ values Output │ │ │ │ +0028de60: 4469 6374 696f 6e61 7279 2020 2020 2020 Dictionary │ │ │ │ +0028de70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028de80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028de90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028de90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0028dea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028deb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028dec0: 0a7c 6f33 203d 207b 342c 2061 7364 6671 .|o3 = {4, asdfq │ │ │ │ -0028ded0: 7765 727d 2020 2020 2020 2020 2020 2020 wer} │ │ │ │ -0028dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028df00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028df10: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0028deb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dee0: 2020 207c 0a7c 6f33 203d 207b 342c 2061 |.|o3 = {4, a │ │ │ │ +0028def0: 7364 6671 7765 727d 2020 2020 2020 2020 sdfqwer} │ │ │ │ +0028df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028df30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028df30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0028df40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028df50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028df60: 0a7c 6f33 203a 204c 6973 7420 2020 2020 .|o3 : List │ │ │ │ +0028df50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028df80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028df80: 2020 207c 0a7c 6f33 203a 204c 6973 7420 |.|o3 : List │ │ │ │ 0028df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028dfa0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028dfb0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0028dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028dfd0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0028dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0028e000: 0a7c 6934 203a 2064 6963 7469 6f6e 6172 .|i4 : dictionar │ │ │ │ -0028e010: 7950 6174 6820 2020 2020 2020 2020 2020 yPath │ │ │ │ -0028e020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e040: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e050: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0028dff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e020: 2d2d 2d2b 0a7c 6934 203a 2064 6963 7469 ---+.|i4 : dicti │ │ │ │ +0028e030: 6f6e 6172 7950 6174 6820 2020 2020 2020 onaryPath │ │ │ │ +0028e040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028e060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e070: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0028e080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e090: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e0a0: 0a7c 6f34 203d 207b 476c 6f62 616c 4469 .|o4 = {GlobalDi │ │ │ │ -0028e0b0: 6374 696f 6e61 7279 7b7d 2c20 4d61 6361 ctionary{}, Maca │ │ │ │ -0028e0c0: 756c 6179 3244 6f63 2e44 6963 7469 6f6e ulay2Doc.Diction │ │ │ │ -0028e0d0: 6172 792c 2056 6172 6965 7469 6573 2e44 ary, Varieties.D │ │ │ │ -0028e0e0: 6963 7469 6f6e 6172 792c 2020 2020 207c ictionary, | │ │ │ │ -0028e0f0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0028e100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e0c0: 2020 207c 0a7c 6f34 203d 207b 476c 6f62 |.|o4 = {Glob │ │ │ │ +0028e0d0: 616c 4469 6374 696f 6e61 7279 7b7d 2c20 alDictionary{}, │ │ │ │ +0028e0e0: 4d61 6361 756c 6179 3244 6f63 2e44 6963 Macaulay2Doc.Dic │ │ │ │ +0028e0f0: 7469 6f6e 6172 792c 2056 6172 6965 7469 tionary, Varieti │ │ │ │ +0028e100: 6573 2e44 6963 7469 6f6e 6172 792c 2020 es.Dictionary, │ │ │ │ +0028e110: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 0028e120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0028e140: 0a7c 2020 2020 2049 736f 6d6f 7270 6869 .| Isomorphi │ │ │ │ -0028e150: 736d 2e44 6963 7469 6f6e 6172 792c 2054 sm.Dictionary, T │ │ │ │ -0028e160: 7275 6e63 6174 696f 6e73 2e44 6963 7469 runcations.Dicti │ │ │ │ -0028e170: 6f6e 6172 792c 2050 6f6c 7968 6564 7261 onary, Polyhedra │ │ │ │ -0028e180: 2e44 6963 7469 6f6e 6172 792c 2020 207c .Dictionary, | │ │ │ │ -0028e190: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0028e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e160: 2d2d 2d7c 0a7c 2020 2020 2049 736f 6d6f ---|.| Isomo │ │ │ │ +0028e170: 7270 6869 736d 2e44 6963 7469 6f6e 6172 rphism.Dictionar │ │ │ │ +0028e180: 792c 2054 7275 6e63 6174 696f 6e73 2e44 y, Truncations.D │ │ │ │ +0028e190: 6963 7469 6f6e 6172 792c 2050 6f6c 7968 ictionary, Polyh │ │ │ │ +0028e1a0: 6564 7261 2e44 6963 7469 6f6e 6172 792c edra.Dictionary, │ │ │ │ +0028e1b0: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 0028e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0028e1e0: 0a7c 2020 2020 2053 6174 7572 6174 696f .| Saturatio │ │ │ │ -0028e1f0: 6e2e 4469 6374 696f 6e61 7279 2c20 456c n.Dictionary, El │ │ │ │ -0028e200: 696d 696e 6174 696f 6e2e 4469 6374 696f imination.Dictio │ │ │ │ -0028e210: 6e61 7279 2c20 436f 6d70 6c65 7865 732e nary, Complexes. │ │ │ │ -0028e220: 4469 6374 696f 6e61 7279 2c20 2020 207c Dictionary, | │ │ │ │ -0028e230: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0028e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e200: 2d2d 2d7c 0a7c 2020 2020 2053 6174 7572 ---|.| Satur │ │ │ │ +0028e210: 6174 696f 6e2e 4469 6374 696f 6e61 7279 ation.Dictionary │ │ │ │ +0028e220: 2c20 456c 696d 696e 6174 696f 6e2e 4469 , Elimination.Di │ │ │ │ +0028e230: 6374 696f 6e61 7279 2c20 436f 6d70 6c65 ctionary, Comple │ │ │ │ +0028e240: 7865 732e 4469 6374 696f 6e61 7279 2c20 xes.Dictionary, │ │ │ │ +0028e250: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 0028e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0028e280: 0a7c 2020 2020 2054 616e 6765 6e74 436f .| TangentCo │ │ │ │ -0028e290: 6e65 2e44 6963 7469 6f6e 6172 792c 2053 ne.Dictionary, S │ │ │ │ -0028e2a0: 696d 706c 6544 6f63 2e44 6963 7469 6f6e impleDoc.Diction │ │ │ │ -0028e2b0: 6172 792c 2052 6565 7341 6c67 6562 7261 ary, ReesAlgebra │ │ │ │ -0028e2c0: 2e44 6963 7469 6f6e 6172 792c 2020 207c .Dictionary, | │ │ │ │ -0028e2d0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0028e2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e2a0: 2d2d 2d7c 0a7c 2020 2020 2054 616e 6765 ---|.| Tange │ │ │ │ +0028e2b0: 6e74 436f 6e65 2e44 6963 7469 6f6e 6172 ntCone.Dictionar │ │ │ │ +0028e2c0: 792c 2053 696d 706c 6544 6f63 2e44 6963 y, SimpleDoc.Dic │ │ │ │ +0028e2d0: 7469 6f6e 6172 792c 2052 6565 7341 6c67 tionary, ReesAlg │ │ │ │ +0028e2e0: 6562 7261 2e44 6963 7469 6f6e 6172 792c ebra.Dictionary, │ │ │ │ +0028e2f0: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 0028e300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0028e320: 0a7c 2020 2020 2050 7269 6d61 7279 4465 .| PrimaryDe │ │ │ │ -0028e330: 636f 6d70 6f73 6974 696f 6e2e 4469 6374 composition.Dict │ │ │ │ -0028e340: 696f 6e61 7279 2c20 4d69 6e69 6d61 6c50 ionary, MinimalP │ │ │ │ -0028e350: 7269 6d65 732e 4469 6374 696f 6e61 7279 rimes.Dictionary │ │ │ │ -0028e360: 2c20 2020 2020 2020 2020 2020 2020 207c , | │ │ │ │ -0028e370: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0028e380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e340: 2d2d 2d7c 0a7c 2020 2020 2050 7269 6d61 ---|.| Prima │ │ │ │ +0028e350: 7279 4465 636f 6d70 6f73 6974 696f 6e2e ryDecomposition. │ │ │ │ +0028e360: 4469 6374 696f 6e61 7279 2c20 4d69 6e69 Dictionary, Mini │ │ │ │ +0028e370: 6d61 6c50 7269 6d65 732e 4469 6374 696f malPrimes.Dictio │ │ │ │ +0028e380: 6e61 7279 2c20 2020 2020 2020 2020 2020 nary, │ │ │ │ +0028e390: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 0028e3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0028e3c0: 0a7c 2020 2020 2050 6163 6b61 6765 4369 .| PackageCi │ │ │ │ -0028e3d0: 7461 7469 6f6e 732e 4469 6374 696f 6e61 tations.Dictiona │ │ │ │ -0028e3e0: 7279 2c20 4f6e 6c69 6e65 4c6f 6f6b 7570 ry, OnlineLookup │ │ │ │ -0028e3f0: 2e44 6963 7469 6f6e 6172 792c 2020 2020 .Dictionary, │ │ │ │ -0028e400: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e410: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0028e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e3e0: 2d2d 2d7c 0a7c 2020 2020 2050 6163 6b61 ---|.| Packa │ │ │ │ +0028e3f0: 6765 4369 7461 7469 6f6e 732e 4469 6374 geCitations.Dict │ │ │ │ +0028e400: 696f 6e61 7279 2c20 4f6e 6c69 6e65 4c6f ionary, OnlineLo │ │ │ │ +0028e410: 6f6b 7570 2e44 6963 7469 6f6e 6172 792c okup.Dictionary, │ │ │ │ +0028e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e430: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 0028e440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0028e460: 0a7c 2020 2020 204c 4c4c 4261 7365 732e .| LLLBases. │ │ │ │ -0028e470: 4469 6374 696f 6e61 7279 2c20 496e 7665 Dictionary, Inve │ │ │ │ -0028e480: 7273 6553 7973 7465 6d73 2e44 6963 7469 rseSystems.Dicti │ │ │ │ -0028e490: 6f6e 6172 792c 2020 2020 2020 2020 2020 onary, │ │ │ │ -0028e4a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e4b0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0028e4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e480: 2d2d 2d7c 0a7c 2020 2020 204c 4c4c 4261 ---|.| LLLBa │ │ │ │ +0028e490: 7365 732e 4469 6374 696f 6e61 7279 2c20 ses.Dictionary, │ │ │ │ +0028e4a0: 496e 7665 7273 6553 7973 7465 6d73 2e44 InverseSystems.D │ │ │ │ +0028e4b0: 6963 7469 6f6e 6172 792c 2020 2020 2020 ictionary, │ │ │ │ +0028e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e4d0: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 0028e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0028e500: 0a7c 2020 2020 2049 6e74 6567 7261 6c43 .| IntegralC │ │ │ │ -0028e510: 6c6f 7375 7265 2e44 6963 7469 6f6e 6172 losure.Dictionar │ │ │ │ -0028e520: 792c 2050 7573 6846 6f72 7761 7264 2e44 y, PushForward.D │ │ │ │ -0028e530: 6963 7469 6f6e 6172 792c 2020 2020 2020 ictionary, │ │ │ │ -0028e540: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e550: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0028e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e520: 2d2d 2d7c 0a7c 2020 2020 2049 6e74 6567 ---|.| Integ │ │ │ │ +0028e530: 7261 6c43 6c6f 7375 7265 2e44 6963 7469 ralClosure.Dicti │ │ │ │ +0028e540: 6f6e 6172 792c 2050 7573 6846 6f72 7761 onary, PushForwa │ │ │ │ +0028e550: 7264 2e44 6963 7469 6f6e 6172 792c 2020 rd.Dictionary, │ │ │ │ +0028e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e570: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 0028e580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0028e5a0: 0a7c 2020 2020 2043 6f6e 7761 7950 6f6c .| ConwayPol │ │ │ │ -0028e5b0: 796e 6f6d 6961 6c73 2e44 6963 7469 6f6e ynomials.Diction │ │ │ │ -0028e5c0: 6172 792c 2043 6c61 7373 6963 2e44 6963 ary, Classic.Dic │ │ │ │ -0028e5d0: 7469 6f6e 6172 792c 2043 6f72 652e 4469 tionary, Core.Di │ │ │ │ -0028e5e0: 6374 696f 6e61 7279 2c20 2020 2020 207c ctionary, | │ │ │ │ -0028e5f0: 0a7c 2020 2020 202d 2d2d 2d2d 2d2d 2d2d .| --------- │ │ │ │ -0028e600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e5c0: 2d2d 2d7c 0a7c 2020 2020 2043 6f6e 7761 ---|.| Conwa │ │ │ │ +0028e5d0: 7950 6f6c 796e 6f6d 6961 6c73 2e44 6963 yPolynomials.Dic │ │ │ │ +0028e5e0: 7469 6f6e 6172 792c 2043 6c61 7373 6963 tionary, Classic │ │ │ │ +0028e5f0: 2e44 6963 7469 6f6e 6172 792c 2043 6f72 .Dictionary, Cor │ │ │ │ +0028e600: 652e 4469 6374 696f 6e61 7279 2c20 2020 e.Dictionary, │ │ │ │ +0028e610: 2020 207c 0a7c 2020 2020 202d 2d2d 2d2d |.| ----- │ │ │ │ 0028e620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -0028e640: 0a7c 2020 2020 2047 6c6f 6261 6c44 6963 .| GlobalDic │ │ │ │ -0028e650: 7469 6f6e 6172 797b 2e2e 2e34 2e2e 2e7d tionary{...4...} │ │ │ │ -0028e660: 2c20 5061 636b 6167 6544 6963 7469 6f6e , PackageDiction │ │ │ │ -0028e670: 6172 797d 2020 2020 2020 2020 2020 2020 ary} │ │ │ │ -0028e680: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e690: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0028e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e660: 2d2d 2d7c 0a7c 2020 2020 2047 6c6f 6261 ---|.| Globa │ │ │ │ +0028e670: 6c44 6963 7469 6f6e 6172 797b 2e2e 2e34 lDictionary{...4 │ │ │ │ +0028e680: 2e2e 2e7d 2c20 5061 636b 6167 6544 6963 ...}, PackageDic │ │ │ │ +0028e690: 7469 6f6e 6172 797d 2020 2020 2020 2020 tionary} │ │ │ │ 0028e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e6b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0028e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e6d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e6e0: 0a7c 6f34 203a 204c 6973 7420 2020 2020 .|o4 : List │ │ │ │ +0028e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e700: 2020 207c 0a7c 6f34 203a 204c 6973 7420 |.|o4 : List │ │ │ │ 0028e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e720: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e730: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0028e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e750: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0028e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0028e780: 0a7c 6935 203a 2070 6565 6b20 4f75 7470 .|i5 : peek Outp │ │ │ │ -0028e790: 7574 4469 6374 696f 6e61 7279 2020 2020 utDictionary │ │ │ │ -0028e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e7c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e7d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0028e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e7a0: 2d2d 2d2b 0a7c 6935 203a 2070 6565 6b20 ---+.|i5 : peek │ │ │ │ +0028e7b0: 4f75 7470 7574 4469 6374 696f 6e61 7279 OutputDictionary │ │ │ │ +0028e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0028e7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e7f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 0028e800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e820: 0a7c 6f35 203d 2047 6c6f 6261 6c44 6963 .|o5 = GlobalDic │ │ │ │ -0028e830: 7469 6f6e 6172 797b 226f 3122 203d 3e20 tionary{"o1" => │ │ │ │ -0028e840: 6f31 7d20 2020 2020 2020 2020 2020 2020 o1} │ │ │ │ -0028e850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e870: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0028e880: 2020 2020 2020 2020 226f 3222 203d 3e20 "o2" => │ │ │ │ -0028e890: 6f32 2020 2020 2020 2020 2020 2020 2020 o2 │ │ │ │ -0028e8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e8b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e8c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0028e8d0: 2020 2020 2020 2020 226f 3322 203d 3e20 "o3" => │ │ │ │ -0028e8e0: 6f33 2020 2020 2020 2020 2020 2020 2020 o3 │ │ │ │ -0028e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e900: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e910: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0028e920: 2020 2020 2020 2020 226f 3422 203d 3e20 "o4" => │ │ │ │ -0028e930: 6f34 2020 2020 2020 2020 2020 2020 2020 o4 │ │ │ │ -0028e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028e950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0028e960: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -0028e970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e840: 2020 207c 0a7c 6f35 203d 2047 6c6f 6261 |.|o5 = Globa │ │ │ │ +0028e850: 6c44 6963 7469 6f6e 6172 797b 226f 3122 lDictionary{"o1" │ │ │ │ +0028e860: 203d 3e20 6f31 7d20 2020 2020 2020 2020 => o1} │ │ │ │ +0028e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e890: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0028e8a0: 2020 2020 2020 2020 2020 2020 226f 3222 "o2" │ │ │ │ +0028e8b0: 203d 3e20 6f32 2020 2020 2020 2020 2020 => o2 │ │ │ │ +0028e8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e8e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0028e8f0: 2020 2020 2020 2020 2020 2020 226f 3322 "o3" │ │ │ │ +0028e900: 203d 3e20 6f33 2020 2020 2020 2020 2020 => o3 │ │ │ │ +0028e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e930: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0028e940: 2020 2020 2020 2020 2020 2020 226f 3422 "o4" │ │ │ │ +0028e950: 203d 3e20 6f34 2020 2020 2020 2020 2020 => o4 │ │ │ │ +0028e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028e980: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 0028e990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0028e9b0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -0028e9c0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2064 ===.. * *note d │ │ │ │ -0028e9d0: 6963 7469 6f6e 6172 7950 6174 683a 2064 ictionaryPath: d │ │ │ │ -0028e9e0: 6963 7469 6f6e 6172 7950 6174 682c 0a0a ictionaryPath,.. │ │ │ │ -0028e9f0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0028ea00: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0028ea10: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0028ea20: 7420 2a6e 6f74 6520 4f75 7470 7574 4469 t *note OutputDi │ │ │ │ -0028ea30: 6374 696f 6e61 7279 3a20 4f75 7470 7574 ctionary: Output │ │ │ │ -0028ea40: 4469 6374 696f 6e61 7279 2c20 6973 2061 Dictionary, is a │ │ │ │ -0028ea50: 6e20 696e 7374 616e 6365 206f 6620 7468 n instance of th │ │ │ │ -0028ea60: 6520 7479 7065 0a2a 6e6f 7465 2047 6c6f e type.*note Glo │ │ │ │ -0028ea70: 6261 6c44 6963 7469 6f6e 6172 793a 2047 balDictionary: G │ │ │ │ -0028ea80: 6c6f 6261 6c44 6963 7469 6f6e 6172 792c lobalDictionary, │ │ │ │ -0028ea90: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ -0028eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028e9d0: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ +0028e9e0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +0028e9f0: 7465 2064 6963 7469 6f6e 6172 7950 6174 te dictionaryPat │ │ │ │ +0028ea00: 683a 2064 6963 7469 6f6e 6172 7950 6174 h: dictionaryPat │ │ │ │ +0028ea10: 682c 0a0a 466f 7220 7468 6520 7072 6f67 h,..For the prog │ │ │ │ +0028ea20: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +0028ea30: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +0028ea40: 626a 6563 7420 2a6e 6f74 6520 4f75 7470 bject *note Outp │ │ │ │ +0028ea50: 7574 4469 6374 696f 6e61 7279 3a20 4f75 utDictionary: Ou │ │ │ │ +0028ea60: 7470 7574 4469 6374 696f 6e61 7279 2c20 tputDictionary, │ │ │ │ +0028ea70: 6973 2061 6e20 696e 7374 616e 6365 206f is an instance o │ │ │ │ +0028ea80: 6620 7468 6520 7479 7065 0a2a 6e6f 7465 f the type.*note │ │ │ │ +0028ea90: 2047 6c6f 6261 6c44 6963 7469 6f6e 6172 GlobalDictionar │ │ │ │ +0028eaa0: 793a 2047 6c6f 6261 6c44 6963 7469 6f6e y: GlobalDiction │ │ │ │ +0028eab0: 6172 792c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ary,...--------- │ │ │ │ 0028eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028ead0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028eae0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0028eaf0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0028eb00: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0028eb10: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0028eb20: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -0028eb30: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0028eb40: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ -0028eb50: 6c61 7932 446f 632f 6f76 5f72 6570 6c2e lay2Doc/ov_repl. │ │ │ │ -0028eb60: 6d32 3a39 3831 3a30 2e0a 1f0a 4669 6c65 m2:981:0....File │ │ │ │ -0028eb70: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ -0028eb80: 6e66 6f2c 204e 6f64 653a 2063 6c65 6172 nfo, Node: clear │ │ │ │ -0028eb90: 416c 6c2c 204e 6578 743a 2063 6c65 6172 All, Next: clear │ │ │ │ -0028eba0: 4f75 7470 7574 2c20 5072 6576 3a20 4f75 Output, Prev: Ou │ │ │ │ -0028ebb0: 7470 7574 4469 6374 696f 6e61 7279 2c20 tputDictionary, │ │ │ │ -0028ebc0: 5570 3a20 746f 7020 6c65 7665 6c20 6c6f Up: top level lo │ │ │ │ -0028ebd0: 6f70 0a0a 636c 6561 7241 6c6c 202d 2d20 op..clearAll -- │ │ │ │ -0028ebe0: 666f 7267 6574 2065 7665 7279 7468 696e forget everythin │ │ │ │ -0028ebf0: 670a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a g.************** │ │ │ │ -0028ec00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -0028ec10: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -0028ec20: 2020 2020 2063 6c65 6172 416c 6c0a 0a44 clearAll..D │ │ │ │ -0028ec30: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0028ec40: 3d3d 3d3d 3d3d 0a0a 636c 6561 7241 6c6c ======..clearAll │ │ │ │ -0028ec50: 2069 7320 6120 636f 6d6d 616e 6420 7468 is a command th │ │ │ │ -0028ec60: 6174 2061 7474 656d 7074 7320 746f 2072 at attempts to r │ │ │ │ -0028ec70: 656c 6561 7365 206d 656d 6f72 7920 6279 elease memory by │ │ │ │ -0028ec80: 2063 6c65 6172 696e 6720 7468 6520 7661 clearing the va │ │ │ │ -0028ec90: 6c75 6573 0a72 6574 6169 6e65 6420 6279 lues.retained by │ │ │ │ -0028eca0: 2074 6865 206f 7574 7075 7420 6c69 6e65 the output line │ │ │ │ -0028ecb0: 2073 796d 626f 6c73 2061 6e64 2061 6c6c symbols and all │ │ │ │ -0028ecc0: 2074 6865 2075 7365 7220 7379 6d62 6f6c the user symbol │ │ │ │ -0028ecd0: 732e 0a0a 5365 6520 616c 736f 0a3d 3d3d s...See also.=== │ │ │ │ -0028ece0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0028ecf0: 2075 7365 7253 796d 626f 6c73 3a20 7573 userSymbols: us │ │ │ │ -0028ed00: 6572 5379 6d62 6f6c 732c 202d 2d20 6120 erSymbols, -- a │ │ │ │ -0028ed10: 6c69 7374 206f 6620 7468 6520 7573 6572 list of the user │ │ │ │ -0028ed20: 2773 2073 796d 626f 6c73 0a20 202a 202a 's symbols. * * │ │ │ │ -0028ed30: 6e6f 7465 2063 6c65 6172 4f75 7470 7574 note clearOutput │ │ │ │ -0028ed40: 3a20 636c 6561 724f 7574 7075 742c 202d : clearOutput, - │ │ │ │ -0028ed50: 2d20 666f 7267 6574 206f 7574 7075 7420 - forget output │ │ │ │ -0028ed60: 7661 6c75 6573 0a0a 466f 7220 7468 6520 values..For the │ │ │ │ -0028ed70: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0028ed80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0028ed90: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0028eda0: 636c 6561 7241 6c6c 3a20 636c 6561 7241 clearAll: clearA │ │ │ │ -0028edb0: 6c6c 2c20 6973 2061 202a 6e6f 7465 2063 ll, is a *note c │ │ │ │ -0028edc0: 6f6d 6d61 6e64 3a20 436f 6d6d 616e 642c ommand: Command, │ │ │ │ -0028edd0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ -0028ede0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028edf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028eae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028eaf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028eb00: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0028eb10: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0028eb20: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0028eb30: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0028eb40: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0028eb50: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +0028eb60: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ +0028eb70: 6163 6175 6c61 7932 446f 632f 6f76 5f72 acaulay2Doc/ov_r │ │ │ │ +0028eb80: 6570 6c2e 6d32 3a39 3831 3a30 2e0a 1f0a epl.m2:981:0.... │ │ │ │ +0028eb90: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ +0028eba0: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2063 oc.info, Node: c │ │ │ │ +0028ebb0: 6c65 6172 416c 6c2c 204e 6578 743a 2063 learAll, Next: c │ │ │ │ +0028ebc0: 6c65 6172 4f75 7470 7574 2c20 5072 6576 learOutput, Prev │ │ │ │ +0028ebd0: 3a20 4f75 7470 7574 4469 6374 696f 6e61 : OutputDictiona │ │ │ │ +0028ebe0: 7279 2c20 5570 3a20 746f 7020 6c65 7665 ry, Up: top leve │ │ │ │ +0028ebf0: 6c20 6c6f 6f70 0a0a 636c 6561 7241 6c6c l loop..clearAll │ │ │ │ +0028ec00: 202d 2d20 666f 7267 6574 2065 7665 7279 -- forget every │ │ │ │ +0028ec10: 7468 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a thing.********** │ │ │ │ +0028ec20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028ec30: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +0028ec40: 0a20 2020 2020 2020 2063 6c65 6172 416c . clearAl │ │ │ │ +0028ec50: 6c0a 0a44 6573 6372 6970 7469 6f6e 0a3d l..Description.= │ │ │ │ +0028ec60: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 636c 6561 ==========..clea │ │ │ │ +0028ec70: 7241 6c6c 2069 7320 6120 636f 6d6d 616e rAll is a comman │ │ │ │ +0028ec80: 6420 7468 6174 2061 7474 656d 7074 7320 d that attempts │ │ │ │ +0028ec90: 746f 2072 656c 6561 7365 206d 656d 6f72 to release memor │ │ │ │ +0028eca0: 7920 6279 2063 6c65 6172 696e 6720 7468 y by clearing th │ │ │ │ +0028ecb0: 6520 7661 6c75 6573 0a72 6574 6169 6e65 e values.retaine │ │ │ │ +0028ecc0: 6420 6279 2074 6865 206f 7574 7075 7420 d by the output │ │ │ │ +0028ecd0: 6c69 6e65 2073 796d 626f 6c73 2061 6e64 line symbols and │ │ │ │ +0028ece0: 2061 6c6c 2074 6865 2075 7365 7220 7379 all the user sy │ │ │ │ +0028ecf0: 6d62 6f6c 732e 0a0a 5365 6520 616c 736f mbols...See also │ │ │ │ +0028ed00: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0028ed10: 6e6f 7465 2075 7365 7253 796d 626f 6c73 note userSymbols │ │ │ │ +0028ed20: 3a20 7573 6572 5379 6d62 6f6c 732c 202d : userSymbols, - │ │ │ │ +0028ed30: 2d20 6120 6c69 7374 206f 6620 7468 6520 - a list of the │ │ │ │ +0028ed40: 7573 6572 2773 2073 796d 626f 6c73 0a20 user's symbols. │ │ │ │ +0028ed50: 202a 202a 6e6f 7465 2063 6c65 6172 4f75 * *note clearOu │ │ │ │ +0028ed60: 7470 7574 3a20 636c 6561 724f 7574 7075 tput: clearOutpu │ │ │ │ +0028ed70: 742c 202d 2d20 666f 7267 6574 206f 7574 t, -- forget out │ │ │ │ +0028ed80: 7075 7420 7661 6c75 6573 0a0a 466f 7220 put values..For │ │ │ │ +0028ed90: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +0028eda0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0028edb0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +0028edc0: 6f74 6520 636c 6561 7241 6c6c 3a20 636c ote clearAll: cl │ │ │ │ +0028edd0: 6561 7241 6c6c 2c20 6973 2061 202a 6e6f earAll, is a *no │ │ │ │ +0028ede0: 7465 2063 6f6d 6d61 6e64 3a20 436f 6d6d te command: Comm │ │ │ │ +0028edf0: 616e 642c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d and,...--------- │ │ │ │ 0028ee00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028ee10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028ee20: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0028ee30: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0028ee40: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0028ee50: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0028ee60: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -0028ee70: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0028ee80: 2f70 6163 6b61 6765 732f 0a4d 6163 6175 /packages/.Macau │ │ │ │ -0028ee90: 6c61 7932 446f 632f 6f76 5f72 6570 6c2e lay2Doc/ov_repl. │ │ │ │ -0028eea0: 6d32 3a36 3535 3a30 2e0a 1f0a 4669 6c65 m2:655:0....File │ │ │ │ -0028eeb0: 3a20 4d61 6361 756c 6179 3244 6f63 2e69 : Macaulay2Doc.i │ │ │ │ -0028eec0: 6e66 6f2c 204e 6f64 653a 2063 6c65 6172 nfo, Node: clear │ │ │ │ -0028eed0: 4f75 7470 7574 2c20 4e65 7874 3a20 746f Output, Next: to │ │ │ │ -0028eee0: 704c 6576 656c 4d6f 6465 2c20 5072 6576 pLevelMode, Prev │ │ │ │ -0028eef0: 3a20 636c 6561 7241 6c6c 2c20 5570 3a20 : clearAll, Up: │ │ │ │ -0028ef00: 746f 7020 6c65 7665 6c20 6c6f 6f70 0a0a top level loop.. │ │ │ │ -0028ef10: 636c 6561 724f 7574 7075 7420 2d2d 2066 clearOutput -- f │ │ │ │ -0028ef20: 6f72 6765 7420 6f75 7470 7574 2076 616c orget output val │ │ │ │ -0028ef30: 7565 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ues.************ │ │ │ │ -0028ef40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028ef50: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -0028ef60: 6765 3a20 0a20 2020 2020 2020 2063 6c65 ge: . cle │ │ │ │ -0028ef70: 6172 4f75 7470 7574 0a0a 4465 7363 7269 arOutput..Descri │ │ │ │ -0028ef80: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0028ef90: 3d0a 0a63 6c65 6172 4f75 7470 7574 2069 =..clearOutput i │ │ │ │ -0028efa0: 7320 6120 636f 6d6d 616e 6420 7468 6174 s a command that │ │ │ │ -0028efb0: 2061 7474 656d 7074 7320 746f 2072 656c attempts to rel │ │ │ │ -0028efc0: 6561 7365 206d 656d 6f72 7920 6279 2063 ease memory by c │ │ │ │ -0028efd0: 6c65 6172 696e 6720 7468 6520 7661 6c75 learing the valu │ │ │ │ -0028efe0: 6573 0a72 6574 6169 6e65 6420 6279 2074 es.retained by t │ │ │ │ -0028eff0: 6865 206f 7574 7075 7420 6c69 6e65 2073 he output line s │ │ │ │ -0028f000: 796d 626f 6c73 2e0a 0a53 6565 2061 6c73 ymbols...See als │ │ │ │ -0028f010: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ -0028f020: 2a6e 6f74 6520 636c 6561 7241 6c6c 3a20 *note clearAll: │ │ │ │ -0028f030: 636c 6561 7241 6c6c 2c20 2d2d 2066 6f72 clearAll, -- for │ │ │ │ -0028f040: 6765 7420 6576 6572 7974 6869 6e67 0a0a get everything.. │ │ │ │ -0028f050: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0028f060: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0028f070: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0028f080: 7420 2a6e 6f74 6520 636c 6561 724f 7574 t *note clearOut │ │ │ │ -0028f090: 7075 743a 2063 6c65 6172 4f75 7470 7574 put: clearOutput │ │ │ │ -0028f0a0: 2c20 6973 2061 202a 6e6f 7465 2063 6f6d , is a *note com │ │ │ │ -0028f0b0: 6d61 6e64 3a20 436f 6d6d 616e 642c 2e0a mand: Command,.. │ │ │ │ -0028f0c0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ -0028f0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028ee20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028ee30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028ee40: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0028ee50: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0028ee60: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0028ee70: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0028ee80: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0028ee90: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +0028eea0: 6c61 7932 2f70 6163 6b61 6765 732f 0a4d lay2/packages/.M │ │ │ │ +0028eeb0: 6163 6175 6c61 7932 446f 632f 6f76 5f72 acaulay2Doc/ov_r │ │ │ │ +0028eec0: 6570 6c2e 6d32 3a36 3535 3a30 2e0a 1f0a epl.m2:655:0.... │ │ │ │ +0028eed0: 4669 6c65 3a20 4d61 6361 756c 6179 3244 File: Macaulay2D │ │ │ │ +0028eee0: 6f63 2e69 6e66 6f2c 204e 6f64 653a 2063 oc.info, Node: c │ │ │ │ +0028eef0: 6c65 6172 4f75 7470 7574 2c20 4e65 7874 learOutput, Next │ │ │ │ +0028ef00: 3a20 746f 704c 6576 656c 4d6f 6465 2c20 : topLevelMode, │ │ │ │ +0028ef10: 5072 6576 3a20 636c 6561 7241 6c6c 2c20 Prev: clearAll, │ │ │ │ +0028ef20: 5570 3a20 746f 7020 6c65 7665 6c20 6c6f Up: top level lo │ │ │ │ +0028ef30: 6f70 0a0a 636c 6561 724f 7574 7075 7420 op..clearOutput │ │ │ │ +0028ef40: 2d2d 2066 6f72 6765 7420 6f75 7470 7574 -- forget output │ │ │ │ +0028ef50: 2076 616c 7565 730a 2a2a 2a2a 2a2a 2a2a values.******** │ │ │ │ +0028ef60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028ef70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0028ef80: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0028ef90: 2063 6c65 6172 4f75 7470 7574 0a0a 4465 clearOutput..De │ │ │ │ +0028efa0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +0028efb0: 3d3d 3d3d 3d0a 0a63 6c65 6172 4f75 7470 =====..clearOutp │ │ │ │ +0028efc0: 7574 2069 7320 6120 636f 6d6d 616e 6420 ut is a command │ │ │ │ +0028efd0: 7468 6174 2061 7474 656d 7074 7320 746f that attempts to │ │ │ │ +0028efe0: 2072 656c 6561 7365 206d 656d 6f72 7920 release memory │ │ │ │ +0028eff0: 6279 2063 6c65 6172 696e 6720 7468 6520 by clearing the │ │ │ │ +0028f000: 7661 6c75 6573 0a72 6574 6169 6e65 6420 values.retained │ │ │ │ +0028f010: 6279 2074 6865 206f 7574 7075 7420 6c69 by the output li │ │ │ │ +0028f020: 6e65 2073 796d 626f 6c73 2e0a 0a53 6565 ne symbols...See │ │ │ │ +0028f030: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ +0028f040: 2020 2a20 2a6e 6f74 6520 636c 6561 7241 * *note clearA │ │ │ │ +0028f050: 6c6c 3a20 636c 6561 7241 6c6c 2c20 2d2d ll: clearAll, -- │ │ │ │ +0028f060: 2066 6f72 6765 7420 6576 6572 7974 6869 forget everythi │ │ │ │ +0028f070: 6e67 0a0a 466f 7220 7468 6520 7072 6f67 ng..For the prog │ │ │ │ +0028f080: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +0028f090: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +0028f0a0: 626a 6563 7420 2a6e 6f74 6520 636c 6561 bject *note clea │ │ │ │ +0028f0b0: 724f 7574 7075 743a 2063 6c65 6172 4f75 rOutput: clearOu │ │ │ │ +0028f0c0: 7470 7574 2c20 6973 2061 202a 6e6f 7465 tput, is a *note │ │ │ │ +0028f0d0: 2063 6f6d 6d61 6e64 3a20 436f 6d6d 616e command: Comman │ │ │ │ +0028f0e0: 642c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d d,...----------- │ │ │ │ 0028f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028f110: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -0028f120: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -0028f130: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -0028f140: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -0028f150: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ -0028f160: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -0028f170: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ -0028f180: 7932 446f 632f 6f76 5f72 6570 6c2e 6d32 y2Doc/ov_repl.m2 │ │ │ │ -0028f190: 3a36 3436 3a30 2e0a 1f0a 4669 6c65 3a20 :646:0....File: │ │ │ │ -0028f1a0: 4d61 6361 756c 6179 3244 6f63 2e69 6e66 Macaulay2Doc.inf │ │ │ │ -0028f1b0: 6f2c 204e 6f64 653a 2074 6f70 4c65 7665 o, Node: topLeve │ │ │ │ -0028f1c0: 6c4d 6f64 652c 2050 7265 763a 2063 6c65 lMode, Prev: cle │ │ │ │ -0028f1d0: 6172 4f75 7470 7574 2c20 5570 3a20 746f arOutput, Up: to │ │ │ │ -0028f1e0: 7020 6c65 7665 6c20 6c6f 6f70 0a0a 746f p level loop..to │ │ │ │ -0028f1f0: 704c 6576 656c 4d6f 6465 202d 2d20 7468 pLevelMode -- th │ │ │ │ -0028f200: 6520 6375 7272 656e 7420 746f 7020 6c65 e current top le │ │ │ │ -0028f210: 7665 6c20 6d6f 6465 0a2a 2a2a 2a2a 2a2a vel mode.******* │ │ │ │ -0028f220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028f230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028f240: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -0028f250: 0a20 2020 2020 2020 2074 6f70 4c65 7665 . topLeve │ │ │ │ -0028f260: 6c4d 6f64 6520 3d20 780a 2020 2a20 496e lMode = x. * In │ │ │ │ -0028f270: 7075 7473 3a0a 2020 2020 2020 2a20 782c puts:. * x, │ │ │ │ -0028f280: 2061 202a 6e6f 7465 2073 796d 626f 6c3a a *note symbol: │ │ │ │ -0028f290: 2053 796d 626f 6c2c 2c20 2a6e 6f74 6520 Symbol,, *note │ │ │ │ -0028f2a0: 5465 586d 6163 733a 2054 6558 6d61 6373 TeXmacs: TeXmacs │ │ │ │ -0028f2b0: 2c2c 206f 7220 2a6e 6f74 6520 5374 616e ,, or *note Stan │ │ │ │ -0028f2c0: 6461 7264 3a0a 2020 2020 2020 2020 5374 dard:. St │ │ │ │ -0028f2d0: 616e 6461 7264 2c20 6f72 202a 6e6f 7465 andard, or *note │ │ │ │ -0028f2e0: 2057 6562 4170 703a 2057 6562 4170 702c WebApp: WebApp, │ │ │ │ -0028f2f0: 0a20 202a 2043 6f6e 7365 7175 656e 6365 . * Consequence │ │ │ │ -0028f300: 733a 0a20 2020 2020 202a 2074 6865 2069 s:. * the i │ │ │ │ -0028f310: 6e74 6572 7072 6574 6572 2077 696c 6c20 nterpreter will │ │ │ │ -0028f320: 7072 6f64 7563 6520 696e 7075 7420 616e produce input an │ │ │ │ -0028f330: 6420 6f75 7470 7574 2070 726f 6d70 7473 d output prompts │ │ │ │ -0028f340: 2061 7070 726f 7072 6961 7465 2066 6f72 appropriate for │ │ │ │ -0028f350: 0a20 2020 2020 2020 2074 6865 206d 6f64 . the mod │ │ │ │ -0028f360: 652c 2061 6e64 2077 696c 6c20 666f 726d e, and will form │ │ │ │ -0028f370: 6174 206f 7574 7075 7420 7661 6c75 6573 at output values │ │ │ │ -0028f380: 2061 7070 726f 7072 6961 7465 6c79 0a0a appropriately.. │ │ │ │ -0028f390: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -0028f3a0: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 7661 =======..This va │ │ │ │ -0028f3b0: 7269 6162 6c65 2069 7320 696e 7465 6e64 riable is intend │ │ │ │ -0028f3c0: 6564 2066 6f72 2069 6e74 6572 6e61 6c20 ed for internal │ │ │ │ -0028f3d0: 7573 6520 6f6e 6c79 2e0a 0a2a 204d 656e use only...* Men │ │ │ │ -0028f3e0: 753a 0a0a 0a4b 6579 7320 666f 7220 746f u:...Keys for to │ │ │ │ -0028f3f0: 7020 6c65 7665 6c20 6d6f 6465 733a 0a2a p level modes:.* │ │ │ │ -0028f400: 2053 7461 6e64 6172 643a 3a20 2020 2020 Standard:: │ │ │ │ -0028f410: 2020 2020 2020 2020 2020 2020 2020 2074 t │ │ │ │ -0028f420: 6865 2073 7461 6e64 6172 6420 746f 7020 he standard top │ │ │ │ -0028f430: 6c65 7665 6c20 7072 696e 7469 6e67 206d level printing m │ │ │ │ -0028f440: 6574 686f 640a 2a20 5765 6241 7070 3a3a ethod.* WebApp:: │ │ │ │ -0028f450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028f460: 2020 2020 2020 7468 6520 7765 6220 6170 the web ap │ │ │ │ -0028f470: 7020 746f 7020 6c65 7665 6c20 7072 696e p top level prin │ │ │ │ -0028f480: 7469 6e67 206d 6574 686f 640a 2a20 5465 ting method.* Te │ │ │ │ -0028f490: 586d 6163 733a 3a20 2020 2020 2020 2020 Xmacs:: │ │ │ │ -0028f4a0: 2020 2020 2020 2020 2020 2020 7468 6520 the │ │ │ │ -0028f4b0: 5465 586d 6163 7320 746f 7020 6c65 7665 TeXmacs top leve │ │ │ │ -0028f4c0: 6c20 7072 696e 7469 6e67 206d 6574 686f l printing metho │ │ │ │ -0028f4d0: 640a 0a4b 6579 7320 666f 7220 6d6f 6465 d..Keys for mode │ │ │ │ -0028f4e0: 2d64 6570 656e 6465 6e74 2070 7269 6e74 -dependent print │ │ │ │ -0028f4f0: 696e 6720 6d65 7468 6f64 733a 0a2a 2050 ing methods:.* P │ │ │ │ -0028f500: 7269 6e74 3a3a 2020 2020 2020 2020 2020 rint:: │ │ │ │ -0028f510: 2020 2020 2020 2020 2020 2020 2074 6f70 top │ │ │ │ -0028f520: 206c 6576 656c 206d 6574 686f 6420 666f level method fo │ │ │ │ -0028f530: 7220 7072 696e 7469 6e67 2072 6573 756c r printing resul │ │ │ │ -0028f540: 7473 0a2a 204e 6f50 7269 6e74 3a3a 2020 ts.* NoPrint:: │ │ │ │ -0028f550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028f560: 2020 2074 6f70 206c 6576 656c 206d 6574 top level met │ │ │ │ -0028f570: 686f 6420 666f 7220 6e6f 6e2d 7072 696e hod for non-prin │ │ │ │ -0028f580: 7469 6e67 2072 6573 756c 7473 0a2a 2041 ting results.* A │ │ │ │ -0028f590: 6674 6572 4576 616c 3a3a 2020 2020 2020 fterEval:: │ │ │ │ -0028f5a0: 2020 2020 2020 2020 2020 2020 2074 6f70 top │ │ │ │ -0028f5b0: 206c 6576 656c 206d 6574 686f 6420 6170 level method ap │ │ │ │ -0028f5c0: 706c 6965 6420 6166 7465 7220 6576 616c plied after eval │ │ │ │ -0028f5d0: 7561 7469 6f6e 0a2a 2042 6566 6f72 6550 uation.* BeforeP │ │ │ │ -0028f5e0: 7269 6e74 3a3a 2020 2020 2020 2020 2020 rint:: │ │ │ │ -0028f5f0: 2020 2020 2020 2074 6f70 206c 6576 656c top level │ │ │ │ -0028f600: 206d 6574 686f 6420 6170 706c 6965 6420 method applied │ │ │ │ -0028f610: 6265 666f 7265 2070 7269 6e74 696e 6720 before printing │ │ │ │ -0028f620: 7265 7375 6c74 730a 2a20 4166 7465 7250 results.* AfterP │ │ │ │ -0028f630: 7269 6e74 3a3a 2020 2020 2020 2020 2020 rint:: │ │ │ │ -0028f640: 2020 2020 2020 2020 746f 7020 6c65 7665 top leve │ │ │ │ -0028f650: 6c20 6d65 7468 6f64 2061 7070 6c69 6564 l method applied │ │ │ │ -0028f660: 2061 6674 6572 2070 7269 6e74 696e 670a after printing. │ │ │ │ -0028f670: 2a20 4166 7465 724e 6f50 7269 6e74 3a3a * AfterNoPrint:: │ │ │ │ -0028f680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0028f690: 746f 7020 6c65 7665 6c20 6d65 7468 6f64 top level method │ │ │ │ -0028f6a0: 2061 7070 6c69 6564 2061 6674 6572 206e applied after n │ │ │ │ -0028f6b0: 6f74 2070 7269 6e74 696e 670a 0a46 6f72 ot printing..For │ │ │ │ -0028f6c0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -0028f6d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0028f6e0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -0028f6f0: 6e6f 7465 2074 6f70 4c65 7665 6c4d 6f64 note topLevelMod │ │ │ │ -0028f700: 653a 2074 6f70 4c65 7665 6c4d 6f64 652c e: topLevelMode, │ │ │ │ -0028f710: 2069 7320 6120 2a6e 6f74 6520 7379 6d62 is a *note symb │ │ │ │ -0028f720: 6f6c 3a20 5379 6d62 6f6c 2c2e 0a0a 2d2d ol: Symbol,...-- │ │ │ │ -0028f730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028f740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028f750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028f110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028f130: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +0028f140: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +0028f150: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +0028f160: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +0028f170: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +0028f180: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +0028f190: 7932 2f70 6163 6b61 6765 732f 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*******.. * Usa │ │ │ │ +0028f270: 6765 3a20 0a20 2020 2020 2020 2074 6f70 ge: . top │ │ │ │ +0028f280: 4c65 7665 6c4d 6f64 6520 3d20 780a 2020 LevelMode = x. │ │ │ │ +0028f290: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +0028f2a0: 2a20 782c 2061 202a 6e6f 7465 2073 796d * x, a *note sym │ │ │ │ +0028f2b0: 626f 6c3a 2053 796d 626f 6c2c 2c20 2a6e bol: Symbol,, *n │ │ │ │ +0028f2c0: 6f74 6520 5465 586d 6163 733a 2054 6558 ote TeXmacs: TeX │ │ │ │ +0028f2d0: 6d61 6373 2c2c 206f 7220 2a6e 6f74 6520 macs,, or *note │ │ │ │ +0028f2e0: 5374 616e 6461 7264 3a0a 2020 2020 2020 Standard:. │ │ │ │ +0028f2f0: 2020 5374 616e 6461 7264 2c20 6f72 202a Standard, or * │ │ │ │ +0028f300: 6e6f 7465 2057 6562 4170 703a 2057 6562 note WebApp: Web │ │ │ │ +0028f310: 4170 702c 0a20 202a 2043 6f6e 7365 7175 App,. * Consequ │ │ │ │ +0028f320: 656e 6365 733a 0a20 2020 2020 202a 2074 ences:. * t │ │ │ │ +0028f330: 6865 2069 6e74 6572 7072 6574 6572 2077 he interpreter w │ │ │ │ +0028f340: 696c 6c20 7072 6f64 7563 6520 696e 7075 ill produce inpu │ │ │ │ +0028f350: 7420 616e 6420 6f75 7470 7574 2070 726f t and output pro │ │ │ │ +0028f360: 6d70 7473 2061 7070 726f 7072 6961 7465 mpts appropriate │ │ │ │ +0028f370: 2066 6f72 0a20 2020 2020 2020 2074 6865 for. the │ │ │ │ +0028f380: 206d 6f64 652c 2061 6e64 2077 696c 6c20 mode, and will │ │ │ │ +0028f390: 666f 726d 6174 206f 7574 7075 7420 7661 format output va │ │ │ │ +0028f3a0: 6c75 6573 2061 7070 726f 7072 6961 7465 lues appropriate │ │ │ │ +0028f3b0: 6c79 0a0a 4465 7363 7269 7074 696f 6e0a ly..Description. │ │ │ │ +0028f3c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ +0028f3d0: 7320 7661 7269 6162 6c65 2069 7320 696e s variable is in │ │ │ │ +0028f3e0: 7465 6e64 6564 2066 6f72 2069 6e74 6572 tended for inter │ │ │ │ +0028f3f0: 6e61 6c20 7573 6520 6f6e 6c79 2e0a 0a2a nal use only...* │ │ │ │ +0028f400: 204d 656e 753a 0a0a 0a4b 6579 7320 666f Menu:...Keys fo │ │ │ │ +0028f410: 7220 746f 7020 6c65 7665 6c20 6d6f 6465 r top level mode │ │ │ │ +0028f420: 733a 0a2a 2053 7461 6e64 6172 643a 3a20 s:.* Standard:: │ │ │ │ +0028f430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028f440: 2020 2074 6865 2073 7461 6e64 6172 6420 the standard │ │ │ │ +0028f450: 746f 7020 6c65 7665 6c20 7072 696e 7469 top level printi │ │ │ │ +0028f460: 6e67 206d 6574 686f 640a 2a20 5765 6241 ng method.* WebA │ │ │ │ +0028f470: 7070 3a3a 2020 2020 2020 2020 2020 2020 pp:: │ │ │ │ +0028f480: 2020 2020 2020 2020 2020 7468 6520 7765 the we │ │ │ │ +0028f490: 6220 6170 7020 746f 7020 6c65 7665 6c20 b app top level │ │ │ │ +0028f4a0: 7072 696e 7469 6e67 206d 6574 686f 640a printing method. │ │ │ │ +0028f4b0: 2a20 5465 586d 6163 733a 3a20 2020 2020 * TeXmacs:: │ │ │ │ +0028f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028f4d0: 7468 6520 5465 586d 6163 7320 746f 7020 the TeXmacs top │ │ │ │ +0028f4e0: 6c65 7665 6c20 7072 696e 7469 6e67 206d level printing m │ │ │ │ +0028f4f0: 6574 686f 640a 0a4b 6579 7320 666f 7220 ethod..Keys for │ │ │ │ +0028f500: 6d6f 6465 2d64 6570 656e 6465 6e74 2070 mode-dependent p │ │ │ │ +0028f510: 7269 6e74 696e 6720 6d65 7468 6f64 733a rinting methods: │ │ │ │ +0028f520: 0a2a 2050 7269 6e74 3a3a 2020 2020 2020 .* Print:: │ │ │ │ +0028f530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028f540: 2074 6f70 206c 6576 656c 206d 6574 686f top level metho │ │ │ │ +0028f550: 6420 666f 7220 7072 696e 7469 6e67 2072 d for printing r │ │ │ │ +0028f560: 6573 756c 7473 0a2a 204e 6f50 7269 6e74 esults.* NoPrint │ │ │ │ +0028f570: 3a3a 2020 2020 2020 2020 2020 2020 2020 :: │ │ │ │ +0028f580: 2020 2020 2020 2074 6f70 206c 6576 656c top level │ │ │ │ +0028f590: 206d 6574 686f 6420 666f 7220 6e6f 6e2d method for non- │ │ │ │ +0028f5a0: 7072 696e 7469 6e67 2072 6573 756c 7473 printing results │ │ │ │ +0028f5b0: 0a2a 2041 6674 6572 4576 616c 3a3a 2020 .* AfterEval:: │ │ │ │ +0028f5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0028f5d0: 2074 6f70 206c 6576 656c 206d 6574 686f top level metho │ │ │ │ +0028f5e0: 6420 6170 706c 6965 6420 6166 7465 7220 d applied after │ │ │ │ +0028f5f0: 6576 616c 7561 7469 6f6e 0a2a 2042 6566 evaluation.* Bef │ │ │ │ +0028f600: 6f72 6550 7269 6e74 3a3a 2020 2020 2020 orePrint:: │ │ │ │ +0028f610: 2020 2020 2020 2020 2020 2074 6f70 206c top l │ │ │ │ +0028f620: 6576 656c 206d 6574 686f 6420 6170 706c evel method appl │ │ │ │ +0028f630: 6965 6420 6265 666f 7265 2070 7269 6e74 ied before print │ │ │ │ +0028f640: 696e 6720 7265 7375 6c74 730a 2a20 4166 ing results.* Af │ │ │ │ +0028f650: 7465 7250 7269 6e74 3a3a 2020 2020 2020 terPrint:: │ │ │ │ +0028f660: 2020 2020 2020 2020 2020 2020 746f 7020 top │ │ │ │ +0028f670: 6c65 7665 6c20 6d65 7468 6f64 2061 7070 level method app │ │ │ │ +0028f680: 6c69 6564 2061 6674 6572 2070 7269 6e74 lied after print │ │ │ │ +0028f690: 696e 670a 2a20 4166 7465 724e 6f50 7269 ing.* AfterNoPri │ │ │ │ +0028f6a0: 6e74 3a3a 2020 2020 2020 2020 2020 2020 nt:: │ │ │ │ +0028f6b0: 2020 2020 746f 7020 6c65 7665 6c20 6d65 top level me │ │ │ │ +0028f6c0: 7468 6f64 2061 7070 6c69 6564 2061 6674 thod applied aft │ │ │ │ +0028f6d0: 6572 206e 6f74 2070 7269 6e74 696e 670a er not printing. │ │ │ │ +0028f6e0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0028f6f0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0028f700: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0028f710: 6374 202a 6e6f 7465 2074 6f70 4c65 7665 ct *note topLeve │ │ │ │ +0028f720: 6c4d 6f64 653a 2074 6f70 4c65 7665 6c4d lMode: topLevelM │ │ │ │ +0028f730: 6f64 652c 2069 7320 6120 2a6e 6f74 6520 ode, is a *note │ │ │ │ +0028f740: 7379 6d62 6f6c 3a20 5379 6d62 6f6c 2c2e symbol: Symbol,. │ │ │ │ +0028f750: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ 0028f760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028f770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0028f780: 6865 2073 6f75 7263 6520 6f66 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Standard -- the │ │ │ │ -0028f860: 7374 616e 6461 7264 2074 6f70 206c 6576 standard top lev │ │ │ │ -0028f870: 656c 2070 7269 6e74 696e 6720 6d65 7468 el printing meth │ │ │ │ -0028f880: 6f64 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a od.************* │ │ │ │ -0028f890: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028f8a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028f8b0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ -0028f8c0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -0028f8d0: 5468 6973 2073 796d 626f 6c20 6973 2075 This symbol is u │ │ │ │ -0028f8e0: 7365 6420 2874 656e 7461 7469 7665 6c79 sed (tentatively │ │ │ │ -0028f8f0: 2920 6173 2074 6865 2066 6972 7374 2065 ) as the first e │ │ │ │ -0028f900: 6c65 6d65 6e74 206f 6620 6120 7061 6972 lement of a pair │ │ │ │ -0028f910: 2074 6f20 7370 6563 6966 790a 7661 7269 to specify.vari │ │ │ │ -0028f920: 6f75 7320 746f 7020 6c65 7665 6c20 696e ous top level in │ │ │ │ -0028f930: 7465 7270 7265 7465 7220 6d65 7468 6f64 terpreter method │ │ │ │ -0028f940: 732e 0a0a 5365 6520 616c 736f 0a3d 3d3d s...See also.=== │ │ │ │ -0028f950: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0028f960: 2054 6558 6d61 6373 3a20 5465 586d 6163 TeXmacs: TeXmac │ │ │ │ -0028f970: 732c 202d 2d20 7468 6520 5465 586d 6163 s, -- the TeXmac │ │ │ │ -0028f980: 7320 746f 7020 6c65 7665 6c20 7072 696e s top level prin │ │ │ │ -0028f990: 7469 6e67 206d 6574 686f 640a 2020 2a20 ting method. * │ │ │ │ -0028f9a0: 2a6e 6f74 6520 5072 696e 743a 2050 7269 *note Print: Pri │ │ │ │ -0028f9b0: 6e74 2c20 2d2d 2074 6f70 206c 6576 656c nt, -- top level │ │ │ │ -0028f9c0: 206d 6574 686f 6420 666f 7220 7072 696e method for prin │ │ │ │ -0028f9d0: 7469 6e67 2072 6573 756c 7473 0a20 202a ting results. * │ │ │ │ -0028f9e0: 202a 6e6f 7465 204e 6f50 7269 6e74 3a20 *note NoPrint: │ │ │ │ -0028f9f0: 4e6f 5072 696e 742c 202d 2d20 746f 7020 NoPrint, -- top │ │ │ │ -0028fa00: 6c65 7665 6c20 6d65 7468 6f64 2066 6f72 level method for │ │ │ │ -0028fa10: 206e 6f6e 2d70 7269 6e74 696e 6720 7265 non-printing re │ │ │ │ -0028fa20: 7375 6c74 730a 2020 2a20 2a6e 6f74 6520 sults. * *note │ │ │ │ -0028fa30: 4265 666f 7265 5072 696e 743a 2042 6566 BeforePrint: Bef │ │ │ │ -0028fa40: 6f72 6550 7269 6e74 2c20 2d2d 2074 6f70 orePrint, -- top │ │ │ │ -0028fa50: 206c 6576 656c 206d 6574 686f 6420 6170 level method ap │ │ │ │ -0028fa60: 706c 6965 6420 6265 666f 7265 2070 7269 plied before pri │ │ │ │ -0028fa70: 6e74 696e 670a 2020 2020 7265 7375 6c74 nting. result │ │ │ │ -0028fa80: 730a 2020 2a20 2a6e 6f74 6520 4166 7465 s. * *note Afte │ │ │ │ -0028fa90: 7250 7269 6e74 3a20 4166 7465 7250 7269 rPrint: AfterPri │ │ │ │ -0028faa0: 6e74 2c20 2d2d 2074 6f70 206c 6576 656c nt, -- top level │ │ │ │ -0028fab0: 206d 6574 686f 6420 6170 706c 6965 6420 method applied │ │ │ │ -0028fac0: 6166 7465 7220 7072 696e 7469 6e67 0a20 after printing. │ │ │ │ -0028fad0: 202a 202a 6e6f 7465 2041 6674 6572 4e6f * *note AfterNo │ │ │ │ -0028fae0: 5072 696e 743a 2041 6674 6572 4e6f 5072 Print: AfterNoPr │ │ │ │ -0028faf0: 696e 742c 202d 2d20 746f 7020 6c65 7665 int, -- top leve │ │ │ │ -0028fb00: 6c20 6d65 7468 6f64 2061 7070 6c69 6564 l method applied │ │ │ │ -0028fb10: 2061 6674 6572 206e 6f74 0a20 2020 2070 after not. p │ │ │ │ -0028fb20: 7269 6e74 696e 670a 0a46 6f72 2074 6865 rinting..For the │ │ │ │ -0028fb30: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -0028fb40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -0028fb50: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -0028fb60: 2053 7461 6e64 6172 643a 2053 7461 6e64 Standard: Stand │ │ │ │ -0028fb70: 6172 642c 2069 7320 6120 2a6e 6f74 6520 ard, is a *note │ │ │ │ -0028fb80: 7379 6d62 6f6c 3a20 5379 6d62 6f6c 2c2e symbol: Symbol,. │ │ │ │ -0028fb90: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ -0028fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028fbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028f770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028f780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028f790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028f7a0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0028f7b0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0028f7c0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0028f7d0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0028f7e0: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +0028f7f0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +0028f800: 7061 636b 6167 6573 2f0a 4d61 6361 756c packages/.Macaul │ │ │ │ +0028f810: 6179 3244 6f63 2f6f 765f 7265 706c 2e6d ay2Doc/ov_repl.m │ │ │ │ +0028f820: 323a 3431 343a 302e 0a1f 0a46 696c 653a 2:414:0....File: │ │ │ │ +0028f830: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ +0028f840: 666f 2c20 4e6f 6465 3a20 5374 616e 6461 fo, Node: Standa │ │ │ │ +0028f850: 7264 2c20 4e65 7874 3a20 5765 6241 7070 rd, Next: WebApp │ │ │ │ +0028f860: 2c20 5570 3a20 746f 704c 6576 656c 4d6f , Up: topLevelMo │ │ │ │ +0028f870: 6465 0a0a 5374 616e 6461 7264 202d 2d20 de..Standard -- │ │ │ │ +0028f880: 7468 6520 7374 616e 6461 7264 2074 6f70 the standard top │ │ │ │ +0028f890: 206c 6576 656c 2070 7269 6e74 696e 6720 level printing │ │ │ │ +0028f8a0: 6d65 7468 6f64 0a2a 2a2a 2a2a 2a2a 2a2a method.********* │ │ │ │ +0028f8b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028f8c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0028f8d0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 *********..Descr │ │ │ │ +0028f8e0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +0028f8f0: 3d3d 0a0a 5468 6973 2073 796d 626f 6c20 ==..This symbol │ │ │ │ +0028f900: 6973 2075 7365 6420 2874 656e 7461 7469 is used (tentati │ │ │ │ +0028f910: 7665 6c79 2920 6173 2074 6865 2066 6972 vely) as the fir │ │ │ │ +0028f920: 7374 2065 6c65 6d65 6e74 206f 6620 6120 st element of a │ │ │ │ +0028f930: 7061 6972 2074 6f20 7370 6563 6966 790a pair to specify. │ │ │ │ +0028f940: 7661 7269 6f75 7320 746f 7020 6c65 7665 various top leve │ │ │ │ +0028f950: 6c20 696e 7465 7270 7265 7465 7220 6d65 l interpreter me │ │ │ │ +0028f960: 7468 6f64 732e 0a0a 5365 6520 616c 736f thods...See also │ │ │ │ +0028f970: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0028f980: 6e6f 7465 2054 6558 6d61 6373 3a20 5465 note TeXmacs: Te │ │ │ │ +0028f990: 586d 6163 732c 202d 2d20 7468 6520 5465 Xmacs, -- the Te │ │ │ │ +0028f9a0: 586d 6163 7320 746f 7020 6c65 7665 6c20 Xmacs top level │ │ │ │ +0028f9b0: 7072 696e 7469 6e67 206d 6574 686f 640a printing method. │ │ │ │ +0028f9c0: 2020 2a20 2a6e 6f74 6520 5072 696e 743a * *note Print: │ │ │ │ +0028f9d0: 2050 7269 6e74 2c20 2d2d 2074 6f70 206c Print, -- top l │ │ │ │ +0028f9e0: 6576 656c 206d 6574 686f 6420 666f 7220 evel method for │ │ │ │ +0028f9f0: 7072 696e 7469 6e67 2072 6573 756c 7473 printing results │ │ │ │ +0028fa00: 0a20 202a 202a 6e6f 7465 204e 6f50 7269 . * *note NoPri │ │ │ │ +0028fa10: 6e74 3a20 4e6f 5072 696e 742c 202d 2d20 nt: NoPrint, -- │ │ │ │ +0028fa20: 746f 7020 6c65 7665 6c20 6d65 7468 6f64 top level method │ │ │ │ +0028fa30: 2066 6f72 206e 6f6e 2d70 7269 6e74 696e for non-printin │ │ │ │ +0028fa40: 6720 7265 7375 6c74 730a 2020 2a20 2a6e g results. * *n │ │ │ │ +0028fa50: 6f74 6520 4265 666f 7265 5072 696e 743a ote BeforePrint: │ │ │ │ +0028fa60: 2042 6566 6f72 6550 7269 6e74 2c20 2d2d BeforePrint, -- │ │ │ │ +0028fa70: 2074 6f70 206c 6576 656c 206d 6574 686f top level metho │ │ │ │ +0028fa80: 6420 6170 706c 6965 6420 6265 666f 7265 d applied before │ │ │ │ +0028fa90: 2070 7269 6e74 696e 670a 2020 2020 7265 printing. re │ │ │ │ +0028faa0: 7375 6c74 730a 2020 2a20 2a6e 6f74 6520 sults. * *note │ │ │ │ +0028fab0: 4166 7465 7250 7269 6e74 3a20 4166 7465 AfterPrint: Afte │ │ │ │ +0028fac0: 7250 7269 6e74 2c20 2d2d 2074 6f70 206c rPrint, -- top l │ │ │ │ +0028fad0: 6576 656c 206d 6574 686f 6420 6170 706c evel method appl │ │ │ │ +0028fae0: 6965 6420 6166 7465 7220 7072 696e 7469 ied after printi │ │ │ │ +0028faf0: 6e67 0a20 202a 202a 6e6f 7465 2041 6674 ng. * *note Aft │ │ │ │ +0028fb00: 6572 4e6f 5072 696e 743a 2041 6674 6572 erNoPrint: After │ │ │ │ +0028fb10: 4e6f 5072 696e 742c 202d 2d20 746f 7020 NoPrint, -- top │ │ │ │ +0028fb20: 6c65 7665 6c20 6d65 7468 6f64 2061 7070 level method app │ │ │ │ +0028fb30: 6c69 6564 2061 6674 6572 206e 6f74 0a20 lied after not. │ │ │ │ +0028fb40: 2020 2070 7269 6e74 696e 670a 0a46 6f72 printing..For │ │ │ │ +0028fb50: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0028fb60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0028fb70: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0028fb80: 6e6f 7465 2053 7461 6e64 6172 643a 2053 note Standard: S │ │ │ │ +0028fb90: 7461 6e64 6172 642c 2069 7320 6120 2a6e tandard, is a *n │ │ │ │ +0028fba0: 6f74 6520 7379 6d62 6f6c 3a20 5379 6d62 ote symbol: Symb │ │ │ │ +0028fbb0: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ol,...---------- │ │ │ │ 0028fbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0028fbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0028fbe0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0028fbf0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0028fc00: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0028fc10: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0028fc20: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ -0028fc30: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0028fc40: 7061 636b 6167 6573 2f0a 4d61 6361 756c packages/.Macaul │ │ │ │ -0028fc50: 6179 3244 6f63 2f6f 765f 7265 706c 2e6d ay2Doc/ov_repl.m │ │ │ │ -0028fc60: 323a 3432 303a 302e 0a1f 0a46 696c 653a 2:420:0....File: │ │ │ │ -0028fc70: 204d 6163 6175 6c61 7932 446f 632e 696e Macaulay2Doc.in │ │ │ │ -0028fc80: 666f 2c20 4e6f 6465 3a20 5765 6241 7070 fo, Node: WebApp │ │ │ │ -0028fc90: 2c20 4e65 7874 3a20 5465 586d 6163 732c , Next: TeXmacs, │ │ │ │ -0028fca0: 2050 7265 763a 2053 7461 6e64 6172 642c Prev: Standard, │ │ │ │ -0028fcb0: 2055 703a 2074 6f70 4c65 7665 6c4d 6f64 Up: topLevelMod │ │ │ │ -0028fcc0: 650a 0a57 6562 4170 7020 2d2d 2074 6865 e..WebApp -- the │ │ │ │ -0028fcd0: 2077 6562 2061 7070 2074 6f70 206c 6576 web app top lev │ │ │ │ -0028fce0: 656c 2070 7269 6e74 696e 6720 6d65 7468 el printing meth │ │ │ │ -0028fcf0: 6f64 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a od.************* │ │ │ │ -0028fd00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028fd10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0028fd20: 2a2a 0a0a 4465 7363 7269 7074 696f 6e0a **..Description. │ │ │ │ -0028fd30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -0028fd40: 7320 7379 6d62 6f6c 2069 7320 7573 6564 s symbol is used │ │ │ │ -0028fd50: 2028 7465 6e74 6174 6976 656c 7929 2061 (tentatively) a │ │ │ │ -0028fd60: 7320 7468 6520 6669 7273 7420 656c 656d s the first elem │ │ │ │ -0028fd70: 656e 7420 6f66 2061 2070 6169 7220 746f ent of a pair to │ │ │ │ -0028fd80: 2073 7065 6369 6679 0a76 6172 696f 7573 specify.various │ │ │ │ -0028fd90: 2074 6f70 206c 6576 656c 2069 6e74 6572 top level inter │ │ │ │ -0028fda0: 7072 6574 6572 206d 6574 686f 6473 2c20 preter methods, │ │ │ │ -0028fdb0: 696e 2063 6f6e 6e65 6374 696f 6e20 7769 in connection wi │ │ │ │ -0028fdc0: 7468 2074 6865 2075 7365 206f 6620 7468 th the use of th │ │ │ │ -0028fdd0: 650a 2863 7572 7265 6e74 6c79 2064 6576 e.(currently dev │ │ │ │ -0028fde0: 656c 6f70 6564 2920 7765 6220 6170 7020 eloped) web app │ │ │ │ -0028fdf0: 7769 7468 2028 4b61 2954 6558 206f 7574 with (Ka)TeX out │ │ │ │ -0028fe00: 7075 7420 6173 2066 726f 6e74 2065 6e64 put as front end │ │ │ │ -0028fe10: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -0028fe20: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0028fe30: 5374 616e 6461 7264 3a20 5374 616e 6461 Standard: Standa │ │ │ │ -0028fe40: 7264 2c20 2d2d 2074 6865 2073 7461 6e64 rd, -- the stand │ │ │ │ -0028fe50: 6172 6420 746f 7020 6c65 7665 6c20 7072 ard top level pr │ │ │ │ -0028fe60: 696e 7469 6e67 206d 6574 686f 640a 2020 inting method. │ │ │ │ -0028fe70: 2a20 2a6e 6f74 6520 5072 696e 743a 2050 * *note Print: P │ │ │ │ -0028fe80: 7269 6e74 2c20 2d2d 2074 6f70 206c 6576 rint, -- top lev │ │ │ │ -0028fe90: 656c 206d 6574 686f 6420 666f 7220 7072 el method for pr │ │ │ │ -0028fea0: 696e 7469 6e67 2072 6573 756c 7473 0a20 inting results. │ │ │ │ -0028feb0: 202a 202a 6e6f 7465 204e 6f50 7269 6e74 * *note NoPrint │ │ │ │ -0028fec0: 3a20 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696e 7469 6e67 d after printing │ │ │ │ -0028ffa0: 0a20 202a 202a 6e6f 7465 2041 6674 6572 . * *note After │ │ │ │ -0028ffb0: 4e6f 5072 696e 743a 2041 6674 6572 4e6f NoPrint: AfterNo │ │ │ │ -0028ffc0: 5072 696e 742c 202d 2d20 746f 7020 6c65 Print, -- top le │ │ │ │ -0028ffd0: 7665 6c20 6d65 7468 6f64 2061 7070 6c69 vel method appli │ │ │ │ -0028ffe0: 6564 2061 6674 6572 206e 6f74 0a20 2020 ed after not. │ │ │ │ -0028fff0: 2070 7269 6e74 696e 670a 0a46 6f72 2074 printing..For t │ │ │ │ -00290000: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00290010: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00290020: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00290030: 7465 2057 6562 4170 703a 2057 6562 4170 te WebApp: WebAp │ │ │ │ -00290040: 702c 2069 7320 6120 2a6e 6f74 6520 7379 p, is a *note sy │ │ │ │ -00290050: 6d62 6f6c 3a20 5379 6d62 6f6c 2c2e 0a0a mbol: Symbol,... │ │ │ │ -00290060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00290070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00290080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028fbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028fbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0028fc00: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0028fc10: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0028fc20: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0028fc30: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0028fc40: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +0028fc50: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +0028fc60: 6179 322f 7061 636b 6167 6573 2f0a 4d61 ay2/packages/.Ma │ │ │ │ +0028fc70: 6361 756c 6179 3244 6f63 2f6f 765f 7265 caulay2Doc/ov_re │ │ │ │ +0028fc80: 706c 2e6d 323a 3432 303a 302e 0a1f 0a46 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ion.===========. │ │ │ │ +0028fd60: 0a54 6869 7320 7379 6d62 6f6c 2069 7320 .This symbol is │ │ │ │ +0028fd70: 7573 6564 2028 7465 6e74 6174 6976 656c used (tentativel │ │ │ │ +0028fd80: 7929 2061 7320 7468 6520 6669 7273 7420 y) as the first │ │ │ │ +0028fd90: 656c 656d 656e 7420 6f66 2061 2070 6169 element of a pai │ │ │ │ +0028fda0: 7220 746f 2073 7065 6369 6679 0a76 6172 r to specify.var │ │ │ │ +0028fdb0: 696f 7573 2074 6f70 206c 6576 656c 2069 ious top level i │ │ │ │ +0028fdc0: 6e74 6572 7072 6574 6572 206d 6574 686f nterpreter metho │ │ │ │ +0028fdd0: 6473 2c20 696e 2063 6f6e 6e65 6374 696f ds, in connectio │ │ │ │ +0028fde0: 6e20 7769 7468 2074 6865 2075 7365 206f n with the use o │ │ │ │ +0028fdf0: 6620 7468 650a 2863 7572 7265 6e74 6c79 f the.(currently │ │ │ │ +0028fe00: 2064 6576 656c 6f70 6564 2920 7765 6220 developed) web │ │ │ │ +0028fe10: 6170 7020 7769 7468 2028 4b61 2954 6558 app with (Ka)TeX │ │ │ │ +0028fe20: 206f 7574 7075 7420 6173 2066 726f 6e74 output as front │ │ │ │ +0028fe30: 2065 6e64 2e0a 0a53 6565 2061 6c73 6f0a end...See also. │ │ │ │ +0028fe40: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +0028fe50: 6f74 6520 5374 616e 6461 7264 3a20 5374 ote Standard: St │ │ │ │ +0028fe60: 616e 6461 7264 2c20 2d2d 2074 6865 2073 andard, -- the s │ │ │ │ +0028fe70: 7461 6e64 6172 6420 746f 7020 6c65 7665 tandard top leve │ │ │ │ +0028fe80: 6c20 7072 696e 7469 6e67 206d 6574 686f l printing metho │ │ │ │ +0028fe90: 640a 2020 2a20 2a6e 6f74 6520 5072 696e d. * *note Prin │ │ │ │ +0028fea0: 743a 2050 7269 6e74 2c20 2d2d 2074 6f70 t: Print, -- top │ │ │ │ +0028feb0: 206c 6576 656c 206d 6574 686f 6420 666f level method fo │ │ │ │ +0028fec0: 7220 7072 696e 7469 6e67 2072 6573 756c r printing resul │ │ │ │ +0028fed0: 7473 0a20 202a 202a 6e6f 7465 204e 6f50 ts. * *note NoP │ │ │ │ +0028fee0: 7269 6e74 3a20 4e6f 5072 696e 742c 202d rint: NoPrint, - │ │ │ │ +0028fef0: 2d20 746f 7020 6c65 7665 6c20 6d65 7468 - top level meth │ │ │ │ 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l,...----------- │ │ │ │ 00290530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00290540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00290550: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00290560: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00290570: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -00290580: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -00290590: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ -002905a0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -002905b0: 6163 6b61 6765 732f 0a4d 6163 6175 6c61 ackages/.Macaula │ │ │ │ -002905c0: 7932 446f 632f 6f76 5f72 6570 6c2e 6d32 y2Doc/ov_repl.m2 │ │ │ │ -002905d0: 3a34 3237 3a30 2e0a 1f0a 4669 6c65 3a20 :427:0....File: │ │ │ │ -002905e0: 4d61 6361 756c 6179 3244 6f63 2e69 6e66 Macaulay2Doc.inf │ │ │ │ -002905f0: 6f2c 204e 6f64 653a 2050 7269 6e74 2c20 o, Node: Print, │ │ │ │ -00290600: 4e65 7874 3a20 4e6f 5072 696e 742c 2050 Next: NoPrint, P │ │ │ │ -00290610: 7265 763a 2054 6558 6d61 6373 2c20 5570 rev: TeXmacs, Up │ │ │ │ -00290620: 3a20 746f 704c 6576 656c 4d6f 6465 0a0a : topLevelMode.. │ │ │ │ -00290630: 5072 696e 7420 2d2d 2074 6f70 206c 6576 Print -- top lev │ │ │ │ -00290640: 656c 206d 6574 686f 6420 666f 7220 7072 el method for pr │ │ │ │ -00290650: 696e 7469 6e67 2072 6573 756c 7473 0a2a inting results.* │ │ │ │ -00290660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00290670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00290680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -00290690: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -002906a0: 2020 2058 237b 746f 704c 6576 656c 4d6f X#{topLevelMo │ │ │ │ -002906b0: 6465 2c50 7269 6e74 7d20 3d20 660a 2020 de,Print} = f. │ │ │ │ -002906c0: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -002906d0: 2a20 582c 2061 202a 6e6f 7465 2074 7970 * X, a *note typ │ │ │ │ -002906e0: 653a 2054 7970 652c 0a20 2020 2020 202a e: Type,. * │ │ │ │ -002906f0: 2066 2c20 6120 2a6e 6f74 6520 6675 6e63 f, a *note func │ │ │ │ -00290700: 7469 6f6e 3a20 4675 6e63 7469 6f6e 2c2c tion: Function,, │ │ │ │ -00290710: 2020 7468 6174 2063 616e 2070 7269 6e74 that can print │ │ │ │ -00290720: 2073 6f6d 6574 6869 6e67 206f 6620 7479 something of ty │ │ │ │ -00290730: 7065 2058 0a20 202a 2043 6f6e 7365 7175 pe X. * Consequ │ │ │ │ -00290740: 656e 6365 733a 0a20 2020 2020 202a 2061 ences:. * a │ │ │ │ -00290750: 7420 746f 7020 6c65 7665 6c2c 2077 6865 t top level, whe │ │ │ │ -00290760: 6e65 7665 7220 6974 2069 7320 7469 6d65 never it is time │ │ │ │ -00290770: 2074 6f20 7072 696e 7420 616e 206f 7574 to print an out │ │ │ │ -00290780: 7075 7420 7661 6c75 6520 6f66 2074 7970 put value of typ │ │ │ │ -00290790: 6520 582c 0a20 2020 2020 2020 2074 6865 e X,. the │ │ │ │ -002907a0: 2066 756e 6374 696f 6e20 6620 7769 6c6c function f will │ │ │ │ -002907b0: 2062 6520 6361 6c6c 6564 0a0a 4465 7363 be called..Desc │ │ │ │ -002907c0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -002907d0: 3d3d 3d0a 0a54 6865 2066 756e 6374 696f ===..The functio │ │ │ │ -002907e0: 6e20 6620 6973 2072 6573 706f 6e73 6962 n f is responsib │ │ │ │ -002907f0: 6c65 2066 6f72 2070 7269 6e74 696e 6720 le for printing │ │ │ │ -00290800: 7468 6520 6f75 7470 7574 2070 726f 6d70 the output promp │ │ │ │ -00290810: 7420 616e 6420 666f 7220 6170 706c 7969 t and for applyi │ │ │ │ -00290820: 6e67 0a74 6865 202a 6e6f 7465 2042 6566 ng.the *note Bef │ │ │ │ -00290830: 6f72 6550 7269 6e74 3a20 4265 666f 7265 orePrint: Before │ │ │ │ -00290840: 5072 696e 742c 2061 6e64 202a 6e6f 7465 Print, and *note │ │ │ │ -00290850: 2041 6674 6572 5072 696e 743a 2041 6674 AfterPrint: Aft │ │ │ │ -00290860: 6572 5072 696e 742c 206d 6574 686f 6473 erPrint, methods │ │ │ │ -00290870: 2c0a 6966 2064 6573 6972 6564 2e0a 2b2d ,.if desired..+- 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+002905f0: 6c2e 6d32 3a34 3237 3a30 2e0a 1f0a 4669 l.m2:427:0....Fi │ │ │ │ +00290600: 6c65 3a20 4d61 6361 756c 6179 3244 6f63 le: Macaulay2Doc │ │ │ │ +00290610: 2e69 6e66 6f2c 204e 6f64 653a 2050 7269 .info, Node: Pri │ │ │ │ +00290620: 6e74 2c20 4e65 7874 3a20 4e6f 5072 696e nt, Next: NoPrin │ │ │ │ +00290630: 742c 2050 7265 763a 2054 6558 6d61 6373 t, Prev: TeXmacs │ │ │ │ +00290640: 2c20 5570 3a20 746f 704c 6576 656c 4d6f , Up: topLevelMo │ │ │ │ +00290650: 6465 0a0a 5072 696e 7420 2d2d 2074 6f70 de..Print -- top │ │ │ │ +00290660: 206c 6576 656c 206d 6574 686f 6420 666f level method fo │ │ │ │ +00290670: 7220 7072 696e 7469 6e67 2072 6573 756c r printing resul │ │ │ │ +00290680: 7473 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ts.************* │ │ │ │ +00290690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002906a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +002906b0: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +002906c0: 2020 2020 2020 2058 237b 746f 704c 6576 X#{topLev │ │ │ │ +002906d0: 656c 4d6f 6465 2c50 7269 6e74 7d20 3d20 elMode,Print} = │ │ │ │ +002906e0: 660a 2020 2a20 496e 7075 7473 3a0a 2020 f. * Inputs:. │ │ │ │ +002906f0: 2020 2020 2a20 582c 2061 202a 6e6f 7465 * X, a *note │ │ │ │ +00290700: 2074 7970 653a 2054 7970 652c 0a20 2020 type: Type,. │ │ │ │ +00290710: 2020 202a 2066 2c20 6120 2a6e 6f74 6520 * f, a *note │ │ │ │ +00290720: 6675 6e63 7469 6f6e 3a20 4675 6e63 7469 function: Functi │ │ │ │ +00290730: 6f6e 2c2c 2020 7468 6174 2063 616e 2070 on,, that can p │ │ │ │ +00290740: 7269 6e74 2073 6f6d 6574 6869 6e67 206f rint something o │ │ │ │ +00290750: 6620 7479 7065 2058 0a20 202a 2043 6f6e f type X. * Con │ │ │ │ +00290760: 7365 7175 656e 6365 733a 0a20 2020 2020 sequences:. │ │ │ │ +00290770: 202a 2061 7420 746f 7020 6c65 7665 6c2c * at top level, │ │ │ │ +00290780: 2077 6865 6e65 7665 7220 6974 2069 7320 whenever it is │ │ │ │ +00290790: 7469 6d65 2074 6f20 7072 696e 7420 616e time to print an │ │ │ │ +002907a0: 206f 7574 7075 7420 7661 6c75 6520 6f66 output value of │ │ │ │ +002907b0: 2074 7970 6520 582c 0a20 2020 2020 2020 type X,. │ │ │ │ +002907c0: 2074 6865 2066 756e 6374 696f 6e20 6620 the function f │ │ │ │ +002907d0: 7769 6c6c 2062 6520 6361 6c6c 6564 0a0a will be called.. │ │ │ │ +002907e0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +002907f0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 2066 756e =======..The fun │ │ │ │ +00290800: 6374 696f 6e20 6620 6973 2072 6573 706f ction f is respo │ │ │ │ +00290810: 6e73 6962 6c65 2066 6f72 2070 7269 6e74 nsible for print │ │ │ │ +00290820: 696e 6720 7468 6520 6f75 7470 7574 2070 ing the output p │ │ │ │ +00290830: 726f 6d70 7420 616e 6420 666f 7220 6170 rompt and for ap │ │ │ │ +00290840: 706c 7969 6e67 0a74 6865 202a 6e6f 7465 plying.the *note │ │ │ │ +00290850: 2042 6566 6f72 6550 7269 6e74 3a20 4265 BeforePrint: Be │ │ │ │ +00290860: 666f 7265 5072 696e 742c 2061 6e64 202a forePrint, and * │ │ │ │ +00290870: 6e6f 7465 2041 6674 6572 5072 696e 743a note AfterPrint: │ │ │ │ +00290880: 2041 6674 6572 5072 696e 742c 206d 6574 AfterPrint, met │ │ │ │ +00290890: 686f 6473 2c0a 6966 2064 6573 6972 6564 hods,.if desired │ │ │ │ +002908a0: 2e0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 002908b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002908c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -002908d0: 3120 3a20 636f 6465 2054 6869 6e67 237b 1 : code Thing#{ │ │ │ │ -002908e0: 5374 616e 6461 7264 2c50 7269 6e74 7d20 Standard,Print} │ │ │ │ -002908f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290910: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +002908c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002908d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002908e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +002908f0: 2b0a 7c69 3120 3a20 636f 6465 2054 6869 +.|i1 : code Thi │ │ │ │ +00290900: 6e67 237b 5374 616e 6461 7264 2c50 7269 ng#{Standard,Pri │ │ │ │ +00290910: 6e74 7d20 2020 2020 2020 2020 2020 2020 nt} │ │ │ │ 00290920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00290930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00290940: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00290950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290960: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00290970: 3120 3d20 2f75 7372 2f73 6861 7265 2f4d 1 = /usr/share/M │ │ │ │ -00290980: 6163 6175 6c61 7932 2f43 6f72 652f 726f acaulay2/Core/ro │ │ │ │ -00290990: 6275 7374 2e6d 323a 2020 2020 2020 2020 bust.m2: │ │ │ │ -002909a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002909b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -002909c0: 2020 2020 3133 323a 3235 2d31 3438 3a36 132:25-148:6 │ │ │ │ -002909d0: 3a20 2d2d 736f 7572 6365 2063 6f64 653a : --source code: │ │ │ │ -002909e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -002909f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00290a10: 2020 2020 5468 696e 6723 7b53 7461 6e64 Thing#{Stand │ │ │ │ -00290a20: 6172 642c 5072 696e 747d 203d 2078 202d ard,Print} = x - │ │ │ │ -00290a30: 3e20 2820 2020 2020 2020 2020 2020 2020 > ( │ │ │ │ -00290a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00290a60: 2020 2020 2020 2020 206f 7072 6f6d 7074 oprompt │ │ │ │ -00290a70: 203a 3d20 636f 6e63 6174 656e 6174 6528 := concatenate( │ │ │ │ -00290a80: 696e 7465 7270 7265 7465 7244 6570 7468 interpreterDepth │ │ │ │ -00290a90: 3a22 6f22 2c20 746f 5374 7269 6e67 206c :"o", toString l │ │ │ │ -00290aa0: 696e 654e 756d 6265 722c 2022 7c0a 7c20 ineNumber, "|.| │ │ │ │ -00290ab0: 2020 2020 2020 2020 2073 6176 6520 3a3d save := │ │ │ │ -00290ac0: 2070 7269 6e74 5769 6474 683b 2020 2020 printWidth; │ │ │ │ -00290ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00290b00: 2020 2020 2020 2020 2069 6620 7072 696e if prin │ │ │ │ -00290b10: 7457 6964 7468 2021 3d20 3020 7468 656e tWidth != 0 then │ │ │ │ -00290b20: 2070 7269 6e74 5769 6474 6820 3d20 7072 printWidth = pr │ │ │ │ -00290b30: 696e 7457 6964 7468 202d 2023 6f70 726f intWidth - #opro │ │ │ │ -00290b40: 6d70 743b 2020 2020 2020 2020 7c0a 7c20 mpt; |.| │ │ │ │ -00290b50: 2020 2020 2020 2020 207a 203a 3d20 726f z := ro │ │ │ │ -00290b60: 6275 7374 4e65 7420 783b 2020 2020 2020 bustNet x; │ │ │ │ -00290b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290b90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00290ba0: 2020 2020 2020 2020 2077 7261 7070 6572 wrapper │ │ │ │ -00290bb0: 203a 3d20 6c6f 6f6b 7570 2867 6c6f 6261 := lookup(globa │ │ │ │ -00290bc0: 6c20 5772 6170 2c63 6c61 7373 2078 293b l Wrap,class x); │ │ │ │ -00290bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290be0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00290bf0: 2020 2020 2020 2020 2069 6620 7772 6170 if wrap │ │ │ │ -00290c00: 7065 7220 3d21 3d20 6e75 6c6c 2074 6865 per =!= null the │ │ │ │ -00290c10: 6e20 2820 2020 2020 2020 2020 2020 2020 n ( │ │ │ │ -00290c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00290c40: 2020 2020 2020 2020 2020 2020 2020 6675 fu │ │ │ │ -00290c50: 6e20 3a3d 2028 2920 2d3e 207a 203d 2077 n := () -> z = w │ │ │ │ -00290c60: 7261 7070 6572 207a 3b20 2020 2020 2020 rapper z; │ │ │ │ -00290c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290c80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00290c90: 2020 2020 2020 2020 2020 2020 2020 7472 tr │ │ │ │ -00290ca0: 7920 7469 6d65 6c69 6d69 7428 7072 696e y timelimit(prin │ │ │ │ -00290cb0: 7469 6e67 5469 6d65 4c69 6d69 742c 2066 tingTimeLimit, f │ │ │ │ -00290cc0: 756e 2920 2020 2020 2020 2020 2020 2020 un) │ │ │ │ -00290cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00290ce0: 2020 2020 2020 2020 2020 2020 2020 656c el │ │ │ │ -00290cf0: 7365 2028 2020 2020 2020 2020 2020 2020 se ( │ │ │ │ -00290d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00290d20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00290960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00290970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00290980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00290990: 7c0a 7c6f 3120 3d20 2f75 7372 2f73 6861 |.|o1 = /usr/sha │ │ │ │ +002909a0: 7265 2f4d 6163 6175 6c61 7932 2f43 6f72 re/Macaulay2/Cor │ │ │ │ +002909b0: 652f 726f 6275 7374 2e6d 323a 2020 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ate(interpreterD │ │ │ │ +00290ab0: 6570 7468 3a22 6f22 2c20 746f 5374 7269 epth:"o", toStri │ │ │ │ +00290ac0: 6e67 206c 696e 654e 756d 6265 722c 2022 ng lineNumber, " │ │ │ │ +00290ad0: 7c0a 7c20 2020 2020 2020 2020 2073 6176 |.| sav │ │ │ │ +00290ae0: 6520 3a3d 2070 7269 6e74 5769 6474 683b e := printWidth; │ │ │ │ +00290af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00290b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00290b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00290b20: 7c0a 7c20 2020 2020 2020 2020 2069 6620 |.| if │ │ │ │ +00290b30: 7072 696e 7457 6964 7468 2021 3d20 3020 printWidth != 0 │ │ │ │ +00290b40: 7468 656e 2070 7269 6e74 5769 6474 6820 then printWidth │ │ │ │ +00290b50: 3d20 7072 696e 7457 6964 7468 202d 2023 = printWidth - # │ │ │ │ +00290b60: 6f70 726f 6d70 743b 2020 2020 2020 2020 oprompt; │ │ │ │ +00290b70: 7c0a 7c20 2020 2020 2020 2020 207a 203a |.| z : │ │ │ │ +00290b80: 3d20 726f 6275 7374 4e65 7420 783b 2020 = robustNet x; │ │ │ │ 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7772 6170 7069 6e67 2070 7269 6e74 6564 wrapping printed │ │ │ │ -002915b0: 206f 7574 7075 740a 2a2a 2a2a 2a2a 2a2a output.******** │ │ │ │ -002915c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002915d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -002915e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 *************..D │ │ │ │ -002915f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00291600: 3d3d 3d3d 3d3d 0a0a 5468 6520 6465 6661 ======..The defa │ │ │ │ -00291610: 756c 7420 6d65 7468 6f64 2066 6f72 2070 ult method for p │ │ │ │ -00291620: 7269 6e74 696e 6720 7265 7375 6c74 7320 rinting results │ │ │ │ -00291630: 6f66 2063 6f6d 7075 7461 7469 6f6e 7320 of computations │ │ │ │ -00291640: 2873 746f 7265 6420 696e 0a54 6869 6e67 (stored in.Thing │ │ │ │ -00291650: 237b 5374 616e 6461 7264 2c50 7269 6e74 #{Standard,Print │ │ │ │ -00291660: 7d29 2073 6561 7263 6865 7320 666f 7220 }) searches for │ │ │ │ -00291670: 6120 6d65 7468 6f64 2066 6f72 2077 7261 a method for wra │ │ │ │ -00291680: 7070 696e 6720 7468 6520 6f75 7470 7574 pping the output │ │ │ │ -00291690: 2062 7920 7365 6172 6368 0a69 6e20 7468 by search.in th │ │ │ │ -002916a0: 6520 636c 6173 7320 6f66 2074 6865 2072 e class of the r │ │ │ │ -002916b0: 6573 756c 7420 2861 6e64 2069 7473 2061 esult (and its a │ │ │ │ -002916c0: 6e63 6573 746f 7273 2920 666f 7220 6120 ncestors) for a │ │ │ │ -002916d0: 6675 6e63 7469 6f6e 2073 746f 7265 6420 function stored │ │ │ │ -002916e0: 756e 6465 7220 7468 650a 6b65 7920 5772 under the.key Wr │ │ │ │ -002916f0: 6170 2e20 2054 6865 2070 7265 696e 7374 ap. 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---------------- │ │ │ │ +00291c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00291c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00291c30: 2d2b 0a0a 466f 7220 7468 6520 7072 6f67 -+..For the prog │ │ │ │ +00291c40: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00291c50: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00291c60: 626a 6563 7420 2a6e 6f74 6520 5772 6170 bject *note Wrap │ │ │ │ +00291c70: 3a20 5772 6170 2c20 6973 2061 202a 6e6f : Wrap, is a *no │ │ │ │ +00291c80: 7465 2073 796d 626f 6c3a 2053 796d 626f te symbol: Symbo │ │ │ │ +00291c90: 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d l,...----------- │ │ │ │ 00291ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00291cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00291cc0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00291cd0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document 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756e 6374 696f unction: Functio │ │ │ │ -00291e90: 6e2c 2c20 2074 6861 7420 6361 6e20 6163 n,, that can ac │ │ │ │ -00291ea0: 6365 7074 2073 6f6d 6574 6869 6e67 206f cept something o │ │ │ │ -00291eb0: 6620 7479 7065 2058 0a20 202a 2043 6f6e f type X. * Con │ │ │ │ -00291ec0: 7365 7175 656e 6365 733a 0a20 2020 2020 sequences:. │ │ │ │ -00291ed0: 202a 2041 7420 746f 7020 6c65 7665 6c2c * At top level, │ │ │ │ -00291ee0: 2077 6865 6e65 7665 7220 6974 2069 7320 whenever it is │ │ │ │ -00291ef0: 7469 6d65 2c20 6173 2069 6e64 6963 6174 time, as indicat │ │ │ │ -00291f00: 6564 2062 7920 6120 7365 6d69 636f 6c6f ed by a semicolo │ │ │ │ -00291f10: 6e20 6174 2074 6865 0a20 2020 2020 2020 n at the. │ │ │ │ -00291f20: 2065 6e64 206f 6620 616e 2069 6e70 7574 end of an input │ │ │ │ -00291f30: 206c 696e 652c 2074 6f20 7375 7070 7265 line, to suppre │ │ │ │ -00291f40: 7373 2070 7269 6e74 696e 6720 6f66 2061 ss printing of a │ │ │ │ -00291f50: 6e20 6f75 7470 7574 2076 616c 7565 206f n output value o │ │ │ │ -00291f60: 6620 7479 7065 0a20 2020 2020 2020 2058 f type. X │ │ │ │ -00291f70: 2c20 7468 6520 6675 6e63 7469 6f6e 2066 , the function f │ │ │ │ -00291f80: 2077 696c 6c20 6265 2063 616c 6c65 642e will be called. │ │ │ │ -00291f90: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -00291fa0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -00291fb0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -00291fc0: 6563 7420 2a6e 6f74 6520 4e6f 5072 696e ect *note NoPrin │ │ │ │ -00291fd0: 743a 204e 6f50 7269 6e74 2c20 6973 2061 t: NoPrint, is a │ │ │ │ -00291fe0: 202a 6e6f 7465 2073 796d 626f 6c3a 2053 *note symbol: S │ │ │ │ -00291ff0: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d ymbol,...------- │ │ │ │ -00292000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00292010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00291cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00291cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2c0a ote type: Type,. │ │ │ │ +00291e90: 2020 2020 2020 2a20 662c 2061 202a 6e6f * f, a *no │ │ │ │ +00291ea0: 7465 2066 756e 6374 696f 6e3a 2046 756e te function: Fun │ │ │ │ +00291eb0: 6374 696f 6e2c 2c20 2074 6861 7420 6361 ction,, that ca │ │ │ │ +00291ec0: 6e20 6163 6365 7074 2073 6f6d 6574 6869 n accept somethi │ │ │ │ +00291ed0: 6e67 206f 6620 7479 7065 2058 0a20 202a ng of type X. * │ │ │ │ +00291ee0: 2043 6f6e 7365 7175 656e 6365 733a 0a20 Consequences:. │ │ │ │ +00291ef0: 2020 2020 202a 2041 7420 746f 7020 6c65 * At top le │ │ │ │ +00291f00: 7665 6c2c 2077 6865 6e65 7665 7220 6974 vel, whenever it │ │ │ │ +00291f10: 2069 7320 7469 6d65 2c20 6173 2069 6e64 is time, as ind │ │ │ │ +00291f20: 6963 6174 6564 2062 7920 6120 7365 6d69 icated by a semi │ │ │ │ +00291f30: 636f 6c6f 6e20 6174 2074 6865 0a20 2020 colon at the. │ │ │ │ +00291f40: 2020 2020 2065 6e64 206f 6620 616e 2069 end of an i │ │ │ │ +00291f50: 6e70 7574 206c 696e 652c 2074 6f20 7375 nput line, to su │ │ │ │ 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X#{ │ │ │ │ +00292520: 746f 704c 6576 656c 4d6f 6465 2c42 6566 topLevelMode,Bef │ │ │ │ +00292530: 6f72 6550 7269 6e74 7d20 3d20 660a 2020 orePrint} = f. │ │ │ │ +00292540: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +00292550: 2a20 662c 2061 2066 756e 6374 696f 6e20 * f, a function │ │ │ │ +00292560: 746f 2062 6520 6170 706c 6965 6420 6265 to be applied be │ │ │ │ +00292570: 666f 7265 2070 7269 6e74 696e 6720 6120 fore printing a │ │ │ │ +00292580: 746f 702d 6c65 7665 6c20 6576 616c 7561 top-level evalua │ │ │ │ +00292590: 7469 6f6e 0a20 2020 2020 2020 2072 6573 tion. res │ │ │ │ +002925a0: 756c 7420 7220 6f66 2074 7970 6520 582e ult r of type X. │ │ │ │ +002925b0: 0a20 202a 2043 6f6e 7365 7175 656e 6365 . * Consequence │ │ │ │ +002925c0: 733a 0a20 2020 2020 202a 2054 6865 2076 s:. * The v │ │ │ │ +002925d0: 616c 7565 2072 6574 7572 6e65 6420 6279 alue returned by │ │ │ │ +002925e0: 2066 2069 7320 7072 696e 7465 6420 696e f is printed in │ │ │ │ +002925f0: 7374 6561 642e 0a0a 466f 7220 7468 6520 stead...For the │ │ │ │ +00292600: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +00292610: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +00292620: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +00292630: 4265 666f 7265 5072 696e 743a 2042 6566 BeforePrint: Bef │ │ │ │ +00292640: 6f72 6550 7269 6e74 2c20 6973 2061 202a orePrint, is a * │ │ │ │ +00292650: 6e6f 7465 2073 796d 626f 6c3a 2053 796d note symbol: Sym │ │ │ │ +00292660: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d bol,...--------- │ │ │ │ 00292670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00292680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00292690: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -002926a0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -002926b0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -002926c0: 726f 6475 6369 626c 652d 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2020 2020 207c 0a7c /4|.| |.| │ │ │ │ -00292950: 2020 2020 2033 2020 7c0a 7c6f 3120 3d20 3 |.|o1 = │ │ │ │ -00292960: 2d20 207c 0a7c 2020 2020 2034 2020 7c0a - |.| 4 |. │ │ │ │ -00292970: 7c20 2020 2020 2020 207c 0a7c 6f31 203a | |.|o1 : │ │ │ │ -00292980: 2051 5120 7c0a 2b2d 2d2d 2d2d 2d2d 2d2b QQ |.+--------+ │ │ │ │ -00292990: 0a57 6520 636f 756c 6420 7375 7070 7265 .We could suppre │ │ │ │ -002929a0: 7373 2074 6861 7420 6f75 7470 7574 2066 ss that output f │ │ │ │ -002929b0: 6f72 2061 2073 696e 676c 6520 7479 7065 or a single type │ │ │ │ -002929c0: 2061 7320 666f 6c6c 6f77 732e 0a2b 2d2d as follows..+-- │ │ │ │ -002929d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002929e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -002929f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -00292a00: 2051 5123 7b74 6f70 4c65 7665 6c4d 6f64 QQ#{topLevelMod │ │ │ │ -00292a10: 652c 4166 7465 7250 7269 6e74 7d20 3d20 e,AfterPrint} = │ │ │ 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